Carleson’s Theorem, Variations and Applications Christoph Thiele Kiel, 2010

Translation in horizontal direction Dilation Rotation by 90 degrees Translation in vertical direction

Carleson Operator ( identity op, Cauchy projection) Translation/Dilation/Modulation symmetry. Carleson-Hunt theorem (1966/1968):

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

-Carleson operator Theorem: (Oberlin, Seeger, Tao, T. Wright ’10) provided

Redefine Carleson Operator Truncated Carleson operator

Truncated Carleson as average

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 …and no Schwartz functions

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.

Hilbert Transform / Vector Fields Lipshitz, Stein conjecture: Also of interest are a) values other than p=2, b) maximal operator along vector field (Zygmund conjecture) or maximal truncated singular integral

Coifman: VF depends on 1 vrbl Other values of p: Lacey-Li/ Bateman Open: range of p near 1, maximal operator

Application of -Carleson (C. Demeter) Vector field v depends on one variable and f is an elementary tensor f(x,y)=a(x)b(y), then in an open range of p around 2.

Application of Carleson Maximal truncation of HT along vectorfield Under same assumptions as before Carried out for Hardy Littlewood maximal operator along vector field by Demeter.

Variation Norm Another strengthening of supremum norm

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

Rubio de Francia’s inequality Rubio de Francia’s square function, p>2, Variational Carleson, p>2

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

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 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.