Musielak-Orlicz Hardy Spaces

In this chapter, we first recall the notion of growth functions, establish some technical lemmas and introduce the Musielak-Orlicz Hardy space H-phi(R-n) which generalize the Orlicz-Hardy space of Janson and the weighted Hardy space of Garcia-Cuerva, Stromberg and Torchinsky. Here, phi : R-n x [0,infinity) -> [0,infinity) is a function such that phi(x, .) is an Orlicz function and phi(., t) is a Muckenhoupt A(infinity()R(n)) weight uniformly in t is an element of (0, infinity). A Schwartz distribution f belongs to H-phi(.,H-.) (R-n) if and only if its non-tangential grand maximal function f* is such that x -> phi(x, broken vertical bar f*(x)broken vertical bar) is integrable. Such a space arises naturally for instance in the description of the product of functions in (HRn))-R-1( and BMO(R-n), respectively. We characterize these spaces via the grand maximal function and establish their atomic decompositions. We also characterize their dual spaces. The class of pointwise multipliers for BMO(R-n) characterized by Nakai and Yabuta can be seen as the dual space of (LRn)-R-1() + H-phi(R-n), where phi(x,t) = t/log(e + broken vertical bar x broken vertical bar) + log(e + t), for all x is an element of R-n, for all t is an element of (0, infinity). (1.1)