Isoperimetric weights and generalized uncertainty inequalities in metric measure spaces

TL;DR

This paper extends $L^{1}$ uncertainty inequalities to metric measure spaces by introducing isoperimetric weights, establishing new localized inequalities, and broadening the scope to rearrangement invariant spaces.

Contribution

It introduces isoperimetric weights and localized Poincaré inequalities to generalize uncertainty inequalities in metric measure spaces.

Findings

Established $L^{p}$ uncertainty inequalities for $1 \\leq p < \\infty$

Characterized isoperimetric weights via Marcinkiewicz spaces

Extended uncertainty inequalities to rearrangement invariant spaces

Abstract

We extend the recent $L^{1}$ uncertainty inequalities obtained by Dall'ara-Trevisan to the metric setting. For this purpose we introduce a new class of weights, named *isoperimetric weights*, for which the growth of the measure of their level sets $\mu(\{w\leq r\})$ can be controlled by $rI(r),$ where $I$ is the isoperimetric profile of the ambient metric space. We use isoperimetric weights, new *localized Poincar\'e inequalities*, and interpolation, to prove $L^{p},1\leq p<\infty,$ uncertainty inequalities on metric measure spaces. We give an alternate characterization of the class of isoperimetric weights in terms of Marcinkiewicz spaces, which combined with the sharp Sobolev inequalities we had obtained in an earlier paper, and interpolation of weighted norm inequalities, give new uncertainty inequalities in the context of rearrangement invariant spaces.