Lipschitz regularity of fractional $p$-Laplacian

In this article, we investigate the Hölder regularity of the fractional $p$-Laplace equation of the form $(-{\mathrm{\Delta }}_p{)}^{s}u=f$ where $p>1,s\in (0,1)$ and $f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{\infty }}(\mathrm{\Omega })$. Specifically, we prove that $u\in C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{0,{\gamma }_{\circ }}(\mathrm{\Omega })$ for ${\gamma }_{\circ }=\min \{1,\frac{sp}{p-1}\}$, provided that $\frac{sp}{p-1}\mathrm{\ne }1$. In particular, it shows that $u$ is locally Lipschitz for $\frac{sp}{p-1}>1$. Moreover, we show that for $\frac{sp}{p-1}=1$, the solution is locally Lipschitz, provided that $f$ is locally Hölder continuous. Additionally, we discuss further regularity results for the fractional double-phase problems.