V4B5: Real and Harmonic Analysis (Summer Semester 2016)

Dr. Diogo Oliveira e Silva
Instructor
Dr. Pavel Zorin-Kranich
Assistant

Lectures:

• Tuesday, 14(c.t.) - 16, Zeichensaal (Wegelerstr. 10)
• Thursday, 14(c.t.) - 16, Zeichensaal (Wegelerstr. 10)

Exercise classes:

• Thursday, 18(c.t.) - 20, 0.008 (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. The course divides into three sections: (I) Classical operators in Euclidean space; (II) Sharp inequalities in Fourier analysis; (III) Fourier restriction theory. In part I, we present a sample of classical results that can be loosely grouped into three categories: maximal averages, singular integrals and oscillatory integrals. In part II, we discuss some inequalities whose derivation is not too difficult if optimal constants are not demanded. The fact that these inequalities do not follow from simple convexity turns the determination of the corresponding sharp constants and the cases of equality into formidable problems, which we will address insofar as the theory is known. Finally, part III is meant as an introduction to the fascinating subject of Fourier restriction theory. Time permitting, I will try to highlight the interplay between restriction theory and some challenging research directions in modern harmonic analysis.

Topics:

1. The role of L^p spaces in harmonic analysis: Hilbert transform and Hardy-Littlewood maximal function, Hardy spaces and functions of bounded mean oscillation
2. The Fourier transform on the space of tempered distributions
3. Sharp inequalities: Hardy-Littlewood-Sobolev and Hausdorff-Young
4. Oscillatory integrals: van der Corput lemma, stationary phase, Fourier transform of measures supported on surfaces
5. Fourier restriction and Kakeya type problems: Tomas-Stein and Strichartz estimates, bilinear restriction theory, connections with Kakeya, Bochner-Riesz and Decoupling Theory.

Prerequisites:

Familiarity with basic concepts from measure theory, and occasionally with complex and functional analysis, will be useful.

Course credit and exam:

Students achieving at least half of the points of the homework assignments within the stipulated deadline will be allowed to register for an oral exam. The oral exam will take place on Monday, 2016-07-25 and will typically last 15-30 minutes. An overview of the topics that will be on the exam is in the exam syllabus.

References:

• W. Beckner, Inequalities in Fourier analysis. Ann. of Math. (2) 102 (1975), no.1, 159-182.
• E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann. of Math. (2) 118 (1983), no.2, 349-374.
• P. Mattila, Fourier Analysis and Hausdorff Dimension. Cambridge studies in advanced mathematics, 150. Cambridge University Press, Cambridge, 2015.
• B. Simon, Harmonic analysis. A Comprehensive Course in Analysis, Part 3. American Mathematical Society, Providence, RI, 2015.
• 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 Lectures in Analysis, 4. Princeton University Press, Princeton, NJ, 2011.
• T. Tao, Some recent progress on the restriction conjecture. Fourier analysis and convexity, 217-243, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2004.

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