Hal will be stopped for maintenance from friday on june 10 at 4pm until monday june 13 at 9am. More information
Skip to Main content Skip to Navigation
Journal articles

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 $\mathbb R^n$.
Complete list of metadata

Cited literature [48 references]  Display  Hide  Download

Contributor : Luong Dang Ky Connect in order to contact the contributor
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


Files produced by the author(s)


  • HAL Id : hal-00578059, version 1



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⟩



Record views


Files downloads