Weak Musielak-Orlicz Hardy Spaces

Abstract

In this chapter, we introduce the weak Musielak-Orlicz Hardy space \(W\!H^{\varphi }(\mathbb{R}^{n})\) via the grand maximal function and then obtain its vertical or its non-tangential maximal function characterizations. We also establish other real-variable characterizations of \(W\!H^{\varphi }(\mathbb{R}^{n})\), respectively, by means of the atom, the molecule, the Lusin area function, the Littlewood-Paley g-function or the g λ ∗-function. As an application, the boundedness of Calderon-Zygmund operators from \(H^{\varphi }(\mathbb{R}^{n})\) to \(W\!H^{\varphi }(\mathbb{R}^{n})\) in the critical case is presented.