Weyl asymptotics for fractional-order Dirichlet realizations in nonsmooth cases

Let $P$ be a symmetric $2a$-order classical strongly elliptic pseudodifferential operator with even symbol $p(x,\xi )$ on $R^n$ ($0<a<1$), for example a perturbation of $(-\mathrm{\Delta }{)}^a$. Let $\mathrm{\Omega }\subset R^n$ be bounded, and let $P_D$ be the Dirichlet realization in $L_2(\mathrm{\Omega })$ defined under the exterior condition $u=0$ in $R^n\setminus \mathrm{\Omega }$. When $p(x,\xi )$ and $\mathrm{\Omega }$ are $C^{\mathrm{\infty }}$, it is known that the eigenvalues ${\lambda }_j$ (ordered in a nondecreasing sequence for $j\to \mathrm{\infty }$) satisfy a Weyl asymptotic formula $${\lambda }_j(P_D)=C(P,\mathrm{\Omega })j^{2a\mathrm{/}n}+o(j^{2a\mathrm{/}n})\text{ for }j\to \mathrm{\infty },$$ with $C(P,\mathrm{\Omega })$ determined from the principal symbol of $P$. We now show that this result is valid for more general operators with a possibly nonsmooth $x$-dependence, over Lipschitz domains, and that it extends to $\tilde{P}=P+P^{\mathrm{\prime }}+P^{\mathrm{\prime }\mathrm{\prime }}$, where $P^{\mathrm{\prime }}$ is an operator of order $<\min \{2a,a+\frac{1}{2}\}$ with certain mapping properties, and $P^{\mathrm{\prime }\mathrm{\prime }}$ is bounded in $L_2(\mathrm{\Omega })$ (e.g. $P^{\mathrm{\prime }\mathrm{\prime }}=V(x)\in L_{\mathrm{\infty }}(\mathrm{\Omega })$). Also the regularity of eigenfunctions of $P_D$ is discussed.