Integral Equations and Operator Theory

, Volume 78, Issue 1, pp 115–150 | Cite as

New Hardy Spaces of Musielak–Orlicz Type and Boundedness of Sublinear Operators

  • Luong Dang KyEmail author


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 \({\mathbb{R}^{n}}\).

Mathematics Subject Classification (2010)

Primary 42B35 Secondary 46E30 42B15 42B30 


Muckenhoupt weights Musielak–Orlicz functions BMO-multipliers hardy spaces atomic decompositions Hardy–Orlicz spaces quasi-Banach spaces sublinear operators 


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© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Quy NhonQuy NhonVietnam

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