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

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 Garc'ia-Cuerva, Str"omberg, and Torchinsky. Here, $\varphi: \mathbb R^n\times [0,\infty)\to [0,\infty)$isafunctionsuchthat$\varphi(x,\cdot)$ is an Orlicz function and $\phi (\cdot ,t)$ is a Muckenhoupt $A_{\mathrm{\infty }}$ weight. A function $f$ belongs to $H^{\phi (\cdot ,\cdot )}({\mathbb{R}}^n)$ if and only if its maximal function $f^{\ast }$ is so that $x\mapsto \phi (x,\mathrm{\mid }{f}^{\ast }(x)\mathrm{\mid })$ 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 $\theta (x,t)={\displaystyle \frac{t}{\log (e+\mathrm{\mid }x\mathrm{\mid })+\log (e+t)}}$. Furthermore, under additional assumption on $\phi (\cdot ,\cdot )$ 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 $H^{\phi (\cdot ,\cdot )}({\mathbb{R}}^n)$ to $\mathcal B$\mathrm{.}TheseresultsarenewevenfortheclassicalHardy-Orliczspaceson$\mathbb R^n$.