Strengthened Fractional Sobolev Type Inequalities in Besov Spaces

TL;DR

This paper strengthens fractional Sobolev inequalities in Besov spaces using Lorentz and capacitary Lorentz spaces, establishing their equivalence to Hardy and iso-capacitary inequalities and exploring related embeddings and characterizations.

Contribution

It introduces new strengthened inequalities in Besov spaces via Lorentz and capacitary Lorentz spaces, linking them to Hardy and iso-capacitary inequalities.

Findings

Established equivalence between Sobolev, Hardy, and iso-capacitary inequalities in Besov spaces.

Proved embedding results of capacitary Lorentz spaces into classical Lorentz spaces.

Characterized general Sobolev type inequalities using a new fractional perimeter.

Abstract

The purpose of this article is twofold. The first is to strengthen fractional Sobolev type inequalities in Besov spaces via the classical Lorentz space. In doing so, we show that the Sobolev inequality in Besov spaces is equivalent to the fractional Hardy inequality and the iso-capacitary type inequality. Secondly, we will strengthen fractional Sobolev type inequalities in Besov spaces via capacitary Lorentz spaces associated with Besov capacities. For this purpose, we first study the embedding of the associated capacitary Lorentz space to the classical Lorentz space. Then, the embedding of the Besov space to the capacitary Lorentz space is established. Meanwhile, we show that these embeddings are closely related to the iso-capacitary type inequalities in terms of a new-introduced fractional $(\beta, p, q)$-perimeter. Moreover, characterizations of more general Sobolev type inequalities in Besov spaces have also been established.