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# New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators

Abstract : 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 $\mathbb R^n$.
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Cited literature [48 references]

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Submitted on : Friday, March 18, 2011 - 11:27:26 AM
Last modification on : Tuesday, October 12, 2021 - 5:20:12 PM
Long-term archiving on: : Sunday, June 19, 2011 - 2:38:06 AM

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• HAL Id : hal-00578059, version 1

### Citation

Luong Dang Ky. New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators. Integral Equations and Operator Theory, Springer Verlag, 2014, 36 p. ⟨hal-00578059⟩

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