Carleson’s Theorem,Variations and Applications

Christoph Thiele

Santander, September 2014

Bilinear Hilbert transform

Here is a unit vector perpendicular to (1,1,1), with no two components equal.

1)Translation symmetry

2)Dilation symmetry

3)Modulation symmetry with vector

€

(β1,β2,β3)

€

Λ( f1, f2, f3) := p.v. ( f i(x − βit)i=1

3

∏ )dxdt

t∫∫

€

(α 1,α 2,α 3)

New embedding map

Embedding map using orbit of entire group:

Express bilinear Cauchy transform as

€

F j (y,α jη + β j t−1, t)dηdydt

j =1

3

∏R

∫R

∫0

∞

∫€

F(y,η, t) = f (z)e izη∫ t−1φ(t−1(z − y))dz

€

Modified outer measure space

For alpha, beta coming form vectors as above

Tents:

For small b depending on alpha, beta€

T(x,ξ,s) = {(y,η, t) : y − x < s − t,| α (η − ξ) + βt−1 |< t−1}

€

σ(T(x,ξ,s)) = s

€

A = {(y,η, t) : y − x < s − t,|η − ξ |< bt−1}

€

S(F)(T(x,ξ,s)) =1

sF(y,η, t)

2

T (x,ξ ,s)\ A

∫∫∫ dydηdt ⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

+ sup(y,η ,s)∈T (x,ξ ,s) F(y,η, t)

Embedding theorem

Thm: For 2<p<infty

As Corollary obtain bounds for BHT:

Where and

€

Fφ Lp (X ,μ ,S )≤ C f

p

€

Λ( f1, f2, f3) ≤ C f i p ii=1

3

∏

€

1

p1

+1

p2

+1

p3

=1

€

2 < p1, p2, p3 < ∞

Carleson’s theorem

On almost everywhere convergence of Fourier series (integrals for expository purpose):

For almost every x. Assumptions is that f in Lp. May assume for simplicity that Fourier transform of f is integrable on half lines.

€

f (x) = limη → −∞

ˆ f (ξ)e2πixξ dξη

∞

∫ ⎛

⎝ ⎜

⎞

⎠ ⎟

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

Remarks on Carleson’s operator

1) Simple reflection symmetry and cut and paste arguments allow for two-sided truncations.

2) For we note the Cauchy projection

3) For arbitrary eta, we note modulated C. p.

0

€

Πf (x) = ˆ f (ξ)e2πixξ dξ0

∞

∫

Equivalent variants

Maximal modulated Hilbert transform

Linearized maximal operators, to be estimated uniformly in all measurable functions eta€

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

−∞

∞

∫ dt / t

€

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

−∞

∞

∫ dt / t

€

Cη f (x) = ˆ f (ξ)e2πixξ dξη (x )

∞

∫

Symmetries

Dilation sym: inherited from Cauchy proj.

Translation sym: inherited from Cauchy proj.

Modulation symmetry:

€

Cη +ρ Mρ f (x) = ˆ f (ξ + ρ )e2πixξ dξη (x )+ρ

∞

∫

€

= ˆ f (ξ)e2πix(ξ −ρ )dξη (x )

∞

∫ = Mρ (Cη f )(x)

Carleson with embedding maps

Old

New

Here phi has positive smooth FT with small support. Then

€

F(x,ξ,s) = f (z)e izξ∫ s−1φ(s−1(z − x))dz

€

˜ G (x,ξ,s) = g(z)e iz(ξ +s−1 )∫ s−1φ(s−1(z − x)) ˆ φ (s(ξ −η(z)))dz

€

Cη f ,g = F(x,ξ + s−1,s) ˜ G (x,ξ,s)dxdξdsR

∫R

∫0

∞

∫

Proof of representation

Recall Calderon reproducing/Cauchy proj.

Modulated Cauchy projection (fixed xi)

Averaged version

€

Πf ,g = F(x,0 + s−1,s)G(x,0 + s−1,s)dxds /sR

∫0

∞

∫

€

Π f ,g = F(x,ξ + s−1,s)G(x,ξ + s−1,s)dxds /sR

∫0

∞

∫

€

= F(x,ξ + γs−1 + s−1,s)G(x,ξ + γs−1 + s−1,s) ˆ φ (γ )dxds /sR

∫0

∞

∫ dγ

Proof of representation

Write G explicitly as integral

For Carleson, each z sees frequency eta(z)

Change of variable xi, then integrate gamma€

= F(x,ξ + γs−1 + s−1,s) g(z)∫ φx,ξ +γs−1 +s−1 ,s

(z)dz ˆ φ (γ )dxds /sR

∫0

∞

∫ dγ

€

Cη f ,g = ∫ F(x,ξ + γs−1 + s−1,s) g(z)δ (ξ −η(z))∫ φx,ξ +γs−1 +s−1 ,s

(z)dz ˆ φ (γ )dξdxds /sR

∫R

∫0

∞

∫ dγ

€

= F(x,ξ + s−1,s) g(z)δ (ξ − γs−1 −η(z))∫ φx,ξ +s−1 ,s

(z)dz ˆ φ (γ )dξdxds /sR

∫R

∫0

∞

∫ dγ

€

= F(x,ξ + s−1,s) g(z)∫ φx,ξ +s−1 ,s

(z)dz ˆ φ (s(ξ −η(s)))dξdxdsR

∫R

∫0

∞

∫

Modified size

Analoguous tent spaces for tile embedding

Modified size:

Recall old size

€

A = {(y,η, t) : y − x < s − t,|η − ξ |< bt−1}

€

˜ S ( ˜ G )(T(x,ξ,s)) =1

s˜ G (y,η, t)

T (x,ξ ,s)\ A

∫∫∫ dydη dt +1

s˜ G (y,η, t)

2

T (x,ξ ,s)

∫∫∫ dydηdt ⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

€

S(F)(T(x,ξ,s)) =1

sF(y,η, t)

2

T (x,ξ ,s)\ A

∫∫∫ dydηdt ⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

+ sup(y,η ,s)∈T (x,ξ ,s) F(y,η, t)

Modified Embedding theorem

Old Thm: For 2<p<infty

New Thm. For 1<p<infty

With that proof proceeds as before.

€

F Lp (X ,μ ,S ) ≤ C fp

€

˜ G Lp (X ,μ , ˜ S )

≤ C gp

Comparison BHT/Carleson

1) BHT has simpler proof since more symmetric and needs only on embedding theorem

2) Carleson appears more naturally in applications, since it has a more canonical modulation symmetry that does not depend on an ad hoc vector beta.

Generalizations of Carleson

Recall that Carleson reads as

One way of strengthening of Carleson operator is to replace Linfty by larger norms. Then prove Lp bounds for these stronger operators.€

C* f (x) = ˆ f (ξ)e2πixξ dξη

∞

∫L∞(η )

Variation Norm

rrnn

N

nxxxNV

xfxffN

r/1

11,...,,,

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

A strengthening of supremum norm. Note

That finite variation implies convergence.

Variation Norm Carleson

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

€

CV r f (x) = ˆ f (ξ)e2πixξ dξ

η

∞

∫V r (η )

ppVfCfC r

,1 p )',2max( pr

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

fffM

2M

Coifman, R.d.F, Semmes

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

Provided

Hence variational Carleson implies - Carleson€

m ∞ ≤ m M p≤ C m V r

rp /1|/12/1|

pM

Truncated Carleson Operator

tdtetxfxfCc

txi /)(sup)(],[

)(

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

)(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

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 fca

L

ppL

fcap

)('

|)(|log 21 p

1)(|)(|log fca

L

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