Function Spaces And Operators Associated With Anisotropic Dilation

Areal n×n matrix A with all its eigenvalues ? satisfying|?|>1 is called an anisotrop-ic dilation.L is a general multilinear operator generated by anisotropic Calderon-Zygmund operators.In this paper,we first obtain the boundedness of L from weighted Lebesgue s-paces to the unweighted anisotropic Hardy space;for anisotropic Hardy space HA1(Rn)and weighted anisotropic BMO space BMOAw(Rn).Then,an inclusion relationship is obtained:BMOAw(Rn)(?)(HA1(Rn))*.As an application,for the commutator[T,b]of anisotropic Calderon-Zygmund operator T and b in BMOAw(Rn),?[T,b](f)?Lwp(Rn)?C?b?BMOAw(Rn)?f?Lwp(Rn)is obtained.Furthermore,two equivalent characterizations of frac-tional integral commutators for anisotropic BMO function associated with dilation A are established.Finally,by using the local orthonormal basis associated with dilation A,we obtain the anisotropic Fourier series expansion of local L2 integrable function.All these results are still new even in the classical isotropic setting.