Gradient H\"{o}lder regularity for nonlocal double phase equations

This paper is devoted to investigating the interior $C^{1, \alpha}$ regularity of viscosity solutions to the nonlocal double phase equations $$\int_{\mathbb{R}^d} \left(\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{d+sp}}+a(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{d+tq}}\right)\,dy=0,$$ where $2\le p\le q$, $s, t\in (0, 1)$ with $s\le t$, and $a(x, y)\ge0$. In the degenerate case, we solve the higher regularity issue raised by De Filippis-Palatucci [J. Differential Equations \textbf{267} (2019) 547--586]. By assuming the Lipschitz continuity of the modulating coefficient $a$, we are able to prove that the gradient of solution is H\"older continuous, provided the distance of $tq$ and $sp$ is suitably small. The core challenges consist in precisely characterizing the subtle interaction among the pointwise behaviour of the coefficient $a$, the growth exponents and the differentiability orders.