a real variable method for the cauchy transform, and analytic capacity

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1307

Takafumi Murai

A Real Variable Method for the Cauchy Transform, and Analytic Capacity

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo

Author

Takafumi Mural Department of Mathematics, Col lege of General Education

Nagoya University Nagoya, 464, Japan

Mathematics Subject Classif ication ( t980): Primary 3 0 C 8 5 ; secondary 4 2 A 5 0

ISBN 3-540-19091-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-19091-0 Springer-Verlag New York Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1988 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210

PREFACE

The purpose of this lecture note is to study the Cauchy transform on curves

and analytic capacity. For a compact set F in the complex plane ~, H~(F c)

denotes the Banach space of bounded analytic functions in ~U{o~}-F (= F c) with

supremum norm il-i!H ~. The analytic capacity of F is defined by

~,(r) = sup{If'(~>l; IIflIH~_<_ ~, f~H~<rc)>,

where f'(-) : lim z(f(z)-f(m)). We also define

y+(F) = s u p t ( ] / 2 r r ) / d>; ilcultH~ =< 1, Cu~H'(rC), u >= 0},

where

C~(z) = (1 /2~ i )7 l / ( g - z ) d>(¢) ( z ¢ ( t h e suppor t of ~ ) ) .

We are concerned wi th e s t i m a t i n g y ( - ) and y + ( . ) . To do t h i s , compact s e t s hav ing

finite l-dimension Hausdorff measure are critical. Hence we assume that P is a

finite union of mutually disjoint smooth arcs. Let i'} denote the l-dimension

Hausdorff measure (the generalized length). Let LP(F) (lip! ~) denote the L p

space of functions on F with respect to the length element Idzl, and let

LI(F) denote the weak L 1 space of functions on F. Put

p(r) = i n f ~ (E) / iE I, %(r ) = i n f ~r+(E)/I~i,

where the infimums are taken over all compact sets E in F.

transform on F is defined by

Hrf(z) = ( i / ~ ) p .v . IF f ( ¢ ) / ( ~ - z ) id~] ( z ~ r ) .

Then we see that

p+(F) __< p(F) =< Const p+(F) I/3, Const p+(F) __< I/]IHFIILI(F),LI(F ) <= Const p+(F),

where NHFILI(F),LI(F) is the norm of H F as an operator from LI(F) to LI(F) w

(Theorem D). HenceWthe study of y(F) is closely related to the study of H F-

Here is a history of the study of the Cauchy transform on Lipschitz graphs.

According to Professor Igari, the L 2 boundedness of the Cauchy transform on

Lipschitz graphs was first conjectured by Professor Zygmund in his lecture at

Orsay in 1960's. Let F = {(x,A(x)); xE~}, a(x) = A'(x), where • is the real

line. Let C[a] denote the singular integral operator defined by a kernel

I/{(x-y)+i(A(x)-A(y))}. Then the above conjecture means the following assertion:

C[a] is bounded (from L2(~) to itself) if aEL~(~). The operator C[a] is

formally expanded in the following form: (-~)H + En=0(-i)n Tn[a], where H is

the Hilbert transform and Tn[a] is the singular integral operator defined by a

The Cauchy(-Hilbert)

IV

kernel (A(x)-A(x))n/(x-y) n+l. In 1965, Calderdn [3] showed that Tl[a] is

bounded if aeL~(~) (Theorem A). This theorem is very important and closely

related to the BMO(R) theory, where BMO(~) is the Banach space, modulo constants,

of functions on • of bounded mean oscillation. Coifman-Meyer [8], [9] studied

Tn[a], Calder6n [4] showed that C[a] is bounded if llallL~(~ ) is sufficiently

small, and consequently Coifman-McIntosh-Meyer [7] solved the above conjecture in

the affirmative (Theorem B). David [17] studied H F for continuous curves F. It

is already known [44] that IIC[a]IIL2(R),L2(E) ~ Const(l 4-~BMO(~)) (Theorem C)

and that the square root is best possible [18]. Jones-Semmes gives a simple proof

of Theorem B by complex variable methods. (See Appendix II.)

As a first step of the study of H F for discontinuous curves F, we begin

with a review of the study of C[a]. In CHAP. I, g proofs of Theorem A will be

given. Once this theorem is known, we can easily deduce Theorem B (cf. CHAP. II),

and hence Theorem A is very important in the study of C[a]. As is easily seen,

if f, g E L2(~) have analytic extensions f(z), g(z) to the upper half plane

(such that limy÷~ f(iy) = limy+~ g(iy) = 0), then the Poisson extension of

(fg)(x) to the upper half plane is identical with f(z)g(z). This simple property

of analytic functions is essential in a proof of Theorem A by complex variable

methods. We shall give, in CHAP. I, various interpretations of this property from

the point of view of real analysis (cf. Coifman-Meyer-Stein [13]). These proofs

are, of course, mutually very close, but each proof has proper applications and is

interesting in itself.

In CHAP. II, we shall give the proofs of Theorems B and C by perturbation. Our

method is an improvement of Calder6n's perturbation [4] and David's perturbation

[17]. Put

o(C[a]) = sup(i/lli)/llC[a](xlf)(x)l dx,

where XI is the characteristic function of I and the supremum is taken over all

intervals I and all real-valued functions f with llfllL~(~) ~ i. This quantity

is comparable to IIC[a]IIL~(~),BMO(~) and convenient for our perturbation.

Considering a suitable Calderdn-Zygmund decomposition of a primitive A(x) of a(x)

on I, we obtain an a-priori estimate of CI/iII)/IIC[a](xIf)(x) I dx by moderate

graphs. (See the figure in § 2.2.) Repeating this argument infinitely many times

and estimating infinitely many error terms, we see that the boundedness of C[a]

is consequently reduced to the boundedness of H. For the proof, Theorem A is

necessary. We shall also give a proof of Theorem A by perturbation [45]. Tools

which we use are only the Calder6n-Zygmund decomposition and the covering lemma.

For the proof of Theorem C, we put

= sup(i/IIl)fiIC[a](xif)(x)i 2 f(x) dx, ~(C[a])

where the supremum is taken over all intervals I and all real-valued functions

f with 0 ~ f ~ i. Then o(C[a]) 2 ~ Const $(C[a]). Since

II C[a](xlf)(x)f(x) dx = 0, this quantity behaves like a linear functional of

a(x), and this gives an a-priori estimate better than o(C[a]). Our method is not

short but very simple, and this is applicable to various kernels.

In CHAP. III, we shall study H F for discontinuous graphs F and shall

compare y(-) with integralgeometric quantities. We first give the proof of

Theorem D. As is well-known, planar Cantor sets are useful to construct various

examples (cfo Denjoy [23], Vitushkin [52]). Let Q0 = [0,i]~ [0,I] and let Qn

(n$1) be the union of 4 n closed squares with sides of length 4 -n obtained from

Qn-1 with each component of Qn-i replaced by four squares in the four corners of

the component. Put Q= = ~ n= 0 Qn" Then y(Q=) = 0 and IQ~I > 0 (Garnett [28]).

This shows that two classes of null sets of y(.) and I'I are different. We shall

try to give grounds to this example. We may consider that Qn is a graph. (See the

figure in § 3.3.) Let Tsl,..,Sn (Sl,..,s n6~) be the singular integral operator

defined by a kernel

i/{(x-y)+i(Asl ,..,an(x) - Asl ,..,an(y))},

• = _ _ (x) = 0 where Asl,..,sn~X) = s k ((k-l)/n < x < k/n, igkin) and Asl,..,Sn

(x£ [0,i)). Then we see that

max{o(T ); s I .... s n6~} Sl~-',S n

is comparable to ~ (Theorem G), and, if we neglect constant multiples, 0 0

an n-tuple (Sl,..,s n) obtained from a graph {(x,A 0 0(x)); x E [0,i)} similar ~l,..~Sn

to Qm (m = (the integral part of (log n)/4)) is a solution of this extremal

problem. Hence planar Cantor sets are worst curves in a sense. We shall also

generalize Qn" A segument [0,i) is called a (thick) crank of degree 0 and a

finite union F of segments parallel to the x-axis is called a (thick) crank of n

degree n, if F n is obtained from a crank Fn_ 1 of degree n-i with each

component J of Fn_ 1 replaced by a finite number of segments Jl,..,J2p

(p=p(J)) parallel to the x-axis such that iJk! = 2-PlJ[, the distance between

Jk and J is less than or equal to 2-PIJI (i < k j 2 p) and the projections of

these segments to ~ are mutually disjoint and contained in the projection of J.

We shall show that, for any crank F of degree n, IIHFIIL2(F),L2(F ) ~ Const /~n

and that this estimate is best possible (Theorem E). To prove this, we define n+l

singular integral operators {Tk}~= 0 such that T O = (-z)H,

T n are mutually almost llz~= 0 TkIIL2(~),L2(E ) = liHrIIL2(F),L2(F) and { k}k=0

orthogonal. Hence we see that the meaning of ~nn is the central limit theorem.

We define integralgeometric quantities Cr (.) (0<~<i) as follows. Let

D(z,r) be the open disk of center z and radius r. For a compact set E,

NE(r,0 ) (r>0,101%~) denotes the (cardinal) number of elements of ENL(r,0),

where i(r,0) is the straight line defined by the equation x cos 0 + y sin O = r.

We put

VI

Cr (E) = lima+ 0 Cr~S)(E),

Cr(C)(E) = inf f :{f~ Ns{o~=iD(Zk,rk)}(r,8)~ dr} d~ (g>0)

n where O{Uk=iD(Zk,rk)} is the boundary of U~=iD(Zk,r k) and the infimum is

N taken over all finite coverings {D(Zk,rk)}k= 1 of E with radii less than c.

Since y(E) ~ Const CrI(E), it is interesting to compare y(-) with Cr (')

(cf. Marshall [37]). As an application of Theorem E, we shall show that, for

0<~<i/2, there exists a compact set E such that y(E ) = i and Cr (E) = 0

(Theorem F). For the proof, we use a branching process. Let {Xn}:= 1 be a

sequence of independent random variables on the standard probability space

([O,l),8,Prob) such that Prob(X n = ±i) = 1/2 (n~l), and let S O = O, N

S n = Ek= 1 X k (n~l). We define a Galton-Watson process {Yn}n=O by Y0(X) = i,

Yn(X) = Yn_l(X) + Syn_l(x)(X) (n~l). Then we see that, for n~l, there exists

k ~ Prob(Yn=k). a crank Fn of degree n such that Cr (F n) is comparable to Xk= 0

This quantity is comparable to i/n I-~. Using the difference of order between

i/~nn (the central limit theorem) and i/n 1-e (the Galton-Watson process), we

construct the required set E .

I express my hearty thanks to Professors M.Ohtsuka, R.R.Coifman, P.W.Jones

who gave me the chance to lecture during the academic year 1986-1987, and I am

grateful to Professors S.Kakutani, T.Tamagawa, J.Garnett, S.Semmes, T.Steger,

G.David, C.Bishop for their variable comments and suggestions. I especially

express my appreciation to Professor W.H.J.Fuchs for his encouragement. I also

thank to Mrs. Mel D. for typing the manuscr:ipt. This note is dedicated to the

memory of my mother who died while I was staying at Yale University.

New Haven, July, 1987

CONTENTS

CHAPTER I.

i.i.

1.2.

1.3.

1.4.

1.5.

1.6.

1.7.

1.8.

1.9.

i.i0.

1 .ii.

CHAPTER II.

2.1.

2.2.

2.3.

2.4.

2.5.

2.6.

2.7.

2.8.

2.9.

2.10.

CHAPTER III.

3.1.

3.2.

3.3.

3.4.

3.5.

3.6.

3.7.

APPENDIX I.

APPENDIX II.

The Calder6n commutator.( 8 proofs of its boundedness) ........... 1

Calder6n's theorem .............................................. i

Proof of (1.3) .................................................. 1

Area integral ................................................... 2

Good ~ inequalities ............................................. 4

BMO ............................................................. 6

The Coifman-Meyer expression .................................... 9

A tent space ................................................... ii

The McIntosh expression ........................................ 13

Almost orthogonality ........................................... 15

Interpolation .................................................. 21

Successive compositions of kernels ............................. 24

A real variable method for the Cauchy transform on graphs ...... 31

Coifman-McIntosh-Meyer's theorem ............................... 31

Two basic principles ........................................... 32

o-function ..................................................... 35

A-priori estimates ............................................. 39

Proof of Theorem A by perturbation ............................. 47

Proof of Theorem B by perturbation ............................. 50

Estimates of norms of E[ °] and ,C['] ......................... 53

Proof of (2.38) ................................................ 55

Proof of (2.39) ................................................ 61

Application of (2.38) .......................................... 68

Analytic capacities of cranks .................................. 71

Relation between ¥(.) and H ................................. 71

Vitushkin's example, Garnett's example, Calder6n's problem

and extremal problems .......................................... 79

The Cauchy transform on cranks ................................. 83

Proof of the latter half of Theorem E .......................... 91

Analytic capacities of fat cranks .............................. 99

Analytic capacity and integralgeometric quantities ............ 105

Proof of Theorem F ............................................ 112

An extremal problem ........................................... 117

Proof of Theorem B by P.W.Jones-S.Semmes ...................... 126

REFERENCES ................................................................... 129

SUBJECT INDEX ................................................................ 132

CHAPTER I. THE CALDERON COMMUTATOR

(8 PROOFS OF ITS BOUNDEDNESS)

§i.i. Calder~n's Theorem (Calder~n [3])

Let L p (i ~ p ~ ®) denote the L p space on the real line @ with respect

to the 1-dimension Lebesgue measure I'I- Its norm is denoted by ll'llp° Let BMO

denote the Banach space, modulo constants, of functions f on ~ such that

IIflIBM O = sup(I/II I) 7ilf(x)-(f)ildx is finite, where the supremum is taken over all

L ~ (finite) intervals I and (f)I is the mean of f over I. For a ~ , we

define a kernel

(i.i) T[a](x,y) = {A(x) - A(y)} /(x-y) 2,

where A is a primitive of a. We write simply by

itself defined by the above kernel, i.e.,

T[a] the operator from L 2 to

(1.2) T[a]f(x) = lim 7Jx_y I,~ > e T[a](x,y)f(y)dy.

g+O

Calder~n showed

Theorem A ([3]). For any f ~ L 2, T[a]f(x) exists a.e.

(1.3) IIall" --< Const llT[a]ll2, 2

and

where

(1.4) llT[a]l12,2 ~ Const llalI~ ,

llT[a]l12,2 is the norm of T[a] (as an operator from L 2 to itself).

In §1.2, we show (1.3). In §1.3-i.ii, we show various proofs of (1.4).

§1.2. Proof of (1.3) (Coifman-Rochberg-Weiss [15])

For a set E c ~, X E denotes the characteristic function of E. We put

Ps (x) = 171+ {fl s (A(s)-A(t))dt}dsl (x 6 ~, s > 0), s

= , = (x-s, x). Then lim @ / 3 = Const lal a.e. where I+ s (x, x+s) I_s ~ ~ 0

We have, for almost all x,

P (x) = Ill+ s ~l_s T[a](s,t) {(s-x) 2 + 2(s-x)(x-t) + (x-t) 2} dt} ds I

<= fl+ s [(s-x)2} T[a] XI_ (s)l + 21s-x I IT[a] {(x-')Xl s}(s)l

+ IT[a]{(x--) 2 X I } (s)l] ds -8

< C°nst[s5/211T[a]Xl II + g3/2 llT[a]{(x-')Xl }If -s 2 -e 2

+ s I/2 l{T[a]{(x-.)2~ }If 2] -g

-< Const llT[a]II2,2 {s5/211X I I] + s3/211(x-')X 11 2 _g 2 I_ s

3 + sl/2N(X-')~ I N2 } -<- ConstIIT[a]tI2,2 s ,

-g

and hence

lal = Const lira p /3 ConstHT[a]ll2, 2 s_~0 s

Thus we obtain (1.3).

a.e.

§1.3. Area integral ([3])

• ~ denote the In this section we show the proof of (1.4) by Calderon. Let C O

totality of infinitely differentiable functions with compact support, (',')

the inner product and Y = X (g > 0). Given real-valued functions (_~,~)c in C O and ~ > 0, we estimate

(TE[a]g,f) = f2Te[a]g(x)f(x)dx,

where Te[a] is an operator defined by a kernel Ys(x-y) T[a](x,y).

assume that A(x) = fx a(s)ds. Then A(x) = f ~ e(x-s)a(s)ds, where

e = X[0,~ ). We have

r (x-y) (T~[a]g,f) = f_~a(s)[ f-~f-i

denote

a,f,g

We may

{e(x-s)-e(y-s)}g(y)f(x)dydx]ds. (x-y) 2

Set

f_~ f(x) dx Im zl >0) 1 f+(z) = 2~i x-z - <0

We denote also by f+(x) (x ~ ~) the non-tangential limit of f±(z), respectively.

We define analogously g+(z), g±(x). Then f = f+ - f_, g = g+ - g_,

llf_+II 2 < IIfll 2 and llg±N 2 _-< Ilgil2. Let

K0(x,y,s) = Y (x-y){e(x-s)-e(y-s)} /(x-y) 2, g

+ K~ (x,y,s) = {e(x-s)-e(y-s)} /(x-y ± is) 2,

K2(x,y,s) = s/{(x-s) 2 + (y-s) 2 + s2} 3/2

+ Then IK0(x,y,s) - K~(x,y,s) l ~ Const K2(x,y,s), We have

[(T~[alg,f)I = I f_~ a(s)[ /_i Ko(x,y's){g+(Y) - g-(Y)} f(x)dydx] ds I

+ =< If_; a(s)[ 7_= Kl(X,y,s)g+(Y)f(x)dydx] dsl

+ If_; a(s)[ f_; K~(x,y,s)g_(y)f(x)dydx] dsl

+ Const f_~ la(s) l [f_~ K2(x,y,s) {ig+(Y) I + Ig_(y)I} If(x) l dydx]ds

(= If_; a(s) kl(S)ds I + If_~ a(S)kl(S)dsl + Const 7_; Ia(s) Ik2(s)ds,say)"

+ We now estimate k~(s), k2(s). We have

= _ _ Kl(X,y,s)g+(y)dy} dx

= f ; f(x) {e(x-s) f ; g+(Y) dy - f = - - (x_y_is)2 s (x-y-is)

=-i

=-i

g+(Y) 2 dy } dx

f_; f(x) [ f 0 g+(s+it)/{(x-is)-(s+it)} 2 dt] dx

7~ g+(s+it) [ f_~ f(x)/{(x-is)-(s+it)}2 dx] dt

oo = 2~ f0 f+(s+i(t+s))g+(s+it) dt.

Let

F(z) = -i f; f~(z+i(t+s)) g+(z+it) dt (z E U),

where U = {(x,y); x E ~, Y > 0} . Then F is analytic in U and the non-

k~(s). Here is a main lemma necessary for tangential limit F(s) equals (i/2[i)

the proof of(l.4). Let Py(X) be the Poisson± kernel, i.e., Py(X) = y/{~(x2+y2)}.

For a differentiable function v(x,y) in U, we write

IVv(x,y)] = {)8v/Ox]2 + l~v/Oy12 }i/2.

Lemma i.i ([3]). For v E L I, we define

A(v)(x) = {ff IVv(~,D) l 2 d~ d~} I/2 (x E ~), A(x)

where v~) = P * v~) and A(x) = {~); I~ - xl < n} • Then

llvll I ~ ConstilA(v)ll 1

Once this lena is known, (1.4) is deduced as follows. Since

F'(z) = f$(z + is)g+(z), we have A(F)(s) & A(f+)(s)m(g+)(s)

Const A(f+)(s) M g+(s), where m(g+)(s) = sup{Ig+~)l;(~) 6 A(x)}

M is the non-centered maximal operator (Journ~ [35, p.6]).

We have IIMg+ll2 ~ ConstlIg+II2 , Green's formula shows that

Thus we have, by Lemma 1.1,

and

(See Lemma 2,3.)

IIA(f+)II 2 = Constllf~l 2.

I~ a(s)k~(s)dsl ~ 2~ Ilall~ IIFJI 1

Const Hal< tlA(f+)Lt 2 !m(g+)lt 2

Const IlaIl~ llfIl2 Ilgll2 "

In the same manner, we have

We have

< s I f(x) I [ /Z k2(s) = fl (x_s)2+ ~2

Const M f(s) {Mg+(s) + Mg (s)} ,

ConstrlaH, IIA(F)Ii I

Const Ila!I~ I!f+II 2 !Ig+!I 2

I .~ i a(s)kl(s)ds I =< Const PFaII. ]lfll 2 ilgll 2

V(x-s ) + s 2

(x-s)2+(y-s)2+ s 2 {Ig+ (y) I+Ig-(y)I}dy]dX

and hence

S_~ la(s) Ik2(s) ds <- Const llali~ llfll 2 IIglI2 •

Consequently l(Te[a]g,f) i ~ Const IIall~ Ilfll 2 Ilgll 2 . Since f,g ( c O , g > 0 are

arbitrary, we have (1.4) for a 6 C O • In the general case, we can deduce

• o (1.4) from the boundedness of maximal operators T [b] (b ( C ) and Fatou's lemma.

(See Lemma 2.5.)

§1.4. Good k inequalities ([2], [26], [48])

In this section we give the proof of Lemma i.i by the so-called "good k

inequalities". We put m(v)(x) = sup{Iv(x,y) l; y > 0}. Fixing a sufficiently large

T, We prove

(1.5) ix; ~(v)(x) > ~x , A(v)(x) ~ ~/~ I

(Const/T 2) ix; m(x) > k I (X > 0).

Let W(k) = {x; re(x) > k}, 6(k) = IW(k) l . Then we can write W(k) = U]= I I k

with a sequence M k = {Ik} of mutually disjoint open intervals. It is

sufficient to show that, for each I ( M k ,

(1.6) IE I -<- (Const/2) lli,

where E = {x ~ I; m(v)(x) > ~h , A(v)(x)

A(v)(~) ~ X/~ for some ~ ~I; otherwise

for any x ~ I, y ~ 2111 ,

X/~}. To do this we may assume that

E = ~. Since A(v)(~) ~ X/~ , we have,

(1.7) IV(a,y) - v(x,y)] ~ Const A(v)(~) =< Const k/~ ,

where a is the left endpoint of I. We choose • large enough so that the last

quantity in (1.7) is less than ~ . Since m(v)(a) ~ k, we have

Iv(x,y) l ~ 2X (x E I, y ~ 21If). Hence, for any x E E, there exists 0 < Yx < 21II

such that Iv(X,Yx) I = sup{Iv(x,y) l; y > yx } = ~. Let

J(x) = (x - (Yx/5), x + (Yx/5)), J(x) = {(~,yx); I~- xl < Yx/10} (x ~ E). Then,

for any (~,yx) ~ J(x), we have Iv(~,yx) I ~ Iv(x,Yx) I - Const A(v)(x)

~X - Const X/T ~ • ~/2. There exist a finite number of mutually disjoint

intervals ~{J(x )} such that IEI ~ 5 Z IJ(x )I (See §2.2.) Let

R = QO N U A(x ), where QO = {(~'~); ~ E I, 0 < ~ < 2111}, ~(x ) = {(~,~);

I~ - x~I < ~/i0, ~ > Yx } " Green's formula shows that

(1.8) f { ~n ]v12 - ~ 0-~L~ -L~ } ds = Const ff ~ ]VVl 2 d~ dr], OR R

where O/On is the inner normal derivative and ds is the length element. Let

~(v)(x) = {ff , IVvl 2 d~ d~} i/2 where ~*(x) = {(~,~); l~-xl < q/lO}. (x) N R

Then a geometric observation shows that AR(V)(X) ~ A(v)(x ) ~ k/~ , where x v

is a point which is nearest to x in {x }. Hence the right-hand side of (1.8) is

dominated by:

Const fl ~ (v)(x)2dx~C°nst(X/~)2]I] ~ Const k 2 ]I I.

We divide oR into the following three parts: DR 0 = 8R N U J(x ),

DR I = {(~,D); ~ ~ I, ~ = 2111}, 0R 2 = oR - (oR 0 U oRI). Note that

~IV v(~,~) I ~ Const X/~ on oR. By the definition of Yx (x ~ E), we have, for

any (~,D) ~ oR, Iv(~,~)I ~ ~X + Const X/~ ~ Const ~X • Thus

~-~-[~ds] < Const mlVvllvl ds IfDR ~ 8n = fOR

Const (k/~) ~k /oR ds ~ Const ~2 ill.

Since Iv(~,O) l G Const k on OR I, we have If0R I

Ivl 2 These estimates yield that4R 0UoR 2 On

O~/On ~ 0 on OR 2, 0H/0n = i on 0R 0 and

8~ Ivl 2 ds i ~ Const ~2 iii. 8n

ds G Const k 2 II[. Since

Iv(~,~)l ~ ~k12 on 8R 0, we have

2 k21El = Const~8R 0 8~ Const/oRoU 8~ X2 < ~nn I vl 2ds = < OR 2 ~nn I vl 2ds =<C°nst I II ,

which shows (1.6). Consequently (1.5) holds.

By (1.5), we have, with a constant CO,

(1.9) 5(Tk) ~ ~(~I<) + (Colt 2) 6(k),

where 6(k) = Ix; A(v)(x) > h i . We now choose

quantity in (1.9) by dk from 0 to infinity.

[Im(v)II 1 ~ Const llA(v)lll, which gives llvll 1

proof of Lemma i.i.

= 2 C O and integrate each

Then we obtain

Const IiA(v)ll I. This completes the

§1.5. BMO (Fefferman-Stein [27])

Theorem A is closely related to the theory of BMO [27]. In this section, we show

the proof of Theorem A by Fefferman-Stein. We say that a non-negative measure

d~(x,y) in U is a Carleson measure with constant B if

/7 d~(x,y) ~ BIIl I ×(0,111)

for any interval I c R. The following two facts are elementary.

Lemma 1.2 ([27]). Let a ( BMO. Then yiV a(x,y) l 2~ dx dy 2

with constant Const IIalIBM 0 , where a(x,y) = P * a(x). Y

is a Carleson measure

Proof. Given an interval I, we put

a(1)(x) = (a(x) - (a)i) X ,(x), a(2)(x) = (a(x) - (a)l) X , (x), I I c

where (a)I = (i/IIl) /I a(y)dy and I is the double of I, i.e., the (open)

interval of the same midpoint as I and of length 21I I . Then

* a(1)(x) + P * a(2)(x) + (a) I a(x,y) = Py Y

( = a(1)(x,y) + a(2)(x,y) + (a)i , say).

John-Nirenberg's inequality [32] shows that Ila(1)ll2 ~ Const MalIBMO~--~T •

(See Lemma 2.5.) Hence we have, with ~ = I × (0,1If),

7f^ Y IVa(1)(x,Y)I 2 dx dy ~ f7 y IVa(1)(x,y)l 2 dx dy I U

= Const lla(1) II~ ~ Const IIalI~M 0 IIl.

Note that l(a)i. - (a)iI ~ Const jilaliBMO (j ~ I), where Ij is the interval of J

the same midpoint as I and of length 2JlIl. We have, for (x,y) E

[Va(2)(x,y)l < Const

_-< Const Z j=l

=< Const Z j=l

=< Const( E j=l

and hence

1 a(2) (s)i ds fl *c (x_s) 2

-2 I ljl f l a(y)-(a)iI ds

lj+l-I j

-2 + I (a) - (a) I t} t I j I l l j + l I {ll aII BMO I . 3

j 2 -j) II~IBMO/I I] ,

ff^ y ''IVa(2)(x,y)l 2 dx dy ~ Const(IIallBMo/lll) 2' /f^ y dx dy I I

2 Const Ha]]BMO 111 .

Thus

ff^ y ''IVa(x,y) I 2 dx dy ~ Const {ff^ y ''lva(1)(x,y)I 2 dx dy I I

. 2 + dr^ y IVa(2)(x,y)l 2 dx dy} -<_ Const ! a~IBMO III.

I Q.E.D.

Lemma 1.3 ([35, p. 85]). Let d~(x,y) be a Carleson measure with constant

Then, for any f 6 L 2,

flu If(x'y) 12 d~(x,y) < Const BIIfH 2 (f(x,y) = P * f(x)). = 2 y

B.

Proof. Let W(K) = {(x,y) ~ U; If(x,y) l > k} , 8(k) = ffw(k) d~(x,y) ( k > 0).

Then the left-hand side of our lemma is dominated by

Const f~ kS(k)dk • If (x,y) ~ W(~), then

k ~ sup{If(~,~)l; Ix-El < ~} ~ cM f(x) for some constant C. Hence W(k) is

contained in W0(k) = U I x (0,1If) , where the union is taken over all components

I of {x; M f(x) > CX} • Thus

6(k) ~ ffw0(k ) d~(x,y) & BIx; Mf(x) > CX 1 ,

which gives

f~ X 8(k)dX ~ B f~ XIx; M f(x) > Ckldk

Const B llMflI~ ~ const B IlfIl~ . Q.E.D.

We now prove Theorem A. The Hilbert transform H

f(s) ds . lim fls-xl > ~ s-x Hf(x) = ~ ~ ~ 0

is defined by

For a, f ~ C~, we have

(i.i0) T[a]f(x) = -~ H(af)(x) +~ [A,H]f'(x),

where [A,H]f' = A(Hf') - H(Af'). Since llH(af)II2 ~ llall~IlflI2, it is sufficient

< Const llalI~IifIl2 • we will prove a better inequality. to show that II[A,H]f'II 2 =

(i.ii) II[A,H]f'II 2 ~ Const IIalIBM 0 llfll 2 .

Without loss of generality we may assume that a, f

have, for any real-valued function g E C~,

([A,H]f',g) = ~_~ [A,H]f'(x)g(x)dx = (A,Hf''g + f'Hg)

are real-valued. We

= 4 im(A,f~ g+) = 4 Im(A,F') = -4 Im(a,F),

where

(1.12) F(x) = fx_~ f~(s)g+(s)ds = -i f~ f~(x+is)g+(x+is)ds.

* a(x,y), F(x,y) = Py * F(x). Since f$(z), g+(z) are analytic Let a(x,y) = Py

in U, we have ~0F (x,y) = f$(x+iy)g+(x+iy). Thus Lemmas 1.2, 1.3 and

Parseval's formula yield that

Oa OF l(a,F) l = Const l//U y ~ (x,y) 8~x (x,y) dx dy l

8a = Const Iff Y ~-x (x,y)f~(x+iy)g+(x+iy) dx dy I

U

Const {flU Ylf+(x+iy) 12 dx dy} I/2 {f7 U y IVa(x,y) 121g+(x+iy)12dx dY} I/2

Const IIf+II2 IIalIBM O fig+If 2 ~ Const IIalIBM O IIfll 2 llgll 2 •

This completes the proof of Theorem A.

Fefferman-Stein [27] showed also the following inequality, which is

essentially same as (l.ll).

Lemma 1.4 ([27]). Let a E BMO. Then iI[a,H]ll2,2 ~ Const IIaIIBM 0 •

Proof. Without loss of generality we may assume that a is real-valued.

for any real-valued functions f,g E C O ,

We have,

([a,H]f,g) = (a, Hf'g + fHg) = -4 Im (a,f+g+).

Let G(x) = f+(x)g+(x). Then Parseval's formula shows that, with

G(x,y) = P * G(x), a(x,y) = P * a(x), Y Y

8a 8G l(a,f+g+)l = l(a,G)I = Const Iff y ~- (x,y) 8~x (x,y) dx dy I

U

Const {ff U

Y iVal2 I G I dx dy} 1/2 {77 U Y IVGI 2 I GI-1 dx dy }1/2.

Since log I G(x,y) 1 is subharmonic in U,

2 A log I GI (AIGI PJ-~4"[ = _ ) , ~ ,

and hence

~ 2 = ~,G, + ~ =< 2 A]G[.

This shows that

O,

flu y IVGI2 IGI-I dx dy ~ 2 flu

Since IG(x,y) i I/2 is subharmonic in

Hence Lemmas 1.2 and 1.3 yield that

f lu y IVa(x'y) I21G(x'Y)I dx dy

2 Const llallBM 0 IIGII I.

y &IGI dx dy = Const fIG11 I.

* (IGil/2)(x) U, we have IG(x,y)l ~ Py

flu y IVa(x'y) 12 PY * (IGIl/2)(x)2dx dy

Consequently, we have

l([a,H]f,g) I _-< Const IIalIBM O IIGITI _-< Const IIalIBM 0 llfll 2 llgll 2. Q .E.D.

§1.6, The Coifman-Meyer expression (Coifman-Meyer [8])

It is important to understand Theorem A from the point of view of real

analysis. Coifman-Rochberg-Weiss [15] showed Lemma 1.4 without using analytic

functions. Coifman-Meyer gave the following expression.

