Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces

Let be a pair of expansive dilations and phi: a"e (n) xa"e (m) x[0, a) -> [0, a) an anisotropic product Musielak-Orlicz function. In this article, we introduce the anisotropic product Musielak-Orlicz Hardy space via the anisotropic Lusin-area function and establish its atomic characterization, the -function characterization, the -function characterization and the discrete wavelet characterization via first giving out an anisotropic product Peetre inequality of Musielak-Orlicz type. Moreover, we prove that finite atomic decomposition norm on a dense subspace of is equivalent to the standard infinite atomic decomposition norm. As an application, we show that, for a given admissible triplet (), if T is a sublinear operator and maps all ()-atoms into uniformly bounded elements of some quasi-Banach spaces B, then T uniquely extends to a bounded sublinear operator from to B. Another application is that we obtain the boundedness of anisotropic product singular integral operators from to L (phi) (R (n) x R (m) ) and from to itself, whose kernels are adapted to the action of . The results of this article essentially extend the existing results for weighted product Hardy spaces on a"e (n) x a"e (m) and are new even for classical product Orlicz-Hardy spaces.