The first nonzero eigenvalue of the weighted p-Laplacian on differential forms

TL;DR

This paper introduces a weighted p-Laplace operator on differential forms in metric measure spaces, providing sharp lower bounds for its first nonzero eigenvalue, extending previous estimates to a broader geometric context.

Contribution

It generalizes the p-Laplace operator to differential forms on metric measure spaces and derives sharp eigenvalue bounds, extending prior estimates to new settings.

Findings

Established sharp lower bounds for the first nonzero eigenvalue

Extended eigenvalue estimates to weighted p-Laplacian on differential forms

Generalized previous results from submanifolds to metric measure spaces

Abstract

We introduce the weighted p-Laplace operator acting on differential forms on a metric measure space, which is a natural generalization of the p-Laplace operator defined by Seto [32]. We obtain some sharp lower bounds of the first nonzero eigenvalue for the weighted p-Laplacian. Our results extend an estimate of Seto [32], as well as the eigenvalue estimates derived by Cui-Sun [8] for closed submanifolds.