Optimal Young's convolutions inequality and its reverse form on the hypercube

TL;DR

This paper proves optimal forms of Young's convolution inequality and its reverse on the hypercube, with applications to additive energies and sumsets, and explores conditions for the inequality in non-diagonal cases.

Contribution

It establishes sharp inequalities on the hypercube and analyzes the non-diagonal regime, providing new bounds and necessary conditions for these inequalities.

Findings

Sharp forms of Young's inequality on the hypercube for p=q

Bounds for additive energies and sumsets derived from the inequalities

Necessary conditions for the inequality to hold when p≠q

Abstract

We establish sharp forms of Young's convolution inequality and its reverse on the discrete hypercube $\{0,1\}^d$ in the diagonal case $p=q$. As applications, we derive bounds for additive energies and sumsets. We also investigate the non-diagonal regime $p\neq q$, providing necessary conditions for the inequality to hold, along with partial results in the case $r = 2$.