  The fractional Laplacian $(-\Delta )^a$, $a\in(0,1)$, and its generalizations
to variable-coefficient $2a$-order pseudodifferential operators $P$, are
studied in $L_q$-Sobolev spaces of Bessel-potential type $H^s_q$. For a bounded
open set $\Omega \subset \mathbb R^n$, consider the homogeneous Dirichlet
problem: $Pu =f$ in $\Omega $, $u=0$ in $ \mathbb R^n\setminus\Omega $. We find
the regularity of solutions and determine the exact Dirichlet domain
$D_{a,s,q}$ (the space of solutions $u$ with $f\in H_q^s(\overline\Omega )$) in
cases where $\Omega $ has limited smoothness $C^{1+\tau }$, for $2a<\tau
<\infty $, $0\le s<\tau -2a$. Earlier, the regularity and Dirichlet domains
were determined for smooth $\Omega$ by the second author, and the regularity
was found in low-order H\"older spaces for $\tau =1$ by Ros-Oton and Serra. The
$H_q^s$-results obtained now when $\tau <\infty $ are new, even for $(-\Delta
)^a$. In detail, the spaces $D_{a,s,q}$ are identified as $a$-transmission
spaces $H_q^{a(s+2a)}(\overline\Omega )$, exhibiting estimates in terms of
$\operatorname{dist}(x,\partial\Omega )^a$ near the boundary. The result has
required a new development of methods to handle nonsmooth coordinate changes
for pseudodifferential operators, which have not been available before; this
constitutes another main contribution of the paper.
