Modul:   MAT673  Seminar PDE and Mathematical Physics

On Two Extremal Problems For the Fourier Transform

Vortrag von Prof. Dr. Michael Christ

Datum: 29.01.15  Zeit: 18.10 - 19.10  Raum: Y27H35/36

One of the most fundamental facts about the Fourier transform is the Hausdorff-Young inequality, which states that for any locally compact Abelian group, the Fourier transform maps $L^p$ boundedly to $L^q$, where the two exponents are conjugate and $p \in [1,2]$. For Euclidean space, the optimal constant in this inequality was found by Babenko for $q$ an even integer, and by Beckner for general exponents. Lieb showed that all extremizers are Gaussian functions. This is a uniqueness theorem; these Gaussians form the orbit of a single function under the group of symmetries of the inequality.

We establish a stabler form of uniqueness for $1

Related problems concern the size of Fourier coefficients of indicator functions of sets. Here extremizers had been known only for special exponents. Some partial results will be announced.

Common to the analyses of both problems are precompactness theorems, which guarantee that extremizers exist, and that functions/sets that nearly extremize the inequalities must be close to exact extremizers. The remainder of the analysis is perturbative, and quite different in nature. After stating theorems concerning two extremal problems, I will outline the proofs of the relevant precompactness theorems, at the heart of which lie principles of additive combinatorics.

Slides