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

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.