Topics in geometric Fourier analysis, spring 2012
Geometric Fourier analysis here refers to various topics in modern Fourier analysis with geometric, often geometric measure theoretic, nature. Such are multiplier problems (e.g., Fefferman's result that the characteristic function of the disc is L^p-multiplier if and only if p=2), restriction problems (L^p-inequalities for Fourier transforms on spheres and other surfaces), oscillatory integrals and decay of Fourier transforms of surface measures, spherical maximal functions and differentiation of integrals along spheres, Kakeya (or Besicovitch) sets (i.e., sets of measure zero containing a unit line segment in all directions) and their role in Fourier analysis. The lectures will be based on the books of de Guzman, Stein and Wolff mentioned below and possibly on articles by Stein, Bourgain, Wolff, Tao and others.
Good knowledge of Measure, Integration and Real Analysis, at least corresponding to the courses Mitta ja integraali and Reaalianalyysi I and II,
basic Fourier Analysis will be useful but not necessary.
Weeks 4-9 and 11-18, Tuesday 14-16 in room B322 and Friday 10-12 in room C123. The first lecture will be on Tuesday, January 24.
The last lecture will be on Friday, April 27, 2012
Easter holiday 5-11.4.
M. de Guzman, Real Variable Methods in Fourier Analysis, North-Holland, 1981
E.M. Stein, Harmonic Analysis, Princeton UP, 1993.
Th. W. Wolff, Lectures on Harmonic Analysis, AMS, 2003