Eigenvalue estimates and asymptotics for weighted pseudodifferential operators with singular measures in the critical case

TL;DR

This paper derives eigenvalue estimates and asymptotic formulas for a class of weighted pseudodifferential operators with singular measures supported on Lipschitz surfaces, revealing dimension-independent behavior and additive effects for multiple surfaces.

Contribution

It provides new eigenvalue estimates and asymptotics for operators with singular measures on surfaces, showing independence from surface dimension and additive effects for multiple measures.

Findings

Eigenvalue estimates are dimension-independent.

Asymptotics sum over multiple surfaces with different dimensions.

Results extend to operators with multiple singular measures.

Abstract

In a domain $\Omega\subset \mathbb{R}^{\mathbf{N}}$ we consider a selfadjoint operator $\mathbf{T}=\mathfrak{A}^*P\mathfrak{A} ,$ where $\mathfrak{A}$ is a pseudodifferential operator of order $-l=-\mathbf{N}/2$ and $P=V\mu_{\Sigma}$ is a singular signed measure in $\Omega$ concentrated on a Lipschitz surface $\Sigma$ of dimension $d<\mathbf{N}$, absolutely continuous with respect to the surface measure $\mu_{\Sigma}$ on $\Sigma$. We establish eigenvalue estimates and asymptotics for this operator. It turns out that the order of these estimates and asymptotics is independent of the dimension $d$ of the surface. If there are several surfaces, possibly, of different dimensions, as well as an absolute continuous measure on $\Omega$ the corresponding asymptotic coefficients add up.