Mixed-Norm Herz Spaces and Their Applications in Related Hardy Spaces

TL;DR

This paper introduces mixed-norm Herz spaces, explores their duality and interpolation properties, and applies these to develop Hardy space theory and boundedness results for maximal operators.

Contribution

It generalizes mixed Lebesgue spaces to Herz spaces, establishes their duality and interpolation theorems, and applies these to Hardy space theory and maximal operator bounds.

Findings

Established dual spaces of mixed-norm Herz spaces.

Proved Riesz-Thorin interpolation theorem for these spaces.

Demonstrated boundedness of Hardy-Littlewood maximal operator.

Abstract

In this article, the authors introduce a class of mixed-norm Herz spaces, $\dot{E}^{\vec{\alpha},\vec{p}}_{\vec{q}}(\mathbb{R}^{n})$, which is a natural generalization of mixed Lebesgue spaces and some special cases of which naturally appear in the study of the summability of Fourier transforms on mixed-norm Lebesgue spaces. The authors also give their dual spaces and obtain the Riesz-Thorin interpolation theorem on $\dot{E}^{\vec{\alpha},\vec{p}}_{\vec{q}}(\mathbb{R}^{n})$. Applying these Riesz-Thorin interpolation theorem and using some ideas from the extrapolation theorem, the authors establish both the boundedness of the Hardy-Littlewood maximal operator and the Fefferman-Stein vector-valued maximal inequality on $\dot{E}^{\vec{\alpha},\vec{p}}_{\vec{q}}(\mathbb{R}^{n})$. As applications, the authors develop various real-variable theory of Hardy spaces associated with $\dot{E}^{\vec{\alpha},\vec{p}}_{\vec{q}}(\mathbb{R}^{n})$ by using the existing results of Hardy spaces associated with ball quasi-Banach function spaces. These results strongly depend on the duality of $\dot{E}^{\vec{\alpha},\vec{p}}_{\vec{q}}(\mathbb{R}^{n})$ and the non-trivial constructions of auxiliary functions in the Riesz-Thorin interpolation theorem.