On $L_{p}$- theory for integro-differential operators with spatially dependent coefficients

Abstract

The parabolic integro-differential Cauchy problem with spatially dependent coefficients is considered in generalized Bessel potential spaces where smoothness is defined by L\'evy measures with O-regularly varying profile. The coefficients are assumed to be bounded and H\"older continuous in the spatial variable. Our results can cover interesting classes of L\'evy measures that go beyond those comparable to $dy/\left|y\right|^{d+\alpha}$.