Carleson

1
Carlesons Theorem,Variations and Applications

• Christoph Thiele
• Colloquium, Amsterdam, 2011

2
Lennart Carleson

• Born 1928
• Real/complex Analysis, PDE, Dynamical systems
• Convergence of Fourier series 1968
• Abel Prize 2006

3
Fourier Series

4
Hilbert space methods

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

5
Carlesons theorem

• For f continuous or piecewise continuous,
• converges to f(x) for almost every x in 0,1 .

6
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.

7
Carleson Operator
8
Carleson-Hunt Theorem

• Carleson 1966, Hunt 1968 (1ltp)
• Carleson operator is bounded in .

9
Cauchy projection

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

10
Symmetries

• Translation
• Dilation

11
Invariance of Cauchy projection

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

12
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.

13

14

15

16

17
Wavelets

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

18

19

20

21

22
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.

23
Modulation

• Amounts to translation in Fourier space

24

25

26
Modulated Cauchy projection

• Carlesons operator has translation, dilation,
• and modulation symmetry. Larger symmetry
• group than Cauchy projection (sublinear op.).

27

28

29
Wave packets

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

30

31

32

33

34
Quadratic Carleson operator

• Victor Lies result, 1ltplt2

35

36
Vector Fields

• Lipshitz,

37
Hilbert Transform along Vector Fields

• Stein conjecture
• (Real analytic vf Christ,Nagel,Stein,Wainger 99)

38
Zygmund conjecture

• Real analytic vector field Bourgain (89)

39
One Variable Vector Field

40
Coifmans argument

41
Theorem with Michael Bateman

• Measurable, one variable vector field
• Prior work by Bateman, and Lacey,Li

42
Variation Norm
43
Variation Norm Carleson

• Oberlin, Seeger, Tao, T. Wright, 09 If rgt2,
• Quantitative convergence of Fourier series.

44
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

45
Coifman, Rubio de Francia, Semmes

• Variation norm controls multiplier norm
• Provided
• Hence -Carleson implies -
  Carleson

46
Maximal Multiplier Norm

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

47
Truncated Carleson Operator

48
-Carleson operator

• Theorem (Demeter,Lacey,Tao,T. 07) If 1ltplt2
• Conjectured extension to .

49
Birkhoffs 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 .

50
Harmonic analysis with .

• Compare
• With max. operator
• With Hardy Littlewood
• With Lebesgue Differentiation

51
Weighted Birkhoff

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

52
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 .

53
Return Times Theorem

• After transfer to harmonic analysis and one
• partial Fourier transform, this can be
• essentially reduced to Carleson
• Extended to , 1ltplt2 by D.L.T.T,
• Further extension by Demeter 09,

54
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
  .)

55
Triangular Hilbert transform

• All non-degenerate triangles equivalent

56
Triangular Hilbert transform

• Open problem Do any bounds of type
• hold? (exponents as in Hölders inequality)

57
Again stronger than Carleson

• Specify

58
Degenerate triangles

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

59
Vjeko Kovacs 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.

60
Nonlinear theory

• Exponentiate Fourier integrals

61
Non-commutative theory

• The same matrix valued

62
Communities talking NLFT

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

63
Classical facts Fourier transform

• Plancherel
• Hausdorff-Young
• Riemann-Lebesgue

64
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)

65
Conjectured analogues

• Nonlinear Carleson
• Uniform nonlinear Hausdorff Young

66

• Couldnt prove that.
• But found a really interesting lemma.
• THANK YOU!