Estimates For Weighted Hardy-Littlewood Operator

It is well known that Harmonic analysis has been one of the core research areas of modern mathematics, and is widely used in partial differential equations. The central research contents of Harmonic analysis are the theory of function spaces and operators with their applications. Hardy-Littlewood integrals are one of the most important op-erators in Harmonic analysis. Many mathematicians have researched the properties of Hardy-Littlewood operators and Weighted Hardy-Littlewood operators in different func-tion spaces, for example, LP, BMO, Hp. etc. On the basis of the previous achievements, this paper promote to discuss the boundedness of the Hardy-Littlewood and its commu-tators on some important function spaces. Our results will enrich the theory of Hardy-Littlewood. This paper is organized as follows:In section1, we will introduce the background and current research situation for Hardy-Littlewood operators and the Weighted Hardy-Littlewood operators and state my main work.In section2, We studied the boundedness of the weighted Hardy-Littlewood averages on λ-center Campanoto type spaces and weighted λ-center Morrey type spaces, and then obtained the responding operator norms.In section3, We introduce a class bilinear weighted Hardy operator and obtain that the operator is bounded from Lp1×Lp2to Lp with p,p1,p2∈[1,∞].In the last section, We prove that commutators are bounded on one sided dyadic Morrey spaces generating Hardy operators and one sided dyadic CMO functions.