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

In a domain $Ω\subset \mathbb{R}^{\mathbf{N}}$ 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 $P=Vμ_Σ$ is a singular signed measure in $Ω$ concentrated on a Lipschitz surface $Σ$ of dimension $d<\mathbf{N}$, absolutely continuous with respect to the surface measure $μ_Σ$ on $Σ$. 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 $Ω$ the corresponding asymptotic coefficients add up.

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