Research Of Weighted Multilinear Hardy Operator And Sublinear Operator

The multi-linear operator theory is an important theory in the field of har-monic analysis. This theory was established by Coifman. Meyer and Meklntosh in the seventies of the20th century when they solved the Calderon’s guess. Since the multi-linear operator theory plays an important role in solving the bounded problem of some nonlinear operators, the research of this kind operators is a hot research direction for analysis researchers. It occupies an important position in the harmonic analysis theory. In addition, sublinear operator is another hot research objective for researchers in the past few years. For a sublinear operator, searching conditions that ensure the operator to be bounded in some classic spaces is a very meaningful and challenging task.Based on above research background, this dissertation concerns the bounded problems for multi-linear weighted Hardy operator (MWHO) and sublinear oper-ator on some classic spaces. First, the boundedness of MWHO on the product Lebesgue space and the product central Morrey space are respectively studied. Second, the boundedness of commutator that is generated by MWHO and central BMO function on the product central Morrey space is studied. A new description of the product central Morrey space is given. And then, the bounded estimates of MWHO on two Herz-type spaces (i.e., the product Herz spaces and the product Morrey-Herz spaces) are given respectively. After that, the central BMO estimate of the commutator generated by the MWHO on these two Herz-type spaces is inves-tigated. Finally, the boundedness of sublinear operators and its BMO commutator on weighted grand Morrey spaces are studied.The outline of the paper is organized as follows:In chapter1, the research status at home and abroad is reviewed, and the research objective and significance of this dissertation is presented.In chapter2, both the definition of the MWHO and the sufficient and necessary condition on weighted function are given. This ensures that the MWHO is bounded on the product Lebesgue spaces. In addition, the obtained results are further extended to the product central Morrey spaces. A sufficient condition on weighted function is presented. And it is also a necessary condition under some certain assumptions.In chapter3, the definition of commutator, which is generated by the MWHO and the central BMO function, is first given. And then, the boundedness of the commutator on the product central Morrey spaces is studied. A sufficient condition on weighted function is presented. And the necessity of the condition is also discussed.In chapter4, the bounded estimate of the MWHO on the product Herz space is first studied. And further, the boundedness of the MWHO on the product Herz-Morrey spaces is investigated. A sufficient condition on weighted function is presented. And the condition is also necessary under certain assumptions.In chapter5, the boundedness of the commutator generated by the MWHO and the central BMO functions on the product Herz space is firstly studied. And a sufficient condition on weighted function is given. After that, a constraint condition on weighted function is established. This condition ensures the BMO commutator to be bounded on the product Herz-Morrey space.In chapter6, the boundedness of sublinear operator on the weighted grand Morrey space is firstly studied. The obtained result points out that the sublinear operator is bounded on the weighted grand Morrey space when it satisfies certain conditions and it is bounded on the certain case of the weighted grand Morrey space. And further, the bounded problem for the commutator, which is generated by sublinear operator and BMO function, on the weighted grand Morrey space is discussed. The obtained result points out that the commutator is bounded on the weighted grand Morrey space when the sublinear operator satisfies certain conditions and the commutator is bounded on the certain case of the weighted grand Morrey space.