Discrete nonlinear H\"older--Brascamp--Lieb inequalities: a local approach

We study nonlinear H"older–Brascamp–Lieb inequalities over ${\mathbb{Z}}^n$. Our method embeds the integers into a suitable compactification, over which local H"older–Brascamp–Lieb theory may be applied (e.g heat flow). Our main result is a bound for multilinear H"older–Brascamp–Lieb functionals over suitable sublattices in ${\mathbb{Z}}^n$.