Introduction to modern techinques in Fourier Analysis in the Euclidean setting with emphasis in the study of function spaces and operators acting on them. Topics may vary from year to year and include, among others, distribution theory, Sobolev spaces, estimates for fractional integrals and fractional derivatives, wavelets, and some elements of Calderon-Zygmund theory. Applications in other areas of mathematics, in particular, partial differential equations and signal analysis, will be presented based on the instructor's and the student's interest.
Introduction to Fourier Analysis on Euclidean Spaces, Stein and Weiss, Princeton University Press, 1971.
This course introduces some topics in Fourier analysis on the Euclidean spaces. We will use the book by E. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean spaces". For background material, we recommend the book by G. Folland, "Real Analysis: Modern techniques and their applications". We tentatively plan to cover Chapter 1, 2, 4 in Stein and Weiss' book. More precisely, the topics are:
- The L1 and L2 theory of the Fourier transform.
- The Schwartz class and the tempered distributions.
- Introduction to harmonic and subharmonic functions, and characterization of Poisson integrals.
- The Hardy-Littleowood maximal function and boundary values of harmonic functions.
- Spherical harmonics and a decomposition of L2 into Fourier-invariant subspaces.
- The Fourier transform of P(x) and P(x) is a harmonic polynomial of degree k, and the principal value distributions. If time permits, we will discuss the interpolation theory of operators in Lp spaces and Lorentz spaces.
(Shao 2015 )