The Boundedness Of Hardy Operators And Its Commutators

Since the singular integral operator theory is created by A.P.Calderon and A.Zygmund,the boundedness of various operators in different function spaces has been one of the central problems in the field of classical harmonic analysis.This thesis also focuses on this problem,focusing on the discussion of some singular integral operators,especially the boundedness of Hardy operators and its commutators in correlation function spaces.The first chapter of this paper,as an introduction,introduces the relevant research background,and reviews the related properties of the Hardy operator,non measure space,p-adic function space and the definition of product space,and prepare the full text for knowledge.In the second chapter,let ? be a Radon measure on Rd only satisfying the growth condition,we firstly introduced the definition of two class of fractional Hardy operators.Moreover,the boundedness of these operators on Herz space and Lebesgue spaces are obtained in this paper,and compare the influence of the measure to the two kinds of operators.It should be noted that the conclusion extend the case under Euclid measures.The next chapter is focus on the Higher-order commutators generated by the p-adic fractional Hardy operators and CMO(Qpn)functions,established their boundedness on Herz spaces.The results are corresponding to the conclusions in the classical Euclid space.The last chapter,we focus on the commutators generated by bilinear singular integral operators and bi-Lipschitz functions,and It is proved that the commutators are bounded from Hardy space H1(Rn×Rm)to Lebesgue space Lq(Rn×Rm),and q>1.