Regularity in $L_p$ Sobolev spaces of solutions to fractional heat equations

This work contributes in two areas, with sharp results, to the current investigation of regularity of solutions of heat equations (*) $Pu+{\mathrm{\partial }}_{t}u=f$ on $\mathrm{\Omega }\times I$, where $P$ is a nonlocal operator, and $\mathrm{\Omega }\subset R^n$, $I\subset R$.

1. For $P$ a strongly elliptic pseudodifferential operator ($\psi$do) on $R^n$ of order $d\in R_{+}$, a symbol calculus on $R^{n+1}$ is introduced, that allows showing optimal regularity of solutions in the scale of anisotropic Bessel-potential spaces $H^{(s,s\mathrm{/}d)}$, globally over $R^n\times R$, and locally over $\mathrm{\Omega }\times I$, for $s\in R$, $1<p<\mathrm{\infty }$. Similar results hold in anisotropic Besov spaces $B^{(s,s\mathrm{/}d)}$.
2. Let $\mathrm{\Omega }$ be smooth bounded, and let $P$ equal $(-\mathrm{\Delta }{)}^a$ ($0<a<1$), or its generalizations to singular integral operators with regular kernels, that are infinitesimal generators of stable L'evy processes. With the Dirichlet condition $u=0$ on $R^n\setminus \mathrm{\Omega }$, the initial condition $u{\mathrm{\mid }}_{t=0}=0$, and $f\in L_p(\mathrm{\Omega }\times I)$, (*) has a unique solution $u\in L_p(I, H_p^{a(2a)}(\bar\Omega))$with$\partial_tu\in L_p(\Omega \times I)$. Here $H_p^{a(2a)}(\bar{\mathrm{\Omega }})$ equals ${\dot{H}}_p^{2a}(\bar{\mathrm{\Omega }})$ if $a<1\mathrm{/}p$, and is contained in ${\dot{H}}_p^{2a-\epsilon }(\bar{\mathrm{\Omega }})$ if $a=1\mathrm{/}p$, but contains nontrivial elements from $d^a{\bar{H}}_p^a(\mathrm{\Omega })$ if $a>1\mathrm{/}p$ (where $d(x)=\mathrm{dist}(x,\mathrm{\partial }\mathrm{\Omega })$). The interior regularity of $u$ is lifted when $f$ is more smooth.