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 Bonami et al. in J Math Pure Appl 97:230–241, 2012). 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 ${\theta(x, t) = \frac{t}{{\rm log}(e + |x|) + {\rm log}(e + t)}}$. 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 ${\mathbb{R}^{n}}$.