Lemma 1.5 ([8]). [A,H]f'(x)

= - Const f_~ [a_s,H]fs(X)/(l+s2)ds (a E BMO, f 6 C~),

where a = k * a, f = k * f, ks(X) = ~s/IXl l+is and S S S S .

E s = F((l+is)/2)/ {F(-is/2) IS} .

Proof.

where

We have, for a, f 6 C O,

[A,H]f'(x) = Const i 7 ~ 7_~ ei(~+q)x {sign~]- sign(~-~q)}

^ ^

a, f are the Fourier transform of a, f, respectively.

~(~) i~(~)d~ an

Note that

10

{sign ~] - sign(~+q)} (~]/~) = - {sign ~] - sign(<+~q)} X(0,1)(l~]/~I). Since

I~]/~ I = Constf_~ I~]/ ~I is/(l+s2)ds ( ID/ ~I =< i),

%_s(~) = k_s(~)a(~) = I~I -is a<~) and fs(~) = I~ql is fe), we have

[A,H]f'(x) : - Const f~ [i/~ ~ e i~+~)x {sign~ - sign~+~])}

a_s(~) ~s~) d~ d~]]/(l+s 2) ds = -Const ~ ~ [a s,H]fs(X)/(l+s2)ds.

(In the case of a 6 BMO, f 6 CO, it is necessary to show the convergence of

the quantity in the right-hand side of Lemma 1.5. This will be shown later in the

proof of Theorem A.) Q.E.D.

Here is another lemma necessary for the proof of Theorem A.

Lemma 1.6 ([8]). llasllBMO & Const(l + Isl 3/4) II~IBM O.

Proof. Without loss of generality we may assume that s > 0. We put

a (I) = (a-(a) I) X i,, a (2) = (a - (a) I) XI, c . (See Lemma 1.2.) Then

a = a (I) + a (2), where a (j) = k * a (j) (j = 1,2). John-Nirenberg's inequality S S S S S

shows that IIa(1)II2 ~ ConstlIallBMO~I/~ , and hence

I la(1)(x)Idx & s lla~l)H2 ~V~ = lla<l)H2 ~ ~ Const [IalIBM O llI. I

Note that l~sl ~ Const(l + ~).

with x 0 = (the midpoint of I),

In the same manner as in Lemma 1.2, we have,

f la(2)(x) - a(2)(x0 ) I dx I

1 i I~sl f I fZ {Ix yl l+is l+is } a(2)(y)dyl dx

I - Ix0-Y I

Const {IZsl (i + sl/4)} f iX_xoll/4 {fl, c i I iXo_yi5/2 la(y)-(a)lldY} dx

= Const (I + s 3/4) !IaIIBM O Ill.

Thus we obtain

(la s - (as)ll) I ~ 2(la s - a~2)(Xo)l)l ~ Const (i + s 3/4) IIaIIBM O,

which gives the required inequality. Q.E.D.

Theorem A is deduced from Lemmas 1.4-1.6 as follows. Inequality (i.i0) shows

that it is sufficient to show that II[A,H]f'II 2 ~ Const llall~IIfll 2 • Lemmas 1.4-1.6

11

yield that

II [A,H]f'II 2 ~ Const 7~ II[as,H]f32/(l + s2) ds

Const f£ I] [as,H]ll2, 2 ]Ifsl]2/( I + s2) ds

Const IIfll 2 fflIasllBMo/(l + s 2) ds

Const IIaHBMO llfll 2 ~ Const !lall~!Ifll 2 .

§1.7. A tent space (Coifman-Meyer-Stein [13], [14])

The essential part in the sections 1.3 and 1.5 is the proof of the inequality:

IIFII 1 ~ Const llfll 2 l!gll 2 (f, g E L2), where F(x) = fx ~ f$(s)g+(s) ds

( = - i /0 f$(x + is)g+(x + is)ds). Let R s (s E ~) denote the operator defined

by Rsh = ~s * h, where ~s(X) = s2x/(x 2 + s2) 2. Then we have

F(x) = Const /~ _ Rshs(X) ds/s, where hs(Y) = s f~(y + is)g+(y + is). From this

point of view, Coifman-Meyer-Stein introduces tent spaces and generalizes the above • I

inequality. As seen in the proof of the Tb theorem (Davld-Journe-Semmes [20]),

tent spaces are very useful. The following theorem is a special case of Coifman-

Meyer-Stein's theorem; we rewrite their theorem so that only the proof of

Theorem A can be given.

Theorem 1.7 ([13]). Let T be the Banach space of functions h(y,s) in

with norm llhll T = IIS(h)lll, where S(h)(x) = {7f~(x) lh(y,s)l 2 ayas/s" ~ i 241/2

A(x) = {(y,s); ly-xl < s}. For h E T, we put R(h)(x) = fO Rshs(X)ds/s'

hs(Y) = h(y,s). Then IIR(h)II 1 ~ Const llhll T.

Theorem A immediately follows from this inequality, since

risf~(y + is)g+(y + is)If T ~ IIA(f+)m(g+)I[ I ~ Const JifiI 2 prgll 2 .

Here are two lemmas necessary for the proof. For an interval I, we write

U

and

where

I,

= {(y,s); 0 < s < dis(y,lC)} (dis(',') is the distance).

For an open set ~ c ~, we write ~ = D ~, where the union is taken over all

components I of ~ . We say that p E T is a T-atom if, for some interval

(1.13) supp(p) c~, ff^ Ip(y,s) I 2 dy ds ~ I/iii, S

I

where supp(') is the support.

• < Const. Lemma 1.8 ([13]) For any T-atom p, IIR(p) ll I =

12

Proof. Let p be a T-atom and let I be an interval satisfying (1.13).

for any b E L ~ with llblI~ ~ i, I (R(p),b)I ~ I (R(p),bl)I + I(R(p),b2)I

b 1 = b X i , and b 2 = b X I , c ( I * i s t h e d o u b l e o f I ) . S i n c e

IRsb2(Y) l ~ Const s/Ira I ((y,s) E I), we have

I(R(p),b2)I = Iff^ p(y,s) Rsb2(Y) ~ s I I

Then,

where

=< (Const/III) 7f^ Ip(y,s)l dy ds I

_< (Const/ill) {ff^ ip(y,s)l 2 dy ds }1/2 {ff^ s I I

s dy ds} I/2 ~ Const.

We have

I(R(p),bl)I ~ 77^ IP(Y,S) IIRsbI(Y)I dYadS I

{ff^ ip(y,s)12 ~ }1/2 {ff^ iRsbl(y) 12 d y ds }1/2 s s

I I

(l~/iil) {ff iRsbl(Y) l 2 dy ds}i/2 = Const Ilbli]2/~ s -

U Const.

Consequently we have I(R(p),b)[ ~ Const. Since b is arbitrary as long as

Hb]l® ~ i, we obtain HR(p)]] 1 ~ Const. Q.E.D.

Lemma 1.9 ([13]). Let h E T and let E be a subset of an interval I. Then

/f^ ^ ih(y,s) i 2 dYadS =< /I-E S(h)(x) 2 dx, l-g

where ~ = {x E I; MXE(X) > 1/2} .

Proof. A geometric observation shows that, for any (y,s) E ~ - ~, y c Y

Y O ~c ~ ~ , where Y = [y-s,y+s](C ~). Let x 0 E Y N ~c. Then

I Y N EI/IY 1 --< MXE(X 0) <-- 1/2, and hence I Y N ECI >-_ s. This shows that

fI-E S(h)(x)2dx = fI-E { ffa(x) Ih(y's) I2 dy ds 2 } dx s

> ff^ ]h(y,s)12 dy ds Q.E.D. ^ S

I--~

E k = {x; S(h)(x) > 2 k} ,

i T(k) ~ be k, let ~lj 7j=l

We now prove Theorem 1.7. Given h E T, we put

~k = {x; MXEk(X ) > 1/2} (k = O, ±i, ...). For each

the totality of components of ~k and let

p(k)(y,s)j = (2-k-I/iI(k) i)j h(y,s) xj(k)(y,s)

and

(j >-_ 1),

13

where J J Then we obtain the following T-atomic decomposition of

Each

X! k) is the characteristic function of ~!k) _ ~!k) J

h:

h(y,s) = Z Z 2 k+l I£4~(k)j I~ P~k)(y,s)- k =-~ j =l

p(k)j is a T-atom, since supp(p~ k)) c i (k)j alia

ff~!k) I P~k)(y,s)I2 dYsdS = 2-2k-2 ll~k) l-2

J

~(k) = i!k) N ( j J ~k+l )"

f/^(k) ^(k) lh(y's) 12 ~y ds I, ~Q. s J J

2-2k-2 II!k) I-23 fl (k) E S(h)(x)2 dx =< i/ll~k) l

j - k+l

by Lemma 1.9. Hence Lemma 1.8 shows that

IIR(h)iI I --< k =-~E j=IE 2 k+l I jl(k) IIR(P~ k) )II I

Const Z E 2 k I I~ k) k =-~ j~l

= Const S 2klEk 1 k = -~

< Const llS(h)IIl = Const llhl] T .

This completes the proof of Theorem 1.7. As stated above, Theorem A is deduced from

this theorem.

~1.8. The Mclntosh expression (Coifman-Mclntosh-Meyer [7])

The proof of Theorem A in this section is a version of the method given in [7]

for the proof of Theorem B. (See Chapter II .) Here is an interesting expression

of T[a].

+~ I ds (a E L ~) , Len=na i.i0 ([7]). T[a] = f_Z I isD- Ma I + is D s where I is the identity operator, D = -i(8/8x) and M is the multiplier: a f -~ af.

Proof. Let a(x) = e igx, f(x) = e i~x (~, ~ E ~). Then we have

r[a]f(x) = (-~i) { ~+~ sign(~ + ~) - ~ sign ~ }

and

14

I I J-~ { I + is D Ma I + is D

as f} (x) --

8

I (af) }(x) = 7-Z { I + is D

1 ds i+ is ~ S

1 1 ds J-- 1 + is(a+ ~) 1 + is ~ s

i f~ { i 1 ds = 7 1 + is(e + ~) 1 + is ~ } 2-

s

1 ~ ~ + ~ ~ } d s

= 7 f - ~ { (1 + i s ( ¢ + ~ ) ) 2 (1 + i s ~ )2 s

= (-~i) {~ + B sign(~ + B) - ~ sign $}.

Hence

I I ds T[a]f = f { I + is D M } f -- -~ a I + is D s

iax is complete in the space of functions f with norm Since {e }~ E

f_Zlf(x) 12/(I + x 2) dx < ~ , the required equality holds. (It is necessary to show

the convergence of the integral in the right-hand side. This will be given in

the proof of Theorem A.)

Let Ps = I /(~ + s2D2), %

in Lemma 1.2, we have

Lemma i.ii ([7]). Let a ~ BMO. 2

constant Const IIalIBM O •

Lemma i.i0 shows that

(1.14) T[a] = f - Z {PsMaPs

~ ds = -2i fO ~sMaPs s

Q.E.D.

= sD/(l + s2D 2) (s > 0). In the same manner as

2 dxds Then tQsa(~) I s is a Carleson measure with

- i %MaP s - i PsMaQs - QsMa~s } d Ss

-- - 2i fO PsMa % d Ss (= -2i L I - 2i L 2, say).

we see that

%= Q3 s P = -2 3- . S~s s ~s

Hence the integration by parts shows that

L1 =fO [8Q + S { -% + 2 Ps%}]M a P d s s s

" " 8 ds = 870 ~ Ma Ps d Ss -70 {-% + 2 Ps%} Ma(SS~s Ps) ~ -

" -- 2 ds = 81"O Q3 s M a Ps d Ss + 270 { -% + 2%Ps }Ma Qs s-

( = 8 LII + 2 LI2, say).

QslI " Since IIPsll2,2 -< - Const, II 2,2 =< Const and 70 jIQsfJl Shwartz's inequality shows that ]IL12112, 2 =< Const llaIl~ •

] (g, Lllf) l = If; (~g, %MaPsf) d s is

; 2 2 ds }i/2 ~ 22 ds 1/2 {7 I]~g]12 s- {~0 ]]~MaPsfN -- }s

2 ds }1/2 = Const I]gl] 2 {7; I]%MaPsfN 2 ~- •

d s = Const IIfll 2 s 2

we have, for f,g E L 2,

We see that

{%MaPs}f = (%a)(Psf) + Ps {(Psa)(% f)} - %{(%a)(%f)} .

(To see this,use a(x) = e iax, f(x) = e i~x (~, ~ E R).) Hence we have

II%MaPsfI122 ~ Const {Ilall 2 ll%fII~ + II(%a)(~sf)II~ }.

Lemma I.ii shows that

7; II%MaPsflI22 d Ss =< Const Iiall 2 7; H%fH22 d Ss

+ Const flU ]Qsa(X) Psf(x) 12 dXsdS

_-< Const Ilall 2 llfIl~ + Const /7 ]Qsa(X) Ps * f(x)12 U

2 2 2 _-< Const {llall + IIaIIBMO} I]f]l -<_ Const IIall 2 llfll ,

which gives IILIIII2, 2 ~ Const llall, . Thus

the dual operator of LI, we have IIL 2 I12, 2

(1.4).

dx ds s

IILII]2,2 ~ Const Ilall~ . Since L 2 is

=If LIII2, 2 • Consequently, (1.14) gives

• s §1.9. Almost orthogonality (Davld-Journe [19])

David-Journ~ [19] showed the so-called T1 theorem. David-Journe-Semmes [20]

showed the so-called Tb theorem (cf. McIntosh-Meyer [40]). These theorems give

immediately Theorem A. Given 6 > 0, we use C 6 for various constants depending

only on 5 ; the value of C 6 differs in general from one occurrence to another.

For 0 < 5 ~ i, we say that a kernel K(x,y) (x # y; x,y E ~) is a 8-standard

kernel if there exists B > 0 such that

16

We denote by

inequalities,

t K ( x , y ) l N B / I x - y l ,

I K ( x , y ) - K ( x ' , y ) l -<- BIx-x'l 6 / t x - y l 1+6 (Ix-x'l =< I x - y l / 2 ) ,

IK(x,Y) - K ( x , Y ' ) t =< BIy-Y ' I 5 / Ix -Yi 1+8 ( IY-Y'I =<- / x - y I / 2 ) -

m 8 (K) the minimum of constants B satisfying the above three

For the sake of simplicity, we assume that

(I.15)

(1.16)

K(x,y) is anti-symmetric, i.e., K(x,y) = -K(y,x),

Kl(x) = lim fg K(x,y) dy s ~0 < I x-yl < 1

+ lim fl Ix-yl < i/e K(x,y)dy exists a.e. s~ 0 <

We write simply K the operator defined by the kernel K(x,y). The following

theorem is a special case of the TI theorem; we rewrite the TI theorem so that only

the proof of Theorem A can be given.

Theorem 1.12 ([19]). Let K(x,y) be a 6-standard kernel satisfying (1.15) and

(1.16). Then IlK]f2, 2 ~ C 8 {IIKIlIBM 0 + ~8(K)} .

Integration by parts shows that T[a]l =~ H a. We see that

IIHaIIBM O ~ Const IIail~ and ~l(T[a]) ~ Const !fall . (See eemma 2.5.) Hence this

theorem immediately yields Theorem A.

Lemma 1.13 ([19]). For b ( BMO, there exists an anti-symmetric l-standard kernel

e(x,y) such that El = b, []elI2, 2 ~ Const IIBIIBM 0 and el(L) N Const I]BIIBM O.

Proof. Let L be an operator defined by

ds ef = 2 f; %{(~b)(Psf) } d~Ss + 2 f; Ps{(~b)(Qsf)} °~-

Then its kernel L(x,y) is given by

dt ds L(x,y) = Censt {// Vs(X-t)(Vs*b)(t)Us(t-y)

U s dt ds + flu Us(X-t)(Vs*b)(t)Vs(t-y) T } '

where Us(X) = (i/s)e -]Xi/s , v s(x) = sign(x/S)Us(X) (x ( ~, s > 0) and

L m

U = {(t,s); t ( ~, s > 0}. Then L(x,y) is anti-symmetric and

L1. = 2 f~ QQb d ss = b . U s

2 s i n c e I V s * b ( t ) ' t 2 d t d s / s i s a C a r t e s o n m e a s u r e w i t h c o n s t a n t C o n s t ITbtlBM 0 ,

we h a v e SLIt2, 2 ~ C o n s t HblIBM o .

I t r e m a i n s to p r o v e In t h e same manner a s i n el(L) ~ Const IIbIIBM 0 •

17

Lemma 1.2, we have IIVs*bll = ~ Const IIblIBM O. Since IVs(X)l ~ Us(X),

dt ds [L(x,y)l ~ Const HDIIBM O flu Us(X-t)Us(t-Y) s

d t So i e -I/s ds Const IIBIIBM O S_~ (Ix-tl + it_yl) 2 s3

=< Const II bll BMO/I x-yl

Since lU's(X) I -<- Us(X)/S, IV's<X>I < Us(X)/S (x # 0), we have

dt ds 1 8~ L(x,y) I ~ Const IIBIIBM 0 fS U Us(X-t)Us(Y-t) 2

s

dt ~ i -i/s = Const llbIIBM 0 f~ (ix_tl + it_yl)3 f0 ~es

Const IIbllBMo/IX-yl 2

Thus c01(L) =< ConstllbIIBM O .

ds

Q.E.D.

Here is the main tool for the proof of Theorem 1.12.

operator L from L 2 to itself is anti-self adjoint if

any f, g ~ L 2.

We say that an

(ef,g) = -(f,Lg) for

Lemma 1.14 (Cotlar's Lemma [16]). Let @(t) be a function from [0, ~) to

itself and let {Lk}~=_N be anti-self adjoint operators from L 2 to itself such

that llejLkll2, 2 ~ o(lJ - kl) (j,k = O, il ..... iN). Then

llZ ~=_#kI12,2 < Const E 2N = k= 0 ~o(k) •

N Proof. The following proof is due to Fefferman [25]. Let L = Ek=_~ k. Then,

for any M g i, IIe2MII I/2M 2,2 = IIell2, 2. We have

L 2M = % .. -N~kl,...,k2M ~ N LklLk2 " Lk2M

Since

• 112, 2 • II HLklLk2 "" Lk2 M ~ PILklLk2ll2,2 " "]ILk2M_ILk2M 2,2

p(Ikl-k21) .,. @(Ik2M_I - k2M I)

and

IILkl...Lk2MII2,2 =< IILklIl2, 2 IILk2Lk3112,2 -.- IILk2M_ 2 Lk2M_III2,2 IILk2MII2,2

~ P(Ik2-k3 I) ... @(IkmM_2-k2M_l I) " P~ ,

18

we have

2 IF Lkl..-Lk2 M H 2,2 ~ p(O) p(l kl-k21 )P(I k2-k31 ) ... p(l k2M_l-k2Ml )"

Thus

I}LN2,2 = IIL2~II/2M2,2

{Vp (0) z

-N~ kl,...,k2M

{gp(o) z -N ~ kl,...,k2M_l

• .. ~ { ~p(0)(2N+l)(2

~P(I kl-k21 )P(I k2-k31 ) -.-P(I k2M_l-k2Ml ) }II2M =< N

~P(I kl-k21 ).. ,P(I k2M_2-k2M_iI ) =< N

2N }i/2M ×(2 z Vp(j))

j=0

2N % OV~) 2M} 1 / 2M

j=O

2N < p(0) I/4M (2N+I) I/2M 2 Z I/p(j) .

j=0

Letting M tend to infinity, we obtain the required inequality. Q.E.D.

We now give the proof of Theorem 1.12. We may assume that K(x,y) is real-

valued. Since KI ~ BMO, we can define, by Lemma 1.13, an anti-symmetric kernel

e(x,y) so that KI = LI, llelI2, 2 ~ Const IIKIIIBM 0 and

~8(L) ~ ~I(L) ~ Const IIKII!BM O. Consider the kernel K(x,y) - L(x,y). Then this

is anti-symmetric and satisfies (K-L)1 = 0,

~08(K - L) _-< Const {IIKIIIBM O + ¢08(K) } .

Hence from the beginning, we assume KI = 0, and show IIKII2, 2 ~ C8~8(K).

this, we may assume that ~5(K) = i. Choosing h ( C~ so that

0 =< h(x) < i, h(x) = h(-x), supp(h) c [-i,i], []h[] 1 = i,

To do

we put

Kk(X,Y) = I_: i : K(x-s,y-t){hk(S)hk(t)- hk+l(S)hk+l(t)} ds dt (k = O,tl .... ),

where hk(X) = 2-~(2-kx). Then K k is anti-self adjoint, Kkl = 0 and

K = lim k ~ ~ Zk=_NN K k. We show that

(1.17) IKk(X,y) I & C 8 2-k/{l + Ix-yl2-k} I+6,

19

8 (i.18) I ~-x Kk(X'Y) l < C8 2-2k/{i + Ix-yl2-k} 1+8

If Ix-yl ~ 4 ° 2 k, then we have

IKk(X,Y) l

= I 7_~ /_~{K(x-s,y-t)-K(x,Y)}{hk(S)hk(t)-hk+l(S)hk+l(t)}

Isl 6 + Itl~_ /_: /~ c 8 ix_ylZ+5 {hk(S)hk(t) + hk+l(S)hk+l(t)}

C8 28k/[x_yll+ 8 ~ C8 2-k/ {i + Ix-y[2-k} I+8

ds dt I

ds dt

Let I, J be two intervals in an interval L. Since K(x,y) is anti-symmetric,

we have /IN J fIN J K(s,t) ds dt = 0, and hence

(1.19) I /I {/J K(s't)ds}dtl = I/l-(In J) {$J K(s,t)ds} dt

+ / I N J { /J - ( INJ) K(s,t)ds}dt I < 2 / LI ds dt < Const ILl. = S +t =

Integration by parts and (1.19) show that, if Ix-yl < 4 " 2 k, then

IKk(X,Y) I

v = I /_~ /_~ {/~ /0 K(x-s,y-t)ds dt} {h~(u)h~(v)-h~+l(U>h~+l(V)} du dv I

/_~ / ~ (Const 2 k) {lh~(u)h~(v) I + lh~+l(U)h~+l(V)l} du dv

Const 2 -k < C 8 2-k/{l + Ix-yl2-k} I+8 =

Thus (1.17) holds. If Ix-yl ~ 4 " 2 k, then

18~ Kk(X,Y) l

_~ -~ - ' t dt I If ~ 7 "{K(x-s,y-t)-K(x,y)} {h{(S)hk(t) hk+l(S)hk+l( )} ds

/_~ /_~ C8-Jsl 6 + I t15 {lh~(s) lhk(t ) + lh~+l(S) lhk+l(t)} ds dt Ix_yj 1+5

C 8 2 (8-1)k ~ C 8 2-2k/ {I + Ix-yl2-k} I+8

If Ix-y[ < 4 • 2 k, then

20

[fi ~(x,Y) [

~ V = [f_~ f_~ {fo f0 K(x-s,y-t)dsdt}{h~(u)h~(v) - h~+l(U)h~+l(V )} du dv [

Const 2 -2k ~ C 8 2-2k/{I + ]x-yl2-k} I+6 .

Thus (1.18) holds.

We now show that, for k ~ g,

(1.20)

where

[(KkKe)(x,y)[ -<- C 8 Pk,e(x-Y),

pk,g(s) = 2-(k+~)/2(i + [s[2-g) -(I/2)-8

+2 (~8/2) {2-2k]sll-(8/2) + 2-kls] -8/2} (i + [sl 2-k) -I-8 .

Let I be the interval of midpoint y and of length Ix-yl/2. By (1.17), we have

[fic Kk(X,s)K (s,y)ds[ ~ C 6 fic 2-k 2-g

(i + Jx-sI2-k) I+6 (i + Is-yl2-g) I+8 ds

2 -g/2 ® 2 -k C 6 -~ (x-y). (i + Ix-yl2-g) (I/2)+~ f 2-k)i+8 1/2 ds < (1 + Ix-sl Ix-sl = CsPk'~

By Kgl(y) = 0, we have

If I Kk(X,s)Kg(s,y)ds I ~ If I {Kk(X,s)-Kk(X,y)} Kg(s,y)ds I

+ IKk(X,y) f I K~(s,y)dsl

+ IKk(X,Y) / cK~(s,y)dsl I

= Jfl {Kk(X'S) - Kk(X'Y)} Kg(s,y)dsJ

( = Ll(X,y ) + L2(x,y), say).

By (1.17) and (1.18), we have

2-2k I Ll(X,y) ~ C 8 f I s-Yl

(i + Ix-yle-k) I+8

--< C 8 2-2k+(gS/2)

(i + Ix-yl2-k) I+6

2-~ (i + Is-yI2-g) I+8

fl Is-y1-6/2 ds _<- C 8 pk,g(x-y)

ds

and

21

2 -k 2 L2(x,y ) -<_ C 6 f 2_Z)146 ds

(i + I x-yl2-k) I+ 5 I c (i + I s-yl

2-k+ (g 6 / 2 ) C 8 (i + Ix-yl 2-k) 146

flC Is-yl -I-(5/2) ds =< C 6 Pk,~(x-y)-

Thus (1.20) holds. Since llpk,~II 1 ~ C 8 2 -(k-g)/2, we have

!IKkKgll2, 2 ~ C 6 2 -(k-Z)/2 Hence Lemma 1.14 gives IIZ~=_N Kkll2, 2 ~ C 6 (N ~ I).

Letting N tend to infinity, we obtain IIKII2, 2 ~ C 5 . This completes the proof

of Theorem 1.12.

§i. I0. Interpolation (Lemarie [36])

In this section, we give a proof of Theorem 1.12 (in the case of K1 = 0)

by interpolation, which was given by Lemarie. For a 6 ~ with I~I < i,

let E denote the Banach space of distributions obtained from the completion of

C O with respect to the norm lllflll~ = {f_~ I~I ~ I~(~)I 2 d<} I/2, where ~ is the

Fourier transform of f (in the sense of distributions). For 0 < a < I, let

[E ,E_~] denote the Banach space of distributions f in E ~ E_~ with norm

llfIllnt,~ = {~ B(s,f)2 d s 2 }1/2 < = , $

where

B(s,f) = inf{(IIlgill2+s2ilihllI2 )i/2;_ f = g + h, g ~ E , h E E_~ }-

Lemarie showed the following two facts.

Lemma 1.15 ([36]). Let K(x,y) be a 6-standard kernel satisfying (1.15), (1.16)

~6(K) where and El = 0. Then, for any 0 < a < min{l,26}, IllKIll~,~ C , 6 ,

IIIKI~I~,~ is the norm of K as an operator from E to itself and C 6 is a

constant depending only on ~ and 6 .

Proof.

and 6. We may assume that ~8(K) = i. Note that

- I f (x)-f(y)~ INfHI~ = c a f_: f ~ ix_yil+~ dx dy

Given x, y E [, we write by I the interval of midpoint

21x-y I . By KI = O, we have

Throughout the proof, we use C 8 for various constants depending only on

(f E E a) .

(x+y)/2 and of length

22

IKf(x) - Kf(y)l = I f~{K(x,s) - K(y,s)} f(s)dsl

I flK(x,s)(f(s)-f(x)) ds - f : K(y,s)(f(s)-f(y))dsl

ifI K(x,s)(f(s)-f(x))ds f I K(y,s)(f(s) - f(y))ds

+ f {K(x,s) - K(y,s)} (f(s) - f(x))ds - (f(x) - f(y)) f K(y,s)dsl i c i c

fI If(s) - f(x) I/Is-xl ds + fI If(s) - f(Y) I/Is-Yl ds

+ c~ ~ If(s)-f(x)l Ix-yl~lls-~l 1+s ds + If(x)-f(y)i Fie K(y,s)dsl

( = Ll(X,y) + L2(x,y) + C 5 L3(x,y) + L4(x,y) , say).

Hence

2 C f : f - : iKf(x)_Kf(y ) [,2 IllKfilla = a ix_y]l+~ dx dy

2 4 Lk(X ,y)

< C Z f ~ f : dx dy = a , 5 k=l -~ - I x-Y i I+~

Choosing 0 < ~ < 1/2 so that a + 2~ > i, we have

Ll(X,y) 2 =< Ca fls-xl < 21x-yl If(s)-f(x)12ls_×12(l-~ ), ds

and hence

' =< C a f ~ f " L 1 _® -~

= c N1flll~ .

I f ( s ) - f ( x ) l 2

Is_~12(1-~5

In the same manner D

2(1 + 6) Y > 1 and

{ J'21x-Yl > I~-xl

L~ ~ C IIIflll~ • Choosing 0 < T t~-yl12

and hence

, ~ If(s)-f(x)I 2 e 3 < Ca, 6 f ® f ~ i s_x -- - 12(1+6) (l-Y)

= %,5 I I If l l l~ •

[f(s)-f(x)l 2 25-2(i+~)~ + i Ix-yl is_xi2(l+6)(l-y)

{flx_yl/2< Is_xllX-y125-2(l+6)Y-~dY} dsdx

23

Since lSlC K(y,s)dsl ~ i $1y-~l > Ix-yll2 K(y's)dsl + Const, we have

' < K(y,s)ds i + 1} . L 4 = C o n s t t l l f l l la ( sup i f l y - s t > y E ~t,~ >0

Given Y0 ( R, s > 0, we have, with J = (Y0 - s, Y0 + g) and J = (Y0-2g,Y0 + 2g),

+ !_, Sj $j l~c K(Y0's)dsl = lSjc K(Y0>s)ds ij I

= i ~ Sj {Sjc (K(Y0,S) - K(y,s))ds} dy

ds < (Const/IJi) Sj {S , ~ }dy

J -J

ly0-yl ~ + (Cs/iJi) Sj { Sj, c i1+6 as}

ly-s

2 Consequently we have Hence L 1 =< Ca, 8 lllfill~ .

gives the required inequality.

K(y,s)ds dyl

i fJI l f j { S j , _ j + S *c } l J

dy & C 8 .

IIIKflll~ ~ ca, ~ I l l f l l l~ , which

Q.E.D.

Lemma 1.16 ([36]). For any 0 < a < i, [E ,E_¢] = L 2. More precisely,

(I/C)NflI 2 ~ llfilint, ¢ ~ c a llflI 2 (f (L2).

Proof. Given f 6 L2~ s > O, we put

2 D f dt, h = 2 s -I/a D = -- S®-I/~ t2D 2 s SO t2D 2 gs ~ s r + "~ r +

f dt.

Then f = gs + hs and

IIfI!2Int,a = S; B(s,f) 2 ~ --< S 0 {Iilgslii2a S

I ~I dt) 2 4 E I_; S°_li= a s i + t2E

s-i/a _ill at)2 1 } l~(~)I 2 d E ] ds + ( S0 i + t2~ 2 IE I a

= c So E s_Z i'li dr) a s i + t2E 2 s

-lla = _ I~f d t ) 1 } I7 (< ) I 2 d E ] ds

+ ( SO 1 + t2~ ~ I~1 a

ds : c a S_l 11(O12 {7o (I'_~ -7 ) t2E2

t s i +

ds + S 2 liihsNi2a } --~ S

2 S

t -c~ - - I ~ I 1 - L dt} dK dt +dO(f0 ds) l+teiGi2

2 = C a d'_] If(E) l 2 d~ = C a IIfll 2 .

24

Let f ~ [E ,E_~].

f = gs + hs and

For each s > 0, we choose gs ~ Ea' hs

lllgslll2 + s 2 lllhslll2 < 2B(s,f)2 Then -(l = °

E E so that -<l

2 II fll 2 = Const 7 0 II tD fll dt

2 ~- I + (tD)

Const J" { II tD g-a tl + II tD h _all } ~- I + (tD) 2 t r + (tD) 2 t

Const fO [ j'-~ (t~)2 (i + (t~)2) 2 {lgt_~(~)12 + lht- a(~)12 } d~ ] d tt

oo Const 70 [7_~ ~ {t a l~lalg _a(~)l 2 + t -a I~I -a IE _a(~)l 2} d~] d tt

t t

dt ca fo ( IIIg _JII 2 + t -2c~ ]llh -~1112~- ) 1-~ t t t

O -- < ~ = C-IIfi[2 Cc ~ f ([ilgsii[2 + s 2 iiihsiii2a ) ds Ca B(s f)2 as 2 = f0 ' --2 ~ Int,a " s s

Q.E.D.

