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LITTLEWOOD–PALEY CHARACTERIZATIONS OF ANISOTROPIC WEAK MUSIELAK–ORLICZ HARDY SPACES

Part of: Harmonic analysis in several variables Linear function spaces and their duals

Published online by Cambridge University Press:  16 March 2018

BO LI ,
RUIRUI SUN ,
MINFENG LIAO  and
BAODE LI
BO LI
Affiliation:
College of Mathematics and System Science, Xinjiang University, Urumqi 830046, PR China email bli.math@outlook.com
RUIRUI SUN
Affiliation:
College of Mathematics and System Science, Xinjiang University, Urumqi 830046, PR China email 1565343765@qq.com
MINFENG LIAO
Affiliation:
College of Mathematics and System Science, Xinjiang University, Urumqi 830046, PR China email 2434042446@qq.com
BAODE LI*
Affiliation:
College of Mathematics and System Science, Xinjiang University, Urumqi 830046, PR China email 1246530557@qq.com
*
*Corresponding author.
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Abstract

Let $A$ be an expansive dilation on $\mathbb{R}^{n}$ and $\unicode[STIX]{x1D711}:\mathbb{R}^{n}\times [0,\infty )\rightarrow [0,\infty )$ an anisotropic growth function. In this article, the authors introduce the anisotropic weak Musielak–Orlicz Hardy space $\mathit{WH}_{A}^{\unicode[STIX]{x1D711}}(\mathbb{R}^{n})$ via the nontangential grand maximal function and then obtain its Littlewood–Paley characterizations in terms of the anisotropic Lusin-area function, $g$-function or $g_{\unicode[STIX]{x1D706}}^{\ast }$-function, respectively. All these characterizations for anisotropic weak Hardy spaces $\mathit{WH}_{A}^{p}(\mathbb{R}^{n})$ (namely, $\unicode[STIX]{x1D711}(x,t):=t^{p}$ for all $t\in [0,\infty )$ and $x\in \mathbb{R}^{n}$ with $p\in (0,1]$) are new. Moreover, the range of $\unicode[STIX]{x1D706}$ in the anisotropic $g_{\unicode[STIX]{x1D706}}^{\ast }$-function characterization of $\mathit{WH}_{A}^{\unicode[STIX]{x1D711}}(\mathbb{R}^{n})$ coincides with the best known range of the $g_{\unicode[STIX]{x1D706}}^{\ast }$-function characterization of classical Hardy space $H^{p}(\mathbb{R}^{n})$ or its weighted variants, where $p\in (0,1]$.

MSC classification

Primary: 42B25: Maximal functions, Littlewood-Paley theory 42B30: $H^p$-spaces 42B35: Function spaces arising in harmonic analysis 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Type
Article
Information
Nagoya Mathematical Journal , Volume 237 , March 2020 , pp. 39 - 78
Copyright
© 2018 Foundation Nagoya Mathematical Journal  

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Footnotes

This project is partially supported by the National Natural Science Foundation of China (Grant Nos. 11461065 and 11661075).

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