A Berezin-Li-Yau type inequality for the fractional Laplacian on a bounded domain

TL;DR

This paper improves a Berezin-Li-Yau type inequality for the fractional Laplacian operator on bounded domains, providing sharper bounds relevant for spectral analysis of fractional differential operators.

Contribution

It presents a novel, sharper inequality for the fractional Laplacian on bounded domains, extending classical spectral bounds to fractional orders.

Findings

Established a new inequality for fractional Laplacian spectra

Extended classical bounds to fractional operators

Provided sharper spectral estimates for bounded domains

Abstract

An improvement to a Berezin-Li-Yau type inequality for $(-\Delta)^{\alpha/2}|_{\Omega},$ the fractional Laplacian operators restriced to a bounded domain $\Omega\subset \mathbb{R}^d$ for $\alpha\in(0,2],$ $d\ge 2,$ is proved.