Theorem 1.12 is deduced as follows. We may assume that K1 = 0 and

~05(K) = i. We use Lemmas 1.15 and 1.16 with a = 6/2. Let f ~ L 2. Then,

for each s > 0, we can choose gs E E a, hsE E_a so that f = gs + hs'

Iligs III2 + IIIhs IiI2-a -<- 2 B(s,f) 2, by Lemma 1.15. Thus Lemmas 1.15 and 1.16 show that

[[Kfi[22 -< C6 [[Kfll~nt,a = 2 C6 /0 B(s'Kf)2 d s s

=< C6 J'o (INKgs [[12 + s2 [IIKhs Iii2-a ) ~ds s

2 + s 2 [IlhslIi2a)--2as < C fo B(s'f)2 __ds -<- C6 fO ([[[gsIiia - = 6 2 s s

= C 6 II fII2nt,c~ < c6 IlfIl~.

§i.ii Successive compositions of kernels

Meyer [41] also gave a proof of Theorem 1,12 from the point of view of

composition of kernels. Using his method, we show the following lemma which also

yields Theorem 1.12.

Lemma 1.17.

K1 = 0 and

(1.21)

Let K(x,y) be a 6-standard kernel satisfying (1.15), (1.16),

sup IK(x,y) I (i + Ix-yI) I+5 < ~.

x,y E

25

We define kernels {K(n)(x,y)}~=l by

K(1)(x,y) = K(x,y), K(n)(x,y) = f_~ K(x,s)K(n-l)(s,y)ds (n a 2),

and define

~(K (n)) = sup { ~ I~ I K (n) Xl(X)dx I ; I interval} (n a I).

Then, for any 0 < s < 6,

~(K (n)) + ~8_s(K (n)) ~ C n ~6(K) n 8~g

where Cs,g is a constant depending only on 8, s.

Postponing the proof later, we now deduce Theorem 1.12 (in the case of

KI = 0) from this lemma. (This lemma plays the role of Cotlar's lemma.)

Without loss of generality, we may assume that K(x,y) is real-valued,

and K1 = 0. Using Kk(X,y) in §1.9, if necessary, we may assume that

satisfies (1.21). We put

i (1.22) ~(K) = sup { ~ o(l,K,f); f 6 Lreal,l I interval},

where, in general,

(1.23) Lreal,~ = {f 6 L~; llflI~ --< ~ , f real-valued } (6 > 0)

(1.24) o(l,K(n),f) = fl IK(n)(xI f)(x)l dx (n ~ i).

and

Then

IIKH2,2 <_ Const {o(K) + C8~08(K)} -<_ Const {o(K) + C 5} •

(See Lemma 2.5 in Chapter II.) For n >-_ I, f ~ Lreal,l and an interval

we have

1 n-i 1 / =IK(2n-l)(xif)(x)l 2 dx}i/2 ~f~ ~(I'K(2 )'f) =< {4 -=

_2 n 12 = {~ 7i f(x)~( )(Xlf)(x)dx} / --< { ~ o(l,K(2n),f)} I/2

and hence

i i n -n Tf ~ ~(l,K,f) _<- { ~ o(I,K (2) f)}2 .

~06(K) = I

K(x,y)

I,

26

For a while, we assume that Xlf is a step function, saying

XI f = Zk=IN ~k Xlk (l~kl ~ I) I k interval, I k DIg = ~ (k # g)).

inequality and Lemma 1.17 show that

{ ~- ~(l,K(2n),f)} 2 ~ ~ 171 f(x) K <2n+l) (Xlf)(x)dxl

Then Shwartz's

1 N K(2~+I) ~T % I: I f(x) Xlk(X)dxl

k=l

N :on+l, :^n+l. N d

< i kEllflk Kk~ :X (x) dx I + ~5(K ~Z )) ~ k%l:l_ik{flk ~ }dx = ~f~ = I k =

~(K(2n+l)) (2 n+l) 2 n+l + Const N ~6(K ) ~ Const N C 6 ,

2n+l 2-n which shows that (i/If I) ~(l,K)f) ~ {Const N C 5 } . Letting n tend to

infinity, we have (i/II I) ~(l,K)f) ~ C 6. Since this inequality holds for any

N ~ i, ~k' Ik (i ~ k ~ N), we can remove the above assumption, i.e.,

(i/III) ~(l,K,f) ~ C 6 for the given f. Taking the supremum over all f ( e~eal,l

and all intervals I, we have ~(K) ~ C6, which implies Theorem 1.12.

We now give the proof of Lemma 1.17. Assuming that ~6(K) = i, we

inductively show that

~(K(n)) (K(n)) n (n g I) + ~8-e ~ CO '

C O is a constant depending only on 6, ~, and is determined later. Since

is anti-symmetric and ~6(K) = I, we have

~(K (I)) + ~5_8(K (I)) ~ 0 + ~5(K (I)) = I.

Suppose that the required inequality holds for n-l. For the sake of simplicity,

we use, from now, C for various constants depending only on 5, 8. First we

show that ~(K (n)) ~ CC~ -I For an interval I, we have

:I K(n) XI(x)dx = 7~ {:I :I K(x's)K(n-I)(s)Y) dx dy} ds

wheT e

K(x ,y)

7I {:I :I } + :ic {II fI } = LI + L2 '

dx

and

ILl1 ~ { I I IKXi(x) I2dx} I/2 {I I IK (n-l) ki(x) l 2 dx} I/2 .

27

Since K(n-l)l = 0, we have

fl IK(n-I) El (x) 12 dx = fl IK(n-l) XlC(X) I2 dx

where

< =

< =

I

2 71 IK (n-l) X (x) l 2 dx + 2 71 IK (n-l) X , (x) l 2 dx I *c I -I

_2n-2 2 /I IK(n-l) Xl*c(X) 12dx + 2 C 0 71 (f * dx~_y T )2 dx

I -I

i,c(X) _2n-2, + C 2n-2 i i I, 2 fl I K(n-l) X I 2 dx + C C 0 Ill = 2 LI0 C O

is the double of I. Since

171 K (n-l) Xl,c(X)dx I ~ If I K (n-l) Xlc(x ) dx I + 171 K (n-l) X , (x)dx I I-I

n-l If I K(n-l) Xl(X)dxl + C O 71 ( 7 , ) dx

I -I

n-1 n-i ~(K (n-l)) III + C C O III < C C O III,

we have

2n-2 El0 ~ 2 71 IK (n-l) X ,c(X) - (K (n-l) X ,c)112 dx + C C O III

I I

< 2 _2n-2 = ii12 71 { fl ( fl*C IK(n-l)(x'y)-K(n-l)(s'y)Idy)ds}2 dx + C C 0 II

(cc~n-2/lli2) fl {fl ( /l*C - Ix~slS-¢ 2n-2 ii 1 ix_yll+8_e ) ds} 2 dx + C C O

CC~ n-2 l If.

2n-2 Thus fl IK(n-l) Xl (x) 12 dx -<_ C C O III. In the same manner,

fl IKXI (x) 12 dx ~ CIII, and hence Iell =< CCo -I iii. Consequently we have

If I KXi(x)dx I <= C C0-1 III. Since I is arbitrary, we obtain ~(K (n)) < CC0 -I

° -I Next we show that ~o6_e(K(n)) -<_ C C In the same manner as in the

estimate of ILII, we have

sup Iflx_y I > g K(n-l)(x,y)dyl ~ C CO-I , x ~ ~,e> 0

sup Iflx-y I > s K(x'y) dyl < C. ~E ~,c >0

28

For x, y ( ~, we have

K(n)(x,y) = f ~ K(x,s)K(n-l)(s,y)ds

i + i + l : f I i + f l 2' + f I~ = LI L2 L3 '

I t 12, 1 ! is the interval of midpoint x and of length I x-yl 12 where I 1 ! =

interval of midpoint y and of length I x - y l / 2 and 13 (1~ U I~) c

have

is the

We

- ds n-i flx-yI IL~I ___ Co l Si ~ - -pT--~--~S ---~-~[- ~ c c o

= K (n-l) (s, y) dsl IL~I dsl + IK(x,Y)IIII~

....... n-1 o-l/l~_yl I s-yj ~ i i c c o ~ c c n-i fl ~ _-< c c o l x-yl 1+5 -T~_y T ds + ] ~ i ~ -

and

IL i l <-- Isq n-1

n-i 1 I s - x l S - s ds + C C 0

=< c c o $i i ~ ix_yll+~_ ~

I%-1 x T , Thus IK(n)(x,y>l _<- c c 0 llx-yl For x, y (

we have

K(x,s) {K (n-l) (s,y)-K (n-l) (x,y) }dsl + IK (n-l) (x,Y) I IfiiK(x,s)dsl

n-I c -_< c C 0 /[x-yl.

I ! = (X -- where I 1

31xTyl 13 (x - 4

with IX-X'I ~ t x - y I / 2 ,

K (n)(x,y) - K (n)(x',y) = f_®{K(x,s)-K(x ,s)} K (n-l)(s,y) ds

6 6 = II

= Z fl~ Z L k , k=l k=l

Ix-x'i Ix-xii i, = (x' Ix-x'L x,+ i~i[) 2 , x+ 2 )' I2 - 2 ' 2

3 x ~ 4 ,, I x - x ' l , y + ~ ' l , x + ) - (I~' U 12), I~ = (y - i0 i0 "

I'~ = (y - Ix-Y~4 ' y + Ix?--~i)4 - I~ and I~ = (I~ U...U I~) e. We have

- ~ i n-i , Ie~l < C C 0 1 fl~ ix_sll+8 -T-~ ds < c c O x-x 18/Ix-yl I+6

_n-i , 8-6 / 1+6-6 _-< c ~o Ix-x I I ~ - y l ,

29

and

Since

= n-i ,,l~-x'l 8 f ~ ]-~cFfds IL~I < c c o ix_ytl_ ~

n-i _ l~-x'l 5 C C O Ix_yll+6 log(C ) C C~ -I Ix-x'18-g/Ix-yl l+8-s,

IL~I ~ Ifl~ {K(x,s)-K(x',s)-K(x,y) + K(x',y)} K(n-i)(s,y) ds j

+ IK(x,y) - K(x',y) l Ifl~ K(n-l)(s,y)dsl

n-~ s~ { _Isnyl ~ l~-yl ~ l C C 0 ix_yll+8 + } ds Ix'_yl I+8

n-i ix_x,18/ix_yll+8 + +

I x-y

-_< C C 0n-I ix_x,18-S/ix_yjl+8-s

EL = fl~ {K(x,s)-K(x',s)} {K(n-l)(s,y) - K(n-l)(x',y)} ds

+ K (n-l)(x',y) ~I~ {K(x,s) - K(x',s)} as = ELI + e~0 ,

L~ = ]I~ {K(x,s)-K(x',s)} {K(n-l)(s,y) - K(n-l)(x',y)} ds

+ K(n-l)(x''Y) fl~ {K(x,s)-K(x',s)} ds = e21 + L20

L~ = fI~ {K(x,s)-K(x',S)} {K (n-l)(s,y) - K (n-l)(x,y)} ds

+ {K(n-I)(x'Y) - K(n-l)(x''Y)} fI~' {K(x,s) - K(x',s)} ds

+ K(n-l)(x''Y) 7I~' {K(x,s) - K(x',s)} ds = LII + n12 + L"10.

IL~z I S C C~ -I $Z~ "Ix-x'j8 Js-x'jS-c " ix,_sl l+~ ix_yl I+6-~

c c o~-I l~_~,ls-~/ix_y11+8-~

,, i Is-x'l ~-~ Ie211 ~ C Cg -I fl~ ~ --ix_yll+8_g as

n-i 18-g/ l+8-g C C 0 Ix-x' Ix-yl ,

ds

30

,, n-i i I s-xJ 6-s ]~111 _-< c c O Ii~, ]-~ i x_y l l+~_d

n-i I x-x' 16-81 i+s-5 s C C 0 I x-yl

ds

and

IL~21 ~ c~ -I Ix-x'16-8 {lli~ K(x,s)ds I +I ,, ds Ix-yP I+5-8 Ii I~'-sl

C C 0n-I Ix-x' 16-e/Ix-yl I+6-8

" = " O " U I~ , we have, with I 0 I 1 12

. . . X-X t 18-8 IK(n)(x,Y) - K(n)(x',y)I $ ILl0 + L20 + L30 I + C C~ -I x_yll+6_s

n-I _!x-x'[ 8-g = IK(n-l)(x',y)IIYI~ {K(x,s)-K(x',s)}dsl + C C O

18-g i n-i Jx-x[14_6_8 -< C Co -I ~_y~ I 71~ c {K(x,s)-K(x',s)}dsl + C C O Ix-Y

1 ..... Ix-x'18 ds + C n-i Ix-x'l 6-8

n-i ix_x,18-S/ix_yll+8-¢ C C O

Since K(n)(x,y) is either anti-symmetric or symmetric, we have also

n-i ly_y,18-g/ix_yil+8-g IK(n)(x,y) - K(n)(x,y')l ~ C C O

n-i Consequently, for x, y, y' ( IR with IY-Y'I ~ Ix-Yl/2. Thus ~8_g(K (n)) ~ C C O •

~(K(n)) ~8_¢(K(n) n-i ~8 s (K(n) n if + ) ~ C C O This shows that ~(K (n)) + _ ) a C O

C O is large enough. This completes the proof of Lemma 1.17.

In Chapter I, we showed 8 proofs of the boundedness of the CalderSn

commutator T[']. Since the Calder~n commutator is closely related to analycity

of functions, it seems necessary to give more proofs and to have a unified

understanding.

CHAPTER II. A REAL VARIABLE METHOD FOR THE CAUCHY

TRANSFORM ON GRAPHS

§2.1. Coifman-Mclntosh-Meyer's Theorem ([7])

For a real-valued locally integrable function a, we define a kernel by

(2.1) C[a](x,y) = i/{(x-y) + i(A(x) - A(y))},

where A is a primitive of a. We write simply by C[a] the singular integral

operator defined by the kernel (2.1). This is called the Cauchy transform of

Calder~n on a graph {(x, A(x)); x E ~}. We put

Lreal~ = ~ >U 0 Lreal,~ = {a E L®; a is real-valued}.

(See (1.23).) Coifman-McIntosh-Meyer showed

Theorem B ([7]). The norm IIC[a]II2,2 is bounded if a E Lreal"

The operator C[a] is expressed formally in the following form

C[a] (-~)H + Z (-i) n = Tn[a], n=l

where Tl[a] = T[a] (the Calder~n commutator) and Tn[a] is the n-th Coifman-

Meyer commutator (n ~ 2), i.e., Tn[a ] is an operator defined by

(2.2) Tn[a](x,y) = (A(x) - A(y))n/(x-y) n+l.

Prior to this theorem, the following three theorems were shown. CalderSn showed

that llTl[a]II2, 2 ~ Const Ilall= (a E L~), Coifman-Meyer [9] showed that

(2.3) IITn[a]II2,2 ~ Const n! llalI~ (a E L ~, n ~ 2)

and Calder~n showed that

(2.4) IIC[a]ll2, 2 is bounded if llall~ (a E ereal) is small enough.

At present, there are three proofs of Theorem B; the original proof, a proof by the

Tb theorem [40] and a proof by perturbation. In this chapter, we show a self-

contained proof by perturbation. A proof by perturbation was first given by

Calder~n [4] and David [17]. Improving their methods and repeating a simple

perturbation method, we shall deduce Theorem B only from the boundedness of H

([17], [42], [45]). (See APPENDIX II.)

32

§2.2. Two basic principles (Zygmund [54])

Here are two basic principles in real analysis.

Coverin$ Lemma. Let {~}X E A

IUxEA ~I < = • Then there exists a sequence

intervals such that

iux~ A ~I-< 5 k=iE lIXkl.

be a family of intervals in ~ such that

{I~k}k=la of mutually disjoint

The proof is as follows. Let IXI be an interval such that 211XII is

larger than the supremum of llxI over all X E A . Suppose that

~i' "''' ~k-i have been chosen. Let IXk be an interval such that 211~kl

larger than the supremum of II~I over all X E Ak_ I, where A 0 = A and

Ak_ 1 = {X E A ; Ik n IX. = ~ (i ~ j ~ k-l)} (k ~ 2). (If Ak_ 1 = ~, we stop our J

induction at k-l.)

Now we show that {IXk } is the required sequence. We first assume that

{IXk } is an infinite sequence. Since the intervals are mutually disjoint and

IUk= I IXk I < ~ , we have lim k ~ ~ llXk I = 0. For IX, there exists IX. 3

such that [IxI > 211X. I, which implies that X ~ A.. Hence j J

{j; X ~ Aj} # ~ . Let k be the smallest integer in the set. Then

IIxI ~ 211kk I , according to the definition of our choice. Since X ~ Ak,

we have 1% n Ixk # ~, which gives that I X c IXk , where IXk is the interval of

the same midpoint as and of length 51 I" Thus I~ k IX k

I UhE21 Ikl --< I U IXk I --< 5 % llXkl • k=l k=l

is

If {IXk } is a finite sequence, each I X intersects with U IXk" Hence,

in the same manner, we have the required inequality. This completes the proof of

this lemma.

Risin$ Sun Lemma. Let a be a function in an interval I such that

~ a(x) ~ ~ for any x E I, where ~ ~ O. Let A be a primitive of a. For

( ~ ~ T ~ ~ ), we define a function B in I by B(x) = inf ~(x), where

the infimum is taken over all functions ~ such that ~ ~ A, ~' ~ Y a.e. on I.

33

Let b = B' and ~ = {x 6 I; A(x) # B(x)} = Uk= IIk, where {Ik}k= I

components of ~ . Then

(2.5) Y _-< b(x) _-< ~ a.e. on I,

(2.6) b(x) = Y (x 6 ~),

I (2.7) (a)l k _-< Y ((a)l k = ~ fl k a(s)ds, k >-_ i),

- (b)l i (2.8) I~I--< ~- T III ((b)l = "I~[ ~I b(s)ds).

are the

I

I

i z

Inequalities (2.5)-(2.7) are easily seen. We have

(b) I III = 71 b(s)ds = 71_62 + 7~

which gives (2.8). For the sake of convenience, we call this rising sun lemma RSL

of Type i (7-r~y,8-~e£~t); we shall use later various rising sun lemmas. For

an open set ~ , we denote by {I~ ,k}k=l its components. The following two lemmas

are also the rising sun lemmas for integrable functions.

Lemma 2.1 (The Calder~n-Zygmund decomposition [35, p. 12]). Let f E L I and

k > 0. Then there exists an open set ~ such that

= = ~c. I~I < llflll/k , (Ifl)l~,k k (k => i), If(x) l < k a.e. on

X TO see this, we put A(x) = f0 If(s) Ids (x > 0), and define a function B

in (0,~) by B(x) = sup ~(x), where the supremum is taken over all functions

such that ~ & A, ~' ~ k a.e. on (0,~). Let ~I = {x > 0; A(x) # B(x)}. Then

1 f~ If(s) Ids (Ifl) = ~ (k ~ i), l~ll ~ ~ • l~l,k

If(x) l ~ X a.e. on (0,®) - ~I "

34

Considering f(-x), we obtain, in the same manner, an open set

Then ~i U {-x; x E ~2 } is the required open set.


~2 in (0,~).

Lemma 2.2. Let f be an integrable function in an interval I and let k > 0

satisfy k > (Ifl)l. Then there exists an open set ~ in I such that

i I~I _-< ~ 71 If(x)I ds, (Ifl) I =<

~,k

If(x) l < k a.e. on I

(k>= i),

The (non-centered) maximal operator M is defined by Mf(x) = sup(Ifl) I, where

the supremum is taken over all intervals I containing x. For p > i, NMIIp,p

denotes the norm of M as an operator from L p to itself. The following lemma

is deduced from Covering Lemma.

Lemma 2 . 3 ([35, p.7]). IIMIIp,p ~ Cp (p > i).

For f ( L I, X > O, we put Ek = {x; Mf(x) > ~}. For each x ( EX, we

can choose an interval I x containing x so that (Ifl) I > ~ . Covering Lemma X

shows that there exists a sequence {Ixk}k=l of mutually disjoint intervals such

that IE~) ~ 5 %k= I Ilxk), which yields that

5 5 /i k if(s) ids ~ ~ [iflll " Ix; Mf(x) > ~I ~ ~ k~ I

For f ( L p and k > O, we define fx by fk(x) = f(x) if If(x) l > k/2

fk(x) = 0 if If(x) I ~ k/2. Then

HM f[] = Cp f0 xP-l]x; Mf(x) > k]dx

and

Cp 15 k p-I {Ix; Mfx(x) > X/21 + Ix; M(f-fk)(x) > k/2 I} dX

= Cp 75 xp-I ix ; MfN(x) > X / 2 ) dX

Cp /5 xp-2 llfkll I dX = Cp f~ xp-2 {fk/2 Ix;If(x) I > s I ds} dk

® 2s = Cp f0 Ix;If(x)] > sl { /0 xp-2 d~} ds = Cp llfll ,

which gives that IIMIIp,p Cp.

At last we note John-Nirenbergts inequality, which was used in Chapter I.

This is deduced from RSL. (For the proof of Theorem B, this is not necessary.)

Lemma 2.4 ([32]). Let f ( BMO and I be an interval. Then

35

Ix E I; If(x)-(f)l I > k I =< exp(- Const >OIIl (7~_> l).

§2.3.

the sake of simplicity, we deal with only kernels

(See §i.9.) We use the notation ~(K), ff(l,K,f)

standard kernel, we define an operator K by

= K(x,y)f(y)dy I . K f ( x ) sup I f l x _ y 1 ' ' > g E > 0

-function ([8], [35], [54])

In this section, we show a fundamental inequality for standard kernels.

K(x,y) satisfying (1.21).

in (1.22), (1.24). For a

For

We show

Lemma 2.5 ([35], p. 49). Let K(x,y)

Then IIK I12, 2 ~ Const o(K) + C 6 ~5(K).

We begin by showing

be a 6-standard kernel (satisfying (1.21)).

(2.9) ~(K ) ~ Const if(K) + C 6 ~6(K),

where ~(K ) is the supremum of (l/Ill) /I K (Xlf)(x)dx over all f E Lreal,l

and intervals I. For ¢ > O, f E Lreal,l, an interval I and a point x on

I, we put J' = (x - s/2, x + s/2), J = (x - s, x + g), g = %1 fl J f and

h = XI_ J f. If 0 < ~ < IiI, we have, for any s E J' ,

Iflx-y I > s K(x'y)(XIf)(y)dyl = IKh(x) J ~ IKh(s) J + IKh(x)-Kh(s)J

IKh(s) I + C 5 ~5(K) ~ IK(Xlf)(s) I + IKg(s) l + C 6 ~6(K)

= IX ,(s)K(Xlf)(s)l + IKg(s) I + C O ~b(K), I

where I is the double of I. Taking first the square roots of the first

quantity and the last three quantities, and taking next their means over J' with

respect to s, we obtain

Iflx-Y I > s K(x'y)(XIf)(y)dyII/2

M(I X , K(XIf)II/2)(x) + (IKgll/2)j, + C 6 ~6(K) I/2.

I

If S = > III, then /Ix-yl > s K(x'Y)(XIf)(y)dy = 0. Hence this inequality holds

for all s > 0, which shows that this inequality holds with the first quantity

* (x)i/2 replaced by K (Xif) . Taking the squares of both sides of the resulting

inequality, and using Shwartz's inequality, we obtain

36

K (Xif)(x) _-< Const M(IX , K(Xif ) I1 /2) (x) 2 I

1/2 2 + Const(iKg I )fl, + C 6 ~6(K).

Since

( M(IX ,K(Xlf) iI/2) < i IK(X (x) idx I 2) I = Const ~ fl* If)

< Const ~ {~(l,K,f) + ¢05(K) f , ( fl ~ dx} I -I

Const {~(K) + ~6(K)}

and

1/2.2 (IKgl )j, ~ (IKgl)j, & (IK(kj,g) l)j, + (IK(Xj_j,g))j,

we have

Let

~(K) + Const ~5(K),

(K*(XIf)) I ~ Const ~(K) + C 6 ~6(K), which implies (2.9).

f E L 2, K > 0. We show the following good k inequality:

* i (2.10) Ix; K f(x) > 3k, I@f(x) =< ~k I ~_ -~ Ix; K*f(x) > k I

where ~ > 0 is determined later. To prove this, it is sufficient to show that,

for each component I of {x; K f(x) > X} ,

* i Ix E I; K f(x) > 3X, Mf(x) ~ ~Ikl =< -~ llI.

If Mf(x) > ~k on I, this inequality evidently holds. Assuming that

M f(<) ~ ~k for some ~ E I, we prove

(2.11) Ix E I; K*f(x) > 3X} < 1 i1 l = - i O '

(See §1.4). Let g = Xjf and h = Xjcf, where J = (x 0 - 2II I, x 0 + 2III)

(x 0 is the left endpoint of I). Then we have

Ix E I; K*f(x) > 3X 1 k I

+ Ix E I; K*h(x) > 2k I (= L I + L2, say).

First we estimate L 2. Note that K h(x 0) ~ >~. For ~ > 0 and x E I, we have

37

Iflx-y I > e K(x'y)h(y)dy -71x0_Y 1 > g K(x0'Y)h(y)dy I

~ I K(x,y) - K(x0,Y)IIN(Y)I dy + Const oos(K) Mf~)

=< C6 oos(K) Mf(~) =< C 5 oo 6(K) ~.

Since g > 0 is arbitrary, we have, with a constant C8,1 depending only on 5,

(2.12) K*h(x) <= K*h(x 0) + C 6,1 c°5(K) ~X

-<_ {i + CS,IOOs(K)~}k (x E I).

This shows that L 2 = 0 if C5 ,I oo5 (K) D < i. Next we estimate L I. By

Lemma 2.1, there exists an open set ~ = Uk= 1 I k (I k = ~,k ) such that

I~I < Ilglll/(100DX), (Igl)l k = 100~X

I g(x)I _<- i00~)~ a.e. on ~c

(k > I),

We define a function g by

f J g(x) (x ~ ~e),

\ L (x ~ I k, k ~ i).

(g) l k

Then llgll~ i00~. Put ~* ~ * * = Uk= I I k , where I k is the double of I k. Then

I~*I < 21~I =< llglll/(50 "qX) --< Mf(~)IJl/(50 ~X) =< 111/15"

*c For s > 0 and x ~ ~ , there exist at most two intervals (saying I I and 12 )

which intersect with the boundary of (x - s, x + s). We have, with

x k = (the midpoint of Ik),

I71x-y I > s K(x,y)(g(y) - g(y))dy I = If(i IN 12) n (x-s,x+g) c K(x,y)(g(y)-g(y))dY

+ % {K(x,y)-K(X,Xk)} (g(Y) - g(y))dy I I kc (x-s, x+s) e flk

< + (Igl)12} = Const oo6(K) {(Igl)ll

38

+ C 0 co 6(K) Z (Igl) ~ C 6 co 6(K) ~ (i + A(x)). k=l (Ix-Xkl + Ilkl) I~5 I k

where &(x) : Sk= I "'[IkIl'~/(IX-Xk I + ''Ilkl) I+6. Since s > 0 is arbitrary, we

have

Since

J of

K g(x) ~ K g(x) + Cs~o6(K) q%(l + &(x)) (x e C'c).

supp(g) c J and I~l ~ III, the support of g is contained in the double

J. Hence (2.9) shows that

f , K g(x)dx =< f , K (X , g)(x)dx I-~ J J

--< ~(K ) llgll~ IJ*l =< {Const o(K) + C 0 ~8(K)} n~ Ill.

We have easily

I , {C 6 ~8(K) NX (i + 8(x)) } dx I-~

-<- c 6~8( K)~{l~i + I~i} ~_ c 0 ~8(K)~l~ l~J.

Consequently, we have, with an absolute constant C O and a constant C6, 2

depending only on 6,

(2.13) L I --< Ix E I - @ ; K g(x) > k I + I~*I

i * / , K g(x)dx + iII/15 I-~

i {K*~ T 7 , g(x) +c 0 ~8(K)n~(l +Mx))} dx + I~I/15 I-~

{(C O c(K) + C6, 2 ~8(K))~ + (1/15)} III .

Let

C6,1~6(K))-I ~6(K)) -I} = min {(2 , (30 C O c(K) + 30 C8, 2 •

Then (2,12) and (2.13) show that

• 1 l x E I; K f(x) > 3X l ~ L 1 + L 2 = L 1 ~ To III*

Thus (2.11) holds, which implies (2.10).

In the same manner as in §1.4, (2.10) yields that

llK*fll 2 ~ (Const/~) IIMfll 2 ~ {Const ~(K) + C 5 ~8(K)} llfH 2,

3g

which implies the required inequality in our lemma.

~2.4. A-priori estimates

In this section, we show some inequalities which play important roles later.

For an operator T from L 2 to itself, we put

where

(2.14) ~0(T) = sup {~ $(I, T, XI); I interval},

(2.15) $(T) = sup { $(I, T, f); f ~ Lreal,l, I interval},

(2.16) ~(T) = sup {~ ~(I, T, f); 0 =< f <= i, I interval},

(2.17) ~(I, T, f) = fl IT(XI f)(x)I2 dx,

(2.18) $(I, T, f) = fl IT(Xlf)(x)I 2 f(x)dx.

For an open set ~ with I~I < ~ , we put

(2.19) ~(T;~) = sup { ~(l,T,f); f ( Lreal,l, I component of ~} .

For a 8-standard kernel K(x,y), we have easily o(K) ~ ~(K) I/2 ~- IIKII2, 2 ,

and hence, by Lemma 2.5,

(2.20) ~(K) I/2 -_< IIKII2, 2 ~_ Const ~(K) I/2 + C 8 0~8(K).

For a non-negative measure ~ on ~, we denote by (.,.) the inner product with

respect to ~ , i.e., (f,g) = f~ fg d~. (In the case of the 1-dimension

Lebesgue measure, we omit the suffix.) Here is an inequality necessary for the

proof of Theorem B.

Lemma 2.6. Let I be an (open) interval, ~ = Uk= 1 I k (I k = I2,k) be an

open set in I and let K(x,y), T(x,y) be two 8-standard kernels such that

K(x,y) = T(x,y) for any x, y ~ I - ~, x # y. Then, for any u, v ~ L 2

supported on ~ and a non-negative measure ~ with d~/dx ~ Lreal,l,

l(Ku,v) I =< I(ru,v) I + y~ I((K-r)(Xlk u), XlkV) 1 k=l

+ C8(co8(K) + cos(T)) HuPI~2 Ilvll,~2 ,

where

" 2 }1/2 (2.21) llwll,~2 = IIwll 2 + { E ]IXI~ wll 2

k=l (w = u,v; I k is the double of Ik).

Proof. Since XI_ ~ (K-T)(XI_~U) = 0, we have

40

L 1

+ Z k=l

(= Z k=l

Let x k be an endpoint of I k

*¢ y ( (I k ) N Ij, k < j, then

I (Ku,v)~I -- i),

=< Z ]((K-T)(XkU), XkV)~l k=l

® ~ k-I

Z l((K-T)(XkU), Xjv)~ I + Z Z j =k+l k=2 j =I

]((K-T)(XkU), XkV)~] + LII + LI2, say).

such that x k ( I -~ (k >-- i).

IIii _>- 1121 ->- . . . . We have,

l((K-T)~kU), Xjv)~I

If x ( I k,

IK(x,y) - T(x,y) I = IK(x,y) - K(Xk,X j) + T(Xk,X j) - T(x,y) l

C 8 llklS/[x-yl I+6

* c * then we have evidently (Here we assume xj ( (I k) . If xj ( I k,

IK(x,y) - T(x,y) l ~ C 8 llklS/Ix-yll+8.) Hence

LII --< Z f , l(K-T)(XkU)(X)V(x)Idx k=l Ik-I k

+ Z Z f *c I (K-T) (XkU) (x)v(x) ] dx k=2 j=k+l (I k ) N Ij

=< C8(¢o8(K) + ~6(T)) [ Z f * {flk dy} k=l Ik-I k

+ Z fl {fie I Ik]6 k= 2 k e ix_yll+8 lu(y) Idy} Iv(x) l dx ]

C8(¢o8(K) + cos(T)) { >~ k= l

+ Z lu(Y) l Mv(y)dy} k=2 fI k

I v ( x ) l dx

f , IH(X k u)(x)v(x)I dx Ik-I k

~- C6(es(K) + ¢o8(T)) lluII 2 llvIl,~ 2

_-< C8<¢o8(K) + ~8(T)) Ilull,~2 Iivli,22 •

41

Using the adjoint kernel of

LI2 ~ C8(~8(K) + ~8(T)) HulI~2 llvll~2 *e

If x E I k, y ~ (I k ) n (I - ~), then

IK(x,y) - T(x,y) l = IK(x,y) - K(Xk,Y ) + r(xk,Y) - T(x,y) l

c 8 IIkIs/Ix-yI 1+8

Hence we have, in the same manner as in LII,

L 2 a Z f , [(K-T)(XkU)(X)V(x)Idx k=l Ik-I k

+ Z f * c l(K-T)(XkU)(X)V(X) I dx k=l (I k) n (i-~)

C8( ~8(K) + ~8(T)) lluIl,~2 IIvII,~2 .

using the adjoint kernel, we have also

n 3 ~ C8(~8(K)+ ~8(T)) lluN,~ 2 Ilvll*~2 •

Thus the required inequality holds.

