Spectral Estimates and Asymptotics for Integral Operators on Singular Sets

Journal article, 2022

For singular numbers of integral operators of the formu(x)↦∫F1(X)K(XYX−Y)F2(Y)u(Y)u(dY) with a measure μ singular with respect to the Lebesgue measure in ℝN we obtain ordersharp estimates for the counting function. The kernel K(X, Y, Z) is assumed to be smooth in X, Y, Z ≠ 0 and to admit an asymptotic expansion in homogeneous functions in the Z variable as Z → 0. The order in the estimates is determined by the leading homogeneity order in the kernel and geometric properties of the measure μ and involves integral norms of the weight functions F1 and F2. In the selfadjoint case, we obtain asymptotics of the eigenvalues of this integral operator provided that μ is the surface measure on a Lipschitz surface of some positive codimension.