Classes of Holomorphic Functions of Several Variables in Circular DomainsAuthor(s): S. BochnerSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 46, No. 5 (May 15, 1960), pp. 721-723Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/70906 .

Accessed: 04/05/2014 10:30

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

National Academy of Sciences is collaborating with JSTOR to digitize, preserve and extend access toProceedings of the National Academy of Sciences of the United States of America.

http://www.jstor.org

This content downloaded from 130.132.123.28 on Sun, 4 May 2014 10:30:10 AMAll use subject to JSTOR Terms and Conditions

CLASSES OF HOLOMORPHIC FUNCTIONS OF SEVERAL VARIABLES IN CIRCULAR DOMAINS*

BY S. BOCHNER

PRINCETON UNIVERSITY

Communicated March 24, 1960

We will extend a theorem of (F. and M. Riesz and) Hardy-Littlewoodl from holo-

morphic functions f(z) in the unit disk [ z| < 1 to holomorphic functions of severa- variables f(zl, ..., Zk) in general circular domains.

THEOREM 1.1 If f(z) is holomorphic in | zl < 1 and, for some X > 0,

sup fo l f(re2'ri?) X dO C < oo, 0 <u < 1

then

f,1 sup If(re2?ri) I dO < acCX, O<r<

where ax depends only on X and nothing else. Also, for rl, r2 < 1,

fol If(rie2ri9) - f(r2e2"i) I X dO -- 0 as (ri, r2) -> (1, 1),

and there exists a measurable function F(O) on the boundary e = e2i?0 such that, for

0< r < 1,

fol If(re2"iO) - F(0) I dO -> 0 as r -- 1.

The quantity

M (r) = fo' f(re2"i) x dO

is monotonely increasing in r. Using this, it is not hard to verify that Theorem 1

gives rise to the following generalization of itself.

THEOREM 2. If our holomorphic function depends on a parameter ~, f(z) =

o(z; )); if is the point of a Lebesgue measure space {X: (k), dp(5) } and o(z; ~) is measurable in (z; 5); if for almost all p, p(z; ~) is holomorphic in |zl < 1; and if, for X > 0,

sup fx d5 Sf' p(re2ris; ) X dO CX < o, O<r<

then, for the same ax,

Sx djfo sup | p(re2urio; )l d < xCX; 0 < r 1

also,

fx d f,1 I Ip(re2i; ,) - ?(r2e2"; ) 1 dO --> 0 as (r, r2) -- (1, 1)

and there exists a function ,(0; /) measurable in (0; s) such that

x dE fo | p(re2"i ; ~) - b(0; I) x dO'-- 0.

721

This content downloaded from 130.132.123.28 on Sun, 4 May 2014 10:30:10 AMAll use subject to JSTOR Terms and Conditions

722 MATHIEM'ATICS: S. BOCHiNER Pioc. N. A. S.

Now, in the space of k complex variables (z1, ..., zk) let D be a (bounded) circular domain containing the origin. Thus, if (a,, .. ., a) e D, then also (zal, ..., zak) e D for I z\ < 1. Let B denote the boundary of B with relative topology. We take on B any finite Borel measure c(Bo), Bo c B, subject only to the conditioni that it be circularly invariant. By this we mean that if we replace all points (1,, .. ., ~k) of Bo by (e2 riO 1, . ., e2'iO 'k) and denote the new set by Boo then w(Bo0) = w(Bo). The leading feature of such a measure is the following. If 4(t', .. ., k) is a (non- holomorphic) function on B summable with respect to c(Bo) then

fB t ( ", ..., r)dw = fo1 dO fB e(e2 ri ,..., e2riO rk)dw.

Because of this, Theorem 2 implies as follows.

THEOREM 3. If f(zl, . .., Z) is holomorphic in a bounded circular domain D with origin, and if

sup fBSf(rl, . . ., rrk) X dw(D) CX < o, 0 < r < 1

X > 0, relative to a finite circularly invariant Borel measure w(Bo) on the boundary B of D; then

fB sup If(rD1, ..., r',) (x dco(D) < axC. 0 < r < 1

A. lso

SB If(r,?j) - f(r2j) I dw() -> 0 as ( 1, r2) - (1, 1)

and there exists a function F(gj) = F(1l, o. ., r) on B, measurable relative to w(Bo) sudh that

SB f ^(rj) - F(r ) Xd( -0as r -> 1.

