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

TL;DR

This paper refines the Berezin-Li-Yau inequality for a broad class of nonlocal elliptic operators, including fractional Laplacians, providing optimal bounds and connecting to classical Laplacian results as a limit case.

Contribution

It introduces a sharper inequality for nonlocal elliptic operators, extending previous results and establishing the classical case as a limit of the new bounds.

Findings

Refined Berezin-Li-Yau inequality for fractional Laplacians

Optimal bounds consistent with Weyl's asymptotic formula

Connection to classical Laplacian inequality as a limit

Abstract

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 $\sm\to 2^-$.