The Stein-Weiss inequality in variable exponent Morrey spaces

TL;DR

The paper establishes a weighted Stein–Weiss inequality for fractional integrals in variable exponent Morrey spaces on bounded domains, extending known results from variable Lebesgue and classical Morrey settings. The authors develop a Morrey-space framework with variable exponents, prove the main weighted bound via a modular-to-norm approach and a careful decomposition of the fractional integral, and leverage maximal-operator bounds to control each component. As important applications, they derive Poincaré-type inequalities and a suite of Sobolev-type inequalities (Hardy–Sobolev, fractional Hardy–Sobolev, Gagliardo–Nirenberg) within these spaces. The results broaden the toolkit for analysis in nonstandard function spaces and pave the way for weighted integral inequalities in variable-exponent contexts, with open questions remaining for unbounded domains.

Abstract

In this paper we prove the Stein-Weiss inequality in variable exponent Morrey spaces over a bounded domain. Our work extends earlier results in the variable exponent Lebesgue and Morrey settings, and utilizes new proof techniques applicable to Morrey spaces. We build on the foundational paper by Almeida, Hasanov, and Samko, which introduced Morrey spaces of variable exponents. As an application of our main result, we prove Poincaré-type inequalities using the approach of a recent paper by the first and third authors.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8
• Remark 2.1
• Lemma 2.2