Local Musielak-Orlicz Hardy Spaces

Abstract

In this chapter, we introduce a local Musielak-Orlicz Hardy space \(h^{\varphi }(\mathbb{R}^{n})\) by the local grand maximal function, and a local BMO-type space \(\mathrm{bmo}^{\varphi }(\mathbb{R}^{n})\) which is further proved to be the dual space of \(h^{\varphi }(\mathbb{R}^{n})\). As an application, we prove that the class of pointwise multipliers for the local BMO-type space \(\mathrm{bmo}^{\phi }(\mathbb{R}^{n})\), characterized by E. Nakai and K. Yabuta, is just the dual of \(L^{1}(\mathbb{R}^{n}) + h^{\Phi _{0}}(\mathbb{R}^{n})\), where ϕ is an increasing function on (0, ∞) satisfying some additional growth conditions and \(\Phi _{0}\) a Musielak-Orlicz function induced by ϕ. Characterizations of \(h^{\varphi }(\mathbb{R}^{n})\), including the atoms, the local vertical or the local non-tangential maximal functions, are presented. Using the atomic characterization, we prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of \(h^{\varphi }(\mathbb{R}^{n})\), from which, we further deduce some criterions for the boundedness on \(h^{\varphi }(\mathbb{R}^{n})\) of some sublinear operators. Finally, we show that the local Riesz transforms and some pseudo-differential operators are bounded on \(h^{\varphi }(\mathbb{R}^{n})\).