Carleson’s Theorem,Variations and Applications

Christoph Thiele

Kiel, 2010

x

• Translation in horizontal direction

• Dilation

• Rotation by 90 degrees

• Translation in vertical direction

)()( yxfxfTy

2/1/)/()( xfxfD

dxexff ix 2)()(ˆ

ixexfxfM 2)()(

Carleson Operator

( identity op, Cauchy projection)

Translation/Dilation/Modulation symmetry.

Carleson-Hunt theorem (1966/1968):

defxfC ix2)(ˆsup)(

pppfcfC

p1

0

Multiplier Norm

- norm of a function f is the operator normof its Fourier multiplier operator acting on

- norm is the same as supremum norm

qM

)(1 FgfFg

)(RLq

)(sup2

fff

M

2M

-Carleson operator

Theorem: (Oberlin, Seeger, Tao, T. Wright ’10)

provided

qM

)(

2 ||)(ˆ||)(

qq M

xi

MdefxfC

pqpp

MfcfC

q,

qp /1|2/1/1|

Redefine Carleson Operator

Truncated Carleson operator

tdtetxfvpxCf txi /)(..sup)( )(

tdtetxfxfCc

txi /)(sup)(],[

)(

Truncated Carleson as average

tdttetxfR

txi /)/()(sup )(

tdtdeetxfR R

titxi /)(ˆ)(sup )(

dtdteetxf

R R

titxi /)(sup)(ˆ )(

Maximal Multiplier Norm

-norm of a family of functions is the

operator norm of the maximal operator on

No easy alternative description for

)(sup 1 FgfFg

)(RLp

pM f

2M

-Carleson operator

Theorem: (Demeter,Lacey,Tao,T. ’07)

Conjectured extension to , range of p ?

Non-singular variant with by Demeter 09’.

2M

)(],[

)(*2

*2

||/)(||)(

M

txi

MtdtetxfxfC

c

pppM

fcfC *2

pMpM

Birkhoff’s Ergodic Theorem

X: probability space (measure space of mass 1).

T: measure preserving transformation on X.

: measurable function on X (say in ).

Then

exists for almost every x .

)(2 XL

)(1

lim1

xTfN

N

n

n

N

f

Harmonic analysis with …

Compare

With max. operator

With Hardy Littlewood

With Lebesgue Differentiation

…and no Schwartz functions

)(1

lim1

xTfN

N

n

n

N

)(1

sup1

xTfN

N

n

n

N

00

)(1

lim dttxf

0

)(1

sup dttxf

Weighted Birkhoff

A weight sequence is called “good” if the

weighted Birkhoff holds: For all X,T,

Exists for almost every x.

na

)(1

lim1

xTfaN

nN

nnN

)(2 XLf

Return Times Theorem

(Bourgain, ‘88) Y probability space, S measure preserving transformation on Y, . Then is good for almost every y.

Extended to , 1<p<2 by Demeter, Lacey,Tao,T. Transfer to harmonic analysis, take Fourier transform in f, recognize .

)(2 YLg)( ySga n

n

)(YLg p

*2M

C

Hilbert Transform / Vector Fields

Lipshitz,

Stein conjecture:

Also of interest are a) values other than p=2, b) maximal operator along vector field (Zygmund

conjecture) or maximal truncated singular integral

1

1

/))(()( tdttxvxfxfHv

22: RRv yxCyvxv )()(

22fCfH vv

Coifman: VF depends on 1 vrbl

Other values of p: Lacey-Li/ Bateman

Open: range of p near 1, maximal operator

),(2

/))(,(yxLR

tdttxvytxf

),(

)(

2

/),(ˆ

yxLR

txiv

R

iy dtdtetxfe

2),(),(

)(2

2

),(ˆ/),(ˆ fxftdtetxfxL

xLR

txiv

Application of -Carleson (C. Demeter)

Vector field v depends on one variable and f

is an elementary tensor f(x,y)=a(x)b(y), then

in an open range of p around 2.

pM

ppv fCxfH )(

),(

)( /)()(ˆ

yxLR

txiv

R

iy

P

dtdtetxabe

Application of Carleson

Maximal truncation of HT along vectorfield

Under same assumptions as before

Carried out for Hardy Littlewood maximal operatoralong vector field by Demeter.

