Sharp Estimates For Weighted Hardy Operators On The Heisenberg Groups

As we all known, operators boundedness theory and its application in some space is to reconcile central to harmonic analysis of the field. The famous mathe-matician, American academy of sciences, Princeton University Stein [2] professor of the harmonic analysis of operator attributed to three categories, namely based on Hardy-type operators, represented by the average operator; to Hilbert transform as the basic form of the singular integral operator; in Fourier transform of the proto-type of the shock-type integral operators. Since the average operator to Hardy-type operators, represented by the theory since its inception, will be in harmonic anal-ysis in an important position, we will study the sharp estimate for the weighted problem situations Hardy-type integral operators on the Heisenberg group. This ar-ticle discusses the boundedness of weighted Hardy operators in Lp(Hn), BMO(Hn),Morrey spaces, the boundedness of multilinear weighted Hardy operators in Lp(Hn),BMO(Hn), Morrey spaces, and the estimate for weighted Hardy operators and mul-tilnear weighted operators in Heisenberg group. The main content of this paper is organized as follows:In section 1, fristly, we introduce the background and research status of the Hardy-type average operators, then present the definition and propertition of the Heisenberg group, thus, we establish the definition of the weighted Hardy operators in Heisenberg, futhermore, extended to the case of multilinear, Next we focus on several types of paper used in the c lassic space and necessary lemma, Finally, a brief description of the main research work of this article.Section 2, we find sufficient and neccessary conditions of the weighted Hardy operators are bounded in the Lp(HIn)?BMO(Hn) and Morrey spaces, and to deter-mine the corresponding operator norm.Section 3, we establish sufficient and neccessary conditions of the multilinear weighted Hardy operators are bounded in the products of Lp(Hn) and also Morrey spaces, in addition, to determine the corresponding operator norm.Section 4, firstly, we give the definition of the weighted Cesaro operator and multilinear weighted Cesaro operator in Heisenberg group, then, we introduce the Cesaro operator is the adjoint operators of weighted Hardy operators and focus on the propertition of it, this can also be estended to the multilinear Cesaro operator,finally, according to the contents of the section 2 and 3 gives the boundness theo-rems of the weighted Cesaro operator and the multilinear one in Heisenberg group's associated function space.