  When $P$ is the fractional Laplacian $(-\Delta )^a$, $0<a<1$, or a
pseudodifferential generalization thereof, the Dirichlet problem for the
associated heat equation over a smooth set $\Omega \subset{\Bbb R}^n$:
$r^+Pu(x,t)+\partial_tu(x,t)=f(x,t)$ on $\Omega \times \,]0,T[\,$, $u(x,t)=0$
for $x\notin\Omega $, $u(x,0)=0$, is known to be solvable in relatively
low-order Sobolev or H\"older spaces. We now show that in contrast with
differential operator cases, the regularity of $u$ in $x$ at $\partial\Omega $
when $f$ is very smooth cannot in general be improved beyond a certain
estimate. An improvement requires the vanishing of a Neumann boundary value.
--- There is a similar result for the Schr\"odinger Dirichlet problem
$r^+Pv(x)+Vv(x)=g(x)$ on $\Omega $, $v(x)=0$ for $x\notin \Omega $, with
$V(x)\in C^\infty $. The proofs involve a precise description, of interest in
itself, of the Dirichlet domains in terms of regular functions and functions
with a $dist(x,\partial\Omega)^a$ singularity.
