Intrinsic Structures of Certain Musielak-Orlicz Hardy Spaces

For any p. (0, 1], let H Phi p (Rn) be the Musielak-Orlicz Hardy space associated with the Musielak-Orlicz growth function Phi p, defined by setting, for any x. Rn and t. [0, 8), Phi p(x, t) := {t/log (e + t) + [t(1 + vertical bar x vertical bar)n]1-p when n(1/p-1) is not an element of N boolean OR {0}, {t/log (e + t) + [t(1 + vertical bar x vertical bar)n]1-p when n(1/p-1) is an element of N boolean OR {0}, which is the sharp target space of the bilinear decomposition of the product of the Hardy space H p(Rn) and its dual. Moreover, H Phi 1 (Rn) is the prototype appearing in the real-variable theory of general Musielak-Orlicz Hardy spaces. In this article, the authors find a new structure of the space H Phi p (Rn) by showing that, for any p. (0, 1], H Phi p (Rn) = Hf0 (Rn) + H p Wp (Rn) and, for any p. (0, 1), H Phi p (Rn) = H1(Rn)+ H p Wp (Rn), where H1(Rn) denotes the classical real Hardy space, Hf0 (Rn) the Orlicz-Hardy space associated with the Orlicz function f0(t) := t/log(e + t) for any t. [0,8), and H p Wp (Rn) theweighted Hardy space associated with certain weight function Wp(x) that is comparable to Phi p(x, 1) for any x. Rn. As an application, the authors further establish an interpolation theorem of quasilinear operators based on this new structure.