Higher Hölder regularity for the fractional \(p\)-Laplace equation in the subquadratic case (Q6634474)

The non-local equation\N\[\N(-\Delta_p)^s u = f \ \text{ in }\Omega\N\]\Nis studied in the singular case \(1<p<2\), also called the ``subquadratic'' case. Here\N\[\N(-\Delta_p)^s u(x) \equiv 2 \int_{\mathbb{R}^N} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}} dy\N\]\Nis the fractional \(p\)-Laplace operator, the integral being understood as a principal value. (For \(p=2\) and \(s=1/2\) we have the operator \(\sqrt{-\Delta}\).)\N\NLet \(f\in L_{loc}^7(\Omega)\), where \(\Omega\) is a bounded domain in \(\mathbb{R^N}\). An \textit{explicit} local Hölder regularity exponent is provided for the weak solutions of the equation; \(1<p<2\), \(0<s<1\). The result \(u\in C_{loc}^\Theta(\Omega)\), \(\Theta=\min\{1, \frac{1}{p-1}(sp-\frac Nq)\}\) is given. The exponent \(\Theta\) is sharp, at least in a certain parameter range.\NIn contrast, no such explicit exponent seems to be known for \(p>2\).\N\NIn the homogeneous case \(f=0\), the proof is based on a sophistic version of the Moser iteration. Then the inhomogeneous case is reached through a perturbation of the homogeneous result.\N\NProposition 3.1 yields improved Besov regularity.