Carleson’s Theorem,Variations and Applications

Christoph Thiele

Santander, September 2014

Lennart Carleson

• Born 1928• Real/complex

Analysis, PDE, Dynamical systems

• Convergence of Fourier series 1968

• Abel Prize 2006

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’s Operator

Closely related maximal operator

Carleson-Hunt theorem (1966/1968):

Can be thought as stepping stone to Carl. Thm

defxfC ix2)(ˆsup)(

€

C∗ fp

≤ c p fp

p1

Other forms of Carleson operator

€

˜ C * f (x) = p.v. f (x − t)e itη

−∞

∞

∫ dt / tL∞ (η )

€

˜ C η f (x) = p.v. f (x − t)e itη (x )

−∞

∞

∫ dt / t

€

Cη f (x) = ˆ f (ξ)e2πixξ dξη (x )

∞

∫

€

C* f (x) = ˆ f (ξ)e2πixξ dξη

∞

∫L∞(η )

Quadratic Carleson operator

Victor Lie’s result, 1<p<2

tdtetxfvpxQf tiit /)(..sup)(2

,

pppfconstQf

Directional Hilbert transform

In the plane:

Rotate so that u=0, apply HT in first variable and Fubini:

€

Hu f (x) = p.v. f (x + t,y + ut)dt / t∫

€

Hu fp

≤ Cp fp

Alternative description

1)Take the Fourier transform of f

2)Multiply by a certain function constant on half planes determined by (1,u)

3)Take the inverse Fourier transform

Maximal directional Hilbert t.

In the plane:

Turns out unbounded

€

supu p.v. f (x + t, y + ut)dt / t∫

Nikodym set example

Set E of null measure containing for each

(x,y) a line punctured at (x,y). If vector field

points in direction of this line then averages

of characteristic fct of set along vf are one.

“Half” max directional HT

Unbounded:

Bounded:

Bounded (3/2<p<infty)(Bateman, T. 2012)

€

supu Hu f (x, y)Lp (y ) Lp (x )

/ ≤ Cp fp

€

supu Hu f (x, y)Lp (y ) Lp (x )

≤ Cp fp

€

supu Hu f (x,y)Lp (y ) Lp (x )

≤ Cp fp

Direct.HT w.r.t Vector Field

€

Hu f (x) = p.v. f (x + t,y + u(x,y)t)dt / t∫

BT case: One Variable V.F

€

p.v. f (x + t, y + u(x)t)dt / tR

∫

L2:Coifman’s argument

€

f (x + t,y + u(x)t)dt / tR

∫L2 (x,y )

€

= e iyη

R

∫ ˆ f (x + t,η)e iu(x )tη dt / t dηR

∫L2 (x,y )

€

= ˆ f (x + t,η)e iu(x )tη dt / tR

∫L2 (x,η )

2),(2),(ˆ fxf

xL

Coifman’s argument visualized

A Littlewood Paley band

For Lp theory need Littlewood Paley instead FT.

Idea of Lacey and Li: Generalization of Carleson

Further generalization

Vector field constant along suitable family of Lispchitz curves (tangents nearly vertical,vector field nearly horizontal)

Shaming Guo 2014: HT bounded in L2/Lp

Lipschitz conjecture

Conjecture: The truncated Hilbert transform (integral from -1 to 1) along (two variable) vector field is bounded in L2 provided the vector field is Lipschitz with small enough constant

Only known for real analytic vector fields. Christ,Nagel,Stein,Wainger 99, Stein/Street 2013

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 .),(

)/(

Symmetric dual trilinear form

All non-degenerate triangles equivalent by linear transformation. No parameters.

€

Λ( f ,g,h) = ∫∫ p.v. f (r x +

r β 1t)g(

r x +

r β 2t)h(

r x +

r β 3t)dt / t

−∞

∞

∫ dx1dx2

€

Λ( f ,g,h) = p.v. f (x,y)g(y,z)h(z, x)1

x + y + zdxdydz∫∫∫

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 ),(),(),(..

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

mmmM

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

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 fca

L

ppL

fcap

)('

|)(|log

21 p

1)(|)(|log fca

L

Conjectured analogues

Nonlinear Carleson

Uniform nonlinear Hausdorff Young

2)(2

|)(|logsup

fcyaLy

ppfca

'|)(|log 21 p

THANK YOU!