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

We introduce a new class of Hardy spaces H(R), called Hardy spaces of Musielak-Orlicz type, which generalize the Hardy-Orlicz spaces of Janson and the weighted Hardy spaces of Garćıa-Cuerva, Strömberg, and Torchinsky. Here, φ : R × [0,∞) → [0,∞) is a function such that φ(x, ·) is an Orlicz function and φ(·, t) is a MuckenhouptA∞ weight. A function f belongs to H(R) if and only if its maximal function f∗ is so that x 7→ φ(x, |f∗(x)|) is integrable. Such a space arises naturally for instance in the description of the product of functions in H(R) and BMO(R) respectively (see [6]). 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) characterized by Nakai and Yabuta can be seen as the dual of L(R)+H (R) where H (R) 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(R) to B. These results are new even for the classical Hardy-Orlicz spaces on R.