Periodic homogenization of convolution type operators with irregular L\'{e}vy type tails

TL;DR

This paper proves homogenization results for nonlocal convolution operators with irregular Lévy tails, showing convergence to a fractional Laplacian-like operator under specific conditions.

Contribution

It introduces new homogenization results for nonlocal operators with Lévy-type tails and irregular kernels, extending previous theories to more general settings.

Findings

Proves resolvent convergence of nonlocal operators with Lévy tails.

Explicitly identifies the homogenized operator as comparable to fractional Laplacian.

Establishes conditions under which homogenization holds for irregular kernels.

Abstract

We establish the homogenization results for a class of nonlocal operators of convolution type with integrable jumping kernel $p$ multiplied by rapidly oscillating periodic or locally periodic coefficients. The associated measure $p(z)dz$ is assumed to belong to the domain of attraction of a symmetric $\alpha$-stable law. We also assume that $p$ satisfies a pointwise L\'evy type lower bound and an averaged annular upper bound for points bounded away from the origin, and that the local $L^1$ oscillation of $p$ decays faster at infinity than its local $L^1$-norm. Under these assumptions, we prove the resolvent convergence of the nonlocal operators and explicitly determine the corresponding homogenized nonlocal operator, which is shown to be comparable to the fractional Laplacian. The proof relies on compactness arguments and a refined analysis based on the annular integral upper bound and an $\varepsilon$-cube decomposition.