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

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 $r = 2$.