Weighted Estimates For Multilinear Singular Integral Operators And Their Commutators

Let(X,d,μ)be a space of homogeneous type in the sense of Coifman and Weiss with the Borel measure μ satisfying the doubling condition.Based on this space,we define a multilinear strongly singular Calder ón-Zygmund operator whose kernel does not need any size condition and is more singular near the diagonal than that of the standard multilinear Calder ón-Zygmund operator.For such operator,we establish its boundedness on product of weighted Lebesgue spaces by means of the pointwise estimate for the sharp maximal function.In addition,the endpoint estimates of the type L~∞(X)×· · ·× L~∞(X)→BMO(X)are also obtained.Moreover,we prove weighted boundedness results for multilinear commutators generated by multilinear strongly singular Calder ón-Zygmund operators and BMO functions.In the second half of this thesis,we focus on the two-weight estimates for commutators of multilinear Calder ón-zygmund operators in spaces of homogeneous type,and give the two-weight inequalities of general multilinear higher order commutators.The proof involves a pointwise sparse domination of multilinear commutators.These results contribute to the extension of multilinear singular integral operator theory in the Euclidean case to the context of spaces of homogeneous type.61 references.