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 $${\mathbb{R}^{n}}$$ .