The structure of solution spaces for fractional-order operators, with gradient estimates

Abstract

The solution space of the homogeneous Dirichlet problem for the fractional Laplacian $(-\Delta {)}^a$ ($0<a<1$) or a pseudodifferential generalization $P$, on a bounded open set $\Omega \subset R^n$ with $C^{1+\tau }$-boundary, $$Pu=f\text{ on }\Omega ,u=0\text{ on }{R}^n\setminus \Omega ,$$ is analysed in detail. It is shown, both for solutions in Sobolev spaces of Bessel-potential type $H_q^t$ and in Hölder-Zygmund spaces $C_{\ast }^t$, that the solution space for $f$ of regularity $s\in [0,\tau -2a)$ is the direct sum of a component ${\dot{H}}_q^{2a+s}(\bar{\Omega })$ resp. ${\dot{C}}_{\ast }^{2a+s}(\bar{\Omega })$ with full regularity $2a+s$ and a component of the form $d^a$ times a lifting of boundary values by Poisson operators. Here $d(x)=dist(x,\partial \Omega )$. This extends to non-smooth problems results known in the $C^{\infty }$ setting. The knowledge is used to establish gradient estimates for $a>1/2$, e.g. estimating $d^{1-a+s}\nabla (u/d^a)$ in terms of norms of $f$ and $u$, both in $H_q^t$-spaces and $C_{\ast }^t$-spaces. This is entirely new in the case of Bessel-potential spaces; it extends previous results by Fall and Jarohs in Hölder spaces. A new tool is introduced: ${\dot{H}}_q^{s+t}(\bar{\Omega })\subset d^s{\dot{H}}_q^t(\bar{\Omega })$ holds for $s,t\ge 0$ with $s+t<\tau$.

Small corrections and clarifications, Corollary 1.4 added