The sharp $p$-Poincar\'e inequality under the measure contraction property

TL;DR

This paper establishes optimal bounds for the $p$-spectral gap in metric measure spaces with measure contraction property, and characterizes cases of equality, advancing understanding of geometric analysis in these spaces.

Contribution

It provides sharp estimates and rigidity results for the $p$-spectral gap under the measure contraction property, a significant step in geometric analysis.

Findings

Sharp estimate on $p$-spectral gaps in metric measure spaces

Rigidity results for the equality case of the $p$-spectral gap

Advancement in understanding geometric inequalities under measure contraction

Abstract

We obtain sharp estimate on $p$-spectral gaps, or equivalently optimal constant in $p$-Poincar\'e inequalities, for metric measure spaces satisfying measure contraction property. We also prove the rigidity for the sharp $p$-spectral gap.