Mixed-norm Herz spaces and their applications in related Hardy spaces

Abstract

In this paper, the authors introduce a class of mixed-norm Herz spaces, [Formula: see text], which is a natural generalization of mixed-norm 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 [Formula: see text]. 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 [Formula: see text]. As applications, the authors develop various real-variable theory of Hardy spaces associated with [Formula: see text] by using the existing results of Hardy spaces associated with ball quasi-Banach function spaces. These results strongly depend on the duality of [Formula: see text] and the non-trivial constructions of auxiliary functions in the Riesz–Thorin interpolation theorem.