Mathematical Institute of the University of Bonn

RG Analysis and Partial Differential Equations

V5B7: Advanced Topics in Analysis - Geometric aspects of Harmonic Analysis (Winter Semester 2016/2017)

Dr. Diogo Oliveira e Silva

Instructor

Lectures:

• Tuesday, 14(c.t.) - 16, N0.003 (Neubau, Endenicher Allee 60)
• Thursday, 14(c.t.) - 16, N0.007 (Neubau, Endenicher Allee 60)

Overview:

This is a fast-paced course intended for Master and Ph.D. students which surveys a wide array of results in classical and modern Euclidean harmonic analysis that relate to structures with curvature. This course is intended as a follow-up to the "V4B5: Real and Harmonic Analysis" taught in the previous semester.

Topics:

1. Differentiation and integration: Hardy-Littlewood maximal function, Lebesgue differentiation theorem, rectifiable curves and the isoperimetric inequality
2. Hausdorff measure and fractals: Radon transforms, Besicovitch sets and regularity
3. Maximal functions and counterexamples: the multiplier problem for the ball
4. Maximal averages and oscillatory integrals: spherical averages and square functions
5. Introduction to the Kakeya problem and Bochner-Riesz multipliers
6. Fourier restriction theory: applications to PDE and number theory, connections with Radon, Kakeya, Bochner-Riesz and decoupling

Prerequisites:

In order to make the course accessible to students who did not attend V4B5, needed results from that course will be recalled but usually not reproved, and references will be provided. It is however expected that students are familiar with basic concepts from measure theory, complex and functional analysis.

Course credit and exam:

Students delivering at least one successful oral presentation during the course will be allowed to register for an oral exam. A list of possible topics for the oral presentations is here. Students are encouraged to find their own topics, subject to the instructor's approval. The oral exam will take place at the end of the lecture period and will typically last 30 minutes. Here is the exam syllabus.

References:

• L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, 1992.
• L. Grafakos, Classical Fourier Analysis. Second Edition. Springer, 2008.
• L. Grafakos, Modern Fourier Analysis. Second Edition. Springer, 2008.
• P. Mattila, Fourier Analysis and Hausdorff Dimension. Cambridge University Press, 2015.
• C. Muscalu and W. Schlag, Classical and Multilinear Harmonic Analysis. 2 volumes, Cambridge University Press, 2013.
• B. Simon, Harmonic analysis. A Comprehensive Course in Analysis, Part 3. American Mathematical Society, Providence, RI, 2015.
• C. D. Sogge, Fourier Integrals in Classical Analysis. Cambridge University Press, 1993.
• E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, 1971.
• E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Univ. Press, Princeton, NJ, 1993.
• E. M. Stein and R. Shakarchi, Functional analysis. Introduction to further topics in analysis. Princeton University Press, 2011.
• T. H. Wolff, Lectures on Harmonic Analysis. American Mathematical Society, 2003.