In this article, we investigate the Hölder regularity of the fractional \(p\) -Laplace equation of the form \((-\Delta_p)^s u=f\) where \(p > 1, s\in (0, 1)\) and \(f\in L^\infty_{\rm loc}(\Omega)\) . Specifically, we prove that \(u\in C^{0, \gamma_\circ}_{\rm loc}(\Omega)\) for \(\gamma_\circ=\min\{1, \frac{sp}{p-1}\}\) , provided that \(\frac{sp}{p-1}\neq 1\) . 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.
