Hölder regularity of doubly nonlinear nonlocal quasilinear parabolic equations in some mixed singular-degenerate regime

We study local Hölder regularity of bounded, weak solutions for the nonlocal quasilinear equations of the form $$(\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u{)}_t+\text{P.V.}{\int }_{{\mathbb{R}}^n}\frac{\mathrm{\mid }u(x,t)-u(y,t){\mathrm{\mid }}^{p-2}(u(x,t)-u(y,t))}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+sp}}dy=0,$$ with $p\in (1,\mathrm{\infty })$, $q\in (1,\mathrm{\infty })$ and $s\in (0,1)$. Analogous Hölder continuity result in the local case is known 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 $\{q=2,1<p<\mathrm{\infty }\}$. In the nonlocal setting, Hölder regularity is known when the equation is either translation invariant $\{q=2,1<p<\mathrm{\infty }\}$ or scale invariant $\{q=p,1<p<\mathrm{\infty }\}$ or purely degenerate case $\{2<p,q<p\}$. Similar strategy can be used to obtain Hölder regularity in the purely singular case $\{1<p<2,p<q\}$. In this paper, we adapt several ideas developed over the past few years and combine it with a new intrinsic scaling to prove Hölder regularity in the mixed singular-degenerate range $\max \{p,q,2\}<\min \{q+\frac{p-1}{1+\frac{n}{sp}},2+\frac{p-1}{1+\frac{n}{sp}}\}$. The proof explicitly makes use of the nonlocal nature of the problem and as a consequence, our estimates are not stable at $s\to 0$. We note that the analogous regularity in the local problem remains open.