Boundedness Of The Hausdorff Operators On The Weighted Lebesgue Spaces

Here we consider the boundedness of the Hausdorff operators on the power weighted lebesgue spaces.At first,by Minkowski inequality we get a estimated result for general multi-variable Hausdorff operators.Then,we mainly study the 1-dimension case.We reformulate a 1-dimension Hausdorff operator as a convolution operator.By virtue of the weight theory of the singular integral,we obtain a dedicate result.The paper is divided into three chapters:The first chapter introduces the development history of Hausdorff operator and the main content of this thesis.In second chapter,we mainly review some preparatory knowledge,and gives some related theorems and lemmas.The third chapter is the proof of the main theorems.We transform the 1-dimension Hausdorff operator into a convolution operator.By virtue of the weighted theory of singular integrals,we complete the proof of the main theorems.