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As we all know, harmonic analysis and partial differential equations are closely linked. In In 1952, APCalderón and A.Zygmund famous about the origin of the singular integral operator be done the continuous elliptic equations. With the development of harmonic analysis, Calderón-Zygmund singular integral operators including convolution operator, but also including the Cauchy integral operator commutators. These operators are increasingly applied to practical problems. In the first part of this article, we let T be a sublinear operator. We already know that the Beurling (see [1]) and Herz (see [2]) has introduced the characteristics of some function space. These spaces called the Herz space from L p space to promote from. The people study Herz space for a long time, in these studies, Lu Shanzhen and characterizations on R n many important results on the Herz space applications and theory. In order to study the local properties of the second class of elliptic differential equations, Morrey (see [3]) introduced a class Morrey space (with M q λ (R n )). We define a class of space called Herz-Morrey space with the MK p, q α, λ (R n ) to represent two special cases of M q λ (R n ) and K q α, p (R n ). In the specific case, Zhao Xiang Qing establish sublinear operators on Herz-Morrey space bounded (see [4]). Donlin and characterizations of the nature of some sublinear operator of these results generalized to the weighted Herz space (see [5]). In this section, we will define the weighted Herz-Morrey space, and study sublinear operator bounded on the type of space. In the second part of this article, so M Ω rough Hardy-Littlewood maximal operator. And the first part, we were able to get the kind of operators on the weighted Herz-Morrey space bounded. In the third part of this article, so T is a linear or sublinear operator, b ∈ BMO (R n ) commutators [b, T] of T and b, [b, T] is defined as [b, T] f (x) = b (x) Tf (x)-T (bf) (x) by a well-known result of Coifman, Rochberg and Weiss (see [6]) that if T is a standard Calderón-Zygmund operator, b ∈ BMO (R n ), then [b, T] L p (R n bounded, p ∈ (1, ∞)),. Chanillo (see [7]) consider the Calderón-Zygmund operator fractional integral operator in place of such similar problems, Lu Shanzhen and characterizations (see [8]), these results are extended to the Herz space. Here, we see these results extended to weighted Herz-Morrey space.
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