Subcritical Lp bounds on spectral clusters for Lipschitz metrics

TL;DR

This paper derives new asymptotic L^p bounds for spectral clusters of 2D Dirichlet forms with Lipschitz coefficients, especially for p>6, by analyzing energy dispersion in hyperbolic PDEs.

Contribution

It introduces novel L^p bounds for spectral clusters in Lipschitz metric settings, extending previous results to higher p ranges.

Findings

Established asymptotic L^p bounds for p>6

Bounded energy spread rates for hyperbolic solutions

Extended spectral cluster bounds to Lipschitz coefficients

Abstract

We establish asymptotic bounds on the L^p norms of spectrally localized functions in the case of two-dimensional Dirichlet forms with coefficients of Lipschitz regularity. These bounds are new for the range p>6. A key step in the proof is bounding the rate at which energy spreads for solutions to hyperbolic equations with Lipschitz coefficients.