# Sharp Estimates for the Principal Eigenvalue of the p-Operator

## TL;DR

This paper establishes sharp lower bounds for the principal eigenvalue of a generalized p-operator on manifolds, extending classical results and characterizing cases of equality using curvature-dimension conditions.

## Contribution

It introduces a nonlinear p-operator generalizing the p-Laplacian and derives sharp eigenvalue estimates under curvature-dimension conditions using $ ext{Gamma}_2$-calculus.

## Key findings

- Proves a sharp estimate for the principal eigenvalue of $L_p$ under BE$(0,N)$.
- Characterizes equality cases when $L$ satisfies BE$(0,1)$.
- Provides a lower bound for the real part of eigenvalues of non-symmetric operators satisfying BE$(a, ext{infinity})$.

## Abstract

Given an elliptic diffusion operator $L$ defined on a compact and connected manifold (possibly with a convex boundary in a suitable sense) with an $L$-invariant measure $m$, we introduce the non-linear $p-$operator $L_p$, generalizing the notion of the $p-$Laplacian. Using techniques of the intrinsic $\Gamma_2$-calculus, we prove the sharp estimate $\lambda\geq (p-1)\pi_p^p/D^p$ for the principal eigenvalue of $L_p$ with Neumann boundary conditions under the assumption that $L$ satisfies the curvature-dimension condition BE$(0,N)$ for some $N\in[1,\infty)$. Here, $D$ denotes the intrinsic diameter of $L$. Equality holds if and only if $L$ satisfies BE$(0,1)$. We also derive the lower bound $\pi^2/D^2+a/2$ for the real part of the principal eigenvalue of a non-symmetric operator $L=\Delta_g+X\cdot\nabla$ satisfying $\operatorname{BE}(a,\infty)$.