INTRODUCTION TO HARMONIC ANALYSIS
Description

This 16-hour course aims to cover fundamental topics in harmonic analysis, with a selection of subjects particularly useful for the study of partial differential equations. It is designed for graduate students (and possibly advanced undergraduate students), and attendees are expected to be familiar with basic concepts from real, complex, and functional analysis.

Topics

The plan is to cover the list of topics described below (possibly in a different order), but the content may be modified depending on time and feedback from the students.

(1) Convolutions and approximate identities:

• Young’s inequality;
• Lp -convergence of approximate identities;
• Hardy–Littlewood–Wiener theorem;
• Lebesgue differentiation theorem;
• Pointwise convergence of approximate identities.

(2) Real and Complex interpolation:

• Riesz–Thorin interpolation;
• Stein’s interpolation;
• Lorentz spaces;
• Marcinkiewicz interpolation.

(3) The Fourier transform:

• L1 + L2 theory;
• The Schwartz class and tempered distributions;
• Oscillatory integrals;
• Paley–Wiener–Schwartz theorem.

(4) Fractional Integration:

• Definition of the Riesz potentials and its Fourier properties;
• Hardy–Littlewood–Sobolev theorem;
• The fractional maximal function;
• Sobolev spaces and their embeddings.

(5) Singular integrals:

• The Hilbert and Riesz transforms;
• The Calder ́on–Zygmund decomposition;
• Strong and weak Lp boundedness of singular integrals;
• Introduction to Fourier multipliers.

References

[1] J. Duoandikoetxea, Fourier Analysis. Graduate Studies in Mathematics, 29, AMS, Providence, RI 2001.
[2] L. Grafakos, Classical Fourier Analysis. Graduate Texts in Mathematics, 249, Springer, New York, NY 2014.
[3] L. Grafakos, Classical Fourier Analysis. Graduate Texts in Mathematics, 250, Springer, New York, NY 2014.
[4] C. Muscalu, W. Schlag, Classical and Multilinear Harmonic Analysis. Cambridge University Press, New York, NY 2013.
[5] F. Linares, G. Ponce, Introduction to Nonlinear Dispersive Equations. Second edition. Universitext. Springer, NewYork, NY 2015.
[6] E. Stein, Harmonic Analysis. Princeton University Press, Princeton, NJ 1993.
[7] E. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, NJ1970.
[8] E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton,NJ 1971.
[9] T. Wolff, Lectures on Harmonic Analysis. American Mathematical Society, Providence, RI 2003.