Lower Bound Estimates for The First Eigenvalue of The Weighted $p$-Laplacian on Smooth Metric Measure Spaces

TL;DR

This paper derives new lower bounds for the first eigenvalue of the weighted p-Laplacian on smooth metric measure spaces, extending classical estimates under various curvature conditions.

Contribution

It introduces new lower bound estimates for the eigenvalues of the weighted p-Laplacian using novel Bochner and Reilly formulas under curvature assumptions.

Findings

Established Escober--Lichnerowicz--Reilly type estimates.

Provided Li--Yau type lower bounds under curvature conditions.

Derived weighted p-Bochner and p-Reilly formulas as key tools.

Abstract

New lower bounds of the first nonzero eigenvalue of the weighted $p$-Laplacian are established on compact smooth metric measure spaces with or without boundaries. Under the assumption of positive lower bound for the $m$-Bakry--\'{E}mery Ricci curvature, the Escober--Lichnerowicz--Reilly type estimates are proved; under the assumption of nonnegative $\infty$-Bakry--\'{E}mery Ricci curvature and the $m$-Bakry--\'{E}mery Ricci curvature bounded from below by a non-positive constant, the Li--Yau type lower bound estimates are given. The weighted $p$-Bochner formula and the weighted $p$-Reilly formula are derived as the key tools for the establishment of the above results.