Spectral asymptotics and estimates for matrix Birman-Schwinger operators with singular measures

We consider operators of the form $\mathbf{T}={\mathbf{A}}^{\ast }(V\mu )\mathbf{A}$ in ${\mathbb{R}}^{\mathbf{N}}$, where $\mathbf{A}$ is a pseudodifferential operator of order $-l$, $\mu$ is a compactly supported singular measure, order $s>0$ Ahlfors-regular, and $V$ is a weight function on the support of $\mu$. The scalar type operator $\mathbf{A}$ and the weight function $V$ are supposed to be $m\times m$ matrix valued. We establish Weyl type asymptotic formulas for singular numbers and eigenvalues of $\mathbf{T}$ for $\mu$ being the natural measure on a compact Lipschitz surface. For a general Ahlfors-regular measure $\mu$, we prove that the previously found upper spectral estimates are order sharp.