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

Hilbert space methods The Functions with form an orthonormal basis of a Hilbert space with inner product

Carleson’s theorem For f continuous or piecewise continuous, converges to f(x) for almost every x in [0,1].

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

Carleson-Hunt Theorem Carleson 1966, Hunt 1968 (1<p): Carleson operator is bounded in.

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

Symmetries Translation Dilation

Invariance of Cauchy projection Cauchy projection and identity operator span the unique two dimensional space of linear operator with these symmetries.

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.

Wavelets From a carefully chosen generating function with integral zero generate the discrete (n,k integers) collection Can be orthonormal basis.

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.

Modulation Amounts to translation in Fourier space

Modulated Cauchy projection Carleson’s operator has translation, dilation, and modulation symmetry. Larger symmetry group than Cauchy projection (sublinear op.).

Wave packets From a carefully chosen generating function generate the collection (n,k,l integers) Cannot be orthonormal basis.

Quadratic Carleson operator Victor Lie’s result, 1<p<2

Vector Fields Lipshitz,

Hilbert Transform along Vector Fields Stein conjecture: (Real analytic vf: Christ,Nagel,Stein,Wainger 99)

Zygmund conjecture Real analytic vector field: Bourgain (89)

One Variable Vector Field

Coifman’s argument

Theorem with Michael Bateman Measurable, one variable vector field Prior work by Bateman, and Lacey,Li

Variation Norm

Variation Norm Carleson Oberlin, Seeger, Tao, T. Wright, ’09: If r>2, Quantitative convergence of Fourier series.

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

Coifman, Rubio de Francia, Semmes Variation norm controls multiplier norm Provided Hence -Carleson implies - Carleson

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

Truncated Carleson Operator

-Carleson operator Theorem: (Demeter,Lacey,Tao,T. ’07) If 1<p<2 Conjectured extension to.

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.

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

Weighted Birkhoff A weight sequence is called “good” if 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. g: measurable function on Y (say in ). Then Is a good sequence for almost every x.

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,

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.)

Triangular Hilbert transform All non-degenerate triangles equivalent

Triangular Hilbert transform Open problem: Do any bounds of type hold? (exponents as in Hölder’s inequality)

Again stronger than Carleson: Specify

Degenerate triangles Bilinear Hilbert transform (one dimensional) Satisfies Hölder bounds. (Lacey, T. 96/99) Uniform in a. (T., Li, Grafakos, Oberlin)

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.

Nonlinear theory Exponentiate Fourier integrals

Non-commutative theory The same matrix valued…

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 transform Plancherel Hausdorff-Young Riemann-Lebesgue

Analogues of classical facts Nonlinear Plancherel (a = first entry of G) Nonlinear Hausdorff-Young (Christ-Kiselev ‘99, alternative proof OSTTW ‘10) Nonlinear Riemann-Lebesgue (Gronwall)

Conjectured analogues Nonlinear Carleson Uniform nonlinear Hausdorff Young

Couldn’t prove that…. But found a really interesting lemma. THANK YOU!