Carleson

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Carlesons Theorem,Variations and Applications

• Christoph Thiele
• Kiel, 2010

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

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Carleson Operator

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

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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

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-Carleson operator

• Theorem (Oberlin, Seeger, Tao, T. Wright 10)
• provided

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Redefine Carleson Operator

• Truncated Carleson operator

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Truncated Carleson as average
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Maximal Multiplier Norm

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

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-Carleson operator

• Theorem (Demeter,Lacey,Tao,T. 07)
• Conjectured extension to , range of p ?
• Non-singular variant with by Demeter
  09.

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Birkhoffs 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 .

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Harmonic analysis with

• Compare
• With max. operator
• With Hardy Littlewood
• With Lebesgue Differentiation
• and no Schwartz functions

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Weighted Birkhoff

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

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Return Times Theorem

• (Bourgain, 88) Y probability space, S measure
  preserving transformation on Y,
  . Then is good for
  almost every y.
• Extended to , 1ltplt2 by Demeter,
  Lacey,Tao,T. Transfer to harmonic analysis, take
  Fourier transform in f, recognize .

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Hilbert Transform / Vector Fields

• Lipshitz,
• Stein conjecture
• Also of interest are a) values other than p2,
• b) maximal operator along vector field (Zygmund
  conjecture) or maximal truncated singular integral

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Coifman VF depends on 1 vrbl

• Other values of p Lacey-Li/ Bateman
• Open range of p near 1, maximal operator

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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.

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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.

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Variation Norm
Another strengthening of supremum norm
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Variation Norm Carleson

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

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Rubio de Francias inequality

• Rubio de Francias square function, pgt2,
• Variational Carleson, pgt2

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Coifman, R.d.F, Semmes

• Application of Rubio de Francias inequality
• Variation norm controls multiplier norm
• Provided
• Hence variational Carleson implies -
  Carleson

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Nonlinear theory

• Fourier sums as products (via exponential fct)

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Non-commutative theory

• Nonlinear Fourier transform, other choices of
  matrices lead to other models, AKNS systems

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Incarnations of NLFT

• (One dimensional) Scattering theory
• Integrable systems, KdV, NLS, inverse scattering
  method.
• Riemann-Hilbert problems
• Orthogonal polynomials
• Schur algorithm
• Random matrix theory

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Analogues of classical facts

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

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Conjectured analogues

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

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Picard iteration, exp series

• Scalar case symmetrize, integrate over cubes

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Terry Lyons theory

• Etc. If for one value of rgt1 one
  controls all with nltr, then bounds
  for ngtr follow automatically as well as a bound
  for the series.

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Lyons for AKNS, rlt2, n1

• For 1ltplt2 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

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Lyons for AKNS, 2ltrlt3, n1,2

• Now estimate for n1 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.