In this talk, We will focus on the regularity theory of bounded, weak solutions for the nonlocal quasilinear equations of the form \begin{equation} \partial_t (|u|^{q-2}u) + (-\Delta)^s_p(u) = 0, \end{equation} where $p\in (1,\infty)$, $q\in (1,\infty)$ and $s \in (0,1)$.
Analogous H"older continuity result in the local case is known only in the purely singular case ${1<p<2, p<q}$, purely degenerate case ${2<p, q<p}$, scale invariant case ${p=q}$ and translation invariant case ${q=2,1<p<\infty}$. In the nonlocal setting, H"older regularity is known when the equation is either translation invariant ${q=2, 1<p<\infty}$ or scale invariant ${q=p, 1<p<\infty}$. For general ${p,q}$, the equation is neither translation invariant nor scale invariant.

In this talk, we will start with an overview of the regularity theory and discuss the H"older regularity result in the mixed singular - degenerate range $\max\{p,q,2\} < \min\{q + \frac{p-1}{1+\frac{n}{sp}}, 2 + \frac{p-1}{1+\frac{n}{sp}}\}$. We note that the analogous regularity in the local problem remains open.