Local Hardy spaces of Musielak-Orlicz type and their applications

Abstract

Let φ: ℝ n × [0,∞) → [0,∞) be a function such that φ(x, ·) is an Orlicz function and $$\phi ( \cdot ,t) \in \mathbb{A}_\infty ^{loc} \left( {\mathbb{R}^n } \right)$$ (the class of local weights introduced by Rychkov). In this paper, the authors introduce a local Musielak-Orlicz Hardy space h φ(ℝ n ) by the local grand maximal function, and a local BMO-type space bmo φ (ℝ n ) which is further proved to be the dual space of h φ(ℝ n ). As an application, the authors prove that the class of pointwise multipliers for the local BMO-type space bmo φ (ℝ n ), characterized by Nakai and Yabuta, is just the dual of $$L^1 \left( {\mathbb{R}^n } \right) + h_{\Phi _0 } \left( {\mathbb{R}^n } \right)$$ , where ϕ is an increasing function on (0,∞) satisfying some additional growth conditions and Φ0 a Musielak-Orlicz function induced by ϕ. Characterizations of h φ (ℝ n ), including the atoms, the local vertical and the local nontangential maximal functions, are presented. Using the atomic characterization, the authors prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of h φ (ℝ n ), from which, the authors further deduce some criterions for the boundedness on h φ (ℝ n ) of some sublinear operators. Finally, the authors show that the local Riesz transforms and some pseudo-differential operators are bounded on h φ (ℝ n ).