Multilinear embedding estimates for the fractional Laplacian

TL;DR

This paper introduces three new multilinear embedding inequalities for the fractional Laplacian, extending classical inequalities like Hardy-Littlewood-Sobolev, Gagliardo-Nirenberg, and Pitt's inequality, with sharp trace integral estimates.

Contribution

The paper presents novel multilinear embedding estimates for the fractional Laplacian expressed through trace integrals, extending key classical inequalities.

Findings

Derived three sharp multilinear inequalities for the fractional Laplacian.

Extended classical inequalities to multilinear trace integral forms.

Provided new tools for analysis involving the fractional Laplacian.

Abstract

Three novel multilinear embedding estimates for the fractional Laplacian are obtained in terms of trace integrals restricted to the diagonal. The resulting sharp inequalities may be viewed as extensions of the Hardy-Littlewood-Sobolev inequality, the Gagliardo-Nirenberg inequality and Pitt's inequality.