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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 naturalgeneralization of mixed Lebesgue spaces and some special cases of whichnaturally appear in the study of the summability of Fourier transforms onmixed-norm Lebesgue spaces. The authors also give their dual spaces and obtainthe Riesz-Thorin interpolation theorem on$\dot{E}^{\vec{\alpha},\vec{p}}_{\vec{q}}(\mathbb{R}^{n})$. Applying theseRiesz-Thorin interpolation theorem and using some ideas from the extrapolationtheorem, the authors establish both the boundedness of the Hardy-Littlewoodmaximal 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 associatedwith $\dot{E}^{\vec{\alpha},\vec{p}}_{\vec{q}}(\mathbb{R}^{n})$ by using theexisting results of Hardy spaces associated with ball quasi-Banach functionspaces. These results strongly depend on the duality of$\dot{E}^{\vec{\alpha},\vec{p}}_{\vec{q}}(\mathbb{R}^{n})$ and the non-trivialconstructions of auxiliary functions in the Riesz-Thorin interpolation theorem.

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