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 New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators
 (2011-03-18)
 We introduce a new class of Hardy spaces $H^{\varphi(\cdot,\cdot)}(\mathbb R^n)$, called Hardy spaces of Musielak-Orlicz type, which generalize the Hardy-Orlicz spaces of Janson and the weighted Hardy spaces of García-Cuerva, Strömberg, and Torchinsky. Here, $\varphi: \mathbb R^n\times [0,\infty)\to [0,\infty)$ is a function such that $\varphi(x,\cdot)$ is an Orlicz function and $\varphi(\cdot,t)$ is a Muckenhoupt $A_\infty$ weight. A function $f$ belongs to $H^{\varphi(\cdot,\cdot)}(\mathbb R^n)$ if and only if its maximal function $f^*$ is so that $x\mapsto \varphi(x,|f^*(x)|)$ is integrable. Such a space arises naturally for instance in the description of the product of functions in $H^1(\mathbb R^n)$ and $BMO(\mathbb R^n)$ respectively (see \cite{BGK}). We characterize these spaces via the grand maximal function and establish their atomic decomposition. We characterize also their dual spaces. The class of pointwise multipliers for $BMO(\mathbb R^n)$ characterized by Nakai and Yabuta can be seen as the dual of $L^1(\mathbb R^n)+ H^{\rm log}(\mathbb R^n)$ where $H^{\rm log}(\mathbb R^n)$ is the Hardy space of Musielak-Orlicz type related to the Musielak-Orlicz function $\theta(x,t)=\displaystyle\frac{t}{\log(e+|x|)+ \log(e+t)}$. Furthermore, under additional assumption on $\varphi(\cdot,\cdot)$ we prove that if $T$ is a sublinear operator and maps all atoms into uniformly bounded elements of a quasi-Banach space $\mathcal B$, then $T$ uniquely extends to a bounded sublinear operator from $H^{\varphi(\cdot,\cdot)}(\mathbb R^n)$ to $\mathcal B$. These results are new even for the classical Hardy-Orlicz spaces on $\mathbb R^n$.
 1: Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO) Université d'Orléans – CNRS : UMR7349
 Subject : Mathematics/General MathematicsMathematics/Functional AnalysisMathematics/Classical Analysis and ODEs
 Keyword(s): Muckenhoupt weights – Musielak-Orlicz functions – BMO-multipliers – Hardy spaces – atomic decompositions – Hardy-Orlicz spaces – quasi-Banach spaces – sublinear operators
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 hal-00578059, version 1 http://hal.archives-ouvertes.fr/hal-00578059 oai:hal.archives-ouvertes.fr:hal-00578059 From: Luong Dang Ky <> Submitted on: Friday, 18 March 2011 11:27:26 Updated on: Friday, 18 March 2011 16:44:06