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 $-(-{\mathrm{\Delta }}^{\text{sm}\mathrm{/}2})$ restricted to a bounded domain $D\subset {\text{BR}}^n$ for $n\ge 2$ and $\text{sm}\in (0,2]$, 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 $\mathrm{\Delta }$ as the limit case of our result as $\text{sm}\to 2^{-}$.