The following three lemmas are corollaries of Lemma 2.6.

Lemma 2.7. Let I, ~, K(x,y) and

for any f E Lreal,l,

Proof.

we have

K(x,y) - T(x,y), we have, in the same manner,

Q.E.D.

T(x,y) be the same as in Lemma 2.6. Then,

~(I,K,f) 5 (~(I,T,f) + Z ~(Ik,K,f) k=l

+ C8(~(T;~)I/2 + oos(K) + o)8(T)) Ill.

using Lemma 2.6 with u = klf, v = X I K(XIf)/IK(XIf) I and d~ = dx,

~(l,K,f) = l(Ku,v) l <-- I(Tu,v) I + Z [((K-T)(XkU), XkV) I k=l

+ C8(~08(K) + ~os(T)) III _-< ~(l,T,f) + Z ~(l,K,f) k=l

+ C8(~(T;~)I/2 +~8(K) +~8(T)) lIl.

Lemma 2.8. Let

anti-symmetric

Q.E.D.

I, ~ be the same as in Lemma 2.6. Let K(x,y), T(x,y) be two

8-standard kernels such that K(x,y) = T(x,y)

42

(x, y 6 1 - ~, x # y). Then, for any f 6 Lreal,l with 0-<- f < i,

(l,K,f) < ~(l,T,f) + Z k='i

~(Ik,K,f) + C 5 AI(K,T;~)III

where

AI(K,T;~) = {~(T~52) I/2 +~os(K) +~os(T)} {a(K) +a(T) +o~8(K) +~Os(T)}

Proof. Without loss of generality we may assume that supp(f) c I. Using

Lemma 2.6 with u = f, v = XIKf and d ~ = f dx, we have

~(l,K,f) < l(rf, XiKf)fdxI + Z ]((K-T)(Xkf), XkKf)fd x ] k=l

+ C8(~8(K ) + c08(T)) [Ifll,~ 2 11XiKfIl,~ 2 (= L 1 + L 2 + L3, say).

We have easily

L 3 =< C8( ~6(K) + ~8(T)) IIKII2, 2 t I]

_-< C6(~o6(K) + ~08(T))(a(K ) +¢o6(K))II I .< C 8 AI(K,T;~) ] I [ .

(See (2.22)). Since K(x,y), T(x,y) are anti-syn~netric, (Xkf(K-T)~k f))Ik

, ' = (the midpoint of Ik), (k ~ i). Hence we have with x k

L 2 = Z Iflk f(x)(K-T)(Xkf)(x) K{(X k + X , + X *c )f} (x)dx 1 k=l Ik-I k I k

<= z If k=l I k

+ ~o8(K) k=iZ flk l(K-r)(Xkf)(x) I (

+ k=IZ Iflk f(x)(K-r)(Xkf)(x) { K(X *e I k

=< Z $(Ik,K,f) + Z llXkT(Xkf)II 2 llK(Xkf)ll 2 k = l k= l

+ ~6(K) Z ll(K-r)(Xkf)II2 {fie ( f , k = l I k - I k

+ C 8 ~o6(K) k=iZ flk I (K-T) (Xkf) (x) I dx

--< Z ~(Ik,K,f) + Cs(~(T;a) I/2 + ~8(K)) (]IKII2, 2 k = l

--< Z ~(Ik,K,f) + C 8 AI(K,T;~) Ill. k = l

f(x)(K-T)(Xkf)(x) K(Xkf)(x) dxl

f. ~ ) dx I k- I k

f) (x) - K(X (~) dx Ik ef ) } 1

--~)2dx}i/2

+ IPTIF2, 2) I~I

= 0

43

Using Lemma 2.6 with u = f, v = EiTf and d~ = f dx, we have

L I = l(Kf, XiTf)fdxl ~ ~(l,T,f) + Z I((K-T)(Xkf), XkTf)fd x 1 k=l

+ C6(~Os(K) + ~o6(T)) llfll,~ 2 IIXITflI,~ 2 (= ~(l,T,f) + Ell + LI2, say).

We have

LI2 -<- C8(~06(K) + ~5(T)) IITII2, 2 II! :< C 6 AI(K,T;~) Ii~. ~

In the same manner as in L 2,

LII = Z Ifik f(x)(K-T)(Xkf)(x ) T{(X k + X , + X ,c)f}(x) k=l Ik-I k I k

dx p

=< k=iZ I71k f(x)(K-T)(Xkf)(x) T(Xkf)(x) dx 1 + C 8 AI(K,T;~) Ill

+ liTIl2, 2) ~(T;~) I/2 III + C 8 AI(K,T;~) III (IIKII2, 2

c 8 AI(K,T;~) III.

Thus the required inequality holds.

Lemma 2.9. Let I, ~, K(x,y) and

E c [, we put K E = ~KME, where M E

define inductively K~ k) = ~4k-l)(k

define T~ k) in the same manner as

@(l'K~2)'f){1 ~ ~(I, TI(2), f)

Q.E.D.

T(x,y) be the same as in Lemma 2.8. For

where

is a multiplier: g ÷ XEg. We

i; K~ 0) is the identity operator).

K~ k). Then, for any f ( LTeal,l,

+ Z $(I k, K (2) f) + C 6 A3(K,T;~), k=l I k '

We

A3(K,T;~ ) = {$0(K) I/2 + $0(T) I/2 + ~(T;9) I/2 + ~6(K) +

× {o(K) + o(T) + ~8(K) + ~8(T)} 3.

ms(T)}

Proof. We divide the proof into several steps.

(First Step). We begin by showing that

(J) < C6(~(x ) + ~8(x))JlI 1 (2.22) fIX I fll,~ 2 =

For any g ( L 2 supported on I, we have, with

(i -<_ j ~ 3, X = K, T).

I k = (the double of I k) ,

44

fl~ IXg(x) I 2 dx ~ 3 / , !X~ ** g)(x)I 2 dx

I k I k

+ 3 f , IX~ **c g)(x) - (X~ **c g))Ik I2 dx I k I k I k

+ 3 Ii~I l(X(Xlk**C g))Ik I2 = < 3 IlxiI~,2 IiXlk** glI~

2 )ikl 2 + C6 ~6(X) 2 ![Xk Mgll 2 + 6 llkl ](X(X **c g) I k

and

Thus

IXkl I(X(X **c g))Ik 12 ~ 211kl l(xg)Xkl2 + I k

=< 2 [Ikl l(Xg)ikl2 + C6(~(K ) +~6(K)) 2

211kl [(X(X ** g))ik 12 I k

fIX ** gll~" I k

% f , IXg(x)I 2 dx =< Const % Ilkl l(Xg)ikl2 k=l I k k=l

)2 ~g]122 + 2 + CS(~(K) + 0~6(K) % (ilk k IlK ** g]I 2 ) k =I I k

=< C6((~(K) + ~o6(K))2 {Hgt122 + % Ilk ** gll~ }, k=l I k

which shows that

HgiI,~ 2 _-< C6(a(K) + ~6(K)) llgIl**~2 ,

where II'II**~2 is defined by (2.21) with I~ replaced by I k . If j = i, we

put g = Xif. Then this inequality gives (2.22). If j = 2, we put g = Xif and

use the above argument. To estimate IIXifll**~2 , we use again the above argument

with I k replaced by the double of I k . Then we obtain consequently (2.22).

If j = 3, we put g = Xi(2)f and use the above argument 3 times. Then we obtain

(2 .22) .

(Second Step) . We show tha t , for any u, v E L 2 supported on I ,

(2.23) E I(XkU, Yiv) I ~ % I(Xk u, YkV) l + C 6 { ~6(Y)(~(X) + 0~6(X)) k=l k=l

+ $0(xll/2(~(Y) + o~6(Y))} Ilull 2 Ilvli,~ 2 (X = K, T; Y = K, T),

where X k ' Yk = Xik = Yik.

We have

45

+

Z k=l

Z k=l

(XkU, YIV) I ~ Z l(XkU, YkV) l + Z l(XkU, Y(X , v))l k=l k=l Ik-I k

(XkU, Y(X *c v))I ( = Z I (XkU, YkV) I + L 1 + L 2, say). I k k=l

Put <k = (XkU)ik (k ~ i). Then

l~kl = I(uX k Xk)ikl ~ $0(X) I/2 { ~ ~I k lu(Y) l 2 dy} I/2 (k ~ i)

We have, with x k (the midpoint of Ik)

L 2 = Z Ill k (XkU(X) - ~k ) {f *c (Y(x,y) - Y(x{,y))v(y)dy} dx k=l I k

Y(X *c v)(x) dx I + ~k fl k ik

--< C b ~o6(Y) Z (IIXk(XkU) ~vll I + -}~kl fix k Mvll I) k=l

+ z l<kl(Hx k YvH 1 + [l×kY(× , v)HI) k=l Ik-I k

=< C 6 c°6(Y) (IIXli2, 2 + ~o(X) I/2) ilull 2 Iivll 2

+ C 6 $0(X) I/2 (NY[]2, 2 + c06(Y)) [lull 2 J[vH,e2

C 6 { ~b(Y)(o(X) + cob(X)) + ~0(X)I/2(o(Y) + cob(Y))} IIull 2 !Ivli,~22

and

L I e6(Y) Z Ilk IXkU(X) l ( I , dy) k=l Ik-I k

= Const e6(Y) Z I , Iv(Y) llHIXkUl(Y)IdY k=l Ik-I k

C 5 ~6(Y)(o(X) + ~b(X)) llull 2 Ilv![,~2 •

dx

Hence (2.23) holds.

(Third Step).

(2.24)

We show that

. (3):. ~ (2) l l(Kk f, t< I z) i -<- Z %(Ik' ~k ' f) k=l k=l

+ C 6 A3(K) Ill,

where

A3(K) = ($0(K) I/2 + ~6(K))(~(K) + ~6(K)) 3.

46

. (2)f X = Y = K and using (2.22) we have Using (2.23) with u = Xlf, v = ~I '

Z I (Kk f, K~3)f) 1 -< Z" I (Kkf, K k K~2)f) I k=l k=l

+ % ($0(K) I/2 +~o6(K))(~(K) +o08(K))

(2)f)l + C 6 A3(K) III Z I (Kk (2)f, K I k=l

Using (2.23) with

fix if!I 2 II K~2) fll ~2 2

e o

u = Zk= 1 Kkf, v = Klf , X = Y = K, we have

Z I(Kk(2)f, K~2) f) I < l I(Kk(3)f, K I f)l + C 8 A3(K)IIl- k = l k = l

Using (2.23) 42)f, with u = Zk= I v = Xlf, X = Y = K, we have

% I " (3)~ (~k ~' Klf)l -<- % ~(Ik' 4 2)'f) + C8 A3(K) III" k = l k = l

Thus (2.24) holds.

(Fourth Step).

(2.25)

We show that

Z l(ZkT~J)f, K~3-J)f) I =< C 8 A3(K,T;~)III k=l

(i ~ j =< 3, Z = K, T).

C6{~6(T) (~(Z) + ~6(Z)) + ~0(Z)I/2(~(T) + ~8(T))} ]IK~2)fll 2 I]fll,~ 2

)2 C 6 a(T;e)I/2(~(Z) + ~o6(Z))(a(K) + ~Os(K) IIl

C 8 {~os(T)(o(Z ) + o~8(Z)) + $o(Z)I/2(o(T) + ~8(T))} (<~(K) + ~o8(K))2 III

If j = i, we use (2.23) with u = K~2)f,{ v = Xlf, X = Z, Y T. Then

Z I(ZkTI f,K~2)f) I = % I(ZkK~2)f, Tlf)I =< % l(ZkK~ 2)f, Tkf) l k=l k=l k=l

+

+

< C 8 AB(K, T; e) III.

If j = 2, we use (2.23) with u = Kif, v = Tif, X = Z, Y = T.

Z l(ZkT~2)f, Klf) I = Z l(ZkKlf, T~2)f)l k=l k=l

=< Z I(ZkKI f, TkTlf) I + C 8 AN(K , T; ~) III. k=l

Then

47

Using (2.23) with u = Ek= I ZkKlf , v = f, X = Y = T, we have

Z I(Z k Klf, TkTlf)I = Z I(Tk Z k Elf, rlf) 1 k=l k=l

_-< Y [(rkZkKlf , Tkf) [ + C 6 A3(K,T;~ ) III =< C6A3(K,T; ~) [I[. k=l

If j = 3, we use (2.23) 3 times. Then, in the same manner, we obtain (2.25).

(Final Step). We now show the required inequality in our lemma. Without loss of

generality, we may assume that supp(f) c I. Using Lemma 2.6 with u = f,

v = K$3)f," d~ = dx and using (2.22), (2.24), we have &

(2.26)

+ Z I CCKk- Tk)f, K~3)f)[ + C8(~6(K) + ~6(T)) NfH,~ 2 l]K~3)fli,~2 k=l

I(rl f, K~3)f) I + % I(Kkf,K~B)f)I + C 6 A3(K,T;~) Ill k=l

< l(rlf ' K~3)f) I + ~ ~ (2) = ~(Ik, ~ ,f) + C 5 A3(K,T;~) [I[. k=l

Using Lemma 2.6 with u =Tlf , v = K~2)f and using (2.22), (2.25), we have

+ Z I((Kk - Tk)Tlf , K~2)f)l + C 8 A3(K,T;~) [I I k=l

~l(T~2)f, K~2) f) I + C 5 A3(K,T;~)l!I.

Repeating this argument 2 times~ we obtain

(2.27) i(T 2)f, K 2)f)i < 2), f) ÷ c6 A3(K,T )

Thus (2.26) and (2.27) give the required inequality in our lermna.

§2.5. Proof of Theorem A by perturbation ([45])

In this section, we deduce Theorem A from the boundedness of the Hilbert

transform and Lemma 2.9. (We do not use Cotlar's lemma nor the Fourier transform.)

Fixing 0 < s < 1/2, we define

S[a](x,y) = ks(x-y ) T[a](x,y),

48

where

(s) = g

(IsIl~) - i ( ~ < Isl --< 2~)

(2S < !s I --< ll(2s))

(i - slsl) (1/(2s) < Isl <_- l/s).

We shall show that IIS[a]ll2, 2 $ Const flail, . Once this is known, Fatou's lemma

gives Theorem A. Without loss of generality, we may assume that a ( Lreal,l.

, ~ Const{o(S[a]) + I} Since ~l(S[a]) ~ Const Lemma 2.5 shows that Iis[a]II2, 2

Hence it is sufficient to show that

(2.28) ~S = sup {o(S[a]); a ( Lreal,l } < Const .

Let

- i (2) f); OS(2) = sup { ~[" ~(I, S[a]i ,

where S[a]~ 2) is defined in the same manner as K~ 2) in Lemma 2.9.

Shwartz's inequality shows that ~ <~ From the definition of = S(2)"

is finite. For a, f ( Lreal,l and an interval I, we show that

a, f ( Lreal,l , I interval},

Then

S[.], @ (2) S

2 )4 3 ~ 3 (2.29) i~ I ~(I, S[a]~ 2), f) = < {( ~ + 7 } ~S(2) + Const {~S + i}.

Considering -a if necessary, we may assume that (a) I $ O. RSL of Type 1

(-1/3-&.,l-a.) shows that there exists b E Lreal,l such that, with

= {x ( I; A(x) # B(x)} (B(x) = A(x0) + f~O b(s)ds, x 0 is the left endpoint of

I),

(2.30) -1/3 ~ b(x) ~ 1 a.e. on I, b(x) = -1/3

l-(b)I l-(a)I 3 (2.31) I~I ~ 1+(1/3) Ill ~ i+(i-7~ Ill ~ 7 III"

on ~,

(The function b obtained from RSL is defined only on I. Since (2.30), (2.31) are

independent of the behavior outside I, we may put b(x) = 1 (x (IC).) Since

A(x) = B(x) (x ( I -~), we have S[a](x,y) = S[b](x,y) (x, y E I -~). Using

Lemma 2.9 with K = S[a], T = S[b], 6 = i, we have

~(I, S[a]~2),f)= < $(I, S[b]~ 2), f) + Z ~(I k, S[a]~2 k), f) k = l

+ Const A3(S[a] , S[b]; ~) [I I,

49

where ~ = Uk= I I k (I k = l~,k)-

dominated by (3/4) ~S(2) Ill . Put

S[D] = S[b] + (~/3)H, and hence

The second quantity in the above inequality is

= b - (1/3). Then II~II~ ~ 2/3 and

~ ( I , S[b]~2) , f ) =<- ~ ( I , S["~]~ 2) , f) + Const IIsEE]ll~, 2 III

3 =~ (2)4~s(2) i! I +Const {%+1} IIl.

Thus

, 2 )4 3 $ [I[ $(I, S[a]~ 2) f) ~ {( ~ + ~} S(2)

3 + Const {oS + i}III + Const A3(S[a], S[b]; ~) IIl.

It is necessary to estimate

for any interval J and x E J,

k (x-y) ] SIal Xj(x) I -<- Ifj ~x-y

Lemma 2.5 shows that

k (x-y) fj ifj g x-y

A3(S[a], S[b]; ~). Integration by parts shows that,

a(y)dy I + Const.

a(y)dy I dx ~ Const NH*(Xja)II 2 IJl I12

Const (~(H) + 1) IJl E Const IJl.

Hence we have (i/IJ I) ~(Ji S[a], Xj) ~ Const. Taking the supremum over all

~0(S[a]) & Const . In the same manner, ~0(S[b]) ~ Const. Since

b(x) = -1/3 on ~, we have, for g ~ Lreal,l ,

J,

and hence

~(I k, S[b], g) = Const $(I k, H, g) & Const Ilkl (k a i),

~(S[b]; ~) ~ Const. Consequently,

AB(S[a], S[b]; ~) ~ Const {~ (S[a]) + ~(S[D]) + 1} 3

3 Const (~S + i) ,

which gives (2.29).

Taking the supremum of (i/IIl) 5(I, S[a]~ 2), f)

and all intervals I, we obtain, by (2.29),

over all a, f 6 Lreal,l

2 )4 ¼ ~ 3 8S(2) ~ {( ~ + } ~S(2) + Const {o S + i}.

50

Since (2/3) 4 + (3/4) < I and ~ ~ ~ ~S(2) , this inequality yields (2.28).

This completes the proof of Theorem A.

§2.6. Proof of Theorem B by perturbation ([17])

In this section, we deduce Theorem B from the boundedness of

two lemmas necessary for the proof.

H. Here are

Lemma 2.10 (Calder~n [4]). IITn[a]ll2,2 $ (Const) n llalI~ (n ~ i).

Proof. Fixing 0 < s < 1/2, we put

Since

Sn[a](x,y ) = X (x-y) Tn[a](x,y) g

el(Sn[a]) ~ Const n JPalI~, it is sufficient to show that

n+l (2.32) ~S = sup {~(Sn[a]); a ( ereal,l } ~ C O

n

for some absolute constant

So[-] = (-~)H.

(0 ~ k ~ n-l).

4

<~Sn -<- ~S (2) " n

C O (which will be determined later). Let k+l

Then ~So = ~(-[ H) ~ ~. Suppose that ffSk ~ C O

We d e f i n e ~S(2) in the same manner as ~S(2)" Then n

Let a, b, f, I, ~ be the same as in §2.5. Using Lemma 2.9 with

K = Sn[a], T = Sn[b] , 6 = i, we have

(2.33) ~(I, Sn[a]~ 2), f)= < ~(I, Sn[b]~ 2), f)+ Z ~(I k, Sn[a]~2), f) k=l

+ Const A3(Sn[a] , Sn[b];~ )llI.

Put b = (3/2)(b- (1/3)). Then

2 b* i Sn[b ] = Sn[ 3 + ~] =

I]b I[~ =< 1 and

n i n-k * (2)k Sk[b I k=0

By Lemma 2.5 and the assumption of our induction, we have

IISk[d]II2, 2 ~ Const {~Sk + ~l(Sk[d])}

Const (C~ + n) (d ( e~eal,l, 0 ~ k ~ n-l).

Hence, in the same manner as in §2.5, we obtain, by (2.33),

51

~(I, Sn[a] 2) f) ~ { (~)4n + 4 } S (2) n

+ Const (C~ + n)~ S + n) 3 + Const (C~ + n) 4. n

Since a, f E Lreal,l and I

quantity replaced by @S(2) .

4 n ~S ~ ~ we have

n S (2) ' Sn n

are arbitrary, this inequality holds with the first

Since (2/3) 4n + (3/4) ~ 0.99 and

' Let =< C 0' (C + n) for some absolute constant C O •

C O max {2 C~, ,} Then ~S _n+l = • ~ Gn • This shows that (2.32) holds for all n

n ~ 0. This comple t e s the p roof of Lemma 2 .10 . Q.E.D.

Remark 2.11. It is known that

IITn[a]II~,BM 0 ~ Const {IiTn[a]II2, 2 + ~l(Tn[a])}

¢onst {I!Tn[a]!12, 2 + (n+l)} (n ~ O, a E Lreal,l),

IITn[a]II~,BMO L ~ where is the norm of Tn[a ] from to BMO. (See Lemma 2.5.)

The proof of Lemma 2.10 by Theorem 1.12 is as follows. Let a E Lreal,l

and n ~ i. Integration by parts shows that Tn[a]l = Tn_l[a]a. Hence

Theorem I shows that

IITn[a]H2, 2 ~ Const {IITn[a]IIIBM 0 + n}

= Const {liTn_l[a]aIIBM O + n} ~ Const {IITn_I[a]II~,BM O

Const {Hrn_l[a]II2,2 + n} ,

+ n}

which gives Lemma 2.10.

For a E Lreal, we define a kernel

(2.34) E[a](x,y) = 1 exp {i A(x)-A(y)_} , x-y x-y

where A is a primitive of a. The following lemma was first shown by

Coifman-McIntosh-Meyer [7]. A proof by perturbation was given by David [17].

Lemma 2.12.

Proof. Since

There exists an absolute constant N O such that

N O IIE[a]II2,2 ~ Const(l + llall=) (a E L~eal).

~l(E[a]) ~ Const (i + llall~), it is sufficient to show that

N O ~(E[a~) ~ Const (i + llall )

52

for some absolute constant N O ~ i. We put

(2.35) aE(~) = sup {o(E[a]); a ( ereal,6} ( ~ > 0)

and show that

N O (2.36) ~E(~) -<_ Const(l + 6) ( ~ > 0).

Lemma 2.10 shows that, for any a ( Lreal,

i n = i

~(E[a]) = ~( Z nl Tn[a]) =< Z ~-f ~(Tn[a]) n=O n=O

-<- Z i n! (C°nst)n 11alln <= exp {Const(l + Iiaii®)} • n=0

Hence OE(~) < ~ for all ~ > 0 and ~E(1) ~ Const. Let

> i, a ( L~eal,~ , f ( Lreal,l and I be an interval. RSL of

Type i (-~/3- A.,B- a.) shows that there exists b ( Lreal such that, with

= {x ( I; A(x) # B(x)},

- 613 ~ b(x) ~ 6 a.e. on I, b(x) = - ~13 on ~ ,

6 - (b) I 3

Using Lemma 2.7 with K = E[a], T = E[b], 5 = i, we have

~(I, E[a], f) _~ a(l, E[b], f) + Z O(Ik, E[a], f) k=l

+ Const {$(E[b];e) I/2 + ~01(E[a]) + ~Ol(E[b])} Ill,

where I k = I~, k (k >_- i). Put ~ = b - (6/3). Then II{II~ =< 2~/3 and

~(I, E[b],f) = ~(I, E[~], f). We have 001(E[a]) + O~l(E[b]) =< Const ~ .

Since b(x) = -~/3 on ~, we have

D(E[b]; ~) = ~2~(H;~) =< Const.

Thus

~(I, E[a], f) ~ OE ( ) + ~E(~) + Const 6 ,

which yields that

~E(6) _-< OE( ) +~ aE(~) + Const ~ ,

that is,

53

(2.37) OE(~) < 4 ~E(23 ~) + Const ~ .

Let n be the minimum of integers k->- 1 such that (2/3)k~ ~ I.

n =< (log ~)/(log 3/2) + Const. Inequality (2.37) shows that

n-i OE(~) =< 4n~E((2)~) + Const % 4 k (2)k3

k=0

-_< 4 n {~E(1) + Const (2)n ~}

_-< Const 4n_- < Const (i + ~) N0 ,

Then

where N O = (log 4)/(log 3/2). Thus (2.36) holds. This completes the proof of

Lemma 2.12. Q.E.D.

e-iXs e-S We now give the proof of Theorem B. Since i/(I + ix) = f0 ds

(x ~ ~), we have

C[a] = f~ E[-sa]e -s ds.

By Lemma 2.12, we have

IIC[a]I12,2 =< f0 IIE[-sa]ll2,2 e-s ds

N O N O Const f0 (I + sllall ) e -s ds _-< Const (i + IIall~)

This completes the proof of Theorem B.

§2.7. Estimates of norms of E[.] and C[.]

In this section, we show

Theorem C ([44]). For any real-valued function a in BM0,

(2.38) NE[a]II2,2 ~ Const (i + IIalIBMO ),

(2.39) IIC[a]I12,2 ~ Const (i +~IIalIBMO).

9 In [7], (2.38) was given with IIalIBM O replaced by IIalIBM O, and (2.39)

8 with __~MO replaced by IIaIIBM O. This theorem was established in was given

[44] via [42], [43], [50]. In this note, we deduce Theorem C from (2.36). We

use RSL repeatedly. (The sun also rises!) RSL in §2.2 is called of

Type i (y - r.,$ - a.) The lower bound of a(x) is independent of (2.5)-(2.8).

Let a(x), I, ~, ~ and A(x) be the same as in RSL in §2.2. For 0 ~ T ~ ~ ,

we define analogously B(x) by using the sun at the right upper infinity of angle

aretan y. Then

54

~ b(x) ~ 7 a.e. on I, b(x) = 7 (x ( I k, k ~ i),

(a)ik ~ T (k g i), I~I ~ (-~) * (b)l III. (-~) + 7

This RSL is called of Type 2 (y-hay,~-dg6ee~). In this case ~ is

independent of the above estimates. For 0 ~ 7 ~ ~ , we define B(x) = sup @(x),

where the supremum is taken over all functions @ such that • ~ A, ~' ~ T

a.e. on !. We define b, ~ in the same manner. Then

--<- b(x) --< 7 a.e. on I, b(x) = 7 (x ( I k, k >_- i),

(-~) + (b) (a)i k I Ill. > y (k ~ i), I~I =< (-~) + 7

This RSL is called of Type 3 (y-r., ~-d.) . In this case ~ is independent

of the above estimates. For ~ ~ 7 ~ 0, we define B(x), in the same manner

as in Type 3, by using the sun at the lower right infinity of angle - arctan 171'

Then

7 ~ b(x) ~ ~ a.e. on I, b(x) = 7 (x ( I k, k ~ i),

- (b)l

(a)ik ~ T (k ~ i), I~I ~ ~ _ 7 llI.

This RSL is called of Type 4 (Y-~.,B - a.).

of the above estimates.

In this case, ~ is independent

Type 1 Type 2

Type 3 Type 4

RSL of Type j is reduced to RSL of Type i by a suitable affine transformation.

55

§2.8. Proof of (2.38)

(First Step). We divide the proof into three steps. Recall ~E(~) defined by

(2.35). Put

~E(~) = sup {~(E[a]);a E Lreal,~} (~ > 0).

(See (2.16).) In this step, we show that, for ~ ~ i, 0 < 5 ~ i,

(2.40) ~E(~) ~ ~E(@~) + ~ ^ .l+e ~E~T ~) + (Cs/e) ~5{~E(~) + ~5} ,

where e (0 < e < i) is determined later. For a E Lreal,~, f E Lreal,

0 ~ f ~ i and an interval I, we study an a-priori estimate with respect to

~(I, E[a], f). Since ~(l,E[a],f) = ~(l,E[-a],f), we may assume that

L ~ > 0. RSL of Type i (- e~- ~., ~ - ~.) shows that there exists b ~ real (a) I = such that, with ~ = {x E I; A(x) # B(x)} = Ok= I I k (I k = l~,k).

- 6~ ~ b(x) =< ~ a.e. on I, b(x) = - e~ (x E I k, k > i),

(a)ik ~ - e~ (k = > i), I~I =< ~ +- ~ III ( ~ = (b)l)-

Using Lemma 2.8 with K = E[a], T = E[b], we have

~(I, E[a], f) -<_ ~(I, m[b], f) + % ~(Ik, E[a], f) k=l

+ C 6 AI(E[a] , E[b]; ~) IIl.

Since ~o6(E[a] ) _< C5 ~5, 0~6(E[b] ) < C5 ~6 and

~)i/2 $(E[b]; = Const $(H;~) I/2 < Const =

we have

and hence

AI(E[a ],E[b];~) = {&(E[b];~)I/2 + ~5(E[a]) + ~5(E[b]) }

× {~(E[a]) + ~(E[b]) + ~5(E[a]) + ~5(E[b])}

_<_ C5 F5 {~E(F) + ~5 } ,

~(I,E[a],f) < ~(I,E[b],f) + % k=l

$(ik,E[a ],f) + C5 ~5{~E(~) + ~5} ii1.

56

< For each I k, we use~ RSL of Type 2 (e~ -&.,-6-d.).~ Since (a)~ - e~

t he re e x i s t b k ~ L rea l and an open se t ~k = U g = l l k , g in I k such tha t

-~ & bk(X) ~ e~ a.e. on I k, bk(X) = 8~ (x E ~k ),

(a) I ¢ e~ ( ~ I), k,g

-(-6 )+(bk) Ik ~+ (a)ik

=< -(-~) + e~ I Ikl < ~+ e~ i- e

IXkl --< "i + e IXk I"

Let bk = bk - (i -e)~/2). Then ll%k[l= =< (l +e)~/2 and

~(Ik,E[%k],f) = $(Ik,E[bk],f). Hence, by Lermma 2.8, we have

g(ik,E[a],f) < ~(Ik,E[bk],f) + Y $(Ik, g, E[a], f) g=l

l+e + % AI(E[a],E[bk]; g~k ) llkl =< ~E (T~) Ilkl

+ ~ g(Ik,g, E[a] ,f) + C 5 ~5 {OE(~) + ~5} i xkl ' g=l

which yields that

g(l,E[al,f) N $(l,E[b],f) + l+O ~E ( T ~) z l~kt

k=$

+ k=l% g=i% o(Ik' g, E[a],f) + C 5 ~5 {~E(~) + ~6}{ii I + k=lY llkl}.

For each Ik,~, we use RSL of Type i (-e~-&.,~-4.) Since (a) Ik,g

= in such that we obtain an open set ~k,~ U~=I Ik,g,m Ik,~

(a) ~ - e~, Z llk,~,ml < 1 - e I " Ik,g, m m= 1 = I + e llk,g

>--eF,

In the same manner as above,

i+@ ~(l,E[a],f) ~ ~(l,E[b],f) + ~E ( T ~)

^

% % % ~(Ik,g,m, E[a], f) k=Ig=l m=l

{ £ llkl + E Z llk,~l} k=l k=l ~=i

+ C5 ~5{aE(~) + ~5} {lll+ % llkl + % Z Ilk,~I} • k=l k=l g=l

57

^

Since (a) I -<- - 8 ~ , c~(Ik,g,m, E[a], f) ^ k,& ,m

in ~(Ik,E[a],f). Repeating this argument,

is estimated in the same manner as

.i+@ l-e i-8 2 ~(l,E[a] ,f) =< ~(l,E[b] ,f) + E(~ - ~) I~I {1 + i~ ÷ (i~) + "}

l-e I-8 2 +C 8 p8 {~E(~) +~8} {JZj + I~I (I+~ + (i-~-) + ...)}

^ l+e~. ~-n ~(l,E[b],f) + OE(2-~-p) ~ + 8~

^ ^ ,l+e ~ - n = o(l,E[b] ,f) + OE<-- ~- ~) 2e~

l+e J iJ + (C5/8)~5{aE(~) + ~6} J xl 28

-I~I + (%/e) ~8{C~E(~) + ~8} I~I.

To estimate

There exist

~(l,E[b],f), we use RSL of Type 3 (85-r.,-e~-d.) to b(x)

r Oj=l ' in I such that c ( L eal and an open set ~' = ~ Ij

and I.

