Carleson’s Theorem, Variations and Applications

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

New embedding map Embedding map using orbit of entire group: Express bilinear Cauchy transform as

Modified outer measure space For alpha, beta coming form vectors as above Tents: For small b depending on alpha, beta

Embedding theorem Thm: For 2<p<infty As Corollary obtain bounds for BHT: Where and

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.

Carleson’s Operator Closely related maximal operator Carleson-Hunt theorem (1966/1968): Can be thought as stepping stone to Carl. Thm

Remarks on Carleson’s operator • Simple reflection symmetry and cut and paste arguments allow for two-sided truncations. • For we note the Cauchy projection • For arbitrary eta, we note modulated C. p.

Equivalent variants Maximal modulated Hilbert transform Linearized maximal operators, to be estimated uniformly in all measurable functions eta

Symmetries Dilation sym: inherited from Cauchy proj. Translation sym: inherited from Cauchy proj. Modulation symmetry:

Carleson with embedding maps Old New Here phi has positive smooth FT with small support. Then

Proof of representation Recall Calderon reproducing/Cauchy proj. Modulated Cauchy projection (fixed xi) Averaged version

Proof of representation Write G explicitly as integral For Carleson, each z sees frequency eta(z) Change of variable xi, then integrate gamma

Modified size Analoguous tent spaces for tile embedding Modified size: Recall old size

Modified Embedding theorem Old Thm: For 2<p<infty New Thm. For 1<p<infty With that proof proceeds as before.

Comparison BHT/Carleson • BHT has simpler proof since more symmetric and needs only on embedding theorem • 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.

Variation Norm A strengthening of supremum norm. Note That finite variation implies convergence.

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

Multiplier Norm - norm of a function f is the operator norm of its Fourier multiplier operator acting on - norm is the same as supremum norm

Coifman, R.d.F, Semmes Application of Rubio de Francia’s inequality: Variation norm controls multiplier norm Provided Hence variational Carleson implies - Carleson

Truncated Carleson Operator

Maximal Multiplier Norm -norm of a family of functions is the operator norm of the maximal operator on No easy alternative description for

-Carleson operator Theorem: (Demeter,Lacey,Tao,T. ’07) Conjectured extension to , range of p ? Non-singular variant with by Demeter 09’.

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 .

Harmonic analysis with … Compare With max. operator With Hardy Littlewood With Lebesgue Differentiation

Weighted Birkhoff A weight sequence is called “good” if the weighted Birkhoff holds: For all X,T, Exists for almost every x.

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 .

Nonlinear theory Fourier sums as products (via exponential fct)

Non-commutative theory Nonlinear Fourier transform, other choices of matrices lead to other models, AKNS systems

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 facts Nonlinear Plancherel (a = first entry of G) Nonlinear Hausdorff-Young (Christ-Kiselev) Nonlinear Riemann-Lebesgue (Gronwall)

Conjectured analogues Nonlinear Carleson Uniform nonlinear Hausdorff Young Both OK in Walsh case, WNLUHY by Vjeko Kovac

Picard iteration, exp series Scalar case: symmetrize, integrate over cubes

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.

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

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.

Carleson’s Theorem, Variations and Applications