Improved Hölder regularity of fractional $(p,q)$-Poisson equation with regular data

Abstract

We prove a quantitative Hölder continuity result for viscosity solutions to the equation $$(-{\Delta }_p{)}^{s}u(x)+\mathrm{PV}{\int }_{{\mathbb{R}}^n}|u(x)-u(x+z){|}^{q-2}(u(x)-u(x+z))\frac{\xi (x,z)}{|z{|}^{n+tq}}dz=f\text{in}{B}_2,$$ where $t,s\in (0,1),1<p\le q,tq\le sp$ and $\xi \ge 0$. Specifically, we show that if $\xi$ is $\alpha$-Hölder continuous and $f$ is $\beta$-Hölder continuous then any viscosity solution is locally $\gamma$-Hölder continuous for any $\gamma <{\gamma }_{\circ }$, where \[ γ_\circ=\left\{\begin{array}{lll} \min\{1, \frac{sp+α\wedgeβ}{p-1}, \frac{sp}{p-2}\} & \text{for}\; p>2, \\ \min\{1, \frac{sp+α\wedgeβ}{p-1}\} & \text{for}\; p\in (1, 2]. \end{array} \right. \] 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 Hölder 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].

25 pages, 1 figure