It is well-known that on a bounded domain $\Omega\subset\mathbb{R}^n$, by considering Dirichlet Problem: $$ \begin{array}{cc} \Delta u=0 & in\ \Omega\\ u=0 & on\ \partial\Omega \end{array} $$ we can obtain a sequencee of eigenvalues $0<\lambda_1\leq\lambda_2\cdots$ for Laplace operator.
Also classical theory tells us $\lambda_k\rightarrow\infty\ as\ k\rightarrow\infty$. Moreover, Weyl's law tells us the explicit blowup speed for this sequence of eigenvalue: $$ \dfrac{\lambda_k^{\frac{n}{2}}}{k}\rightarrow\dfrac{(2\pi)^n}{|\Omega|\cdot\omega_n}\ as\ k\rightarrow\infty. $$ Now when I consider for regional fractional Laplacian operator, defined as: $$ (-\Delta_{\Omega})^{\sigma}u(x):=p.v.\int_{\Omega}\dfrac{u(x)-u(y)}{|x-y|^{n+2\sigma}}\mathrm{d}y. $$ It seems there aren't any result considering the blowup speed for this operator. (Maybe some reslut using the language of probability and I didn't understand it properly).
I wonder is there exists any (non-local, because fractional is nonlocal operator and it seems we cannot extension to a higher dimension here) way that we can prove that the eigenvalues blowup slower than a polynomial degree (say $\lambda_k\leq Ck^{\alpha}$) and faster than another polynomial degree (say $\lambda_k\geq k^{\beta},\ \forall$ k large enough.)
Appreciate for the reading and thanks for any idea or reference on this type of regional fractional laplacian.