Math 633. Harmonic Analysis: Fourier Analysis with Applications to Partial Differential Equations
- Spring 26 - Hans Lindblad

Math 633 Harmonic Analysis, Spring 2026 Mondays and Wednesdays 10.40-11.55 in Krieger 411

Harmonic analysis is the analysis of functions and operators in terms of simpler harmonic components. It originated in the revolutionary idea of Fourier over 200 years ago that one can solve the heat and the wave equation by decomposing into a series of sines and cosines. The first part will be classical harmonic analysis such as function space inequalities and decompositions. The second part will deal with estimates for operators defined as Singular Integrals or Fourier multipliers. These are solution operators for Linear Partial Differential Equations. The techniques will be further developed to deal with nonlinear equations using Para Differential Operators and Multilinear Fourier Multiplier estimates. A goal is to develop the harmonic analysis techiques used to study Euler's equations of Fluid Mechanics and other nonlinear partial differential equations, with in particular applications to water waves.

It will be based mostly on the lecture notes of Terry Tao available online at Math 247A and Math 247B. We will be covering material from notes 0-8, we will cover the first part about function spaces (notes 0-3) mostly as needed for the second part about operators (notes 4-8). We will also pick material from the text books of Muscat and Schlag 'Classical and Multilinear Harmonic Analysis Vol I and II' that will be available on Canvas.

Prerequisite is one semester of graduate real analysis or two semesters of undergraduate real analysis. There will not be any exams but there will be a few homework problems.

Addintional References:
  • Classical Harmonic Analysis and Fourier Integral
    • Sogge 'Fourier Integrals in Classical Harmonic Analysis'
  • Singular Integral Operators and Multlinear Operators
    • Christ 'Lectures on Singular Integral Operators'
    Meyer and Coifman 'Wavelets, Calderón-Zygmund and multilinear operators'
  • Pseudo and Para differential operators with applications to Nonlinear differential equations.
    • Hormander 'Nonlinear Hyperbolic differential equations'
    Taylor 'Pseudo Differential Operators and Nonlinear PDE'
  • Fluids and the free boundary water wave problem
    • Wu 'Almost Global Wellposedness of the 2-D water wave problem'
    Germain, Masmoudi and Shatah 'Global solutions for the gravity water waves equation in dimension 3'
    Alazard 'Paralinearization of Free Boundary Problems in Fluid Dynamics'