Improved H\"older regularity of fractional $(p,q)$-Poisson equation with regular data

Abstract

We prove a quantitative H\"{o}lder continuity result for viscosity solutions to the equation $$ (-\Delta_p)^{s}u(x) + {\rm PV} \int_{\mathbb{R}^n} |u(x)-u(x+z)|^{q-2}(u(x)-u(x+z))\frac{\xi(x,z)}{|z|^{n+ tq}} dz=f \quad \text{in}\; B_2, $$ where $t, s\in (0, 1), 1<p\leq q, tq\leq sp$ and $\xi\geq 0$. Specifically, we show that if $\xi$ is $\alpha$-H\"{o}lder continuous and $f$ is $\beta$-H\"{o}lder continuous then any viscosity solution is locally $\gamma$-H\"{o}lder continuous for any $\gamma<\gamma_\circ $, where \[ \gamma_\circ=\left\{\begin{array}{lll} \min\{1, \frac{sp+\alpha\wedge\beta}{p-1}, \frac{sp}{p-2}\} & \text{for}\; p>2, \\ \min\{1, \frac{sp+\alpha\wedge\beta}{p-1}\} & \text{for}\; p\in (1, 2]. \end{array} \right. \] Moreover, if $\min\{\frac{sp+\alpha\wedge\beta}{p-1}, \frac{sp}{p-2}\}>1$ when $p>2$, or $\frac{sp+\alpha\wedge\beta}{p-1}>1$ when $p\in (1, 2]$, the solution is locally Lipschitz. This extends the result of [20] to the case of H\"{o}lder continuous modulating coefficients. Additionally, due to the equivalence between viscosity and weak solutions, our result provides a local Lipschitz estimate for weak solutions of $(-\Delta_p)^{s}u(x)=0$ provided either $p\in (1, 2]$ or $sp>p-2$ when $p>2$, thereby improving recent works [9, 10, 24].