Hölder regularity of solutions to parabolic equations associated with fractional $p$-Laplacian type operators

In this paper, we study Hölder regularity of local weak super-solutions to the parabolic equation associated with fractional $p$-Laplacian type operators $${\mathrm{\partial }}_{t}u(t,x)-\text{p.v.}{\int }_{{\mathbb{R}}^d}\mathrm{\mid }u(t,y)-u(t,x){\mathrm{\mid }}^{p-2}(u(t,y)-u(t,x))J(t;x,y)dy=0,(t,x)\in \mathbb{R}\times {\mathbb{R}}^d,$$ where $\text{p.v.}$ denotes the principal value, $p\in (1,\mathrm{\infty })$ and the kernel $J(t;x,y)$ is comparable (uniformly with respect to $(t;x,y)\in \mathbb{R}\times {\mathbb{R}}^d\times {\mathbb{R}}^d$) to the fractional $p$-Laplacian kernel $\mathrm{\mid }x-y{\mathrm{\mid }}^{-d-sp}$ with $s\in (0,1)$. Our contribution extends the fundamental work of Kassmann and Weidner to the nonlinear case $p\mathrm{\ne }2$. In particular, we provide an explicit nonlocal parabolic tail condition in terms of an $L^{1+\epsilon }$-norm in the time variable (for a suitable $\epsilon >0$) for all $p\in (1,\mathrm{\infty })$, under which Hölder regularity holds. Our proof is based on the method of expansion of positivity, which is the key ingredient in the reduction-of-oscillation argument.