Spectral estimates and asymptotics for integral operators on singular sets

TL;DR

This paper derives sharp spectral estimates and asymptotic behavior for integral operators with singular measures and kernels exhibiting homogeneous asymptotics, especially on Lipschitz surfaces of positive codimension.

Contribution

It provides order sharp estimates for the counting function of such operators and asymptotics of eigenvalues in the self-adjoint case, considering geometric and kernel properties.

Findings

Established sharp estimates for the counting function.

Derived eigenvalue asymptotics for operators on Lipschitz surfaces.

Linked spectral behavior to measure geometry and kernel homogeneity.

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.