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

We prove a quantitative Hölder continuity result for viscosity solutions to the equation $$(-{\mathrm{\Delta }}_p{)}^{s}u(x)+\mathrm{P}\mathrm{V}{\int }_{{\mathbb{R}}^n}\mathrm{\mid }u(x)-u(x+z){\mathrm{\mid }}^{q-2}(u(x)-u(x+z))\frac{\xi (x,z)}{\mathrm{\mid }z{\mathrm{\mid }}^{n+tq}}dz=f\text{in}{B}_2,$$ 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 $${\gamma }_{\circ }=\{\begin{array}{ll}\min \{1,\frac{sp+\alpha \wedge \beta }{p-1},\frac{sp}{p-2}\}& \text{for}p>2,\\ \min \{1,\frac{sp+\alpha \wedge \beta }{p-1}\}& \text{for}p\in (1,2]\mathrm{.}\end{array}$$ 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 $(-{\mathrm{\Delta }}_p{)}^{s}u(x)=0$ provided either $p\in (1,2]$ or $sp>p-2$ when $p>2$, thereby improving recent works [9, 10, 24].