On the Continuity of Some Integral Operators and Commutators

Abstract

In this thesis, we consider the problem of boundedness of di erent Hausdor operators and their commutators on function spaces. However, from the point of view of applica- tions and complexity, the major part of this thesis is devoted to prove the boundedness of commutator operators generated by Hausdor operator with locally integrable functions on di erent function spaces. The function spaces include Lp function spaces, Morrey type function spaces, Herz type function spaces and Triebel Lizorkin type function spaces with the Euclidean space Rn or the Heisenberg group Hn as underlying spaces. We also con- sider the weighted boundedness of Hausdor operator and commutators on these function spaces de ned on Rn or Hn: Since Hardy integral operators are special cases of Hausdor operator, therefore, we also include some results regarding weighted boundedness of com- mutators of Hardy operators on Morrey type spaces. Almost all work has been published in well reputed mathematical science journals and is enlisted in the reference section of this thesis.