Gradient estimates for a class of anisotropic nonlocal operators

TL;DR

This paper extends gradient estimates to a broad class of anisotropic nonlocal operators combining classical and fractional derivatives, leading to new Hölder regularity results for solutions.

Contribution

It introduces a novel approach to estimate derivatives of solutions for anisotropic nonlocal equations with mixed classical and fractional operators.

Findings

Gradient of solutions is controlled linearly with a logarithmic correction.

Hölder estimates for solutions are established.

Results apply to a general class of anisotropic nonlocal operators.

Abstract

Using a classical technique introduced by Achi E. Brandt for elliptic equations, we study a general class of nonlocal equations obtained as a superposition of classical and fractional operators in different variables. We obtain that the increments of the derivative of the solution in the direction of a variable experiencing classical diffusion are controlled linearly, with a logarithmic correction. From this, we obtain H\"older estimates for the solution.