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

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, $$Pu=f \text{ on }\Omega ,\quad u=0 \text{ on }R^n\setminus \Omega ,$$ is analysed in detail. It is shown, both for solutions in Sobolev spaces of Bessel-potential type $H_q^t$ and in H\"older-Zygmund spaces $C_*^t$, 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. $\dot C_*^{2a+s}(\bar\Omega)$ 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 $C_*^t$-spaces. This is entirely new in the case of Bessel-potential spaces; it extends previous results by Fall and Jarohs in H\"older spaces. A new tool is introduced: $\dot H^{s+t}_q(\bar\Omega)\subset d^s\dot H^{t}_q(\bar\Omega)$ holds for $s,t\ge 0$ with $s+t<1+\tau$.