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

Limited regularity of solutions to fractional heat and Schrödinger equations