On Operator Estimates in the Homogenization of Lévy-Type Operators with Periodic Coefficients

Abstract. The paper deals with homogenization of self-adjoint operators in [Formula: see text] of the form [Formula: see text], where [Formula: see text], and [Formula: see text] is a small parameter. It is assumed that the function [Formula: see text] is [Formula: see text]-periodic in each variable, [Formula: see text] for all [Formula: see text] and [Formula: see text], and [Formula: see text]. Under these assumptions, we show that the resolvent [Formula: see text] converges, as [Formula: see text], in the operator norm in [Formula: see text] to the resolvent [Formula: see text] of the limit operator [Formula: see text] given by [Formula: see text], where [Formula: see text] is the mean value of [Formula: see text]. We also show that the operator norm of the discrepancy [Formula: see text] can be estimated by [Formula: see text] if [Formula: see text], by [Formula: see text] if [Formula: see text], and by [Formula: see text] if [Formula: see text].