-8~ -_< C(X) K e8 a.e. on I ,

-(-e~) + (c) x P~ +n jlJ. ]e'I-<- ./-'(-e~) + e~ Jxj _-< 2ep

In the same manner as above,

~(l,E[b],f) =< ~(l,E[c],f) + Z j=l

^ ,l+e e~ + q, ~E (e~)Ill + ~E<--2 - ~) 2e~

~(l],E[b],f) + C 8 ~8 {OE(~) + ~8} ii I

Consequently,

i $(l,E[a],f) < {~E(e~) + SE (--2- ~) 2e~ + C8 ~8(~E(~) + ~6)} 7U =

+ { ~E<T ~ ) ^ ,i+8 ~28~- ~ + (C8/8)~5(~E(~) + ~5)}

1+8 ^ .l+e SE(e~) + -~- OEi_T ~ ) + (C8/8) ~6{~E(~) + ~8},

which gives (2.40).

(Second Step). In this step, from (2.40), we deduce

(2.41) ~E(~) ~ Const ~ ( ~ g i).

Let

1+8 l+e)k hX(e) = e ) ~ + - ~ - (----f-- (o< e< 1, ~.> o).

58

Then ~(e) > i for any 0< e <!, k > 2 and

Hence we choose e = 1/3

an interval I, we have

i = h2(i/3 ) = min h2(0 ). 0<0<1

C in (2.40). For any a 6 eal,~' f E Lreal,l

~(l,E[a],f) _-< 3 ~(l,E[a], - - f + 2~ 2

3 ) + 3 ~(l,E[a], ~XI)

and

and hence

^ f + 2~ 1/2 ^ 2 ~i)i/2} 1/2 Const {q(l,E[a], ~ ) + a(l, E[a], ~ I I I

Const DE(7)I/21II, qE(~) ~ Const ~E(~) I12. Inequality (2.40), 0 = 113 shows that

(2.42) SE(~) =< ~E ( + 2 DE( ) + C5 ~8 + ~5} ( ~ ->_ i).

By (2,36),

(2.43) ~E(~) ~ Const {~E(~) + ~}2 _ ~ Const ~N ( ~ ~ i),

where N = 2N 0 + 2. Suppose that N ->_ 3. We put

N+I

= sup{ ~E(~) ~ 2 ~m ; i ~ ~ =< (3)m } (m = 3,4 .... ).

Then -c 3 -<_ Const. For any m >_- 4 and (3/2) m-I < ~ < (3/2) m, we have,

by (2.42), N+I N+I

~E(~ ) ~ 2 _<_ ~ 2 {~E(~3 ) + 2 ~E(23 -~) + C8( ~ + ~28)}

N+I N+I N+I N 2 2 + 2 + ~:m-1 + c8 (~ 2 +8 p28) }

3

N+I N+I N+I -(li2)

_<,{( ) 2 + 2( ) 2 } .~m_i + C6(~ +

8-(1/2) 2 (m-i)((i/2)-8)

~m-i + C5 ~ ~ Tm-i + C8(3)

Hence ~m ~ ~m-I C 8 (2/3) (m-I)((1/2)-8) We choose 8 + 1/4.

m-i ~m <= ~3 + Const Z (_~)k/4°3 <= Const,

k=3

)}

Then

59

which gives that

~E(~)

i.e., (2.43) holds with N

obtain

~E(~) = <

Put

Then

above,

N+I 2 Const ~ ( ~_>- i),

replaced by (N+I)/2. Repeating this argument, we

Const 9 3 ( ~ ~ i).

~' = sup {~ ~-2; 1 = = m E(~) < 9 < (~)m } (m = 3,4 .... )

' < Const. Since (1/3) 2 + 2(2/3) 2 i, we have, in the same manner as ~3 = =

~'m -<- ~tm_l + C6 {(2)(m-I)((I/2)-6) + (3)(m-I)(2-6)}

T'm_l + C6 (Q)(m-l)((I/2)-6)9~ (m >= 4).

We put 6 = 1/4. Then ~t ~ Const, which yields that m

Since

(Final Step).

that

(2.44)

We have evidently

Ix-x'l ~ Ix-yl/2

~E(~) ~ Const 92 ( ~ a i)°

~E(~) ~ Const ~E(9) I/2, this shows (2.41).

At last, we deduce (2.38) from (2.41). Let a E Lreal.

~5(E[a]) ~ C 6 (i + ~25) ( ~ = IlalJBM 0, 0 < 5 ~ 1/2)

IE[a](x,y) I ~ 1/Ix-y] Let x, x', y ~ IR satisfy

We may assume that y < x < x'. We have

We show

fyX la(s ) _ (a)(x,x,) ] ds = < Const ~(x-y) log x'-y

which gives that

- X' - X X (a(s) -(a)(x,x,))ds I A(x)-A(Y)x-y A(x')x'- yA(y) I = I (~_y~(~,_y) /y

i X'-y f : ' (a(s) - (a)(x,xt))ds ]

80

Thus

Xt--X Const

x-y {i + log ~ } ~ Const

xt_X

IE[a](x,y) - E[a](x',y) l X ! -- X < =

"(x-y)(x''y~

~(x'-x)i/2/(x-y) I/2.

1 I exp { i A(x),A(y) } - exp + x'-y x-y

{i A(x')-A(y) }I xV-y

x' - x 1

(x-y)(x'-y) + C6 I exp {i A(x)-A(Y-~)} - e X P x - y {i A(X'x!-A(J~)}I25

< x' - x i A(x)-A(y) _ A(x')-A(y) I 2 5 = (x-y)(x'-y) + C 6 ~ I x-y x'-y

x' - x (x' - x) 6 Const + C6 ~26

(x_y)2 (x_y)l + 6 C5(I + ~28)(x'-x)6/(x-y) I+6 .

For f E Lreal,l and an interval I, we estimate o(l,E[a],f). Let

a = a - (a) I. Then o(l,E[a],f) = ~(l,E[a],f). Le~=na 2.2 shows that there exists

an open set ~ = Uk= I I k (I k = IE, k) in I such that

 l ) ( x ~ ~).

e = Then b E real,2~ Using Lemma 2.7 with K = E[a], T = E[b], we have

~(l,E[a],f) = o(l,E[a],f)

<= o(l,E[b],f) + Z ~(Ik,E[a],f) + C 6 (i + ~25) ii I k=l

Z ~(Ik,E[a],f) + C 8 (I + ~26) iii. ~- ~E(2~) III + k= I

Repeating this discussion, we For each Ik, we use Lemma 2.2 with a - (a)ik.

have

81

Thus

i 1 ~(I, E[a], f) =< qE(2~) ]I I {i +~+~+ ... }

i i + C 8 (i + ~ 26) III {i +~+~ + ... } =<_ {2 ~E(2~) + C 8 (i +~ 28) } IiI

o(E[a]) ~ 2 ~E(2~) + C 8 (i + ~28)

Const (i+ ~) = Const(l + IIaIIBMO).

Consequently, Lem~na 2.5 and (2.44) yield (2.38) in the ease where

a be a real-valued function in BMO. We put a E Lreal. Let

iax 1 an(X) = (x) Ia(x) I =< n (n->_ I).

- a(x) < -n

Since a n ~ Lreal , we have

IiE[an]II2, 2 ~ Const(l + IlanllBMO ) ~ Const(l + IIalIBMO).

Letting n tend to infinity, we obtain (2.38).

§2.9. Proof of (2.39)

Put

$C(~,~) = sup {~(C[a]); a ~ Lreal, ~ = a -<_ ~} (e _-< 0, ~ _-> 0).

We show that, for ~ ~ I, 0 < 8 ~ i,

^

(2.45) ~C(-~,~) ~ 2 ~C (- ~ , ~) + C 8 ~8{~C(~) + ~8}

where

~C(~) = sup {o(C[a]); a E LTeal,~} •

(Tchamitehian [51] showed that

¢~C(-~,~) -< 2 OC (- ~2' ~2 ) + C6 B6 (~ ->_ i).

Inequality (2.45) is an improvement of his inequality.) Here is a lemma necessary

for the proof of (2.45).

62

Lemma 2.13. <~c(O,~)= ~C(--~,O) _-<- C 5 ~5 {Crc(~ ) + ~5} (# >_-- i , 0 < 6 - 0 by

6 = log 2/log(i/4q). For a E Lreal, 0 _-< a--< [3, f ~ eal,l' 0 -< f =< 1

interval I, we estimate ~(l,C[a],f). If (a)l < 2q~ , we use RSL of

Type 3 (4qB-/t.,0-d.). There exists b 6 L~eal such that, with

= {x 6 I; A(x) ~ B(x)} ,

and an

0 =< b(x) =< 4Q# a . e . on


~(l,C[a],f) _-< $(l,C[b],f) +

+ c~ ~8 {~c(~) + ~ } IT I

I , I~1 < 0+2r]~3 { i I = 2q~ ,

such that~ with

we use RSL of Type i (qB-A.,{3-a.).

= {x E I; A(x) # B(x)} ,

~ =< b(x) =< ~ a.e. on


~(l,C[a],f)_~$(l,C[b],f) +

+c6 ~6{~c (~) +~8} II

There exists b ~ Lreal

- 2 ~ t I ] = 1 - ~ t i t .

ee

Z ~(I~, k, C[a],f) k=l

We have

- l _ ~ q

i f(y)dyl 2 dx ~(l,C[b],f) ~71 If I (x-y)+ i(B(xiiBiy))

I f=B-l(t) dtl2 ds

= fB(1) IfB(1) (B-l(s)-B-l(t)) + i(s-t) B'°B-I(t) B'°B-I(s)

(Const/~) ~(B(1),C[(B-I)'], foB-i/B'oB -I)

N O (Const/~) Const {i +If(B-I) ' XB(1)II ~ } II(foB-I/B'°B-I)XB(1)IIIIB(1)I

C6 ~-31B(1) I ~ C6 ~-2 i1 I ~ C6111 ,

63

where B(1) = {B(x); x ~ I} and N O is the absolute constant in Lemma 2.12.

Hence, in this case,

~( i , C[a] , f ) < l-zq ~c(O,~) l l l + c5 p~{~c(~) + ~ } I I l • = l - ' q

Thus we have, in both cases,

i ~(I, C[a] f)

1 z_n_2_2n ~c(O,~)} + c5p6{~c(~) + ~5} , --< maX{~c(0'4~) + 25C (0'~)' i-~

which yields that

i ~c(O,~), ~ ~c(O,~)} $C (0'13) --< max{~;c(O'4"q~3) + "2 l -q

+ c ~ { ~ c ( ~ ) + ~ } .

If the second quantity in max {.,.} is larger than the first quantity, then

~c(O,~ ) ~ z_,_-_-n c8 ~ { ~ c ( ~ ) + ~6} = ca ~5{~c(~) + ~5} .

If the first quantity is larger than the second quantity, then

(2.46) $C(0,~) ~ 2 ~C(0,4~) + C6~6{~C(~) + ~6} .

Thus, in both cases, (2.46) holds. Let m be the smallest integer of

k such that (4~]) k ~ =< i. Then m _-< {log ~/log(I/4~)} + i. Inequality (2.46)

yields that

m-i ~C(0,~) --< 2 m ~c(O,(4~)m~) + C 6 Z 2k(4~])kS~8{~C((4~)k~) + (4~)k8 ~6}

k=0

2 m $C(0,i) + C 5 {I + 2m(4~) m6} ~6 {aC(~) + ~5}

C5(~8 + 2 TM) {OC(~) + ~6} ~ C6 ~6 {OC(~) + ~6 } . Q.E.D.

We now prove (2.45). For a ~ e~eal,~, f E e~eal, 0 ~ f ~ i and an

interval I, we estimate $(l,C[a],f). Since $(l,C[a],f) = $(l,C[-a],f), we

may assume that (a) I ~ 0. RSL of Type i (- e~-~.,6- ~.) (e = 1/2) shows

that there exists b E L~eal such that, with ~ = {x E I; A(x) # B(x)}

= U~= 1 I k (I k = l~,k),

64

- ~ ~ b(x) ~ ~ a.e. on I, b(x) = - 85 (x (~),

- (b) I (a)ik ~ - 8~ (k > i) i~ I < ~ ~ @~ ii I ~ I I~I,

= ' = - l+e (e = I/2).


g(l,C[a],f) ~ ~(l,C[b],f) + Z $(Ik,C[a],f) + C5~8{~C(B) + ~5} ii I k=l

gC(-6~,~)Ill + Z g(Ik,C[a],f) + C5~5{ffC(~) + ~8}III . k=l

For each k ~ I, we use RSL of Type 3 (0-&.,-~- d.). Since (a)l ~ - @~, there ~ k

exist b k 6 Lreal and an open set ~k = U~=I Ik,~ (Ik,g = l~k~ ~) in I k such that

- ~ ~ bk(X) ~ 0 a.e. on I k, bk(X) = 0 (x ( ~k ),

(a)ik, ~ 0 (~ ~ I),

Lemmas 2.8 and 2.13 show that

+ (bk)ik

B + 0

~(Ik,C[a],f) ~ ~(Ik,C[bk],f) + Z $(Ik,g,C[a],f) + C8~5{~C(~) + ~5} llkl g=l

--< ~C(-~,0) Ilkl + Z ~(Ik,g,C[a],f) + C5~5{~C(~) + ~5} llkl ~=i

--< Z $(ik,e,C[a],f) + C5 ~5 {OC(~) + ~5} iikl ' g=l

Thus

~(l,C[a],f) ~ $C(-8~,~)II] + Z Z $(Ik,g,C[a],f) k=l ~=I

Since

we have

+ csFS{Oc(F) + ~8}II j,

(a)ik,~ => 0 (k,~ >_- i), k=1% ~=IZ Ilk,~l = < ii +- 8e iii.

(a) I ~ 0, we can apply the above argument to Ik, 3 ° Repeating this, k,g

1

Tff c8~5(~c(~) l-e .l-e.2 $(l,C[a],f) ~ {$C(-@~,~) + + ~6)}{I + i~+ ~# + ...}

~!~ ~c(_ep,F ) + c5~5 {~c(~) + ~5},

65

which gives

(2.47) ~c(_~,~) =< z+i2e ~c (-e~'~) + c~ ~ {Oc (~) + ~} (e = ~/2).

To estimate ~C(-8~,~), we study ~(l,C[a],f) for a ~ Lreal,

- 8~a ~ ~, f £ L~eal, 0 & f & i and an interval I. First we assume that

(a) I ~ 0. RSL of Type 3 (8~-r.,-@$-a.) shows that there exists b 6 Lreal

such that, with ~ = {x 6 I; A(x) # B(x)} = Uk= I I k (I k = l~,k),

- e~ = b(x) =< e~ a.e. on I, b(x) = e~ (x ~ ~),

e~ + (b) I e~_ ½ (a)l k a . . . . e~ (k > 1) I~I < e~ + g I~I ~ e~+e~ I~I = I~I'


$(I,C[a],f) ~ $(l,C[b],f) + E k=l

$(ik,C[a],f) + C8~8{~C(~) + ~8} i i I

~c(-el3,el3) I~1 + k=l

~(Ik,C[a],f) + C858{~C(~) + ~8}II I.

For each k ~ i, we use RSL of Type I (O-r.,~-a.). Since (ak)ik a e~,

t h e r e e x i s t b k 6 L r e a l and an open s e t ~k = Ug=l Ik ,~ ( I k , g = I~ k ,g ) )

such that

0 ~ bk(X) & ~ a.e. on Ik, bk(X) = 0 (x 6 ~k ), (a) I ~ 0 k,i

- (bk) Ik II ~ ~ - e~ llkl = (l-e) llkl.

(9~=> l),


~(ik,C[a],f ) _<_ $(Ik,C[bk],f ) + Z ~(Ik,&, C[a],f) g=l

+ C8~8 {C~C(~) + ~8}llkl --< Z ~(Ik,~,C[a],f) + C8~8{~C(~) + ~8}llkl • g=l

Thus

66

$(l,C[a],f) ~ $C(-8~,@#) Ill + z z ~(Ik, g, C[a], f) k=l g=l

(a) I ~ 0 (k,g ~ i), % Z ilk,gl ~ i_~8 ill. k,g k=l g=l

Since (a) I ~ 0, we can apply the above argumen~ to Ik, ~. Repeating this, k,g

we have

1 (2.48) ~ G(I,C[a],f) _-_- {~-C(-@l~,e#) +C6~35(0C({3) +~6)}{1+ t2----~8 + (~)2+. . . }

Next we assume that

there exists b E Lreal

(I k : l~,k),

< 2 ~c(_e#,e#) + csp~{Cc(~) + #6} . = 1+0

(a) I > 0. RSL of Type4 (0- A.,6-G.) shows that

such that, with ~ = {x E I; A(x) # B(x)} = Uk= I I k

0 ~ b(x) ~ ~ a.e. on I, (a)Ik ~ 0 (k ~ 0).


$(l,C[a],f) =< ~(l,C[b],f) + % k=l

--< Z k=l

~(ik,C[a],f) + C5~6{GC(~) + ~5} ]I]

$(Ik,C[a],f) + C5~5 {OC(~) + ~6}]I[.

Since (a) I ~ 0, we can apply the argument in the case of

estimate of k $(ik,C[a],f) ; we have

(a) ~ 0 to the I

2 $C(_@~,@~) I + C5~5 {GC(~) + ~5} llkl " ~(Ik'C[a]'f) ~ i-~ Ilk

Thus, in this ease also, (2.48) holds. Consequently,

2 ~c (-e~'~) ~i-~ $c (-°~'e~) + c6~{~c(~) + ~} (e = 1/2).

This estimate and (2.47) immediately yield (2.45).

In the same manner as in (Second Step) of the proof of (2.38), (2.45)

shows that G(C[a]) ~ Const(l +~/ilaIl~) (a ~ LTeal). From this inequality, we

now deduce

(2.49) G(C[a]) ~ Const(l + H~aI~BM O) (a is real-valued).

67

For a E L~eal, f E L~eal,l and an interval I, we study ~(l,C[a],f).

Let 8 = IIaNBMO. Inequality (2.44) holds with E[.] replaced by C[-].

Lemma 2.2 shows that there exists an open set ~ = U" = k=l Ik (Ik I~, k) in

such that

121 ~ --~ fI la(s) - (a)II ds ~ III/2, (a - (a)i)Ik ~ 2~

la(x) - (a)iI ~ 2~ aoe. on I - ~.

(k a i),

We put

b(x) = (x { I k, k >_- i)

(a) I (x ~ R).

Then

Recall that

(See Lemma 2.10).

~(l,C[a],f) =< ~(l,C[b],f) + Z ~(Ik,C[a],f) k=l

+ c5(i + 825 ) Ill.

n in IITn[']ll2, 2 --< C O If" I. (n >= i) for some absolute constant C O .

If I(a)iI ~ 4 c O 8, then IIbll. ~ (2 + 4C0)~, and hence

~(l,C[a],f) ~ ~C(I + (2 + 4C0) ~) Ill ~ Const (i +v~) Ill.

If l(a)ll > 4c0~, we put b = b - (a) I. Then II~II. ~ 2 ~. We have

~(l,C[b],f) = fi ]I I f<Y~ dyldx X

(x-y)- i(a)l(X-y) + i fy b(s)ds

I -i ..... li-i(a)iI fl l(-~)H(Xlf)(x) + n= 1Z (l_i(a) I

-~ {~ + Z (i + l(a)l12) -n/2 c0n ii~ii~ } ii I =< Const llI. n=l

--)nTn[~](Xlf)(x)Idx

Thus

~(l,C[a],f) _~ Const(l + V~) III + Z ~(Ik,C[al,f). k=l

We can apply this argument to I k. Repeating this, we have

1 -[-+q-a(l, C[a], f) =< Const (i +V~),

88

which gives (2.49). Lemma 2.5 and (2.49) yield (2.39). This completes the proof

of (2.39).

§2.10. Application of (2.38)

As is well-known, Theorems B and C are applicable to the higher dimensional

Neumann problem~ pseudo-differential operators and the estimate of analytic

capacity ([6], [I0]). (See Chapter III.) In this section, we show an immediate

application of (2.38). For a locally rectifiable curve F in the complex plane

¢, LP(F) denotes the Banach space of functions f on F with norm

IIfIILP(F ) = { IF f(z)IPldzl} I/p (lap<-).

The Cauchy(-Hilbert) transform on is defined by

f(~) Id~l • 1 lim f <-z (2.50) H F f(z) = ~ l<-zl > s S ~0

The norm of H F as an operator from LP(F) to itself is denoted by

IIHFI I We say that F is a chord-arc curve with constant M if, LP(F),LP(F) •

for any z, ~ 6 F, g(z,~) a Mlz-~l, where ~(z,~) is the length of F

between z and ~ . We show

Corollary 2.14. Let F be a chord-arc curve with constant M. Then

IIHFIIL2(F ),L2(F) a Const M 2.

Proof. Fixing z 0 6 F, we parametrize F so that

Let p

r= {z(t); t Em} , ~(z0,z(t)) = Jtf.

be an even function in C O such that

3 p(x) = 1 (la xaM), ~ Hp(k)rl =< Const,

k=O

supp(p) c [-M-l, -1/2] U [1/2, M+l].

Put h(z) = p(Izl)/z (z 6 ¢). Since Is-tl/M a Iz(s)-z(t)I ~ Is-tl (s,t 6 ~),

we have, with a(t) = Mz'(t)

1 M (Mz(s)-Mz(t))-I z(s)-z(t) s-t s-t

= ~s-t h(s-~it fts a(u)du) ( = M T¢[a,h](s,t), say).

69

We have Iz'(t) I = 1 a.e. and

HFf(z(s)) = (-~) lira fls_~>g s -~ 0

z(s)-z(t)

= (-~) M T¢[a,h] {(faz)z'}(s)

Hence II~IIL2(F),L2(F ) = n M IIT~[a,h]II2, 2. Let

f(z(t))z'(t)dt

a.e.

F h(s + it) = f~= /~ e-i(sx+tY)h(x + iy)dx dy •

Then

Tc[a,h ] = Const ~_~ / ~ Fh(s+it) E[Re {a(s-it)}] ds dt

= Const ~_~ f_=Fh(s+it) {E[Re {a(s-it) } ] + ~ H} ds dr,

oo oo

since ~_~ /_ Fh(s+it)ds dt = Const h(0) = 0. Lemma 2.10 and (2.38) show that

II E[Re {a(s-it)} ] + ~ H!I2, 2 Const fiRe a(s-it)llBMO

Const M Is+itl ,

and hence

llTc[a,h ] N2,2 ~ Const M /_~ f_~ I Fh(s+it) I Is+itl ds dt.

Integration by parts shows that, for n = 2, 4,

I Fh(s+it) I n

= l(_is)-n f_~ f_~e-i(sx+ty) B n 8x

Const/Isl n .

h(x+iy) dx dy I

In the same manner, I Fh(s + it) I & Const/It[ n (n = 2, 4). Thus

[ Fh(s + it) l ~ Const/Is + itl n (n = 2,4 ) . We have

® f ® IFh(s+it)[ Is+itj ds dt = Ills+itl ~ i + ffls+itl > i

ds dt +ff ds dt Const {ffls+itl ~ 1 Is+itl Is+itpl Is+itl3 } ~ Const.

Consequently,

70

lIH r = ~ M lIT¢[a,h]II2 llL2(r),L2(r)

Const M 2 f_Z f_Z IFh(s+it) I Is+itl as at

This completes the proof of Corollary 2.24,

Const M 2 .

Q.E.D.

CHAPTER III. ANALYTIC CAPACITIES OF CRANKS

§3.1. Relation between ~'(.) and H

In this chapter, we study analytic capacity y(.) from the point of view of

integralgeometry and the Cauchy transform on graphs. We shall estimate 7(.) of

so-called cranks. For a compact set E in the complex plane ~, H'(E c) denotes

the Banach space of bounded analytic functions in ~ U {=} - E( = E c) with

supremum norm l!'il ~. The analytic capacity of E is defined by H

(3.1) T(E) = sup{If'(~) I ; Ilfll ~ l, f E H®(EC)}, H"

where f'(®) = lim z ~ ~ z(f(z) - f(=)), i.e., f'(') is the 1/z-coefficient of

the Taylor expansion at ~ ([29, p.6]). If f E H~(EC), II fll ~ ~ i, then H

g(z) = (f(z) - f(®))/(l - f(®) f(z)) ( E H=(Ee))

satisfies g(~) = 0, IigIl ~ & i and H

Ig'(®)I = I f'(~)I /(i - I f(~)l 2) _>_ If,(= ) I.

Hence, to estimate y('), we can restrict our attention to functions vanishing at

~. The Cauchy transform of a complex measure ~ in ¢ is defined by

i f~ i d~(~) (z ~ supp(~)). C ~(Z) = 2~i ~ ----q-"z--

We put

i (3.2) y+(E) = sup{-~- / d~; IIC~H ~ i, ~ ~ 0, supp(~)c E} .

H

Since (C~)'(') i f d~ we have Y(E) g T+(E). Let D(z,r) denote 2~i

the open disk of center z and of radius r. For s > 0, we put

IEIg = 2 inf Zk= I rk, where the infimum is taken over all coverings

{D(Zk,rk)}k= 1 of E with radii less than s. The generalized length of E

is defined by IEI = lim s ~ oIEls" If E c e, then the generalized length of E

equals its 1-dimenslon Lebesgue measure. We shall compare T(-), Y+(-) and

If. A set F c ¢ is called a locally chord-arc curve with constant M, if, for

any z £ F, there exists E > 0 such that Fn D(z,e) is a chord-arc curve with

constant M. A locally chord-arc curve is not, in general, connected. Let F be

a locally chord-arc compact curve with constant i00. We define

72

(3.3) p(F) = inf y(E)/IEI, p+(r) = inf y+(E)/IEI,

where the infimums are taken over all compact sets E on F . Let LP(F),

N'II (i ~ p < ~) be the same as in §2.10. Let L~(F) be the L" space on L p (r)

r with supremum norm li "IIL~(F ) and let L (r) be the space of functions f

on F with norm

NflIL (F) = sup If( )l> ; >

The Cauchy transform H F on F is defined by (2.50). The norm of

operator from LI(F) to LI(F) is denoted by w HHFIIL1 (F) ,L~(F) "

relations among p(F), o+(F) and IIHFIILI(F),L I(F)"

Theorem D.

H F as an

Here are

(3.4) C°nst/IIHFIrLZ(F),L~(F) ~ p+(F) ~ Const/llHrHLl(F), L~(F),

1/3 (3.5) p+(F) ~ p(r) ~ Const p+(F) .

We begin by showing the second inequality in (3.4). Let f E L2(F),

llf H = i. For k > 0, - ~ < e ~ ~, we put Ll ( r )

Ek, e = {z ~ F; H F f (z ) ~ D(ke i e , k /4 )} ,

There exists a compact set Fk~ 8 in EX, ~ such that IFx,el ~ IEx,el/2. exists a non-negative measure

H k,e

Since IIC~II ~ i, we can write H ~

show that IIHFhN ~ Const. L~(F)

For z O ~ F such that

choose first ¢ > 0 so that

i00.

on FX, e such that

d~ ~ y+(Fx,e)/2.

d~ = hldz I with h 6 L~(F),

There

where

so that

0 <_- h <_- 2~. We

H F h(zo) exists (in the sense of (2.50)), we

F n D(Zo,¢ ) is a chord-arc curve with constant

Choose next 0 < e' < s so that, for any z E F c n D(Zo,¢'),

IH F h(z 0) - 2i C~(z)[ g ]H F h(z O) - 2i C(hId~])(z)[ + i,

is the restriction of h to F n D(Zo,g). Choose at last 0 < ¢" < g'

73

[(<) IHr 7,(,o)-~- ~" I < i i <-- J--

r n D(z0,~") c ~ - z0

' ~¢ Since F n D(z0,¢) is a chord-arc curve with constant i00, there exists z 0

such "'4 < = - =< " = ~. Thus we have that s ~ I~ o z~'l<, ~"i2 and r n D(%,i0 -i° ~')

IH r h(z O) - 2i C ~(z~) I ~ IH r ~(z o) - 2i c<~Id~l)<z6>l + i

--< I ~- m IdOl r r r n D(Zo,¢")C ~ - z 0 ~- ~ - z~

I < 1 1 + 2

=< 2 s t=o - =6t r n (D(=o,~)-D(~o,~")) t ~ - =oi I~ - %t Idol

+ 21 F N D(Zo,~")

Since IIC~II =< i, we have H"

r, TIH r h}i ~ Const. L~(F)

Sinc e

1 Id~l + 2 _<- Const.

IH F h(zo) [ -<- Const. Since ~F h(z) exists a.e. on

we have

1 1 7+(Fk,e) ~ -~- I F hldzl k,e - T+(Fk '9 ) '

k ~ Y+(Fx, e) =< ~ I I F k e i0 hldzll

k,e

i iiFk,@ hldz 1 ~-- (H F f) h Idzll + ~ IFk,@

k < ~ II F f(H F h)Idzll + 7 %(Fx,e) = 2n

k Const IIfIILl + ~ T+(Fk,e) ,

(r)

which gives

y+(Fk, e) ~ Const llfIILi(F)/k ~ Const/k .

Since

IEk,@l ~ 21Fk,eI ~ 2 y+(Fk ,e ) /p+( r ) ~ Cons t / (k p+( r ) ) ,

we have

74

Since

Iz ~ r; l~ r f(z) l > k I ~_ I U n=0

.4)n = ~ {const/(X p+(r))} Z (g n=O

k > 0 is arbitrary,

i00 u E I

k=l (5/4)nk, 2nk/100

Const/(k p+(F)).

IIH r flIL~(r ) ~ Const/ o+(r) (f ~ L2(F), llfNLl(r) = i).

A standard argument shows that this inequality holds with f replaced by any

g E LI(F), IIglILI(F ) = i. Hence the second inequality in (3.4) holds.

The proof of the first inequality in (3.4) was essentially given by Davie [21],

Marshall [37] and Davie-¢ksendal [22]. Here is a tool for the proof.

Lemma 3.1 (the separation theorem [53, p. 108]). Let P, Q be two compact

convex sets in the Banach space C(F) of continuous functions on F with norm

II'IIL.(F). Then there exists a complex measure ~ on F such that, for any

f E P, g E Q,

Re fF

Since F

~0 > O

z E F.

f d~ > Re fF g d~.

is a locally chord-arc compact curve with constant i00, there exists

such that F N D(z,s 0) is a chord-arc curve with constant i00 for any 8 Let H F (0 < s < SO/2) be an operator defined by

H r~ f(z) = ~i f f(~l Id~I (f e El(F)) r n D(z,c) c ~ - z

We show that there exists an absolute constant C O such that

H s (3.6) II FII ~ ~ C 0 (= C0m 0, say). LI(F) ,e~(F) llHrllnl(r) ,L~(F)

2g be a maximal operator defined by Let M F

i if(~) i id~l MF2g f(z) = 0 < nsup~ 2s ~ /F(z,n)

where F(z,~) = F N D(z,~). Then

IIMISlILI(F ) i ~ ~ Const. ,n (F)

For f ~ LI(F) with IIftlLl(F) ~ i, we put

(f E LI(F)),

E = {z ~ F; IH F f(z) I = < k ' M2~F f<z) = < k} •

75

Then

l r - El ~ {tlHrltLl(r),L!(D

(m 0 + Const)/k

+ NM~lld(r),L~(r) Const mO/k.

} / x

Let z 0 E E. Then, for any z ~ E N D(ZO,e/2),

IH~ f(z) I ~ IH~ f(z) - Hrf(z) 1 + IH r f(z) I

< 1 l f (~ ) Id~II + k

=< 1w[ ] fF(Zo,g) - f (~)~- z IdYll + Const M2F g f ( z )

i ifF(zo,8 ) f(<) ]d~] I + Const k . -< ~ ~-z

+k

Hence we have

[H~ f(z)[= lHr(~0,~ ) fo(Z)[ + C~ k (z E E n r(z0,~/2)),

' is an absolute constant. where fo is the restriction of f to F(zo,S) and C O

Since F(zo,g) is a chord-arc cu=ve with constant i00, Corollary 2.14 shows that

IJHF(zo,s)H LI(F),L~(F ) ~ Const. (See also (2.10).) Thus

Iz ~ E n r(z0,~12); IH~ f(z)I > (% + l)xl

Iz E E 0 £(ZO,S/2); ]HF(zo,s)fo(Z)l > k I ~ (Const/k)IIfOllel(r(Zo,~))

n We choose a finite covering {D(Zk,g/2)}k=l of r so that z k E F

n XF(Zk,~)IIL.(£) < Const, where XF(zk,g ) is the (i ~ k ~ n) and llZk= 1 =

characteristic function of F(Zk,S)° Then

g Iz ~r; IH r f(zll >(%+i) xI

~- Iz ~E;PH~ f(z)I >(%+l)Xl + n

=~ Z ]z ( E fl F(Zk,S/2);IH F f(z) 1 > k=l

n

_<- (Const/k) k=IZ IF(zk,e ) If(~)l Id~l

IF - E l

(% + 1)~J

+

+ Const mo/k

Const mo/k & Const mo/k,

which gives (3.6).

