Math 247A (Harmonic Analysis).

Lectures: MWF 11:00am-11:50am Online-Recorded.

Instructor: Monica Visan, MS 6167. Email address: visan@math.ucla.edu

Office Hours: by appointment.

Topics:

Basic properties of the Fourier transform in R^d.
Lorentz spaces and the Marcinkiewicz interpolation theorem.
Maximal and vector maximal inequalities.
Calderon-Zygmund singular integral operators.
Littlewood-Paley theory.
Fractional chain/product rules.
Stationary phase.
Rearrangement inequalities.

References: We will not be following any single source. The lectures are most strongly influenced by the following:
   Thomas H. Wolff, Lectures on harmonic analysis. American Mathematical Society, Providence, RI, 2003.
   Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton University Press, Princeton, NJ, 1993.
Other good references include:
   Javier Duoandikoetxea, Fourier Analysis. Translated and revised from the 1995 Spanish original by David Cruz-Uribe. Graduate Studies in Mathematics, 29. American Mathematical Society, Providence, RI, 2001.
   Loukas Grafakos, Classical Fourier analysis. Graduate Texts in Mathematics, 249. Springer, New York, NY, 2008.
   Loukas Grafakos, Modern Fourier analysis. Graduate Texts in Mathematics, 250. Springer, New York, NY, 2008.
   Yitzhak Katznelson, An introduction to harmonic analysis. Dover Publications, Inc., New York, NY, 1976.
   Elliott H. Lieb and Michael Loss, Analysis. Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001.
   Camil Muscalu and Wilhelm Schlag, Classical and multilinear harmonic analysis. Vol. I--II. Cambridge Studies in Advanced Mathematics, 137--8. Cambridge University Press, Cambridge, 2013.
   Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces. Princeton University Press, Princeton, NJ, 1971.
The following survey paper is a good reference on Lorentz spaces:
   Richard A. Hunt, On L(p,q) spaces. Enseignement Math. (2) 12 (1966), 249-276.

Grading: Assessment will be based on homework. You may find the homework problems here.