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.

Necessary and Sufficient Conditions for Commutator of the Calderón–Zygmund Operator on Mixed-Norm Herz-Slice Spaces

“…Let (ψ) = lim i→∞ i,i (ψ) and (ψ) be a linear bounded functional on ( KE −β,s u , v ) t (R n ). By using ( [23] Theorem 3.1), we see that there exist φ ∈ ( KE…”

“…) is ball quasi-Banach function space (see [23], Proposition 3.2). We immediately get the following result, which means that T f is well-defined on ( KE…”

“…In 2022, Lu, Zhou and Wang [22] introduced Herz-slice spaces. In 2023, we defined mixed-norm Herz-slice space ( KE β,s u, v ) t (R n ) in [23], and the theory of the Hardy-Littlewood maximal operator is given in this space. Given f ∈ L 1 loc (R n ) and x ∈ R n , the Hardy-Littlewood maximal function M( f ) is defined by…”

“…the supremum is taken over all cubes Q in R n . For a deeper discussion about mixed-norm spaces, we can see [26,[31][32][33] and so on.…”

A new Gammarus species from Xinjiang Uygur Autonomous Region (China) with a key to Xinjiang freshwater gammarids (Crustacea, Amphipoda, Gammaridae)

“…In 2022, Zhang and Zhou [26] introduced the mixed-norm amalgam space (L u , L v )(R n ), and the boundedness of the Hardy-Littlewood maximal operators on the mixed-norm amalgam space (L u , L s )(R n ) was studied, we can see [18]. For more meticulous research about mixed-norm spaces, we refer to [1,15,27] and so on.…”