Persistence of Hölder continuity for non-local integro-differential equations

In this paper, we consider non-local integro-differential equations under certain natural assumptions on the kernel, and obtain persistence of Hölder continuity for their solutions. In other words, we prove that a solution stays in $C^\beta$ for all time if its initial data lies in $C^\beta$. This result has an application for a fully non-linear problem, which is used in the field of image processing. The proof is in the spirit of the paper [18] of Kiselev and Nazarov where they established Hölder continuity of the critical surface quasi-geostrophic (SQG) equation.