Theorem 3 applies in particular2 to the sphere

Sk: z1 2 + ... + Zk2< 1

with "ordinary" (2k-l)-dimensional surface area w(Bo); anid it also applies to the polycylinder

Pt: zi < 1, . . ., IZ < 1,

if the measure w(Bo) on its boundary B is defined in such a manner that

fB ~(r1 , *..., C)dW (C) = o .. ol (e2i ... , e2iOk)d) ... do,.

This measure is concentrated entirely on the k-dimensional subset Ig~ = 1, ..., [ '[ = 1 of the total (2k-l)-dimensional boundary, but it is a finite Borel measure and circularly invariant, and this was all we have required of it.

We note the corollary (X = 1) that if f(zi, . . ., Zk) is holomorphic in Pk and

sup fo . . Jol If(re2i', . . ., re27rk) I dO, . . . dO, < o 0 < r < 1

then there exists a multiperiodic function F(0j) = F(01, . . ., ) of class LI such that

fo' . . fo lf(rOj) - F(Oj) j d, ... dOk -O 0, r -- 1.

This content downloaded from 130.132.123.28 on Sun, 4 May 2014 10:30:10 AMAll use subject to JSTOR Terms and Conditions

VOL. 46, 1960 MATHEMATICS: S. BOCHNER 723

This non-trivial proposition was originally stated and proved by this author,3 and the proof employed the radial maximum of Hardy-Littlewood, but in a much more complicated manner than the present approach suggests. Also an entirely different proof of this "corollary" was recently given by Helson and Lowdenslager,4 and we wish to point out that our original proof and their recent method both yield a somewhat stronger result which cannot be easily stated within the present set-up.

Finally we observe that for X > 1 we have the following theorem again obtainable from its one variable prototype.l

THEOREM 4. For X > 1 all parts of Theorem 3 (and also of Theorem 2) are valid if instead of a holomorphic functicn in D we consider a function u(zl, . . ., k) which is the real part of a holomorphic function.

A.lso, for X = 1 there remains the statement that

sup SB u (r)r,) log+ ju(rjg) dW(i ) C < 0<r< 1

implies the relation

fB sup u(r,j)[dw(') < aCC + 1 0 < < 1

and the existence of a function U(~j) for which

B| u(rj) - U(r) I d() - 0, r -- 1.

Remark: Note that the log factor in the assumption appears in first power for all

complex dimensions k. One can also extend a theorem of M. Riesz5 on conjugate functions and the state-

ment is as follows. THEOREM 5. If f(zj) = u(zj) + iv(zj) is holomorphic in D and v(O) = 0, then, for

p> 1,

sup f v(rLj) Pdco < yp sup f u(r)j) P dwo r r

and, for p = 1,

sup fB V(rrj) dw < y sup fB u(rrj) ) log+ u| dw + a, r r

where yp, y, 8 are fixed constants.

* This research was supported by the United States Air Force through the Office of Scientific Research of the Air Force Research and Development Command.

1 Hardy, G. H., and J. E. Littlewood, "A maximal theorem with function-theoretic applica- tions," Acta Mathematica, 54, 81-116 (1930).

2 Rauch, H. E., "Harmonic and analytic functions of several variables and the maximal theorem of Hardy and Littlewood," Canadian Journal of Mathematics, 8, :171-183 (1956); see also Smith, K. T., "A generalization of an inequality of Hardy and Littlewood," ibid., 157-170 (1956).

3 Bochner, S., "Boundary values of analytic functions in several variables and of almost periodic functions," Annals of Mathematics, 45, 708-722 (1944).

4 Helson, H., and D. Lowdenslager, "Prediction theory and Fourier Series in several variables," Acta Mathematica, 99, 165-202 (1958).

5 Riesz, M., "Sur les fonctions conjugue6s," Mathematische Zeitschrift, 27, 218-244 (1928).

This content downloaded from 130.132.123.28 on Sun, 4 May 2014 10:30:10 AMAll use subject to JSTOR Terms and Conditions