A refinement of the Berezin-Li-Yau type inequality for nonlocal elliptic operators

In this paper, we prove a refinement of the Berezin-Li-Yau type inequality for a wider class of nonlocal elliptic operators including the fractional Laplacians $-(-\Delta^{\sm/2})$ restricted to a bounded domain $D\subset\BR^n$ for $n\ge 2$ and $\sm\in (0,2]$, which is optimal when $\sigma=2$ in view of Weyl's asymptotic formula. In addition, we describe the Berezin-Li-Yau inequality for the Laplacian $\Delta$ as the limit case of our result as $\sm\to 2^-$.