Best Constants For Several Classes Of Hausdorff Type Operators

Hausdorff operators play an important role in harmonic analysis. The study on Haus-dorff operators based on Euclidean spaces Rn is very meaningful and has been fully dis-cussed. In this paper, we consider two classes of questions on Hausdorff operators. Firstly, the underlying space Rn is replaced by the Heisenberg group which is denoted by Hn. We focus on sharp bounds for Hausdorff operators on some classical function spaces, such as Lebesgue spaces Lp(Hn)(1≤p≤∞), Morrey spaces Lq,λ(Hn), central Morrey spaces Bq,λ(Hn), and central BMO spaces CBMO(Hn). Secondly, we are interested to obtain best constants for Hausdorff type q-inequalities which is related on the quantum calculus.This paper is divided into four chapters.The first chapter is the preface. In this section, we mainly introduce some basic defini-tions of Hausdorff operators, the Heisenberg group and q-calculus.In Chapter 2, we calculate exactly best constants for n-dimensional Hausdorff operators on the Lebesgue spaces with the underlying space of Heisenberg group. We also extend these results in the multilinear setting. As applications, we obtain the best estimates for m-linear rectangle Hardy operators.In Chapter 3, we present the best estimates for two classes of n-dimensional Hausdorff operators on (central) Morrey spaces and central BMO spaces with the underlying space of Heisenberg group.In Chapter 4, we discuss the sharp bounds for the Hausdorff type n-inequalities. As applications,we also calculate best estimates for q-inequalities of several special operators.