Gradient estimates for the fractional $p$-Poisson equation
Verena Bögelein, Frank Duzaar, Naian Liao, Kristian Moring
TL;DR
This work studies local weak solutions of the fractional $p$-Poisson equation $(-\Delta_p)^s u=f$ with $p>1$ and $s\in((p-1)/p,1)$. It proves Calderón–Zygmund type gradient estimates: for any $q>1$ and $f\in L^{\frac{qp}{p-1}}_{\mathrm{loc}}$, the gradient has higher integrability $\nabla u\in L^{qp}_{\mathrm{loc}}$, accompanied by a local, quantitative bound that also incorporates a nonlocal tail term. The method combines higher differentiability results for the homogeneous problem, precise comparison estimates between $u$ and the homogeneous Dirichlet problem solution, and a stopping-time/covering approach on super-level sets to obtain decay of the gradient. The results extend gradient regularity theory to the nonlinear, nonlocal fractional regime, linking the integrability of the inhomogeneity to the gradient through nonlocal tail effects and enabling further extensions to variable kernels and broader nonlocal operators. The paper also offers a Gagliardo semi-norm form of the estimate, highlighting versatility in nonlocal regularity frameworks.
Abstract
We consider local weak solutions to the fractional $p$-Poisson equation of order $s$, i.e. $\left( - Δ_p\right)^s u = f$. In the range $p>1$ and $s\in \big(\frac{p-1}{p},1\big)$ we prove Calderón & Zygmund type estimates at the gradient level. More precisely, we show for any $q>1$ that \begin{equation*} f\in L^{\frac{qp}{p-1}}_{\rm loc} \quad\Longrightarrow\quad \nabla u\in L^{qp}_{\rm loc}. \end{equation*} The qualitative result is accompanied by a local quantitative estimate.