S4B1 - Graduate Seminar on Analysis - Harmonic Analysis (winter term 2017/2018)


Instructor:

Dr. Błażej Wróbel


Date:

    Thursday, 16 (c.t.) - 18, place: Endenicher Allee 60 - SemR 0.007
    Preliminary meeting: 12:00, 31.07.2017, place: Endenicher Allee 60 - SemR 0.007

Contents:

    Harmonic Analysis
    In this seminar we will survey various results from both classical and modern harmonic analysis. Some emphasis will be put on dimension-free estimates and sharpness of the constants.

Bibliography:

  • Loukas Grafakos, Classical Fourier Analysis, Graduate Texts in Mathematics Volume 249 (2008)
  • Javier Duoandikoetxea, Fourier Analysis, Graduate Texts in Mathematics Volume 29 (2001)

Requirements:

    Knowledge of Analysis I-II. Familiarity with analysis III, and complex analysis will be helpful but it is not required. Basic functional analysis will be helpful for topics 9-14, which are a bit more advanced.

List of seminar topics

  1. Introduction: definition of the Fourier transform and its basic properties (Inversion formula, Plancherel formula, etc.).
  2. Tempered distributions.
  3. Basics of interpolation of Lp spaces. The Marcinkiewicz and the Riesz-Thorin interpolation theorems.
  4. Hilbert transform.
  5. Calderón-Zygmund theory. Dimension-free estimates for the Riesz transforms.
  6. Hardy-Littlewood maximal operator. Dimension-free estimates in the ball case.
  7. Convolution operators and multiplier operators. Multiplier theorems.
  8. Fefferman's counterexample for the ball multiplier problem and the Bochner-Riesz multipliers.
  9. Spectral theorem and spectral multipiers.
  10. Basics of the theory of strongly continuous semigroups.
  11. Symmetric contraction semigroups. The Lp boundedness of their maximal functions.
  12. Semigroups with Gaussian bounds and spectral multiplier theorems.
  13. Dimension-free estimates in harmonic analysis 1.
  14. Dimension-free estimates in harmonic analysis 2.