Boundedness Of Some Classes Of Operator

The main purpose of this dissertation is to study the boundedness for some classes of operators in harmonic analysis in two different settings.On one hand,we focus on study of the multiple weighted boundedness of commutators generated by multilinear Calderón-Zygmund potential type singular integral operators and B-MO functions.Furthermore,the two weighted norm inequalities of Calderón-Zygmund and potential type singular integral operators are obtained with A? condition.Our another goal is to study the boundedness of multilinear Calderón-Zygmund singular integral operators and their commutators with BMO functions in generalized Morrey spaces over Euclidean space Rn.On the other hand,we establish the norm inequalities for the Intrinsic square functions in Morrey spaces with constant exponent on Euclidean space Rn.This implies that the same norm inequalities for the Lusin area integrals,the Littlewood-Paley operators and the continu-ous square functions since they can be pointwise controlled by the Intrinsic square functions.We also study the boundedness of commutators generated by this class operator and BMO function in Morrey spaces.As application,we get the boundedness for convolution Calderón-Zygmund operators in Morrey spaces with constant exponent.In addition,we also consider the boundedness of Intrinsic square functions in two classes variable exponent Morrey spaces.Next,we study the boundedness of the fractional maximal operator and fractional integral operator on the variable exponent Morrey spaces over spaces of homogeneous type,(X,d,?).At last,we consider the sharp weighted estimate for multilinear maximal functions on spaces of homogeneous type.We define weight class over spaces of homogeneous type,we present if the multilinear Calderón-Zygmund operators are weighted bounded,then the multilinear commutators with BMO functions satisfy the same weighted inequalities.And applying the extrapolation,we can extend the exponents condition.