On a geometric inequality related to fractional integration

TL;DR

This paper introduces a new geometric inequality related to fractional integration, explores its multilinear extensions, and identifies optimal constants and optimizers in Euclidean spaces, advancing the understanding of fractional integral inequalities.

Contribution

It develops a novel geometric inequality linked to fractional integration and extends it to multilinear forms, including the determination of best constants and optimizers.

Findings

Established new geometric inequalities involving bilinear and multilinear forms.

Identified best constants and optimizers for these inequalities in Euclidean spaces.

Extended previous work on fractional integration to more general multilinear settings.

Abstract

In this paper we consider a new kind of inequality related to fractional integration, motivated by Gressman's paper. Based on it we investigate its multilinear analogue inequalities. Combining with the Gressman's work on multilinear integral, we establish this new kind of geometric inequalities with bilinear form and multilinear form in more general settings. Moreover, in some cases we also find the best constants and optimisers for these geometric inequalities on Euclidean spaces with Lebesgue measure settings with $L^p$ bounds.