A sharpened Hausdorff-Young inequality

TL;DR

This paper improves the Hausdorff-Young inequality by providing a stability estimate that quantifies how near functions are to extremizers, with implications for understanding the structure of near-extremizing functions.

Contribution

It establishes a sharpened inequality with a quantitative stability term and proves precompactness of extremizing sequences using additive combinatorics.

Findings

Quantitative stability estimate for near-extremizers

Precompactness of extremizing sequences modulo symmetries

Explicit sharp constant in the stability term

Abstract

The Hausdorff-Young inequality for Euclidean space, in its sharp form due to Beckner, gives an upper bound for the Fourier transform in terms of Lebesgue space norms, with an optimal constant. The extremizers have been identified by Lieb to be the Gaussians. We establish an improved upper bound, for functions that nearly extremize the inequality, with a negative second term roughly proportional to the square of the distance to the set of extremizers. One formulation of this term comes with its own sharp constant. The main step is to show that any extremizing sequence is precompact, modulo the action of the group of natural symmetries of the inequality. This step relies on inverse theorems of additive combinatorial nature.