On two extremal problems for the Fourier transform

Monday, October 6, 2014 -
3:15pm to 4:30pm
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 $1<p<2$: If a function $f$ nearly achieves the optimal constant in the inequality, then $f$ must be close in norm to a Gaussian, with a quantitative bound involving the square of the distance to the nearest Gaussian. Related problems concern the size of Fourier coefficients of indicator functions of sets. Here less seems to be known. Some partial results will be announced. Common to the analyses of both problems are recompactness theorems, which guarantee that extremizers exist, and that functions/sets that nearly extremize the inequalities must be close to exact extremizers. 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. 
Michael Christ
UC Berkeley
Event Location: 
Fine Hall 314