# H\"older regularity for nonlocal double phase equations

## TL;DR

This paper establishes Hölder continuity for solutions to a class of nonlocal double phase equations, which switch between different fractional elliptic operators, extending regularity results to more general, possibly degenerate or singular, integro-differential equations.

## Contribution

It provides the first regularity results for nonlocal double phase problems, showing solutions are Hölder continuous under minimal assumptions on the modulating coefficient.

## Key findings

- Solutions are Hölder continuous with bounded modulating coefficient.
- Results apply to very general classes of measurable kernels.
- First regularity results for nonlocal double phase equations.

## Abstract

We prove some regularity estimates for viscosity solutions to a class of possible degenerate and singular integro-differential equations whose leading operator switches between two different types of fractional elliptic phases, according to the zero set of a modulating coefficient $a=a(\cdot,\cdot)$. The model case is driven by the following nonlocal double phase operator, $$ \int \!\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}\, {\rm d}y + \int \!a(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{n+tq}}\, {\rm d}y, $$ where $q\geq p$ and $a(\cdot,\cdot)\geqq 0$. Our results do also apply for inhomogeneous equations, for very general classes of measurable kernels. By simply assuming the boundedness of the modulating coefficient, we are able to prove that the solutions are H\"older continuous, whereas similar sharp results for the classical local case do require $a$ to be H\"older continuous. To our knowledge, this is the first (regularity) result for nonlocal double phase problems.