Lorentz spaces with variable exponents

TL;DR

This paper introduces and studies Lorentz spaces with variable exponents, establishing their properties, embeddings, and interpolation relations, and addresses a question about Marcinkiewicz interpolation in variable Lebesgue spaces.

Contribution

It defines new Lorentz spaces with variable exponents, explores their fundamental properties, and clarifies their relation to existing spaces and interpolation theorems.

Findings

Spaces arise via real interpolation between variable Lebesgue and $L_im$ spaces.

Established embeddings and identities for these spaces.

Negative answer to Marcinkiewicz interpolation question in variable Lebesgue spaces.

Abstract

We introduce Lorentz spaces $L_{p(\cdot),q}(\R^n)$ and $L_{p(\cdot),q(\cdot)}(\R^n)$ with variable exponents. We prove several basic properties of these spaces including embeddings and the identity $L_{p(\cdot),p(\cdot)}(\R^n)=L_{p(\cdot)}(\R^n)$. We also show that these spaces arise through real interpolation between $L_{\p}(\R^n)$ and $L_\infty(\R^n)$. Furthermore, we answer in a negative way the question posed in Diening, H\"ast\"o, and Nekvinda (2004) whether the Marcinkiewicz interpolation theorem holds in the frame of Lebesgue spaces with variable integrability.