Uncertainty inequalities on groups and homogeneous spaces via isoperimetric inequalities

TL;DR

This paper establishes a family of $L^p$ uncertainty inequalities on various groups and homogeneous spaces, utilizing isoperimetric inequalities to derive key endpoint results in both smooth and discrete contexts.

Contribution

It introduces a novel approach connecting isoperimetric inequalities with $L^p$ uncertainty principles on general groups and spaces, including the discrete case.

Findings

Proves $L^p$ uncertainty inequalities on groups and homogeneous spaces.

Derives the $L^1$ endpoint from a general weak isoperimetric inequality.

Applies results to both smooth and discrete settings.

Abstract

We prove a family of $L^p$ uncertainty inequalities on fairly general groups and homogeneous spaces, both in the smooth and in the discrete setting. The crucial point is the proof of the $L^1$ endpoint, which is derived from a general weak isoperimetric inequality.