Hölder regularity results for parabolic nonlocal double phase problems

In this article, we obtain completely new H\"older regularity results for the weak solutions to parabolic problems driven by the nonlocal double phase operator, align* L u (x): = P. V. ₑ₍ |u (x) -u (y) |^p-2 (u (x) -u (y) ) |x-y|^{N+ps₁}dy P. V. ₑ₍ a (x, y) |u (x) -u (y) |^q-2 (u (x) -u (y) ) |x-y|^{N+qs₂}dy, align* where 1<p q<, 0<s₂, s₁<1 and the modulating coefficient a (, ) is a non-negative bounded function. We also prove higher space-time H\"older continuity results for a particular subclass of the modulating coefficients. We further establish higher (global) H\"older continuity results for weak solutions to the stationary problems involving the operator L.