Eigenvalues of the Birman-Schwinger operator for singular measures: The noncritical case

Abstract

For operators of the form T=TA,P=A⁎PA with pseudodifferential operator A of negative order −l in a domain in RN, 2l≠N, and a singular measure P, an estimate of eigenvalues is found with an order depending on the dimensional characteristics of the measure P and the coefficient depending on an integral norm of the density of P with respect to the Hausdorff measure of an appropriate dimension. These estimates are used to establish asymptotic formulas for the eigenvalues of T for the case when P is supported on a Lipschitz surface of some codimension and on certain sets of a more complicated structure. In one of applications, a version of the CLR estimate for singular measures is proved.