Carleson’s Theorem,Variations and Applications

Christoph Thiele

Colloquium, Amsterdam, 2011

Lennart Carleson

• Born 1928• Real/complex

Analysis, PDE, Dynamical systems

• Convergence of Fourier series 1968

• Abel Prize 2006

Fourier Series

N

Nn

inxn

Nefxf 2ˆlim)(

1

0

2)(ˆ dxexff inxn

Hilbert space methods

The Functions with form an

orthonormal basis of a Hilbert space with

inner product

inxe 2

n

nn gfdxxgxfgf ˆˆ)()(,1

0

Zn

Carleson’s theorem

For f continuous or piecewise continuous,

converges to f(x) for almost every x in [0,1] .

N

Nn

inxn

Nef 2ˆlim

Quote from Abel Prize

“The proof of this result is so difficult that for over thirty years it stood mostly isolated from the rest of harmonic analysis. It is only within the past decade that mathematicians have understood the general theory of operators into which this theorem fits and have started to use his powerful ideas in their own work.”

Carleson Operator

defxfC ix2)(ˆsup)(

dxexff ix

2)()(ˆ

)(

2)(ˆ)(x

ix defxfC

Carleson-Hunt Theorem

Carleson 1966, Hunt 1968 (1<p):

Carleson operator is bounded in .

pppfconstfC

dxxffpp

p

)(:

pL

Cauchy projection

An orthogonal projection, hence a bounded operator in Hilbert space .

0

2)(ˆ)( defxCf ix

2L

Symmetries

• Translation

• Dilation

)()( yxfxfTy

)/()( xfxfD

Invariance of Cauchy projection

Cauchy projection and identity operator span

the unique two dimensional space of linear

operator with these symmetries.

CDCDCTCT yy ,

Other operators in this space

• Hilbert transform

• Operator mapping real to imaginary part of functions on the real line with holomorphic extension to upper half plane.

tdttxfvpxHf /)(..)(

Wavelets

From a carefully chosen generating function

with integral zero generate the discrete

(n,k integers) collection

Can be orthonormal basis.

nkn TD k2,

Wavelets

Properties of wavelets prove boundedness of

Cauchy projection not only in Hilbert space

but in Banach space .

They encode much of singular integral theory.

For effective computations, choice of

generating function is an art.

pL

2L

Modulation

Amounts to translation in Fourier space

ixexfxfM 2)()(

fTf ˆˆ

Modulated Cauchy projection

Carleson’s operator has translation, dilation,

and modulation symmetry. Larger symmetry

group than Cauchy projection (sublinear op.).

def ix2)(ˆ

tdtetxfvp it /)(..

Wave packets

From a carefully chosen generating function

generate the collection (n,k,l integers)

Cannot be orthonormal basis.

nlkln TMD k2,,

Quadratic Carleson operator

Victor Lie’s result, 1<p<2

tdtetxfvpxQf tiit /)(..sup)(2

,

pppfconstQf

Vector Fields

Lipshitz,22: RRv yxcyvxv )()(

/

Hilbert Transform along Vector Fields

Stein conjecture:

(Real analytic vf: Christ,Nagel,Stein,Wainger 99)

1

1

/))((..)( tdttxvxfvpxfHv

22fCfH vv

Zygmund conjecture

Real analytic vector field: Bourgain (89)

22fCfM vv

/))((sup)(

10dttxvxfxfM v

One Variable Vector Field

R

tdttxvytxfvp /))(,(..

Coifman’s argument

),(2

/))(,(yxLR

tdttxvytxf

),(

)(

2

/),(ˆ

yxLR

txiv

R

iy dtdtetxfe

),(

)(

2

/),(ˆ

xLR

txiv tdtetxf 2),(2

),(ˆ fxfxL

Theorem with Michael Bateman

Measurable, one variable vector field

Prior work by Bateman, and Lacey,Li

pvpv fCfH

p2/3

Variation Norm

rrnn

N

nxxxNV

xfxffN

r/1

11,...,,,

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

rVx

fxf )(sup

Variation Norm Carleson

Oberlin, Seeger, Tao, T. Wright, ’09: If r>2,

Quantitative convergence of Fourier series.

