RG Analysis and Partial Differential Equations

V4B5: Real and harmonic analysis

Summer Semester 2018

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

Lectures

  • Tu 14-16, 0.007
  • Th 14-16, 0.008

Topics

A preliminary selection
  1. L^p spaces: Hölder's and Minkowski's inequalities, the dual of Lp, convolution, good kernels and approximations to the identity, interpolation theorems (Riesz–Thorin, Marcinkiewicz and Stein)
  2. Hilbert transform and Hardy-Littlewood maximal function, Hardy spaces and functions of bounded mean oscillation
  3. The Fourier transform on the space of tempered distributions, L^1, and L^2
  4. Littlewood-Paley theory
  5. Calderon–Zygmund theory
  6. Oscillatory integrals: van der Corput lemma, stationary phase, spherical averages, Fourier transform of measures supported on surfaces
  7. Carleson measures, Cauchy integral on Lipschitz curves

Prerequisites

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

Target audience

You might be interested in this course if you have liked one of the courses below:

Exam

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

Literature

  • E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Univ. Press, Princeton, NJ, 1993.
  • E. M. Stein and R. Shakarchi, Functional analysis. Introduction to further topics in analysis. Princeton University Press, Princeton, NJ, 2011.
  • Muscalu and Schlag, Classical and multilinear harmonic analysis, Vol 1, Cambridge studies in advanced mathematics 137, 2013
  • Grafakos, Classical Fourier Analysis, Springer, Graduate Texts in Mathematics 249, 2008
  • Wolff, Lectures on harmonic analysis (maybe for the follow-up course?)
  • T. W. Körner, Fourier analysis (interesting historical notes)