Arbeitsgruppe Analysis und Partielle Differentialgleichungen

V4B5: Real and harmonic analysis

Summer Semester 2018

Dr. Pavel Zorin-Kranich
Instructor
Dr. Olli Saari
Assistant

Lectures

  • Tu 14-16, 1.008
  • Th 14-16, 1.008

Exercise classes

  • Fr 8-10, 1.008
  • Fr 12-14, 0.011

Topics

A preliminary selection
  1. Lp spaces
    1. Hölder and Minkowski inequalities
    2. Banach spaces, duality, Hahn-Banach theorem (if needed)
    3. Dual spaces of Lp spaces
    4. Hardy-Littlewood maximal operator, density of test functions
  2. Fourier transform
    1. Schwartz space
    2. Plancherel theorem
    3. Riesz-Thorin interpolation theorem
    4. Tempered distributions, Hilbert transform
  3. Calderón-Zygmund theory
    1. Marcinkiewicz interpolation
    2. Norm convergence of Fourier integrals in dimension 1
    3. Cotlar-Stein lemma, cancellative CZ kernels
    4. Hardy spaces and functions of bounded mean oscillation
  4. Outer Lp spaces
    1. Littlewood-Paley theory
    2. Carleson measures
    3. Cauchy integral on Lipschitz curves
  5. Ball multiplier
    1. Kahane-Khintchine inequality
    2. Perron tree
  6. Oscillatory integrals: van der Corput lemma, stationary phase, spherical averages, Fourier transform of measures supported on surfaces, Stein interpolation

Lecture notes

We cover classical material that is very well explained in the cited books. Therefore there will be no complete lecture notes. My notes mostly serve the purpose of gauging the amount of material for a lecture, but I might post them here to have a record of which topics we discuss.

Prerequisites

Lebesgue measure and integration, functional analysis (Banach spaces and operators).

Exam

Admission to the (oral) exam is conditional on obtaining at least half of the points on the homework assignments.

Problem sets

Literature

  • E. M. Stein and R. Shakarchi, Functional analysis. Introduction to further topics in analysis. 2011.
  • E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. 1993.
  • Muscalu and Schlag, Classical and multilinear harmonic analysis, Vol 1. 2013
  • Grafakos, Classical Fourier Analysis. 2008
  • Wolff, Lectures on harmonic analysis (maybe for the follow-up course?)
  • T. W. Körner, Fourier analysis (interesting historical notes)