Given 0 < ~ < gO/2 and a compact set E c F, we put

76

F = {f E L'(F); 0 ~ f(z) ~ i,

IE f(z)IdzI ~ IEl/2, supp(f) c E} ,

P = {H i f; f ( F}, Q = {g E C(F); [Igile.(r ) ~ 3 Como},

where mo = NHFIILI(F),L]w(F) and C O is the constant in (3.6). We show that

P n Q # @. Suppose that P N Q = @. Since P, Q are compact and convex in

C(F), Lemma 3.1 shows that there exists a measure ~ on F such that

Re /F HFs f d~ > Re IF g d~ (f E ~ g (Q)

Taking the supremum of Re IF g d~ over all g E Q, we have

Re IF H F ~ f d~ ~ 3 C0m 0 IFid~I (f E F)

which implies that

where

- Re IF f go IdzI { 3 Com 0 (f E F),

1 d~(<). g0(z) = (7 i r Id~l) -I fr, I< - zp > ~ < - z

By (3.6), we have, for any h E LI(F) with IIhIILI(F ) ~ i,

I z ( F;IH ~ h(z) I ~ 2 Como/IEiI ~ IEI/2.

e is uniformly bounded, this inequality holds with Since the kernel of H F

h Idzl replaced by any measure ~ with fFld~I ~ I. Hence

I z E F; Igo(z)I ~ 2 Como/iEiI ~ IEI/2.

Let F = {z E E; Igo(z) I ~ 2 Como/iEl} and let

function. Then X F E F . Hence we have

X F be its characteristic

3 Com 0 _-< - Re fF XF go fdzI --< IF Igo (z) I Idzl

2 Com 0 --< ~ fF I dz] =< 2 Como,

which is a contradiction. Thus P N Q # ~.

Since P N Q # ~, there exists f~ E L'(F) such that

IIH F f¢ile®(F ) <= 3 tom O. Let {On}n= I

= ~i >-- e2 >-- "''' limn -~ ~ Sn 0 and

fs E F, be a sequence of positive numbers such that

gn {f IdzI}~=l converges weakly (as a

77

sequence of measures). Then the limit is absolutely continuous with respect to

Idzl; we write by fOidz I. We have fO 6 ~ We show that

IIH r fOllL~(r ) ~ Const m O. Let z 0 E F and let e k, e e satisfy

0 < 2s e < e k < SO/2. Then, for any z E F(ZO,ek/2),

Sk St fee(K) Id~lt 1% f %)I ~ ! Imr_F<Zo,Ck) -- Z

i imr, 1 1 ) fse(~) IdYll e k < I~ - Z o l < eO/2 ( ~ - Z 0 ~ - z

+ ! ISr, ( 1 1 ) fee(K) IdYll _ F(Zo,CoI2) ~ - z 0 ~ - z

! s O c~ If fee(<) IdYll + Const M F f (Zo)

- ~ F -F(z0,c k) ~ - z

e e / 2 f (~) Idol Const (e k s 0) fF_F(Zo,ao/2 )

! If r fee(~) - r ( z o , C k ) ~ - z

s t Let fk denote the restriction of

z E F(ZO,ek/2),

!~ if r- r(Zo,~k) ~ - z

IdYll + Const {i + (ek/t~)Irl}.

fee to F(Zo,ek). Then, for any

e t s t s t Id~ll = IH r f (z) -H r fk(z) l

~e ~e 3 Com 0 + IH F fk (z) I

which shows that

Ck t~ s~ s~ I HF f (Zo) l ~ IHF fk (z) l + 3 Com 0

+ Const {i + (%l~)Irl} (z E F(Zo,Sk/2)).

By (3.6), we have

s t s t Iz E r; IH r fk (z) I g

10 -3 Ir(Zo,~k) I

s t s t This shows that the generalized length of {z E F(z0,gk/2); IH F fk (z) I

is larger than or equal to IP(Zo,Sk/2) I/2. Hence the mean of

103 Com 0 ] < 10 -3 st = rlf k NLI(F )

1 7 Ir(ZO'Sk/2)I "

< 103Como } =

78

¢g sg IHF fk (z) l over this set is dominated by Const mo, which shows that

[H r f (Zo)l -<_ Const m 0 + 3 C0m 0 + Const {1 + ( 8 k / S 2 ) [ F I }

= / 2 < Const {m 0 + (¢k 80 ) Irl } "

Since z 0 ( F is arbitrary,

ii. " f"Jl Const {m 0 + ( S k / g ~ ) ) F I } .

Letting first 8 tend to infinity, and letting next k tend to infinity, we have

[[HF 0 ~ Const m 0. f IIL.(F )

Now let d~ 0 = f01dz [ . Since f0( F, [[H r f0[[L.(F ) ~ Const mo, the

maximum modulus principle shows that

1 [[c~°lt --<Cxm0' ~ fE d~° ~- [EI/Cl' supp(~ °) c E, H ®

where C 1 is an absolute constant. Let 00 = O/(Clm0). Then

Hence

00 I~ ->_ O, supp(~ 00) c E, tlc~°°ll H ®

< 1. =

d , ° ° ,

which shows that T+(E)/IE 1 ~ Const m 0. Taking the infimum over all compact sets

E c F, we obtain the first inequality in (3.4).

The first inequality in (3.5) is evident. At last we show the second

inequality in (3.5). Let f ~ L2(F), llfIILl(r ) = i. For k > 0, - ~ < O ~ ~,

we put

EX, e = {z ~ F; H F f(z) ( D(ke i0, pk/4)}

There exists a compact set FX, 0 in EX, 0 such that C

exists g (H (Fk, 0) such that

I lg l l . ~ z, g ( - ) = 0, I g ' ( - ) l ~ y ( F k , 0 ) / 2 . H

We can write g = C(hldzl) with h (L~(F) satisfying

(o = ~(r)).

tFk,o l ~ IEk,o I/2" There

llhllL~(r)< 2~, I1~' r hllL~(r) ~ Const, supp(h) c Fk, 0

Since

79

1 Y(Fk, e) -~ [g'(~)l = ~2~ IfF>~,0 hldzt I =< Y(Fk,@)'

we have

k 1 IfFk,e Xe ie hldzl l

l [ fFk ,o p% fFk, 0 -< 2"--~ (Hrf) h Idzll + lhlJdzl

--< 2T Jfr f HF h Idzl l + k

-< Const llfllLl(r ) + E 7(Fk,e),

I Fk, 0 1

which gives y(Fk,o) ~ Const plflILl(r)/k = Const/k. Since

IEx,0] ~ 21Fk,ol ~ y(Fk,0)/p ~ Const/(pk),

we have, with q = (the integral part of 103/p),

Iz ~ r; IH r f(z)l > xl q

[ U U n=0 k=l E{l+(p/4)}n%,2~k/ql

Const/(p3k),

which gives that

= < Const/p(F) 3 II H F ]ILl (F) ,L~(F)

This inequality and the first inequality in (3.4) immediately yield the second

inequality in (3.5).

§3.2. Vitushkin's example, Garnett's example, Calder~n's problem and extremal

problems ([5], [28], [46], [52])

Painlev~ showed that the analytic capacity of a compact set of zero

g e n e r a l i z e d l e n g t h i s equa l to ze ro , For a compact s e t E of f i n i t e p o s i t i v e n

g e n e r a l i z e d l e n g t h , we can choose a f i n i t e c o v e r i n g {D(Zk,rk) }k=l of E so

t h a t Znk=l r k -< IEI" Then, f o r any f ( H~(E c) w i t h IlfllH~ ~ 1,

i I f f (~) d~ [ 1 f' (~) ] = 2--~ 0 {Uk=ID(Zk,rk) }

n Const Z r k _-< Const [El,

k=l

which gives

80

(3.7) Y(E) =< ConstlE 1 .

This inequality immediately yields Palnleve's theorem. Vitushkin [52] constructed

an example P such that Y(P®) = 0, IP I > 0. The set P is defined as

follows. Let P0 = [0,I]. We divide P0 into two non-overlapping closed segments

[0,1/2], [i/2,1]. Fixing their midpoints, we rotate these two segments so that

the resulting two segments are perpendicular to the x-axis. (The midpoints of the

resulting segments are on ~.) Let PI be the union of these two segments. We

divide each segment (of P1 ) into two non-overlapping closed segments of equal

length. Fixing their midpoints, we rotate these four segments so that the

resulting four segments are perpendicular to the y-axis. Repeating this

discussion, we define Pn; Pn is a union of 2 n closed segments of length 2 -n.

We put P nn= 0 Uk=nP k. Garnett [28] also constructed an example Q~ such

that y(Q ) = O, IQ~I > 0. The set Q~ is defined as follows. Let

Q0 = [0,I] x [0,i]. Let QI be the union of four closed squares with sides of

length 1/4 in the four corners of Q0" In the same manner, Qn is defined from

Qn-i with each component of Qn-i replaced by four closed squares with sides of

length 4 -n in the four corners of the component. We put Q~ = Nn=0 Qn"

There are several proofs of y(Q ) = 0. Supposing that the non-trivial

Ahlfors function [29, p. 18] of Q exists, Garnett [28] showed a contradiction.

Using Besicovitch's set theory [i], Mattila [38] also gave an indirect proof. A

direct proof from the point of view of the construction of Garabedian functions

[29, p. 19] is given in [46]. This method is applicable to estimate Y(') of

various sets.

It is sufficient to construct a sequence {(R n,fn )}n=10000~ of pairs of

and f (H=(R~) so that R o fn(~) = i and compact sets R n n n Qn'

78Rn [fn(Z) l Idzl ~ Const/(log n).

If such a sequence exists, we have, for any g (H~(Q~), IIgll ~ i, g(~) = 0, H ~

Ig'(~)l = 2~i I fORn g(z) dz I 2--~i I fSRn g(Z)fn(Z) dz I

i ~2-~ fDR Ifn(Z) lldzl ~ Const/(log n),

n

which shows that y(Qn ) ~ Const/(log n) o Thus y(Q ) = 0. The pair (Rn,f n)

is constructed as follows. We denote by m the integral part of (log n)/2.

For a closed square Q with sides parallel to the coordinate axes, ~(Q)

denotes its lower left corner and g(Q) denotes the length of a side. Let Q_,

Q+ denote two closed squares in Q with sides parallel to the coordinate axes

81

such that 2(Q_) = £(Q+) = 4-mg(Q), g(Q_) = ¢(Q) and the upper right corners of 4 k

Q+ and Q are identical. For k > 0, we can write Qk = U j=l Qk,j with k 4_k. components {Qk,j}4_1. Note that %(;Qk,j ) = We define inductively 4 m + 1

j-1 4m compact sets {Vk}k= 0 by V 0 = (Q0)_U (Q0)+,

V k =U E~ ~k (Qkm'~)-U (Qkm'~)+ (i < k < 4m),

where ~k {i < ~ < 4 km, k-i ~} Let ~0 {i} We now put = - - " Qkm,~N (U j=0 V j) = . = . 4 m 4 TM

(3.8) Rn = {k=0 ~ V k} 0 Q(4m+l) m, fn (z) = k=0~ %E~kH u(4km(z - E(Qkm,%))),

where

u(z) = exp[e i~/4 4 -m { - - - 1 + 1 }]~ z_(e i~/4 4-m/~) z_l_i+(e i~/4 4-m//~)

Then (Rn,fn) satisfies the required conditions. (See [46].)

82

Calder~n [5] suggests to study C[a] for a ~ L °° real" This problem seems very hard;

in effect, Theorem D and Garnett's example immediately yield {IIC[a]II2,2; a E Lreal}

I = = o~ (See Remark 3.16.) Let Qn {(2-i)z/3; z 6 Qn }. (We rotate and contract

! V 4n Qn.) Then Qn is a union of 4 n squares {Qn,j }j=l with sides of length

(/5/3)4-n; the sides of Qn,j are not parallel to the coordinate axes. Let In, j be

the projection of Qn,j to ~R and let Qn,j be the segment in Qn,j whose n

projection to ~ colnsides with In, j (i ~ j ~ 4n). The intervals {In,j}4=l

are mutually non-overlapping, the length of each interval is 4 -n and their 4 n ,,,

union equals [0,i]. Let F n = Uj= I Qn,j , where Qn~,j is the closed sub-segment

of Qn,j of the same midpoint as Qn,j such that IQn,jl = IQn,jl/2. Then F n

is a locally chord-arc compact curve with constant i. Since

IFn I = vr~-/6 and y(Qn ) = (vr~/3)Y(Qn) , Theorem D shows that

IIHFnIL lw ~ Const/p (F n) -_> Const/p(F n) (3.9) l(r n) ,L (F n)

_a Const IFnl/Y(Fn) = Const/y(Fn)

>= Const/Y(Qn) = Const/Y(Qn).

We see that

llH r IIi < Const {l]HFnlIL2(rn) + i} . n L (r n) ,Lwl(rn ) ,L2(Fn )

,,, 4 n (See (3.18).) Since the projections of {Qn,j}j= I to ~ are mutually disjoint,

we can define a graph {(x, An(X)); x 6 ~} containing F n such that

A'n (= an) ~ Lreal." Since an(X) = 1/3 a.e. on the projection of Fn to ~R,

IIHFnIIL2(Fn),L2(Fn ) ~ Const llC[an]II2, 2 .

Thus (3.9) shows that

I/Y(Q n) ~ Const{IiC[an]II2, 2 + i}.

Since lim n ~ Y(Qn ) = y(Q ) = 0, this gives that {IIc[a]II2,2; a E L~eal} =-.

It is very important to give various reasonable grounds to Vitushkin-Garnett's

examples. From this point of view, we consider the following extremal problem.

Let I 0 = [0,I). For s I, ..., s n E ~, we define

TSl ..... sn(X,y) = i/{(x-y) + i(Asl ..... an(X) - Asl ..... sn(Y) )} ,

where

83

As I ..... sn(X) = I 0

s k

(x ¢ I 0)

k-i k 1 <- k < n) (--~-_~ x < n . . . .

Our extremal problem is the following:

ex(~(n) = max {~(Tsl ,...,sn); Sl, ..., s n

We see that

( m} (n _~ i).

Const iV~og(n+l) & ex (n) ~ Const ~og(n+l)

(See Appendix I.) We define a function

A~(x) = E 10m k= I Sk(X) (x 6 I0), where m

and

gk(X) =

(n ~ i).

A 0 on ~ by A~(x) = 0 (x ~ I 0) and n

is the integral part of (log n)/(log i0)

0 (j-l)10 -k ~ x < j i0 -k, i ~ J ~ i0 k j is odd

i0 -k j is even.

Let TAO(X,y) be the kernel associated with A O.n

n

~(T 0 ) ~ Const I/~ = Const ~og(10m). A n

(See (3.14).) Hence r 0 = {(x, A~(x)); x ~ I 0 } n

respect to ex ( n ) . The g r a p h F 0 i s s i m i l a r t o ~ n

that

Then

is one of the worst graphs with

Qm" Hence Theorem D suggests

Problem 3.2. Const T(F~) ~ min {T(Fsl .... ,Sn); s I ..... Sn6 ~}

Const T(F~) (n g i),

= {(x, A (x)); x ( I 0} • where Fsl,...,s n Sl"'''Sn

§3.3. The Cauchy transform on cranks

As a first step of harmonic analysis on discontinuous graphs, it is natural to

begin with worst graphs. We say that a set E c ¢ is thick, if there exists

M > 0 such that, for any z ( E, r > 0,

I/M ~ IE n D(z,r) I/r ~ M.

The 1-dimension Caldergn-Zygmund decomposition is applicable to thick sets. Hence

thick sets are also natural objects. From the point of view of §3.2 and

84

"thick sets", we define (thick) cranks.

An interval i in I 0 = [0,i) is called a dyadic interval if I is

expressed in the form I = [(j-l)2 -g, j2 -g) with integers g >-_ 0, 1 = < j = < 2 g. A m

finite sequence R = {Ik}k= 1 of mutually disjoint dyadic intervals is called a m

covering (of I0) if I 0 = Uk= 1 I k, For a positive integer q and two coverings

n ~ m R' = {lj}j=l, = {Ik}k=l, we write by R'< R if each I'~ is expressed as a

q J

union of at least 2 q elements of R of same length. A segment I 0 is called a (thick)

crank of degree 0. For a positive integer n, a graph F = {(x, ~(x));x E I 0 }

is called a (thick) crank of degree n, if there exist n coverings R I, ... R n

and n functions AI, ..., A n on I 0 such that

(3.10) I 0 <q R I ~q ... ~ R for some n tuple 1 2 qn n

(ql .... , qn ) of positive integers,

(3.11) A F = A I + ...+ An,

(3.12) on each element I of Rk,

)~(x) I ~ Ill (i ~ k ~ n).

A k is a constant and

For two positive integers n, q and two real numbers =, ~ less than or equal to

i, we define a crank

F(n,q,e,~) = {(x, ~(n,q,a,~)(x)); x E I 0 }

by

~(n,q,~,~) = A(q'~'~) + .... + A(q'a'~) n

A(q,a,~)(x ) =I ~2-qk

k (~2_qk

(x E [(j-l)2 -qk, j 2-qk), j odd)

(x E [(j-l)2 -qk, j 2-qk), j even).

We show

Theorem E. Let F be a crank of degree n. Then

(3.13) IIHFNL2(F ) s Const V~. ,L2(F)

There exists an absolute constant ~0 such that, if

(3.14) lIHF(n) llL2(F(n ) ~ Const V~n ),L2(F(n))

I~ - ~I 2 ~ D0/q, then

(r(n) = r(n,q,~,~), n ~ i).

85

In this section, we give the proof of the first half of Theorem E. For

a crank F = {(x, AF(x)); x ~ I0}, we put AF(x) = 0 outside I 0 and define a

kernel

TF(X,y) = I/{(x-y) + i(AF(x) - AF(Y)) }

For f E L2(F), we have

(x # y, x, y ~).

H F f(x + i AF(X))

= 1 1 - ~ p.v. f0 TF(x'Y)f(Y + i AF(y))dy a.e. on I 0,

and hence

(3.15) IIHFIIL2(F) ~ ! lIT FI12,2. ,L2(F)

Here are three lemmas necessary for the proof of (3.13).

Le~mma 3.3. Let F be a crank of degree n. Then

Proof.

NTFII2, 2

Evidently,

_-< Const {~(TF) + I} .

(3.16) I TF(x,Y) I =< i/Ix-yl .

For any dyadic interval I, we have

(3.17) ITF(x,Y) - TF(x',Y) I --< Const II I/ Ix-y 12

(x, x' 6 I, y ~ I = (the double of I)).

This is shown as follows. Let R I, ..., R n be the coverings associated with F

and let AI, ..., An be the functions associated with AF. We denote by m I

the smallest integer of k (i =< k ~ n) such that R k has an element contained

in I. Then, for any x, x ~ 6 I,

IAk(X ) - Ak(X')l

< J 0 (i ~ k < m I)

= I 2ml-k+l • III (ml~ k ~ n).

Thus

ITr(x,y) - rr(x',y) 1 ~ Constl(x + i At(x)) - (x' + i Ar(x')) ! /Ix-yl 2

n

<= Const{Ix-x' I + Z IAk(X) - Ak(X')l} /Ix-yl 2 k=m I

ConstllI/Ix-yl 2.

86

The proof of this lemma is analogous to the proof of Lemma 2.5; (3.16) and (3.17)

play the same role as el(.). In the same manner as in the proof of (2.9), we

obtain

~(T~) ~ Const {~(TF) + i}.

J' = (x - (~/2), x + (e/2)) by the largest dyadic interval in J'

Using this inequality, we obtain

(Replace

containing x.)

(3.18) Ix; T~ f(x) > 3 k, J~ f(x) ~ ~ k I

i I x; T~ f(x) > k I (f E L 2, k > O) <

= i00

where ~ = C O {~(TF) + i} and C O is a suitable constant. (Replace

{Ik}k= 1 by a suitable sequence of dyadic intervals.) Inequality (3.18)

immediately yields IITFII2, 2 ~ Const {~(T F) + i}.

For a non-negative integer n, we put

o(n) = sup {~(TF); F crank of degree & n} ,

~(n) = sup {$(I O, TF,f); f E Lreal, 0 ~ f a i,

F crank of degree ~ n} .

Q.E.D.

supp(f) c I 0. Let F be a crank of degree n, R 1 .... , Rn be n coverings

satisfying (3.10) and let A I, ... A be n functions satisfying (3.11) ' n

(3.12). Put

Then

and

F' = {(x, AF,(X)); x ~ I0} , A F, = A 1 + ...+ An_g_l ,

m

Rn_ g = {Ik}k= I.

F' is a crank of degree n -~ - i. We have

$(I 0, T F, f) = (TFf , TFf)fd x = ~(I0, TF,, f)

+ ((T r - Tr,)f, TFf)fd x + (TF,f, (T r - TF,)f)fd x

We show

Lemma 3.4. For two positive integers n, g with g ~ n-l,

(3.19) ~(n) ~ $(g) + ~(n -g - i) + Const ~(n).

Proof. Let f E Lreal, 0 ~ f ~ i. Without loss of generality we may assume that

87

((T F - TF,)f , TFf)fd x m

= Z flk(T F - Tr,)(X k=l

ik f)(x) TF(Xlkf)(x) f(x)dx

m

Z flk(TF - TF')~Ik f)(x) TF~ c f)(x) f(x)dx k=l I k

m E f c (rr - TF')(Xlkf)(x) rrf(x) f(x)dx k=l I k

( = L I + L 2 + L3, say).

Since TF,(x,y) = i/(x-y) (x,y E I k, i _-< k =< m) and ~(n) ->- Const, we have, by

Lemma 3.3,

m m

ILl1 ~ Z ~(I k, T F, f) + ~ % flk IH(Xlkf)(x) Tr(Xlkf)(x)Idx k=l k=l

m

Z ~(I k, k=l

TF, f) + Const IITFII2,2

m

E k=l

^

~(Ik, TF, f) + Const o(n).

Extending coordinates, we see that, for each Ik, there exist

0 ~ fk ~ i and a crank F k of degree g such that

$(I k, T F, f) l lkl ~(I 0 , fk ). = , TFk

L ~ fk ~ real'

Hence

m

ILl1 ~ % llkl ~(I0, fk ) + Const o(n) k= I TF k '

~(~) + Const a(n).

Recall (3.16) and (3.17). Since TF(X,y ) - TF,(x,y) is anti-symmetric, we

* = (the double of Ik) have, with x k = (the midpoint of Ik) , I k

m

IL21 = I Z 7!k(T F - TF,)(Xlkf)(x) k=l

× {TF(× *c f)(x) - TF(× *c f)(xk)} f(x) dx I k I k

88

m

+ Z flk(T F - TF,)(Xlkf)(x ) TF(× , f)(x) f(x)dx I k=l Ik-I k

m =<- Const Z Ilk I(T F - rF,)(Xlkf)(x)l Mf(x) dx

k = l

m

+ k=iZ Ilk I(T F - TF,)(Xlkf)(x) I (/ik_l k ~ ) dx

_-< Const lIT F - TF,II2, 2 =< Const ~(n).

We have

IL31 ~ j,k;j#kZ Ii j {flk ITr(x,y ) - Tr,(x,Y) idY } ITFf(X) I dx

z fi~ {71, c j,k;j#k cN lj j n I k

ITF(X,y) - TF, (x,y)IdY}ITFf(x) I dx

+ Z {I * c I TF(x,y)-TF,(x,Y) IdY} ITFf(x) I dx $1kC* N I. (lj -lj) N I k j ,k;j#k J

+ Z I , {Ilk ITF(X,Y) - TF,(x,y) IdY} ITFf(X) Idx j,k;j#k (Ik-I k) N lj

= L31 + L32 + L33,

m

[L331 ~ z f, k=l Ik-I k

{flk ITF(X'Y) - Tr,(x,y) Idy } ITFf(x) idx

m

2 Z f , (flk ~ ) ITFf(x) idx k=l Ik-I k

m ~ ) 3 dx}i/3 2 Z {f , (flk {I .

k=l Ik-I k Ik-I k

m Const Z Ilk II/3 {7 , ITFf(x) I 3/2 dx} 2/3

k=l I k

m 1/3 m dx}2/3 Const ( Z llkl) { E I . ITFf(x) I 3/2 k=l k=l I k

m llk 12 Const { I_~ ( Z ) ITFf(x) l 3/2 dx} 2/3

k=l (X-Xk)2 +llk 12

ITFf(x) I 3/2 dx} 2/3

89

m

Const {f ( Z k=l (X-Xk)

IIk 12 )4}I/6 {7 ~ITFf(x) l 2 dx} I/2

T llk 12

I[Trf] [ < Const ][TF] [ < Const ~(n), Const 2 = 2~2 =

m

IL3,21 ~ Z {7 , ITF(x,Y) - TF,(x,y) IdY} ITrf(x) Idx j=l flj lj-lj

m

7i j dy i d x =< 2 z (7 , ]~ ) ITrf(x) j=l I.-i.

J ]

Const IITFfll 2 ~ Const o(n)

and

IL311 l~(x) - Ar,(x) I + IAF(y)-AF,(y) I

Z fI~ {f *c ~, n ~k Ix - yl 2 j,k;j#k c N lj ]

m n Z 71. I Z A (x) l (71,c dy ) iTFf(x) idx

j=l 3 ~=n-g ~ . Ix-yl 2 J

dy}ITFf(x) Idx

Const

+ Const

Const /i 0 ITFf(X)]dx + Const f ~ ( Z - k= I iX_Xkl 2

m n

Z fik {Tik I Z A (y) I/Ix-yl 2 dy} ITrf(x) Idx k= I c ~=n-g

m

lljl (fl, c dy ) irFf(x) idx j=l flj • Ix_yl 2

]

m

Z J" *c IIkl (7I k dy ) id x k=l I k Ix-Yl 2 ITrf (x)

m llk 12

+ Ilk 12 )ITF f(x) Idx

Const IITrfll 2 ~ Const ~(n).

Thus

IL31 ~ Censt ~(n).

Consequently,

l((r r _ rF,)f, rFf)fdxl ~ ILII + Ie21 + IL31 ~ ~(g) + Const ~(n).

90


I(TF, f, (T r - TF,)f)fdx[

m

Ik=IZ Ilk TF,(Xlkf)(x) (T F - TF,)(Xlkf)(x) f(x)dxl + Const ~(n).

Since TF,(x,y) = i/(x-y) (x,y E I k, 1 ~ k ~ m), the first quantity in the right-

hand side is dominated by Const ~(n). Thus

3(10, Tr, f) ~ ~(Io, TF, , f) + $(g) + Const ~(n)

~(n -g- I) + ~(g) + Const c(n),

which gives (3.19).

Lemma 3.5. a(n) 2 ~ Const ~(n) (n $ 0).

Proof. We see that, for any crank F , ~(TF) 2 ~ Const $(TF).

(See(Second Step) in §2.8.) Hence it is sufficent to show that

Q.E.D.

(3.20) sup {$(Tr); F crank of degree n} ~ Const $(n).

Since $(0) ~ Const, (3.20) holds for n = 0. Let F be a crank of degree

n g 1 and let f E L~eal, 0 ~ f ~ i. For any dyadic interval I, there exist

a crank F I of degree ~ n and fI ~ LTeal' 0 ~ fI ~ 1 such that

~(I, TF, f) = III $(I0, TFI,fI ). Hence (i/IIl) $(I, T F, f) ~ $(n). For any

non-dyadic interval I c I0, there exist mutually disjoint two dyadic intervals

I I, 12 such that Jill = tI21 ~2JII, ~I IUI 2 Then

~--~i $(I, TF, f) = ~i 3(11 U 12, T r, Xlf)

~ {3(11' T r, Xlf) + $(I 2, T r, Xlf)

+ (if2 dy )2 d i) 7i I ix_y I dx + 7i 2(7I 1 ix_y I

Const {~(n) + i} ~ Const $(n).

2 dx}

For any interval I c ~, we have

Thus

1 $(I 0 N I, TF,f) + Const ~ Const ~(n). 1 $(I, TF,f) ~ ill

$(TF) ~ Const ~(n), which gives (3.20).

In the same manner as in Lemmas 3.4 and 3.5, we have

Q.E.D.

91

Lemma 3.6. o(n) ~ ~(n-l) + Const (n ~ i).

We now give the proof of (3.13). Since ~(0) ~ Const, Lemmas 3.4 and

3.6 show that ~(n) < - for all n ~ i. By Lemmas 3.4 and 3.5, we have, for

any non-negative integer m,

~(2 m) ~ ~(2 m-l) + $(2 m-I - i) + Const ~(2 m)

_<- 2 $(2 m-l) + Const ~(2 m) =<...

m =< 2 m $(i) + Const Z 2 m-k ~(2 k) =< 2 TM $(i) + Const 2 m ~(2 TM)

k=l

_<- Const 2 m ~(2m) I/2,

which shows that $(2 m) -<_ Const 2 2m, Hence

m 2m_k ~(2 m) -_< 2 m ~(i) + Const E ~(k) I/2 k=l

m -_< 2 m $(I) + Const % 2 m-k 2 k = < Const(m+l)2 TM-

k=l

Consequently,

m 2m_k ~(k) 1/2 ~(2 TM) <= 2 m $(i) + Const Z k=l

m 2k/2 2 m. <_- 2 m ~(i) + Const % 2m-k~/-~ -<_ Const

k=l

For an integer n g i, we choose a non-negative integer m so that

= 2 m+l . 2 m < n < Then

$(n) ~ $(2 m+l) & Const 2 m+l ~ Const n,

Consequently, Lemmas 3.3 and 3.5 yield (3.13).

§3.4. Proof of the latter half of Theorem E

The following idea is essentially due to David [18, Chap. III].

q, ~, ~, we write

F(n) = F(n, q, e, ~) A n = A (q'~'~) (n e i). n -

We put

T (x,y) = TF(n)(X,y) x-y '

~0 (n) = ~(I0' T0n' XI O) (ng i).

Fixing

92

Here are two lemmas necessary for the proof.

Lemma 3.7. ~0(I) ~ [~ - ~12/1o0. Proof. Let x ( [(k0-l)2-q, k02-q) (k 0 is odd). Then

0 ( ~ ) [ = I T 1 ×i0 1710 1 i ~} dy I { (x-y) + i(Al(X)-Al(Y)) x-y

Z f dy I k even [(k-l)2-q,k2 -q) {(x-y) + i(a-~)2-q}(x-y)

>= ~ - ~12 -q I Re Z I I k even [(k-l)2-q,k2 -q)

- #12 -q Z 7 dy ......

k even [(k_l)2-q,k2-q) (x_y)2 + (~_~)2 2-2q

=

f2-q +I - ~12- q dy > I S _ BI/10.

2- q y2 + (~_6)2' 2-2q =

In the same manner, we have, for any x ( [(ko-l)2-q, k02-q) (k 0

IT~ XIo(X) I a Is - ~I/lO.

Thus

~ 0 XI0) > ( ~ )2 dx > Is - ~12/I00. ~°(I) = $(I°' t1' = I[o lO =

is even),

Q.E.D.

Lemma 3.8. For two positive integers n, g with g ~ n - i,

(3.21) T0(n) e ~0(~) + ~0(n - g) - Const q-l~-~-.

Proof. We write

I k = [(k-l)2 -qg, k2 -qg ) (i ~ k ~ 2qg).

We have

= t 2 ~ o ( ~ ) ~ o ( n ) Izo Ir~ ×~o(X) dx =

+ fl0(T~- T~)(x) T 0 (x) dx + 0 (T~ T~) dx XI 0 n XI 0 fl 0 TZ Xl0(X) - XI0(X)

= T0(g) + L 1 + L 2,

2 qg 0 T~(x,y)dy} dx

L I = k=iZ /ik {/ik(T~(x,y) - Tg(x,y))dy}{flk

03

2 qg Z k=l

0 T~(x,y)dy}dx flk {flk (T~(x,y) - T~(x,y))dy} {flo_l k

2 q&

k=IE fl0-1 k {fl k (Tn 0(x'y) - TO(x'y))dY} T0n XI O(x) dx

= LII + LI2 + LI3.

