Weak Musielak-Orlicz Hardy spaces and applications

Let. : R n x [0,8). [0,8) satisfy that.(x, u), for any given x. R n, is an Orlicz function and.(u, t) is a Muckenhoupt A8(R n) weight uniformly in t. (0,8). In this article, the authors introduce the weak Musielak-Orlicz Hardy space WH.(R n) via the grand maximal function and then obtain its vertical or its nontangential maximal function characterizations. The authors also establish other real-variable characterizations of WH.(R n), respectively, in terms of the atom, the molecule, the Lusin area function, the Littlewood-Paley g-function or g*. -function. All these characterizations for weighted weak Hardy spaces WHp w (R n) (namely,.(x, t) := w(x) t p for all t. [0,8) and x. R n with p. (0, 1] and w. A8(R n)) are new and part of these characterizations even for weak Hardy spaces WHp(R n) (namely,.(x, t) := t p for all t. [0,8) and x. R n with p. (0, 1]) are also new. As an application, the boundedness of Calder ' on-Zygmund operators from H.(R n) to WH.(R n) in the critical case is presented.