Lecture, V4B5 - Real and Harmonic Analysis, SS 2013

Introduction

We will discuss aspects of Fourier analysis/Real analysis/Harmonic analysis in more than one dimension that have to do with the rotational symmetry of Euclidean space. This is in part model for more general curvature related phenomena. This area has seen multifaceted developments since approximately 1970, and we plan to follow some of these developments more or less in chronolgical order. This will lead us to a caleidoscope of different techniques: originally these problems were approached mainly with Fourier analytic methods (we discuss work by Fefferman on ball multilier and Bochner Riesz, and work by Stein et al on Restriction theorems), but over time geometric methods became more dominant (we discuss Cordoba, Bourgain, Wolff on Kakeya type problems). Further breakthroughs came through introduction of arithmetic combinatorial methods (work by Bourgain, also Katz, Tao etc) and finally algebraic methods (Dvir, Guth). These topics are fundamental in mathematics and play into many other areas, very prominently into PDE of dispersive equations (consider for example the wave equationin Euclidean space, which has rotational symmetries and can be solved with the aid of the Fourier transform.)
Preparation: All students entering or already in the Master program at Bonn should be well prepared for this course, other students with similar preparation are welcome. We assume a solid understanding of basic Analysis topics from the Bachelor Program (at Bonn for example), in particular basic knowledge about the Fourier transform, Lp spaces. Students unsure about their preparation are welcome to contact me with questions.

Course credit

If you have achieved at least half of the points of the homework assignments in timely fashion, you will be allowed to register for an oral exam on the course. Upon passing the exam you will obtain credit for the course (module).

Books

A survey of the material in lecture series form can be found for example in the bock by Thomas Wolff "Lectures on harmonic analysis" or in recent lecture notes by Michael Bateman, https://www.dpmms.cam.ac.uk/~mdb59/. Obviously the original articles mentioned above are a good source of the material, and we will provide references to these as the course proceeds.

Papers discussed so far

Fefferman, Charles: The multiplier problem for the ball. Ann. of Math. (2) 94 (1971), 330-336.
Fefferman, Charles: A note on spherical summation multipliers. Israel J. Math. 15 (1973), 44-52.
Cordoba, Antonio: The Kakeya maximal function and the spherical summation multipliers. Amer. J. Math. 99 (1977), no. 1, 1-22.
Wolff, Thomas. An improved bound for Kakeya type maximal functions. Rev. Mat. Iberoamericana 11 (1995), no. 3, 651-674.
Bourgain, J. On the dimension of Kakeya sets and related maximal inequalities. Geom. Funct. Anal. 9 (1999), no. 2, 256-282.
Katz, Nets Hawk; Tao, Terence Bounds on arithmetic projections, and applications to the Kakeya conjecture. Math. Res. Lett. 6 (1999), no. 5-6, 625-630.
Dvir, Zeev; On the size of Kakeya sets in finite fields. J. Amer. Math. Soc. 22 (2009), no. 4, 1093-1097.
Bennett, Jonathan; Carbery, Anthony; Tao, Terence; On the multilinear restriction and Kakeya conjectures. Acta Math. 196 (2006), no. 2, 261-302.
Tomas, Peter A.; A restriction theorem for the Fourier transform. Bull. Amer. Math. Soc. 81 (1975), 477-478.
(Not Required for exam: Carbery, Anthony; Valdimarsson, Stefán Ingi: The endpoint multilinear Kakeya theorem via the Borsuk-Ulam theorem. J. Funct. Anal. 264 (2013), no. 7, 1643-1663.