Littlewood-paley Function Characterizations Of Anisotropic Weak Musielak-orlicz Hardy Spaces

Anisotropy means different characterizations in different directions of all or part of the physical or chemical properties of an object.For example,the sound velocity in a fibre shows a pronounced anisotropy.The most useful tool to study anisotropic prop-erty,in mathematics,may be a fairly general discrete group of dilations{A~j:j?Z},where A is a real n×n matrix with all its eigenvalues?satisfying|?|>1.Let?(x,t):R~n×[0,?)?[0,?)be an anisotropic growth function.The weak Musielak-Orlicz Hardy space W H_A~?(R~n)is defined to be the set of all tempered distributions such that their non-tangential grand maximal functions belong to the weak Musielak-Orlicz space W L~?(R~n).By using the atomic decomposition characterization of W H_A~?(R~n),this arti-cle is devoted to establishing Littlewood-Paley function characterizations of W H_A~?(R~n).Even for anisotropic weak Hardy space W H~p(R~n),namely,?(x,t):=t~pfor all(x,t)?R~n×[0,?)with p?(0,1],the Littlewood-Paley function characterizations mentioned above are new.