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

摘要

We introduce a new class of Hardy spaces H~(φ(·,·))(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, Str?mberg, and Torchinsky. Here, φ : R~n × [0,∞) → [0,∞) is a function such that φ(x, ·) is an Orlicz function and φ(·, t) is a Muckenhoupt A∞ weight. A function f belongs to H~(φ(·,·))(R~n) if and only if its maximal function f~? is so that x → φ(x, |f~?(x)|) is integrable. Such a space arises naturally for instance in the description of the product of functions in H~1(R~n) and BMO(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(R~n) characterized by Nakai and Yabuta can be seen as the dual of L~1(R~n) + H~(log)(R~n) where H~(log)(R~n) is the Hardy space of Musielak-Orlicz type related to the Musielak-Orlicz function θ(x, t) = t/(log(e+|x|)+log(e+t)). Furthermore, under additional assumption on φ(·, ·) we prove that if T is a sublinear operator and maps all atoms into uniformly bounded elements of a quasi-Banach space B, then T uniquely extends to a bounded sublinear operator from H~(φ(·,·))(R~n) to B. These results are new even for the classical Hardy-Orlicz spaces on R~n.