0 T~(x,y) = 0 (x, y EIk, 1 ~ k =< 2 q~)- and Note that

n

T (x,y) = [(x-y) + i Z = g+l

(A (x) - A (y))]-i 1 x-y

(x, y E I k, i ~ k ---- 2 qg)

Hence, extending the coordinate axes, we have

2 qg 2 q&

= k=iZ flk [flk T~(x,y)dy] 2 ^ dx = k=iZ Ilk] ~0(n - L) = ~0(n - £). LII

Let p be the integer such that q4 2q4 < p ~ and (log p)/log 2 is an

integer. For each i ~ k ~ 2 qg, we write I k = Ik, 1 U... U I , 2 k,p 2

where P {Ik,j}j= 1 are mutually disjoint dyadic intervals of length p-2 2q~

Let ~k denote the closed interval of the same midpoint as I k and of length

(i + p-l) Ilkl , and let Ik, j denote the closed interval of the same midpoint

as Ik, j and of length (i + p-4) llk,jl . We have, with

x k = (the midpoint of Ik) , Xk, j = (the midpoint of Ik,j) ,

2 qg

LI2 = k=iZ flk {fiE T~(x,y)dy} {fl0_l k T~(x,y)dy} dx

2 qg

k=iE fl k {flk T~(x,y)dy} {f(l 0 N ~k)-Ik T~(x,y)dy}dx

2 qg

z {Ilk k=l fl k % .(T~(x,y)-T~(Xk,Y))dy}dx T~(x,y)dy} {fl0_(10 N i k)

= LI21 + LI22 ,

ILl211 2 qg dy

2 k=iZ fie Iflk Tn0(X,y)dy I (fik_l k ~ ) dx

04

2 q& 2 q8 Z ~(Ik ' T O )}i/2 { Z k= I n' XI k k= I

fl k (f~k_l k ~ )2 dx}i/2

Const {f~ log 2 dx k,j k,j

2q~ p2

Z E Z {/I } {fI0-(l 0 Nik)} k=l j=l f(l k N Ik,j)-Ik, j k,j

2qg p2

Z Z f ~c {fI k } {fI0-(~ 0 n ik)} k=l j=l I k 0 k,j '3

= L1221 + L1222 + L1223 •

Since n 2 IX-Xk,j I +t ~ (A~(x>-A~(Xk, j) ) 1

~=~+I ........... ]TnO(x,y) - T0n(Xk,j, Y)] _-< [x_y I IXk, j - y]

Cons¢ {p-2 2-q8 + 2-q(8+i)} / ]Xk, j _ y12

Const p-2 2-q%/iXk,j _ yi2

(3.13) shows that

2qg p2

IL12211 = I Z Z k=l j=i

(x ( Ik, j, y £ ~),

fik,j{fIk, j T~(x,y)dy}

× {fi0_(i 0 fl I k)

2q8 p2 -2 2-qg

Const Z Z fI Ifik,j T~(x,y)dy] (f~ P dy) dx k=l j=l k,j ~ IXk,j-Y] 2

2q~ p2 Const p-i Z Z fIk, j]fI T~ (x'y)dyl

k=l j =i k,j

95

Const p-i NT~H2, 2 ~ Const p-i ff .

Since

ITnO(x,y) - TOn(xk,Y) I ~ Const p 2-q~llxk-y[ 2 (x E I k, y E ~k ) ,

<_- [-2 [TO(x,y)] =< Const 2-q(g+l)/Ix-yl2 Const 2 -q(e+l) p8 llk, j

~C =< Const pl0 2-q/llk,jl (x 6 leo Ik, j, y ~ Ik, j)

we have

2 qg [L1222] =< Const E

k=l

~_ Const p2 fol

2 P ~-~ P 2 -q~ Z f g (flk'j )(flk IXk -y]2 j=l Ik,j-lk,j

log(l + p-4s-l) ds ~ Const/p

dy)dx

and

2qg p2

[L1223 [ =< Const Z Y f ~c (Const pl02-q)(f~ k=l j=l I k 0 k,j k

2 q~ =~ Const p14 2- q E fl k dx =< Const p14 2- q .

k=l

p 2 -q~ 2 dy) dx

I Xk-Y I

Thus, by (3.13),

ILl21 ~ILI211 + IL12211 + IL12221 + IL1223 I

Const {p-i/2 ~0(n - ~)i/2 + p-i/~ + p-i + p14 2- q }

= Const p-i/2 ~ e)i/2 < {~o(n - + V~ } ~ Const~/q.

Since

iT0n(X,y) 0 2-q(g+l) / - rg(x,y) I _~ Const Ix-yl 2

~C Const p2 2- q llkl/{iX_Xkl2 + llk12 } (x E I k, y EIk),

we have, in the same manner as in the estimate of IL331 in §3.3,

2 qg ILl31 _-< Const k=i% f(lo nik)_Ik ( fi e ~ ) IT0n Xlo(X) lax

96

2 q~ i lkl 2 + Const p2 2-q { % n

sz0 k=l Ix-x~I 2 + tlkl 2 } LT°×I0 (x)l 2 qg

Const { Z ~ ( ~ )3 dx}i/3 k=l fIk-I k IIk

dx

2 qg × { Z f~k IT0 X10(x) 13/2 dx}2/3

k= 1 n

+ 2qg l lkl 2 )

Const p2 2- q {flo ( Z 2 2 k=l I X-Xkl + llkl

Const

2 dx}i/2{/io ITn~IO(x) 12dx}I/2

2 q~ i, ~ ~i/3 {fl0/P log3(l + s)aS~ { Z k=l

n X10(x) 13/2 7~k I T0 dx} 2/3

+ Const p2 2- q 1IT,ll2, 2

Const (p-i/4 + p2 2-q)NT~II2,2 =< Const q-iVan

Thus

L I ~ LII - ILl2 [ - ILl31 ~ ~o(n - g) - Const q-l~/~.

In the same manner,

2 qg IL21 _~ Ik=iZ /ik {flk TO(x,y)dy} {flk(TO(x,y) - TO(x,y))dy} dxl

+ Const q-l~-~n.

Since T~(x,y) ~ 0 (x, y 6 I k, 1 ~ k & 2q&), IL21 & Const q-l~nn. Consequently,

~0(n) ~ ~O(g) + L I - IL21 ~ ~O(g) + ~o(n - g) - Const q-l~-~. Q.E.D.

We now give the proof of (3.14). Lemmas 3.7 and 3.8 show that, for any

positive integer m,

~0(2 m) ~ 2 ~0(2 m-l) - Const q-i 2m/2 ~ ...

-I m~l 2k 2(m_k)/2 2 m ~0(i) - Const q k=0

2 m {I~ - ~12/i00 - COnst/q} .

97

For any integer n ->_ 2, we choose a positive integer m so that

2 m- i 2 m . < n ~ Then

-i ~0(n) >__ ~0(2 m-l) + ~o(n - 2 m-l) - Const q V~

2 m-I {I= - ~12/i00 - Const/q}

~V~{I~ - ~12/200 - Const/q} .

- Const q-iV~

Thus (3.14) holds if ~0 is large enough. This completes the proof of the

latter half of Theorem E.

Corollary 3.9. For any crank F of degree n ~ I, p(F) ~ p+(r) ~ Const/~-~.

If Is - fll 2 ~ q0/q, then p(r(n)) I/3 ~ Const p+(r(n)) ~ Const/~-

(F(n) = r(n,q,~,~), n g i), where ~0 is the constant in Theorem E.

Proof. For any crank F of degree n ~ i, we have

NHrII1 1 ~ const HTFfIL l ,L~(~)" L (F),Lw(r) (~)

Since ~ is a locally chord-arc compact curve with constant i, Theorem D is

valid for F . Thus Theorem D, Lemma 3.3 and (3.13) yield the required

inequalities. The latter half is also deduced from Theorem D, Lemma 3.3 and

(3.14). Q.E.D.

David [18, Chap. III] showed that (2.39) is best possible in the following

sense:

(3.22) sup {IIC[a]II2,2; a 6 Lreal,M } >= Const I/M (M ->_ 1).

This is deduced from Theorem E as follows. We showed that

IITF0(n ) Xl0112 ->_ ConstV~ (n > i) ,

where Fo(n) = F(n, i + (the integral part of ~0 ) , i, 0). Adding some segments

parallel to the y-axis to Fo(n), we define an arc A n with endpoints 0 and

Then IAnl ~ Const n. There exists a Lipsehitz graph

A n = {(x, f0 b (s)ds); x ~ I0} such that IA ~ C O n and

IFo(n) U Amn - Fo(n) n A*n I ~ s, where C O is an absolute constant and

0 < g < 1/2 is determined later. We have

1.

Sio Ib~(~)ldx ~ IA~I ----c o n

98

Lemma 2.2 shows that there exists an open set ~ = Uk= 1 I k (I k = I~, k) in I 0

such that

[~J ~ ~, (Ib~l)ik & Con/8 (k a i),

Ib~(x) I ~ Con/8 a.e. on I 0 - ~ .

We put

C(x>= I * (x ( I k, k ~ i) (bn) Ák

and A n = {(x, f0 bn (s)ds); x ( I0}. Then lib n I]. N C 0 n/g .

projection of F0(n) N A n to ~ and let F - I 0 - E. Since

Let E be the

X :x( :o' (s> s :o C (s>ds) = ),i ..,

we have IF1 ~ 2s. From the definition, C[b n ](x,y) = TF0(n)(X,y) (x,y 6 E).

Hence T h e o r e m E s h o w s t h a t

{rE IC[b~ *] ×E (x)]2dx}l/2 = {rE Irr0(n> ×E (x)12 dx}l/2

=> {rE ITFo(n) ×I O(x) 12 dx}i/2 - {fIoITpo(n) XF (x) 12 dx}i/2

{Const n - /F ITF0(n) Xl0(X) 12 dx}i/2 - Const V~V~.

By (3.13) and (3.18), we have

TF0 )if TF0 )) + l} ~ Const V~, II (n 4,4 ~ Const{o( (n

and hence,

7F ITFo(n ) ×lo(X) l 2 dx _-< IITF0(n)II4) 4 IFI 1/2 =< Const nV~.

Thus

Choosing

{fl0 IC[b~*] XE(X) I 2 dx} I/2 g Const~n- {(i - ConstV~c ) I/2 -

sufficiently small, we have

Const VT}.

** ** 112 Constq-f (n > i), bn (Lreal,C0n/e) llC[bn ] ,2 >= =

99

which yields (3.22).

§3.5. Analytic capacities of fat cranks

For p > 0, z 6 ¢ and E c ¢, we write [pE + z] = {~ + z; ~ 6 E}.

With 0 ~_ ~ -<- i/i00 and a segment J c ¢ parallel to the x-axis, we associate

the closed segment J(~) of the same midpoint as J, parallel to the x-axis

and of length (i + ~) IJl. With a positive integer q ~_ 2, 0 ~ ~ --< I/I00

and a segment J c C parallel to the x-axis, we associate

2q-i

J(q,~) = U {[J2k_l(~) + i 2-qlJl] U J2k(@)} , k=l

2 q where {Jk}k= 1 are mutually non-overlapping segments on J of length 2-qlJl;

these are ordered from left to right. The set J(q,~) is a union of 2 q closed

segments of length 2-q(l + ~) IJl. A segment F 0 = [0,i] is called a crank n = of type O. For a finite sequence {~j }j=0' ~0 0 of non-negative numbers

less than or equal to i/i00, a finite union F of closed segments parallel to n

the x-axis is called a (fat) crank of type {~j }j=O if there exists a crank

n-i such that F' = Uk=l~ Jk ({Jk}k=l£ are components of F') of type {~J }j=0

F = U Jk(qk , ~n ) k=l

for some g-tuple (ql' "''' q~)

2. We write simply F'[(ql ' ...

of positive integers larger than or equal to

T . , q&;~n )

Proposition 3.10. Let F be a crank of type {~j}~=O ' ~0 = 0. Then

n U 1 }-I. (3.23) T(T) ~ Const { Z H

~=I j=0 (l+~j)

At present, the estimate of T(F) from above is unknown. The method in

[29, p. 87] and (3.8) do not yield satisfactory inequalities. The following

n be n+l cranks such that proof of (3.23) is standard. Let {F }~= 0

r 0 [ r I [ ... [ r (. ;~i) (. ;~2) (. ;+n) n

We put

1 0 0

h (z) = j=0 (i + ~j) '

g (z) = Im ~I" hp(z) P

1 = --~ Im p.v. /F

where Im denotes the imaginary part.

I--< ~ = < n,

h (~) Id~I (z E r , 0 = < ~ ~ n),

Then go ~ 0. We show that, for any

(3.24) il g~II N fig ill + const ~ .

L (F) - ~- L (Fp_l) j=0 (i + ~j)

Jk with components We can write F _ I = Uk= 1 {Jk}k= 1 of

= can write F Uk= 1 Jk(qk , ~ ) with some ~-tuple (ql' "''' q~)"

z 0 E Jk0(qk0,~ ) and let z 0 be the point on Jk0 nearest to z 0.

F i'

Let

Then

and

Ig (z 0) - g~_l(Z~)

i

1 +I 7

Im fjk 0

E k#k 0

I i fJk0(qk0,~ ) h (~) ~ Im ~ z0

h~_l(~) , Id~l I

- z 0

h(~) h i(~)

{Im 7jk(qk,@~ ) ~ _ z0-- ]d~I - Im 7Jk ~-±- z~

= L (I) + L (2), say.

Id~l~l

We can write

G 0

U Jk0(qk0 '~) = j=l

G 0 with its components {Tj }j=l ;

x-coordinate of their midpoints are increasing.

qk yj (G 0 = 2 0)

these components are ordered so that the

Without loss of generality, we

may assume that cY0/2

z0 ~ Uj=I Y2j-I "

h i (~) - Id~I Im fj

k 0 ~- z 0

G0/2 Since {Y2j}j=I are disjoint and

h (~) Id~I 0, ......... =

= Im /U°0/2j 1 Y2j-i

I Z 0

we have

101

L (I) h(~) 1

-- IIm I ~012 ~ - z 0 Uj=I T2j

-i (~0 IJkol

=< ! f~ 2 - - (x-Re z0) + (~011Jk01) 2

= ~ 1

( i + _ ® j ) " j = O

Id IL

dx

For 1 ~ k ~ g, 0 ~ v ~ ~ -i, 7k(V) denotes the component of Fv generating

Jk" In particular, Yk(~-l) = Jk (i ~ k ~ g). We put

where

h ( ~ )

L(2)v = k Z( G !fJk(qk'~b ). ~ -~ Zo--- Id~l

hb-l(~)-, Id~II (i ~ v ~ W-I), fJk ~ - z 0

Gv = {i ~ k & ~; k # k 0, Yk(V-l) = Yk0(V-l), Tk(V) # 7k0(V)}.

Then

h (~) h_l(~) - L (2) =< k#k0Z Ifjk(qk,@# > _ b_ z0 Id~l . . . . /Jk ~ -----*--z 0 Id~ll = v=l ~Zl L(2)~ "

We now estimate L(2)'v Note that ITk(V) I = IYk0(V) I (k (Gv) . A geometric

observation s h o w s t h a t , f o r a n y k ( G ,

(3.25) dis(J k U Jk(qk,@ ), Jk0 U Jk0(qk0,@ )) ~ diS(Yk(V), Tk 0(v))

- 2 l~k0(~)l 2-2 2-4 ~f

{(1 + ~v+l ) + (l+ ¢~,+l) (l+ @v+ 2) ...

2-2(~-v) } + Tl (I + %j) j =v+l

=> diS(Yk(V), Yko(V)) - 2 IYko(V) I

3 => diS(Yk(V ), Yk0(~)) - ~ IYko(V) l

1.01 .i.01.2 { --7- + ~--#-~ + "'" }

Since

102

= h_l(~) Id~l /jk(qk,¢~ ) h (~) Jd~ I Ijk

we have, with z k = (the midpoint of Jk )' Qk = Jk U Jk(qk,@ ),

Ifjk(qk,¢~ ) i i L (2) = Z { ~ - Zo , } h (~)Id~I v k (G z k - z 0

v

+ { i * 1 } h l(~ ) idol 1 IJk z k - z 0 ~ - z 0

Const Z k ~G

~-i = Const N

j=O

The segment Tk0(V - i)

{(JJkl + IJk01) dis(Q k, Qk0 )-2 7jk h~-l(~) Id~l}

i Z {(IJk] + ]Jk01)IJkl diS(Qk,Qk0)-2}. (i + Cj) k E G

!

' 2qv generates 2 qv components {km}m= 1 of r of

v

length IYk0(V) I , where q~ = log {(i + ~v) IYk0(V-l) I/ IYk0(V) l} /log 2.

We may assume that k I = 7ko(V). Let

q'

G = {k ~ G ; k m } (2 m ~ 2 v ) v,m v = Yk(V) ~ - "

q~

2 v Then G = G . We have

v Urn=2 v,m

Z (IJkl + IJkol)IJkl k(G

v~m

~_ Ikil 2 -2(~-i-v) N (i + ¢j) Z IJkl v< j -< ~-i k (G - v,m

= Ik112 2 -2(~-I-~) ~ (i + @j)2 _-< ikli2 v<j_- < ~-i

where N (i + ~j) denotes v< j _~ ~-i

observation and (3.25) show that

i if v =D -i.

2 -(~-I-~) ,

Hence a geometric

103

L (2) =< Const H j=0

~-i -<_ Const

j=O (i + @j)

v

2qv 1 % Z (IJkJ + IJk0l)IJkl diS(Qk,Qk0)-2

(i + ~j) m= 2 k # G v,m

v

2qv 1 % diS(km,)~l)-2 Z (IJkl + IJko I) IJkl

m=2 k ~G v,m

2q~ i)Vl I 2 2-(~-l-v ) y, dis(Xm,kl)-2

m=2

~-i 1 --< Const

j=0 (i + ~j)

~-i i I~i 12 2 -(~-l-v) Z ( 1 2 Const H (i + @j) ) j=O k=l I Xll k

~-i --< Const H 1 2_(~_i_~).

j=O ( i + ~j)

Tbus

* (i) L(2) Ig (z0) I ~-I g _l(zO) I + L + E

v= I V

1 ~-i 1 =< IIg~_IIIL~(F ~ 1 ) + n + Const H (i + ~j)

_ j=0 (l + ~j) j=0

_-< IIg~_lll ~ + Const H (i + ~j) " L (r _ I) j=0

Since z 0 ~ F is arbitrary, we obtain (3.24).

By (3.24) and go m 0, we have

n llIm H F hnll ~ Const E H 1

n L°°(rn ) ~=i j=0 (i + ~j)

(= Const ~n' say).

Evidently,

fF hn(~) Id~I = i, IIhnli n L °~ (Fn)

on r so that Hence we can define a non-negative function h n n

104

Ir hn(<) I d<[ = 80/<n ' n

llhn*NL'(rn) + lllm"r h$1L~(r # S l,

h:(<) = 0

h*(<) n

at endpoints of each component of F , n

is c o n t i n u o u s l y d i f f e r e n t i a b l e a l o n g F n ,

where 6 0 is an absolute constant. Let

h (<) ^* i n I d<T hn(Z) = 2~i fF ~ - z ' - ' '

Ii

u (z) = Im h:(z) Vn(Z) = Re h*(z) n ' n '

(3.26) fn(Z) = {I - exp(h:(z))} /{i + exp(~:(z))} (z ~ Fn).

Then f is analytic outside F and n n

: 1 h:(<) {d~ 1 : 60/(2 <n ) Ifn (')I 2~ fF * n

The nontangential limit of fUn(Z) I at each point on Fn is dominated by

llIm HFn h~llL~O(Fn) _< 1.

Since fUn(Z) I is subharmonic in F c and continuous on n

sup F c fUn(Z) [ =< i. Hence, for any z ~ F n, z (

n

U {~}, we have

I fn(Z) I 2 1 + exp(2 Vn(Z)) - 2 exp(Vn(Z))COS(Un(Z))

i, 1 + exp(2 v (z)) + 2 exp(vn(Z))COS(Un(Z))

n

which shows that IifnllH~ ~ i. Consequently,

n T(r n) a If'(-)I ~ 601(2 ~n ) = Const{ Z H I

n ~=i j=0 (i + _ ~j)

-i.

This completes the proof of Proposition 3.10.

105

§3.6. Analytic capacity and integralgeometric quantities

Let L(r,@) (r > 0, -~ < e ~ ~) denote the straight line defined by the

equation x cos @ + y sin @ = r. For a compact set E C ~, we write

NE(r,@) = #{E N L(r,@)}, where #{E A L(r,O)} is the (cardinal) number of

elements of E n g(r,8). For e > 0 and 0 < a ~ i, we put

Cr(e)(E)~ = i n f f _~ { f~ N a { u ~ = l D ( Z k , r k ) } ( r , @ ) ~ dr} dS,

where t h e in f imum i s t a k e n o v e r a l l f i n i t e c o v e r i n g s { D ( Z k ' r k ) k=l

r a d i i l e s s t h a n e . We pu t

of E with

Cr (E) = lim Cr(e)(E) (0 < ~. < i), ~ + 0

Bu(E) = l i m l i m Cr ( ¢ ) ( E ) . c ÷ O a ÷ O

If E c D(0,1), then Bu(E)/2~ is called the Buffon needle probability; this

is the probability (measured by dr d@/2~) of needles L(r,O) (0 < r < i,

I01 ! ~) intersecting with E. Suppose that E is a locally chord-arc compact

curve. Then Crofton's formula [49, p.13] shows that CrI(E ) = Coust I E I. By

(3.7), we have T(E) ~ Const CrI(E ). From this point of view, it is interesting

to compare ~(.) with Cr (.) (and with Bu(.)). It is known that there exists

a compact set E such that y(E) = 1 and Bu(E) = 0 (cf. Jones-Mural [34]).

We shall show

Theorem F. For any 0 < ~ < 1/2, there exists a compact set E such that

y(E) = 1 and Cr (E) = 0.

Acknowledgement. The author expresses his thanks to Professor Kakutani who

communicated (3.28), and Professors Coifman, Steger who suggested to use the

Galton-Watson process [30] for the estimate of Cr (.). According to Professor

Kakutani, various integralgeometric formulae are used for surgeries, since

X-rays react to, for example, cancers outside bodies. RSL is the first stopping

time of the sun's rays (or needles). Hence it is interesting to try to give an

integralgeometric proof of Theorem C,

In this section, we study Cr (.). For 0 < ~ < I, let (X~}~= I be a

sequence of independent random variables on the standard probabillty space

(PO,S,Prob) (F 0 = [0,1]) such that

eroh(X~ = l ) = e r o b ( X ~ = - 1 ) = / 2 ,

Prob(X~ = 0) = 1 - B (n a 1 ) . H

106

We put

S~0 = 0' S~n = X~l + "''+ X~n (n > I).

This is a model of random walks. We define a Galton-Watson process {Yn}n= 0

(3.27) y0~(x) = i, y~n(X)= y~n_l(X)+ S~y~n_l(x) (X) (n ->- i, x E F0).

by

Then Prob(y~ n >_- 0) = i (n >_- 0). We put

c (n) = 2 ~+I fO/2 { Z k=l

k s b~n)(t(e))}cos Od@ (n >__ l, 0 ~ <~ < i),

where

b(kn)(6) = Prob(Y~n = k),

t(8) = ~ tan e (0 ~ e < arctan i)

Itane- 2j I (arctan (2j-l) i e < arctan(2j+l), j ~ i).

First we show

Lemma 3.11. ca(n ) _-< Const/(~ n I-~)

The proof of this lemma is standard.

{bk(J)(6)}k=O is defined by P~(x) = Zk= 0 b(J)(~) x k.

(3.28) P~ (x) = P~j-I ( ~2 + (i- 6)x + ~2 x2 ).

In effect, (3.27) shows that, for any

(3.29) b(J)(~) = Prob(y~ = k) =

(j-l) (6) = Z b~ ~=0

(n ~ i, 0 < ~ S 1).

The generating function of

Then

k_a0,

w

Prob(y~_ IJ =g) Prob(S p~ = k - g) Z g=O

z(~) ~' (~) h(1-~) ~2(-~) ~3 , el! g2 ! g3 !

where Z(~) is the summation taken over all triples (~i' g2' g3 ) of

non-negative integers such that ~1 + g2 + g3 = g ' g2 + 2 g3 = k. The

xk-eoeffieient of P~(x) is b~J)(x) and the xk-coefficient of J

P~_I ( 3+ (i- 6)x + ~x 2 ) is equal to the last quantity in (3.29).

(3.28) holds. Let

Thus

107

vj(~) = I 1 (i - x) -a 8-- P~ j(x) dx (0 ~ j ~ n) ~j(~) ~x -

where ( n {~j ~)}j=0 are inductively defined by 40(~) = 0,

~j(~) = ~ +(i-9)~j_l(X)+ ~ ~j_l(X) 2

Since

1 1 (i - x)-a(k x k-l) dx a fl-(i/k) (i - x)-~(k x k-l) dx fo

k ~ a_ Const i - ~ (k ~_ 2),

we have

Z k=l

k s bln)(~) ~ Const (i - a) v0(~).

Since

(l-~)-a = {i- (~ +(1-~)x+ ~x 2)}-= (o=~i),

(3.28) shows that

vj(~) = f~j(~) (i- x)-~{(l-~) + ~x} ~ P~n_j_l(~2 +(l-~)x + ~2 x2)dx

~l ~l-(~ + ~ - ~ + ~ ~-°~(~-~+ ~ ~ ~ +(~-~+ ~d~ ~j (~) n-j-l~2

= Vj+l(~),

a n d hence

v0(~) ~ Vl(~) ~ . . . ~ Vn(~)

1 -a % p~0(x)dx = ii = f~n(~ ) (i- x) ~- _

We have easily I 1 - ~n(~) I -<- Const/( ~ n). Thus

Z k a b(kn)(~) <= Const/(~n) l-a k=l

(i - ~n(~)) I-~.

Consequently, we have, with e. = arctan j (j ~ 0), J

108

c (n) _-< Const n ~-I 0/2 f t(@) ~-I cos6 d@

Const n cr-I { 81 I0 (tan Q)a-lcos @ d@ +

Const n ~-I { I I Y dy 0 (i + y2) 3/2

" @2j+l Itan 8- 2j I ~-1 j fl I 82 j-I

" i .... y~-i + Z I_ I 3/2 dy ~ Const/(~ nl-~).

j=l {i + (y + 2j) 2}

cos @ d@}

This completes the proof of Lemma 3.11.

Let F' and F be two cranks such that n = (the degree of

(the degree of F') ~ i. If there exist an n-tuple (A I, ..., A n )

such that

r) -

of cranks

r' = A I [ A 2 [ --- [ A n = r

(0-1; ") (0`2; ") (Qn; ")

for some n-tuple (0`i' ~)' t {q(~) g .... = }k~l of finite sequences

integers, we write F' [[ F and

of positive

~(F',F) = gin {q~);{ 1 s k -< g , i _-< ~ < n} .

Note that, for any crank F ,

Cry(F) = 2 ~ I_~ ~ {I~ NF(r,0)~ dr} dG (0 ~ ~ ~ i).

Lemma 3.12. For 0 ~ g ~ i, 0 ~ ~ ~ i, gO ~ 1

crank F* of degree n such that n

and n ~ i, there exists a

(3.30) ~(F0' r'n) => g0' ICr~(r*n) - C~ (n) l < C ,

where F 0 = [0,i].

Proof. For a finite increasing sequence 0` = {qk}gk=l of positive integers,

we put gap(~ = min {qk - qk-l; 1 =< k =< g} , where qo = O. Let

~n = (0`1 ..... h), ~ = {qi~) }k~l be an n-tuple of finite increasing

sequences of positive integers such that

q~) q(~) q~)

~i = i, ~ = 2 + 2 2 + ...+ 2 ~-i (2 --< ~ --< n),

we put gap(~n) = min{gap(~); i ~ ~ ~ n} . With ~n' we associate n cranks

109

r~ = F(Q 1 . . . . . Q)

* * (i) r I = FI(QI) = r0(q 1

been defined so that

(1 & ~ & n) as follows. Let

,0) ( F 0 = [0,i]). Suppose that rl, r2, ..., r _ I have

r k has gk componen t s (1 N k N g - l ) . Then F _ 1 i s

expressed in the form r _i = Uk ~ j ~1) with its components

{ji~-1))~-i "k=l ; these are ordered so that the x-coordinates of their midpoints

are increasing. We put

, , g~-i ji~_l) (qi~_l),o) " r = r(% . . . . . %) = u

k=l

The set rn(~n) is a crank of degree n. We now s~udy Cr~(rn(~n)). We have,

with 0j = arctan j, 0 ~ 0j < ~/2 (j ~ 0),

" (r,O) ~ dr} de 2-~ Cr~(rn*(@n)) " f-n{fo Nr*(~n)n

- O. -eJ-i 70 } Z {I J I 0 +I j=l 0j -i -Oj

+ i s]

( = Z {dj(~n ) + d j(~n )} j=l.

+ do(~n) , say).

i

tan 0 1-tan 0 tan 0 1-tan 0 4 4 4 4

L(x cos e,e)

For 0 < 6 < @i' we put ~0)(@) = 0,

Ix ~ r0; ~ , (x c o s e , e ) = kl r(Q I . . . . . 0.~)

(k_>- 0, i < ~ =<- n).

110

Then bk(~)(e) = 0 (k>= 2~ + 1 , 0 ~_ ~ ~ n). We have

b~0)(e) = 1 = Prob(y0an 8 = i)

bk(1)(e) = Prob(ylan @ = k) (0- -< k ~ - 2).

Let 1 ~ j ~ 2 ~-I, 2 ~ ~ ~ n. To a component V of {x E F0; N , (x cosS, e)=j},

there correspond j components j(~-l) , ..-, j(~-l) of F* F~-I v I vj ~-I

which intersect with L(x cos 8,8) for all x E V; these are ordered so that the

x-coordinates of their midpoints are increasing. If (~) is sufficiently %1 large, then

Ix ~ V; # {j(~-l) ((~) ~i %1.

,0) fl L(x cos e , e ) } = k

- IV I Prob(Slan 8 = k_l) 1

is sufficiently small for all 0 =< k--< 2. If (~) q(~) - (~) %l' ~2 %1

sufficiently large, then

are

-IvI Prob(S~ an e = k-2) I

is sufficiently small for all 0 ~ k & 4. Repeating this argument, we see that,

- q(~) are sufficiently large, then if ~ ' ~V 2 - qv 1 ..... qVj ~-i

.stan 8 = k-j)l llx v; {r nL<xcos0,e) =kl-lvIProb j is sufficiently small for all 0 ~ k ~ 2j. Hence, if gap(Q) is sufficiently

)

large, then

l lX 6 F0; , (x cos @,e) j, , cos N N (x 8,8) kl F F ~-i

- bj~(~-l)(8) Prob(S tan3 8 = k - J)l

is sufficiently small for all 0 -_< k =< 2j. If gap(Q) is sufficiently large,

then

I x ~ F0; NF, (x cos e,e) = 0, N , (x cos 8,e) >= i I F

~-I

111

( = JJx ~ £0; N , (x cos 8,8) = 0, N , (x cos 8,0) >_ 1 J F £ ~-i

_ ~-i)(8 ) Prob(s;an 0 >__ l) j)

is sufficiently small. Thus there exists a positive integer p~(e)

if gap(%) ~ p (@), then

J lx ~ F0; , (x cos O,O) j, N , (x cos @,O) I N k F_ 1 F

- b(~-l)(0). Prob(S tan. @ = k - j)J =< g 2 -3n3 3 3

such that,

for all 0 _~ k -~ 2j, 0 -~ j =< 2g-l. This yields that

2~-i ~tbk~) (e) _ z ~(~-i) (e). erob(stan O = k - J)Jl K j=O ] 3 I

2~-I

j Z {ix E r0; N , (x cos 8,0) = j, N , (x cos 8,0) = kJ j=0 £ r ~-i

_ ~-i)(%) Prob(s~an 8= k- j)}J

=< s(2 g-I + i)2 -3n3 -_< Const s 2 -2n3 (0 _<- k -<_ 2~).

Put p(O) = max{p (8); 1 _~ ~ =< n} . If

2 n

Z Ibk (n)(0) - bk (n)(tan 0) I k=0 k=0

2 n 2 n-I J ~!n-l)(e) Prob(S. tan O <= Z z

k=0 j 0

2n-i

Z Prob(Yn_itan 8 = j)Prob(sjan 8 = k-j) j=0

2n-i

gap(~ n) ->- p(8), then

2 n = E Ib (n)(o) - Prob(Yn tan 8

=k-j)

_-< (2n+l) ~(n-l) _ , . tan 0 Z Jbj (8) - wroD[Yn_ 1

j=0 =j)

n 2 ~_...=< H (2 ~ + i) Z

~=2 j =0

+ Const

= k) t

+ Const ~ (2 n + i)2 -2n3

+ Const 6(2n+I) 2 -2n3

,. tan e ~!l)(o) _ ~r°DiYl = j)j 3

n

~=2

2_2n3 3 . . . . 2 -n (2 ~ + i) (2 n + i)} < Const ~ ,

112

which shows that

(r,@) a dr - Z k ~ " (n)(tan @)cos @ I If~ NF~(~n) k = l b k

2 n

Iro" N , (x cos @,9) g COS @ dx Z k g b~n)(tan" e)eos e I F k=l n

2 n

= I Z k~ ~k~(n)(@) - b~n)(tan @)} cos @I k=l

3 = 2 ~n 2-n < Const s ~ Const ~ .

