The Boundedness Of Some Operators On Non-doubling Measures

It's well known that the doubling measures plays a important role in Harmonic Analysis.Recently,people have found a more mild condition-growth condition, Let d ,n be some fixed integers with 1 ≤ n ≤ d. Assuming that μ is a Radon measure on Rd satisfying the following growth condition:People found that many results are also right in growth condition[l, 2, 3],and they have proved that the analysis on the non-homogeneous space made an important sense in solving the famous Painleve's problem [4]. The paper will obtain the boundness of some common operators under growth condition.In chapter one,Herz spaces and Herz-Morrey spaces with non-doubling measures are introduced. In 2004,Guo[5] proved the boundedness of some operators and commutators in Herz space with non-doubling measures when μ is a Ran-don measure satisfying the growth condition,the commutators is generated by Calderon-Zygmund operators(CZO) with RBMO(μ) functions.Authors find the boundedness of commutators generated by linear operators with size conditions and RBMO(μ) functions.The proof doesn't depend on the Maximal operators which Guo use.And the T of the commutators' size condition is general than CZO's kernel's condition.In chapter two,In [8],it points out the weak (1,1) and (p,p) type for vector-valued CZO where (p > 1),if it is bounded in L2(μ).The paper will extend the results on Herz spaces,in particular,we obtain the weighted inequlities for vector-valued CZO on L|x|βq(Rd) space,where - n < β < n(q - 1).In chapter three,In 2001,Garcia-Cuerva and Martell [9] proved the boundedness of fractional integral operators from Lp to Lq with non-doubling mea-sures.In 2004,SAWANO and TANAKA[10] obtained the boundedness of vector-valued fractional integral operators from Lp to Lq with non-doubling measures. This paper will extend the boundedness into Herz spaces with non-doubling measures, we give the estimate for fractional integral operators and its vector-valued extensions,specially,(q1, q2) estimate for power weighted fractional integral operators.