)(

2)(ˆ)(

r

r

V

ix

VdefxfC

22fCfC rV

Multiplier Norm

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

- norm is the same as supremum norm

qM

)(1 FgmFg

)(RLq

)(sup2

mmm

M

2M

Coifman, Rubio de Francia, Semmes

Variation norm controls multiplier norm

Provided

Hence -Carleson implies - Carleson

rp VM

mCm

rp /1|/12/1|

pMrV

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 FgmFg

)(RLp

pM m

2M

Truncated Carleson Operator

tdtetxfxfCc

it /)(sup)(],[

-Carleson operator

Theorem: (Demeter,Lacey,Tao,T. ’07) If 1<p<2

Conjectured extension to .

2M

)(],[

*2

*2

||/)(||)(

M

it

MtdtetxfxfC

c

pppM

fcfC *2

qM

Birkhoff’s Ergodic Theorem

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

T: measure preserving transformation on X.

f: measurable function on X (say in ).

Then

exists for almost every x .

)(2 XL

)(1

lim1

xTfN

N

n

n

N

Harmonic analysis with .

Compare

With max. operator

With Hardy Littlewood

With Lebesgue Differentiation

)(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

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.

g: measurable function on Y (say in ).

Then

Is a good sequence for almost every x .

)(2 YL

)( xSga nn

Return Times Theorem

After transfer to harmonic analysis and one

partial Fourier transform, this can be

essentially reduced to Carleson

Extended to , 1<p<2 by D.L.T.T,

Further extension by Demeter 09,

2/3/1/1 pp

)(YLg p

*2M

)(XLf q

Two commuting transformations

X: probability space

T,S: commuting measure preserving transformations on X

f.g: measurable functions on X (say in ).

Open question: Does

exist for almost every x ? (Yes for .)

)(2 XL

)()(1

lim1

xSgxTfN

nN

n

n

N

aTS

Triangular Hilbert transform

All non-degenerate triangles equivalent

tdttyxgytxfvpyxgfT /),(),(..),)(,(

Triangular Hilbert transform

Open problem: Do any bounds of type

hold? (exponents as in Hölder’s inequality)

qpqppqgfconstgfT .),(

)/(

Again stronger than Carleson:

Specify

tdttyxgytxfvp /),(),(..

)(),( xfyxf

yxiNeyxg )(2),(

Degenerate triangles

Bilinear Hilbert transform (one dimensional)

Satisfies Hölder bounds. (Lacey, T. 96/99)

Uniform in a. (T. , Li, Grafakos, Oberlin)

tdtatxgtxfvpxgfB /)()(..))(,(

Vjeko Kovac’s Twisted Paraproduct (2010)

Satisfies Hölder type bounds. K is a Calderon

Zygmund kernel, that is 2D analogue of 1/t.

Weaker than triangular Hilbert transform.

dtdstsKtyxgysxfvp ),(),(),(..

Nonlinear theory

Exponentiate Fourier integrals

dxexfygy

ix

2)(exp)(

)()()(' 2 ygexfyg ix

1)( g ))(ˆexp()( fg

Non-commutative theory

The same matrix valued…

)(0)(

)(0)('

2

2

yGexf

exfyG

ix

ix

10

01)(G

)()( fG

Communities talking NLFT

• (One dimensional) Scattering theory

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

• Riemann-Hilbert problems

• Orthogonal polynomials

• Schur algorithm

• Random matrix theory

Classical facts Fourier transformPlancherel

Hausdorff-Young

Riemann-Lebesgue

22

ˆ ff

ppff

'

ˆ )1/(',21 pppp

1ˆ ff

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

Nonlinear Hausdorff-Young (Christ-Kiselev ‘99, alternative proof OSTTW ‘10)

Nonlinear Riemann-Lebesgue (Gronwall)

2)(2|)(|log fcaL

ppL

fcap

)('

|)(|log

21 p

1)(|)(|log fcaL

Conjectured analogues

Nonlinear Carleson

Uniform nonlinear Hausdorff Young

2)(2

|)(|logsup

fcyaLy

ppfca

'|)(|log 21 p

Couldn’t prove that….

But found a really interesting lemma.

THANK YOU!