# 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.