Resolvent bounds for jump generators

Abstract

The paper deals with jump generators with a convolution kernel. Assuming that the kernel decays either exponentially or polynomially, we prove a number of lower and upper bounds for the resolvent of such operators. In particular we focus on sharp estimates of the resolvent kernel for small values of the spectral parameter. We consider two applications of these results. First we obtain pointwise estimates for principal eigenfunction of jump generators perturbed by a compactly supported potential (so-called nonlocal Schrödinger operators). Then we consider the Cauchy problem for the corresponding inhomogeneous evolution equations and study the behaviourofitssolutions.