There exists a positive integer Pl such that the measure of

2 -n" F = {0 < @ < @i; P(@) > Pl } is less than s If gap(~n) ~- PI'

01 " k ~ . (n)(tan @) cos @ d@ I Idl(Qn)- fO Z b k k=l

@i If0 {/O N , (r,@) ~ dr - % k s bk(n)

F k = l n

(tan @) cos @} d@ I

l{f F + /(0,01)_ F} { } de I

Const 2 ~n IFI + Const s & Const s .

then

For an integer j # I, we can choose, in the same manner as above, a positive

integer pj such that, if gap(~ n) ~ pj, then

81Jl Z k ~ " (n)(t(@)) @ del < s /(i + j2), I dj (~)-Teljl_l k=l o k cos =

where 8_1 = 0. This shows that if gap(~n)

holds.

is sufficiently large, then (3.30)

Q.E.D.

§3.7. Proof of Theorem F ([34])

Here are three lemmas necessary for the proof. From now on, we fix

0< ~< 1/2.

t O = = of degree n Lemma 3.13. For > 1 and n > i, there exist a crank F n

and a non-negative function w on F such that w is a constant on each n n n

component of F n,

113

(3.31)

~(r 0, r[) ~ ~0' cr(r[) ~ ClI(~ n~-~),

llWnll i * I, llWnll ~ ~ C I, film H , w~lle.(F~ ) ~ Cn~'n~, L (F n) L (r[) F n

where C 1 is an absolute constant.

P r o o f . Leamms 3 . 1 1 and 3 . 1 2 show t h a t t h e r e e x i s t s a c r a n k F of d e g r e e n n

s a t i s f y i n g t h e f i r s t two i n e q u a l i t i e s i n ( 3 . 3 1 ) . I n e q u a l i t y ( 3 , 1 3 ) shows t h a t

II~rn, IIL2(Fn),L 2(Fn) ~ ConstV~-,

which yields

tlHFn, fiLl(m) ,L~(Fn ) ~ Const V~.

Thus, in the same manner as in the proof of Theorem D, we obtain a non-negative

F* f u n c t i o n on s a t i s f y i n g t h e l a s t t h r e e i n e q u a l i t i e s i n ( 3 . 3 1 ) . T a k i n g t h e n

mean o v e r e a c h componen t o f F , we o b t a i n t h e r e q u i r e d f u n c t i o n w • Q.E.D, n n

Lemma 3 . 1 4 . L e t ~0 ~ 1 and n ~ 1. L e t r m be a c r a n k of t y p e {6j}~= 0 _

and w be a non-negative function on F such that w is a constant m m m • .m+n

on each component of F m. Then there exists a crank Fm+ n of type 16j~j= 0

with ~j = 0 (ra+l N j N ~ - n ) and a n o n - n e g a t i v e f u n c t i o n wm+ n on Fm+ n

such that Wm+ n is a constant on each component of Fm+ n,

rm [[ rm+n' c(rm' Fm+n) >= gO'

Cr (Fm+ n) -<- C 1 IFml/(~ n l-a ),

IlWm+nllL 1 = IlWml]L1 " , IlWm+nll ~ C 1 llWmll . ' (rm+n) (Fm) L'(Fm+n) L (r m)

(3.32) tl~m H r win+nil ~ lllm HFmWmllL®(Fm) + C2g-~ llwmll . , m+n L'(Fm+n) L (F m)

where

Proof.

g {Zk}k= 1

C 1 is the constant in Lemma 3.13 and C 2 is an absolute constant.

= g Jk with its components {Jk}k= I. Let We can write r m Uk= I

j be the left endpoints of { k}k=l, respectively. We put

114

g

Fro+ n = U A k, A k = [IJkIF n + Zk], k=l

Wm+n(Z ) = Wn*((Z-Zk)/IJkl) Wm(Zk) (z E A k, i -<_ k =< ~),

where F and w are the crank and the function in Lemma 3.13, n n m+n

respectively. Then Fm+ n is a crank of type {~j }j=0 such that

F m [[ Fm+n, L(Fm, Fm+n) m gO" The second inequality in (3.31) shows that

Cr (rm+ n) -~ Z Cr (A k) = Z IJkl Cr (rn*) --< C l Irml/(a nl-~). k=l k=l

In the same manner as in Proposition 3.10, we have (3.32). Q.E.D.


m g0 be a crank of type {6j}j= 0 , w be a Lemma 3,15. Let ~ i, n ~ i, r m m

non-negative function on F m such that w m is a constant on each component m+n

of Fm, and let {~j}j=m+l be non-negative numbers less than or equal to m+n

i/I00, Then there exist a crank of type {~j}j=0 and a non-negative

function Wm+ n on Fm+ n such that Wm+ n is a constant on each component

of Fm+n' Fm [[ Fm+n' ~(Fm' Fm+n) ~ g0'

llWm+nIILl (rm+n) llWmllLl (Fm) '

m+n

llWm+nIIe.(Fm+n ) =<-IIWmI[e,(Fm ) / H #=m+l (I + ~ ),

II Im Wm+nll ~ -~ llIm H r Wmll ~ HFm+n L (Fm+ n) m L (F m)

m+n j + Elwll . z {i/ ~ (I+%)}

L (F m) j=m+l ~=m+l

We now construct the required compact set E. Choose a positive number

~0 and a positive integer n o ->_ 2 so that ~0/2 i. Let Pm be the integral part of (i01/i00) 0 (m >_- I). We

define a sequence {mk}k= 0 of non-negative integers by m 0 = 0, m I = n o

mk+ I non ~ + = Pn0mk

and define a sequence {~j}~=0

(k ~ i),

of non-negatlve numbers by

115

Cj

0 (O < j -<- m I)

i/i00 (m k< j _~ nomk) k->_ i)

0 (n0n k < j =< ink+l, k>= i).

Let [ = {~k}k=i be an increasing sequence of positive integers which will be

determined later. Using Lemma 3.14 with gO = ~i' n = m l, F 0 and w 0 m i, we

obtain a crank rml and a non-negative functlOnn m wml" Using Lemma 3.15 with

- {¢j}j=ml+l, we obtain a crank rnoml gO = &!' n = (n O l)ml, rml, wml and 0 1

= = Pn0ml, • ) and a non-negative function Wn0ml Using Lemma 3.14 with t 0 ~2 n

rn0ml and Wn0ml, we obtain a crank rm2 and a non-negative function Wm2.

Repeating this argument, we obtain a sequence {Fmk}~=l of cranks and a

sequence {wmk}k=l of non-negative functions such that, for k a 2,

r~ [[ r~+ l, ~(r~, r~+ l) >-_ ~k'

llw k = i, ]i® , -< IlLl(r~) ilwmk L (r k)

II Im HF~ wmk Ii L'(r~) =< C0 V~I

k-i nomv j v Z + Z C O

v=l j =mv+l

nomk, 1 k

c o / n (]_ + ¢~) , ~=0

{L / 11 ( i + ¢ ) } + FL=O

n0mk_ I

Cra(r~) =< C O H (I + 4~ )/(~ PnOmk 1 ~=0

k-i nomv v+lDv-6----- .~

Z C 0 Pnomv/ ~=i ~=i

),

(l + ¢~),

where £~ = rmk and C O = max{Cl, C2}.

and show that T(E([)) ~ Const. Let k $ 2.

m nomv_ I (v ~ 2) and m I = no, we have m v

n0mk_ 1 k / n (i + ¢9

Ilw m I1. , ~ c 0 mk-l+l k L (rk) ~=

k I01 )-(no-l)mk-i = C 0 ( ~-~ ~ Const.

We put E([) = Nj= 1

Then Ilwmkllel (r~)

v

n o (v ~ l),

uLj = i. Since

and hence

Since

116

~Pnom---~/

nom ~

n (i + ~) ~=0

.lOl.~onomv/2 .i01 -(no-l)mv uonst(l-- ~) ~)

.i01. -{(l-(i/no))-(~O/2)}nomv Const (I--~) (v >= i),

we have

wmkllL®(F ~ Const. II Im HF~ ~)

Thus, in the same manner as in (3.26), we obtain an analytic function

fk ~ H~(FI c) such that

llfkllH ~ ~ i, lfi(~)l e Const.

Since k ~ 2 is arbitrary, we obtain an analytic function f (H~(E([) c)

such that

llfll _<- l, If'(-)l ->_ Const, H ~

which shows that Y(E([)) ~ Const.

Let

.101.(1-(l-a)~o)n0mk gk = (C0/~) ~i--OO ) (k ~ 2).

Then limk ~ ® gk = 0 and

Cr (F~) i01 n0mk-i .101.-(l-~)~0n0mk-i (2c0/~)(i~) ~

2 gk_ 1 .

We can inductively choose ~0 = {gk}k= I0 ®

Cr(i/k)(E([0))~ ~ 2 gk-l' which shows that

satisfies Y(E) > 0 and Cr (E) = 0.

so that, for any k ~ 2,

Cr (E([0)) = 0. Thus E = E([ O)

Remark 3.16. Throughout the note, we use Theorem D to estimate Y(') from

below. Here is a weaker inequality than Theorem D. Let F be a locally chord-arc

curve. Then

> + llHr(f dz/Idzl)[l 2(r ) Y(F) = Const I 7 r f dzI2/{Irfll 2(F)

(cf. [29, p. 19]). This is also useful to estimate Y(r) from below. In effect,

we can deduce (3.23) and {IIC[a]I12,2; a ( ereal} = ~ from this inequality.

APPENDIX I. AN EXTREMAL PROBLEM

For s I .... , s n E ~, we define

(x,y) = i/{(x-y) + i(Asl ' (x) - A (y)}, Ts I, ..., s n ..., s n s I' --., s n

where

A (x) s I, ..., s n

0 x ~ I 0 : [0,i)

k-i k -- < x < -- 1 < k < n). Sk ( n = n' = =

Put

(4.1) ex (n) = max {~(Tsl ' s n E ~} • ..., s ); Sl' .... n

(See (1.22).) We show

Theorem G. Const ~TOg(n+l) ~ ex (n) ~ Const~g(n+l) (n ~ i).

The first inequality is shown in §3.4. We prove the second inequality. For

a positive integer n, F denotes the totality of sets E c ~ such that n

E c U~=_ [I 0 + ik/n], E has a finite number of components and their projections

to I 0 are mutually disjoint, For E E Fn, we define a function AE(X) on

by x + iAE(X ) E E (x E pr(E)) and AE(X) = 0 (x ~ pr(E)), where pr(E) is

the projection of E to I O. We define a kernel by

TE(X,y) = i/{(x-y) + i(AE(X ) - AE(Y)) } .

Here are three lemmas necessary for the proof.

Lemma 4.1. Let E E Fn and let W I, W 2 be two disjoint subsets of

such that AE(X) ~ 0 on W I and AE(X) ~ 0 on W 2. Then, for any

(4.2) fW 1 IrE(XW2f)(x)I 2 dx ~ Const IIXw2fiI ~ .

pr(E)

f E L 2 ,

' by Proof. We define an operator T E

g ~ f_~ g(y)/{(x-y) - i A(y)} dy.

' Then we have Let T E'' denote the adjoint operator of T E

118

I T"g(x)I < H*g(x) + Const Mg(x),

llp = < C (p > i). Since < C (p > i). Hence lIT~Ilp,p which shows that nT~ ,P P = P

AE(X) - AE(Y) ~ AE(X) ~ 0 (x 6 W I, y 6 W2), we have, in the same manner as in

the proof of (2.9),

f)(x) I < Const {M(T~f)(x) I TE(Xw2 =

+ IIrE[14/3,4/3' M(I Xw2fl 4/3)(x)3/4}

which gives (4.2).

Put

(x £ Wl),

where

1 co

~(n) = sup {T'/UT-6TT'Ip~tLjl ~(E,f); E E u ~-' f 6 Lreal, 0 ! f <- i},

~(E,f) = /pr(E) ITE(Xpr(E) f)(x)I2 f(x)dx.

Lemma 4.2. For any n ~ I, ~(n) < ~.

Proof. For E £ FI, we put G = pr(E) G' = pr(E N {Im z = ~}) '

(D = O, ±i, ...). Then, for any f 6 Lreal, 0 ! f ! I,

f(Y) dy - Z i I TE(XG f)(x) - fG~ x - y ~=-~ 1 + i(k-~) /G'~ f(Y) dyl

Const Z (x 6 G~, k = 0, ±i, ...). ~=-~ (k-~) 2 + 1

Hence

which shows that

For E E Fn,

~(E,f) ~ Const ( Z fG IH(x G f)(x)i 2 dx

i fG' f(y)dy] 2 + + z I G llz k=-~ ~=-~

Const ~ {IGkl + IfG, f(y)dyl 2 k=-~

~(i) ~ Const. co

f ~ Lreal, 0 ! f $ I,

z z } k =-oo U=- °° (k-~) 2 + 1

+ 1 G~I } --< C o n s t I G1,

we put G' = [n pr(E)],

E k = {z - k; z 6 [n El, k ~ Re z < k+l},

~.x + k. fk(x) = ~[--~---) (0 <_ k _< n-I).

Then

119

Since

~(E,f) = _111 fG' I fG' f(y/n) ,d~ (x - y) + i(n AE(--Xn) - n AE(Y))

n 1 { ~ ~(Ek,f k) + C }.

<= n n k = l

co

E k E FI' fk E Lreal, 0 < f < i, we have ~(n) < co.

2 f(~) dx

Q .E .D.

Lemma 4,3.

Proof. For

The following Lemma is analogous to Lemma 3.4.

~(22n) ~ 2 ~(2 n) + Const ~(22n) I/2 (n ~ i).

E E F22 n, we define

F = U (the projection of E N {(k-l)2 -n ~ Im z < k 2 -n}

to the line Im z = (k-l)2-n).

Then F E F . Let 2 n

G = pr(E),

G. = pr(E [~ {(j-l)2 -n _< Re z < j 2-n}), j

Gj, k = pr(E n {(j-l)2 -n _<_ Re z < j 2 -n, (k-l)2 -n <Im z < k 2-n})

(j = 1 ..... 2 n, k = O, +i, ...).

We have, for f 6 Lreal, 0 ~ f $ i,

~(E,f) = ~(F,f)

+ I G (T E - TF)(XGf)(x ) TE(XGf)(x) f(x)dx

+ I G TF(XGf)(x) (T E - TF)(XGf)(x) f(x)dx

= ~(F,f) + L (I) + L (2)

and

2 n L (I) =

j=l 2 n

+Z j=i

IG (T E - TF)(XG f)(x) TE(XG f)(x) f(x)dx J 3 ]

fG (TE - TF)(XG f)(x) TE(XG_G f)(x ) f(x)dx J J J

120

2 n Z

j=l IG_G. (T E -TF)(XG.f)(x) TE(XGf)(x) f(x)dx

J J

L I + L 2 + L 3.

For i <_- j _<- 2 n, there exists E!j ( F2n, F'j E F I and f.3 E L "real, 0 =< fj

such that I Eli = I F ]I = 2niGjl,

IG" ITE(XG. f) (x) 12 J J

fG. I TF(XG. f)(x) 1 2 ] J

f(x)dx = 2 -n <(E],fj)

f(x)dx = 2 -n ~(F],fj).

Hence

ILII _-<

2 n + z

j=l

2 n

Z IG ITE(XG.f)(x)I 2 f(x)dx j=l j j

{IG. ITF(XG f)(x)i2f(X)dx}i/2{IG.ITE(XG.f)(x)i2f(x) dx} I/2 J J J J

2-n

<(2 n)

2 n 2 n % ¢(E',fj)j + 2 -n Z

j =i j =I <(F~,fj) I/2 <(E],fj) I/2

2 n 2 n Z IgjI + ~(i) I/2 ~(2n) I/2 Z IGjI

j =i j =i

= IGI {~(2 n) + ~(i) I/2 <(2n) I/2} .

Since F2n c F22n, we have <(2 n) ~ <(22n). Thus Lemma 4.2 yields that

}LII ~ IGI {~(2 n) + Const <(22n) I/2} .

Let xj = (j-l)2 -n and Gj = Gj_ 1 U Gj U Gj+ I (i ~ j ~ 2n), where G O

Then, for any x E G - G., J

2-n I (T E - T F)(X Gjf)(X) I --< Const fG.3 (x_y)2 + 2-2n

2-niGj I _-< Const - .

(x-xj)2 + 2-2n

For any g E L 2, we have

f(y)dy

=< 1

=~.

121

2 n 2-nlGjl

I~Z Z g(x)dxl j=l (x-xj) 2 + 2 -2n

2 n 2-n Const ~ fG. {f ~ 2-2n

j=l 3 - (x-Y) 2 +

Const ~G M g(x) dx ~ Const ~ llgll 2 ,

Ig(x) Idx} dy

which shows that

Thus

2 n 2-nl G4 j ii Z a 2_2n. iI 2 _<_ Const ~ . j=l (--x j)2 +

2 n

IL31-_< Z j = l

2 n

<= Const

{7 G ~ + ~ } I(TE_TF)(XG f)(X)TE(XGf)(x)l f(x)dx - j fGj-Gj j

2-nlcj j iTE(XGf)(x)if(x)dx 2 2-2n j=l 7G-Gj (x-xj) +

2 n dy_ ) ITE(XGf)(x)I f(x)dx

+ 2 % ~G. -G. (~G 3 j = l J J " Ix-Y[

2 n 2-nlGj I ITE(XGf) (x) I f (x)dx -<_ Const fG { Z 2

j=l (x-xj) + 2 -2n-}

2 n + 2 % {fG.-G. (fG --~ )2dx}i/2 {fG,ITE(Xgf)(x)12 f(x)dx}i/2

j=l 3 3 J 3

2 n =< Const ~(E,f) I/2 + Const E ~IGjl {~ ITm(×Gf)(x)12f( x)dx}I/2

j=l 3

_-< Const ~V~ ~(E,f) I/2 + Const ~ ~(E,f) I/2

_<- Const IGI ~(22n) I/2

Let

Then

G. = Gj U Gj 3 ,k ,k-i ,k U Cj,k+ I (j = 1 .... , 2 n, k = 0, -+i, ...).

2 I%

IL21 =< I E Z j =i k =-~

(TE-TF)(X G f)(x) TE(X G ~ f)(x) f(x)dxl fGj ,k j ,k - J

2 n f)(x) TE(XG_~ f)(x) f(x)dxl + I Z Z ~gj (TE-TF)(XG. U gj

j=l k =-" ,k 3 ,k-i ,k+l 3

122

2 n

I S Z fG (TE-TF)(XG _~" f)(x) TE(XG_~ f)(x) f(x)dx I j=l k =-- j,k J ],k J

2 n

lj=iS /Gj (TE-TF)(XGjf)(x) TE(X~ _G f)(x)j J f(x)dx I

L21 + L22 + L23 + L24°

We have 2 n

IL241 =< 2 Z {7G. I(TE-TF)(XG f)(x)l 2 f(xldx} I/2 j=l 3 3

dy )2 dx}i/2 ~ Const IGI ~(22n) I/2. × {fg. (fG.-G.

3 J J

Note that ~(2 2n) ->_ ~(i) > Const. Since

I(T E - TF)(XG _~" f)(x)l J J,k

" 2nlGj ,~I < Const (x ~ G ), _-< Const S 2 = j , k

g=-~ (k-u) + i

we have 2 n ,

IL23 I ~ Const S S ~G ITE(XG_ ~ f)(x) I f(x)dx j=i k =-- j ,k j

2 n

= Const E fg. ITE {(XG - XG.-G. - XG.)f}(x)l f(x)dx k =I J 3 J J

=< Const Igl {~(22n) I/2 + i} -<_ Const IGl ~(22n) I/2.


2 n f)(x)l 2 dx} I/2

IL221 ~ { E E fGj I(TE-TF)(XG UGj j=l k =-- ,k j,k-i ,k+l

2 n x { Z fG. ITE(XG-G. f)(x)12 f(x)dx}i/2

j=l j j

2 n Const { Z S IGj U Gj f(x)2 dx}i/2

j=l k =-- ,k-i ,k+l

× {~(E,f) + 2 n Z f~, ITE(X~ f)(x)12f(x)dx} I/2

j=l 3 3

123

Const IGI ~(22n) I/2.

Since (TE-TF)(x,y) is anti-symmetric, we have

(TE-TF) (X G f) (x)f (x)dx = 0. fGj ,k j ,k

For Gj, k # ~, we choose a point xj, k on G.j,k. Then

IL211 = I Z (TE-TF)(XC. f)(x) j,k;Gj,k # ~ fGj,k j,k

x {TE(XG_~j f) (x) - TE(XG_~.jf)(xj,k )} f(x)dx

Const Z /Gj,k I(TE-TF)(X G f)(x)If(x)dx j,k;Gj,k# ~ j,k

Const IGl ~(22n) I/2.

Consequently,

IL(1) I ~ Igl {~(2 n) + Const ~(22n) I/2} .

In the same manner,

2 n IL(2) I ~ j=l% 7GJ ITF(XGjf)(X)TE(XGjf)(x)If(x)dx + Const IGI ~(22n) I/2

Since the first quantity in the left hand side of the above inequality is

dominated by Const IGI ~(22n) I/2, we obtain IL(2) l = < Const IGI ~(22n) I/2

Thus

~(E,f) ~ ~(F,f) + IGI {~(2 n) + Const ~(22n) I/2}

IGI {2 ~(2 n) + Const ~(22n) I/2} ,

which yields the required inequality.

We now prove the second inequality in Theorem G.

that

n ~(22n-i) n ~(22 ) =< 2 + Const ~(22 )1/2

~(220) n-i 2k 2 n-k 1/2 2 n + Const E ~(2 )

k=O

n-i 2k ~(22n-k) i/2 } Const {2 n + 7

k=O

Q.E.D.

Lemmas 4.2 and 4.3 show

124

which yields that ~(22n) < Const 2 n. (See the proof of (2.43).)

denote the integer satisfying 2 n -<_ n < 2 n.

n ~(2 n) =< ~(22 ) =< Const 2 n _-< Const n.

Then

For E E F n and f E Lreal, 0 ~ f ~ i, we put F = [n2

g(x) = f(x 2 n/n). Then F E F Hence we have 2gn "

~(E,f) 2 n gn) = -- ~(F,g) ~ Const IF1 <(2 n

Const IEI gn ~ Const IEI log(n+l),

-g n E ] and

Let n

which gives that

(4.3) ~(n) ~ Const log(n+l) (ng i).

For s I, ..., s n E ~ we put

n

E(s I ..... s n) = U k = i

^ k-1 k {x + Sk, - n- =< x < n },

where ~k = (the integral part of nsk)/n. Then we have, for an interval

I c I 0 and f E Lreal, 0 ~ f ~ i,

,f) ~ Const {3(1, TE(~I ' , Sn ), f) +llI} • ~(I, Tsl ..... Sn ...

Hence (4.3) shows that $(I, T ,f) ~ Const [I[ log(n+l). Since s I, ..., s n

Tsl ..... sn(X,y) = i/(x-y) (x, y ~ I0) , this inequality gives

^

~(Tsl ' .--, Sn) ~ Const log(n+l). Consequently,

) ~ Const $(T )1/2 ~ Const~log(n+l). °(Tsl, ..., s n s I, ..., s n

Since s I, ..., s n E ~ are arbitrary, the second inequality in Theorem G holds.

This completes the proof of Theorem G.

Let BMO(F) denote the Banach space of functions f on a finite union F

of segments, modulo constants, with norm

1 IIfIIBMo(£) = sup ~£ N D(z,2r) I- /£ N D(z,r) If(z) - (f)r N D(z,r) IidzI '

is the mean of f over r n D(z,r) with respect to Idzl where (f)r N D(z,r)

and the supremum is taken over all z E ¢, r > O° Put

125

r = {(x, A I 0) Sl' .... Sn Sl' .-., sn(X)); x E .

Corollary 4.4. Const'~og(n+l)

Theorem G immediately yields

max{IIH r II

Sl, ..., s n L (FSl ' ), BM0(Fsl ' ) °°'~ S N °''~ S n

; s I, ..., s n ~ ~ }

=< Const ~log(n+l) (n ~ i) ,

where IIHpII , is the norm of H F L (F),BM0(F)

BM0(F).

as an operator from L'(F) to

APPENDIX II. PROOF OF THEOREM B BY

P. W. JONES-$. SEMMES

Quite recently, P. w. Jones-S.Semmes gave a proof of Theorem B by complex

variable methods. The following note is their lecture in M~y, 1987. (The author

expresses his thanks to P. W. Jones-S.Semmes who permitted the author to write

here their proof (cf. P.W. Jones [33]). Here is a fact obtained by C. Kenig.

Lemma 4.5. Let F = {x + iA(x); x ( ~} be a Lipschitz graph and

= {z 6 ~; Im z > A(Re z)} . Then, for any g (L2(F) having an analytic

extension, say simply g(z) (z (~), to ~ ,

(4.4) IIglIL2(F ) ~ C M {ifzig'(z)I 2 dis(z,F)d~(z)} I/2,

where d a is the area element and C M is a constant depending only on

M = IA'II.

For z ( ~, we write

Cf(z) = C(f d~IF)(z) (z £ ~)

* L2(F) z = z - 2i(Im z - A(Re z)). For f 6 , we put

, i.e.,

1 f(~) d~. ~f(z) = 2hi IF ~ - z

For z ( F, we write by ~f(z) the nontangentlal limit of

at z. For z 6 F U ~, we have

Cf(~) (~ ~ ~ )

~f(z) = - i ~ (Cf) i (z + i t ) d r = ~ (Cf)"(z + i t ) t dt

1 I~ {I F (If)'(< + it/2)t d~} dt 2~i (z+(it/2) - ~)2

2 (Cf)'(w) ~-y IIi --- -- {Im w - A(Re w)} {~ + iA'(Re w)} do(w)

(z - w*) 2

Let {Qk}i= 1 be a sequence of mutually disjoint cubes (with sides parallel to

the coordinate axes) such that ~ = Ok=l Qk' dk/CM ~ Z(Qk ) ~ CMdk (k { i), where

6(Qk ) is the length of a side of Qk and d k = diS(Qk,F). Then

(Cf)'(w) {Im w - A(Re w)}{~ + iA'(Re w)} do(w), - SSQk (= - w*) 2

IfQk Ilm w - A(Re w) I do(w) ~ C M diS(Qk,F) 3 (k ! i).

127

Hence

~ " dk 2 (4.5) ][CfIIL2(F ) --< C M sup I] Z Ck(~f)'(z k) ( , ) I]L2(F),

k=l z - z k

where the supremum is taken over all sequences co

{Ck}k= 1, {Zk}k= 1 such t h a t [Ckl ~ d k, z k E Qk (k ~ i). Let M 2 denote the 2~dimension maximal operator.

For two sequences {Ck}k= I, {Zk}k= 1 satisfying the above condition, we define a

function h(~) on C by

h(~) = Ck(Cf)'(z k) dk I/2 ( ~ E Qk' k ~ i), h(~) = 0 ( ~ E ¢ - ~).

Then Lemma 4.5 shows that

(4.6) " dk )2

l[ Z C k ( ~ f ) ' ( z k) ( -- , k=l z - z k IIL2(F)

2

I k Ck(Cf)'(Zk) --dk * 3 12 =< CM {ff~ =El (z - z k) dis(z,F) do(z)} I/2

C M {ff~ ]ff¢ h(~)] IN ~]I/211m z~ I/2 ]z - ~*[3 da(~)12 da(z)}i/2

CM {ff~ M2 h(z) 2 d~(z)}i/2 ~ CM {ffc lh(z) 12 d~(z)}I/2

" ]2 dk}i/2 " C M { Z ICk(Cf)'(z k) ~ C M { Z k=l k=l

l(~f),(Zk)i 2 , 3 . 1 / 2 a k ) •

Let G

Then

denote the totality of sequences {~k}k= I such that

12 .3~i/2 (4.7) [ Z l(Cf)'(z k) ak# k=l

2j = sup {I Z (Cf)'(Zk) ak de '

k=l

Lemma 4.4 shows that, for any {~k}k=l E G,

(4.8) I k=iZ (Cf)'(Zk) ek dk2 I - 2~i

i < IIftl II z = 2~ L2(F) k= I

< C M llf[IL2(r ) { ff¢_f~

{~k}k=l E G} .

[ f r f(z) Z k=l

2 d k

ek - II L2(F) (z Zk )2

2 dk 12

IkZ=l ~k (z - Zk )3

k=l ~k~2dk = i.

2 d k

dz I ~k 2 (Z - z k)

dis(z,F) do(z) }1/2

128

{ Z I~k 12 d k } l / 2 < IlfllL~(n. CM Ilfllc2(r) k=l = ct~

Thus (4.5)-(4.8> show that Ilcfll ~ cM Ilfll , which yields Theorem B. L2(F) L2(F)

The proof of Lemma 4.5 by P.W.Jones-S.Se~nes is as follows. Put

A = Ilgll~2(r) , B = ff~ Ig'(z)I 2 dis(z,F) do,z). Let ~(z) be a conformal one

to one mapping form the upper half plane U to ~. Then

A = f ~ Ig ° ~(x)1 2 I~'(x) i dx

S = flu Ig'° ¢(z) 12 dis(¢(z),r) l¢'(z) l 2 do(z)

Koebe's i/4-theorem shows that dis(~(z),F) ~ l~'(z) l(Im z)/4 (z 6 U), and hence

flu Ig'° ~(z) 12 l~'(z) I3 y do(z) ~ Const B.

Since larg ~'(x) I ~ g/2 - (i/M) (x ~ ~), Green's formula shows that

A ~ C M [f_~ Ig o ~(x) I 2 ~'(x) dx I

= CM [ffu A(Ig o O(z) I 2 ~'(z)) y do(z) I

i C M {flu [g'° ~(z) 12 [O'(z) 13 y do(z)

+ ffu ](g'o ¢)(z)(g o ¢)(z)]l¢'(z)[ 2 ]¢"(z)] y do(z)}

=< CM[B + Bl/2{fYuig ° ~(z) 12[~'(z)l ~ (z) 2 y do(z) I/2].

We can write ~'(z) = e V(z) with an analytic function V(z) in U. Then

V E BMO since Im V ~ L ~. Thus

I¢"(z)/¢'(z)l 2 y dy dx ( = Iv'(s)I 2 y dx dy )

is a Carleson measure in U. Since g o @(z) e V(z)/2 is analytic in U,

~"(z) 2 }1/2 AI/2. {flu [g ° ~(z) 12]~'(z)I ~(z) y do(z) ~ C M

Thus A g CM(B + BI/2AI/2), which yields the required inequality.

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SUBJECT INDEX

Ahlfors function 80

analytic capacity y(E) 71

Area integral 2

BMO 1

Buffon needle probability 105

Bu(E) 105

Calderdn commutator T[a] 1

Calder6n's problem 82

Calder@n's theorem 1

Calderdn-Zygmund decomposition

Carleson measure 6

Cauchy-Hilbert transform H F 68

Cauchy transform C 71

chord-arc curve 68

Coifman-Meyer expression 9

Coifman-Meyer-Stein's theorem ii

Cotlar's lemma 17

Covering Lemma 32

crank 83

Crofton's formula 105

126

Cra(E) 105

E[a] 51

E 21

ex (n) 83 o

fat crank 99

Galton-Watson process 106

Garabedian function 80

Garnett's example 80

generalized length 71

Good % inequalities 4

Green's formula 5

Hilbert transform 7

integralgeometric quantity 105

33

Interpolation 21

John-Nirenberg's inequality

locally chord-arc 71

LP( - ) 68

Lreal 31

L(r,0) i05

L~(.) 72 maximal operator M 34

Mclntosh expression 13

NE(r,6) 105

Poisson kernel 3

Prob 105

Rising Sun Lemma 32

separation theorem 74

T-atom 1.1

T-atomic decomposition 13

Tb theorem 15

tent space ii

T1 theorem 16

Tn[a] 31

Vitushkin's example 80

¥+(E) 71

8-standard kernel 15

I(F',F) 108

~(E,f) 118

~(n) 118

p(F) 72

p+(F) 72

o-function 35

aC(g) 61

$C(~,B) 61

OE(B) 52

~E(~) 55 o(l,K,f) 25

34

133

$(I,T,f) 39

@(I,T,f) 39

~(K) 25

~(n) 86

$0(T) 39

~(T) 39

@(T) 39

3(T ;~-~) 39

~0(n) 91

@(n) 86

co6(K) 16

a real variable method for the cauchy transform, and analytic capacity

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