Michael Christ University of California, Berkeley (UC Berkeley) Mathematics
For any locally compact Abelian group, the Hausdorff-Young inequality states that the Fourier transform maps $L^p$ to $L^q$, where the two exponents are conjugate and $p \in [1,2]$. For Euclidean space, the optimal constant in the inequality was found 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
achieves the optimal constant in the inequality, then $f$ must be close in norm to a Gaussian. (ii) There is a quantitative bound involving the square of the distance to the nearest Gaussian.
The qualitative form (i) can be equivalently formulated as a precompactness theorem in the style of the calculus of variations. Form (ii) is a strengthening of the inequality.
The proof relies on ingredients taken from from additive combinatorics. Central to the reasoning are arithmetic progressions of arbitrarily high rank.