Spectral estimates and asymptotics for integral operators on singular sets

Abstract

For singular numbers of integral operators of the form $u(x)\mapsto \int F_1(X)K(X,Y,X-Y)F_2(Y)u(Y)\mu(dY),$ with measure $\mu$ singular with respect to the Lebesgue measure in $\mathbb{R}^\mathbf{N}$, order sharp estimates for the counting function are established. The kernel $K(X,Y,Z)$ is supposed to be smooth in $X,Y$ and in $Z\ne 0$ and to admit an asymptotic expansion in homogeneous functions in $Z$ variable as $Z\to 0.$ The order in estimates is determined by the leading homogeneity order in the kernel and geometric properties of the measure $\mu$ and involves integral norms of the weight functions $F_1,F_2$. For the case of the measure $\mu$ being the surface measure for a Lipschitz surface of some positive codimension $\mathfrak{d},$ in the self-adjoint case, the asymptotics of eigenvalues of this integral operator is found.