Let and be expansive dilations, respectively, on and . Let and be the class of product Muckenhoupt weights on for . Suppose that and with for Set and for all . For all , the anisotropic product Lusin area function of is defined as follows:
where and is an open and symmetric convex ellipsoid, respectively, in and for . The authors give the first main result in this paper. Let and . Then if and only and . Moreover, for all , , where denotes the set of all vanishing weakly at infinity.
Let and . The weighted anisotropic product Hardy space is defined by
where is a Schwartz function satisfying other extra conditions.
For the above Hardy space, the authors give an atomic decomposition. They show that if is an admissible triplet, then if and only if in , where and are -atoms. Furthermore, they prove that all finite linear combinations of -atoms is dense in
As an application, they prove that if is a sublinear operator and maps all -atoms into uniformly bounded elements of a quasi-Banach space , then uniquely extends to a bounded sublinear operator from to .
The results of this paper improve the existing results for weighted product Hardy spaces and are new in the unweighted anisotropic setting.