RG Analysis and Partial Differential Equations

V4B5: Real and harmonic analysis

Summer Semester 2018

The second round of exams will take place on September 27, 2018. Please send an e-mail to pzorin@math... for an appointment.
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

We have covered the following topics
  1. (weak) Lp spaces
    1. Hölder, Minkowski, and Young convolution inequalities
    2. Banach spaces, duality, Hahn-Banach theorem (in extension form and in separation form)
    3. Dual spaces of Lp spaces
    4. Hardy-Littlewood maximal operator (weak type and Lp estimates), Lebesgue differentiation
    5. Real interpolation
    6. Complex interpolation (Riesz, Stein, iterated Lp)
    7. Hardy-Littlewood-Sobolev inequality, Sobolev embedding
  2. Fourier transform
    1. Distributions, Schwartz space, tempered distributions
    2. Action on Gaussians, multiplication formula, inversion formula, Plancherel theorem, Hausdorff-Young inequality
    3. Poisson summation formula
    4. Heisenberg uncertainty principle
    5. Hilbert transform
  3. Calderón-Zygmund theory
    1. CZ decomposition
    2. Cotlar-Stein lemma, cancellative CZ kernels
    3. Calderón-Vaillancourt theorem on pseudodifferential operators
    4. Norm convergence of Fourier integrals in dimension 1
    5. Cotlar's inequality
    6. Mihlin-Hörmander multipliers
  4. Littlewood-Paley theory
  5. Cauchy integral on Lipschitz curves
    1. Adapted Haar basis, almost orthogonal expansion in L2
    2. Analytic capacity
    3. Bounded functions that are mapped to bounded functions by CZO
  6. Oscillatory integrals
    1. Fourier transform of surface-carried measures
    2. Lacunary spherical maximal function
  7. Hardy and BMO spaces
    1. Atomic decomposition
    2. Complex interpolation between H1 and Lp
    3. Spherical maximal theorem for d≥3
    4. H1-BMO duality
    5. John-Nirenberg inequality
    6. Div-curl lemma
    7. Sharp maximal function
    8. Perturbation of constant coefficient elliptic PDE
  8. Ball multiplier
    1. Sequence valued extensions of linear operators
    2. Perron tree

Lecture notes

These notes provide a record of which topics we discuss. Many details are omitted, but references are often provided.

Prerequisites

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

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
  • M. Christ, Lectures on singular integral operators. 1990
  • Wolff, Lectures on harmonic analysis
  • T. W. Körner, Fourier analysis (interesting historical notes)