Sharp L^p bounds on spectral clusters for Lipschitz metrics

TL;DR

This paper derives optimal L^p bounds for spectral clusters of elliptic operators with Lipschitz coefficients, improving understanding of eigenfunction concentration in various dimensions.

Contribution

It provides the first sharp L^p bounds for spectral clusters with Lipschitz metrics, including optimal bounds in two dimensions and partial results in higher dimensions.

Findings

Optimal L^p bounds in 2D for all p between 2 and infinity.

Logarithmic losses for certain p in 2D.

Best possible bounds in higher dimensions for specific p ranges.

Abstract

We establish L^p bounds on L^2 normalized spectral clusters for self-adjoint elliptic Dirichlet forms with Lipschitz coefficients. In two dimensions we obtain best possible bounds for all p between $2 and infinity, up to logarithmic losses for $6<p\leq 8$. In higher dimensions we obtain best possible bounds for a limited range of p.