Weak Hardy-type spaces associated with ball quasi-Banach function spaces I: Decompositions with applications to boundedness of Calderón-Zygmund operators

By Yangyang Zhang, Dachun Yang, Wen Yuan, Songbai Wang. Science China Mathematics, 2019

95 citations recorded by the source corpus.

Let X be a ball quasi-Banach function space on ℝn. In this article, we introduce the weak Hardy-type space W HX(ℝn), associated with X, via the radial maximal function. Assuming that the powered Hardy-Littlewood maximal operator satisfies some Fefferman-Stein vector-valued maximal inequality on X as well as it is bounded on both the weak ball quasi-Banach function space WX and the associated space, we then establish several real-variable characterizations of W HX (ℝn), respectively, in terms of various maximal functions, atoms and molecules. As an application, we obtain the boundedness of Calderón-Zygmund operators from the Hardy space HX (ℝn) to W HX (ℝn), which includes the critical case. All these results are of wide applications. Particularly, when X:=Mqp(ℝn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X: = M_q^p({\mathbb{R}^n})$$\end{document} (the Morrey space), X:=Lp→(ℝn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X: = {L^{\vec p}}({\mathbb{R}^n})$$\end{document} (the mixed-norm Lebesgue space) and X:=(EΦq)t(ℝn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X: = {(E_\Phi ^q)_t}({^n})$$\end{document} (the Orlicz-slice space), which are all ball quasi-Banach function spaces rather than quasi-Banach function spaces, all these results are even new. Due to the generality, more applications of these results are predictable.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/feb7b702d72c5eced9529f419e707bbb648c33cc

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Dachun Yang
2. Wen Yuan
3. Xianjie Yan
4. Sibei Yang
5. Ziyi He
6. Fan Wang
7. Jun Liu
8. Yangyang Zhang

In Mathematical Physics

1. Dachun Yang
2. Wen Yuan
3. Sibei Yang
4. Xianjie Yan
5. Fan Wang
6. Ziyi He
7. Yoshihiro Sawano
8. Jun Liu

Citations in the corpus

• Applications of Hardy Spaces Associated with Ball Quasi-Banach Function Spaces (2019, background): “However, weighted Lebesgue spaces, Herz spaces, Morrey spaces and Musielak–Orlicz spaces are not necessarily quasi-Banach function spaces (see, for instance, [49, 55, 59] for more details and examples).”
• Hardy spaces associated with ball quasi‐Banach function spaces on spaces of homogeneous type: Characterizations of maximal functions, decompositions, and dual spaces (2021, background): “Observe that the aforementioned several function spaces are not quasi-Banach function spaces (see, for instance, [64, 65, 74, 80]).”
• Intrinsic square function characterizations of Hardy spaces associated with ball quasi-Banach function spaces (2020, background): “introduced weak Hardy-type spaces associated with ball quasiBanach function spaces and developed a complete real-variable theory of these spaces; see [28,36].”