*pM

ppv fCxfH )(*

tdttxvxfxfHv /))((sup)(

1

*

Variation Norm

rrnn

N

nxxxNV

xfxffN

r/1

11,...,,,

)|)()(|(sup||||10

Another strengthening of supremum norm

Variation Norm Carleson

Thm. (Oberlin, Seeger, Tao, T. Wright, ‘09)

)(

2)(ˆ)(

r

r

V

ix

VdefxfC

ppVfCfC r

,1 p )',2max( pr

Rubio de Francia’s inequality

Rubio de Francia’s square function, p>2,

Variational Carleson, p>2

p

xL

ixN

nNfCdef

p

n

nN

)(

)2/(122

1,...,,,)|)(ˆ|(sup

110

p

xL

ixN

nNfCdef

p

n

nN

)(

2/122

1,...,,,)|)(ˆ|(sup

110

Coifman, R.d.F, Semmes

Application of Rubio de Francia’s inequality:Variation norm controls multiplier norm

Provided

Hence variational Carleson implies - Carleson

rp VM

mCm

rp /1|/12/1|

pM

Nonlinear theory

Fourier sums as products (via exponential fct)

dxexfygy

ix

2)(exp)(

)()()(' 2 ygexfyg ix

1)( g

))(ˆexp()( fg

Non-commutative theory

Nonlinear Fourier transform, other choices of matrices lead to other models, AKNS systems

)(0)(

)(0)('

2

2

yGexf

exfyG

ix

ix

10

01)(G

)()( fG

Incarnations of NLFT

• (One dimensional) Scattering theory

• Integrable systems, KdV, NLS, inverse scattering method.

• Riemann-Hilbert problems

• Orthogonal polynomials

• Schur algorithm

• Random matrix theory

Analogues of classical factsNonlinear Plancherel (a = first entry of G)

Nonlinear Hausdorff-Young (Christ-Kiselev)

Nonlinear Riemann-Lebesgue (Gronwall)

2)(2|)(|log fcaL

ppL

fcap

)('

|)(|log 21 p

1)(|)(|log fcaL

Conjectured analogues

Nonlinear Carleson

Uniform nonlinear Hausdorff Young

Both OK in Walsh case, WNLUHY by Vjeko Kovac

2)(2

|)(|logsup

fcyaLy

ppfca

'|)(|log 21 p

Picard iteration, exp series

Scalar case: symmetrize, integrate over cubes

idGxGxMxG )(),()()('

xy

dyyMidxGidxG )()(,)( 21

...)(!2

1)(1)(

2

xyxy

dyyMdyyMxG

...)()()()(12

1221 xyyxy

dydyyMyMdyyMidxG

Terry Lyons’ theory

Etc. … If for one value of r>1 one controls all with n<r, then bounds for n>r follow automatically as well as a bound for the series.

rr

xyx

N

nxxxNr dyymV

nnN

/1

1,...,,,1, )|)(|(sup

110

rr

xyyx

N

nxxxNr dydyymymV

nnN

/22/1221

1,...,,,2, )|)()(|(sup

12110

nrV ,

Lyons for AKNS, r<2, n=1

For 1<p<2 we obtain by interpolation between a trivial estimate ( ) and variational Carleson ( )

This implies nonlinear Hausdorff Young as well as variational and maximal versions of nonlinear HY.

Barely fails to prove the nonlinear Carleson theorem because cannot choose

pp

L

pp

xyx

iyN

nxxxNfcdyeyf

pnnN

)(

)/(12

1,...,,,'1

10

)|)(|(sup

2L1L

22 r

Lyons for AKNS, 2<r<3, n=1,2

Now estimate for n=1 is fine by variational Carleson.

Work in progress with C.Muscalu and Yen Do:

Appears to work fine when . This puts an algebraic condition on AKNS which unfortunately is violated by NLFT as introduced above.

rr

xyyx

yyiN

nxxxNdyeyfyf

nnN

/22/)(221

1,...,,,)|)()(|(sup

121

2211

10

021