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

TL;DR

This paper develops an $L_{p}$-theory for parabolic integro-differential equations with spatially dependent coefficients, broadening the class of Lévy measures covered, and establishing well-posedness in generalized Bessel potential spaces.

Contribution

It introduces a new $L_{p}$-theory for integro-differential operators with spatially varying coefficients and Lévy measures with O-regularly varying profiles, extending previous frameworks.

Findings

Established well-posedness in generalized Bessel potential spaces.

Covered classes of Lévy measures beyond the standard stable-like measures.

Provided regularity results for solutions 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 $dy/\left|y\right|^{d+\alpha}$.