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

Abstract

In this article, the authors introduce a class of mixed-norm Herz spaces, ${\dot{E}}_{\vec{q}}^{\vec{\alpha },\vec{p}}({\mathbb{R}}^n)$, 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 ${\dot{E}}_{\vec{q}}^{\vec{\alpha },\vec{p}}({\mathbb{R}}^n)$. 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 ${\dot{E}}_{\vec{q}}^{\vec{\alpha },\vec{p}}({\mathbb{R}}^n)$. As applications, the authors develop various real-variable theory of Hardy spaces associated with ${\dot{E}}_{\vec{q}}^{\vec{\alpha },\vec{p}}({\mathbb{R}}^n)$ by using the existing results of Hardy spaces associated with ball quasi-Banach function spaces. These results strongly depend on the duality of ${\dot{E}}_{\vec{q}}^{\vec{\alpha },\vec{p}}({\mathbb{R}}^n)$ and the non-trivial constructions of auxiliary functions in the Riesz-Thorin interpolation theorem.

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