Stein-Weiss-Adams inequality on Morrey spaces

TL;DR

This paper proves a generalized Stein-Weiss inequality on Morrey spaces within homogeneous groups, leading to new fractional inequalities on stratified groups and Euclidean spaces, expanding the theoretical framework of functional inequalities.

Contribution

It introduces Adams type Stein-Weiss inequalities on Morrey spaces for general homogeneous groups, including Euclidean spaces, with new fractional Hardy, Hardy-Sobolev, Rellich, and Gagliardo-Nirenberg inequalities.

Findings

Established Stein-Weiss inequality on Morrey spaces on homogeneous groups.

Derived fractional Hardy, Hardy-Sobolev, Rellich, and Gagliardo-Nirenberg inequalities.

Results are new even for Euclidean space .

Abstract

We establish Adams type Stein-Weiss inequality on global Morrey spaces on general homogeneous groups. Special properties of homogeneous norms and some boundedness results on global Morrey spaces play key roles in our proofs. As consequence, we obtain fractional Hardy, Hardy-Sobolev, Rellich and Gagliardo-Nirenberg inequalities on Morrey spaces on stratified groups. While the results are obtained in the setting of general homogeneous groups, they are new already for the Euclidean space $\mathbb{R}^{N}.$