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

Abstract

In a domain $\Omega \subset {\mathbb{R}}^{\mathbf{N}}$ we consider a selfadjoint operator $\mathbf{T}={\mathfrak{A}}^{\ast }P\mathfrak{A},$ 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.