The Commutators Of Singular Integrals

The main purpose of this dissertation is to study the commutators of singular integral operators by some interpolation theorems and extrapolation theorems. In the first part, we consider the boundedness for commutators generated by BMO functions and singular integral operators with kernels of some weak regularity conditions, and we also give some endpoint estimates for commutators. In the second part, a new extrapolation theorem is established, which is suitable for studying commutators. The paper is divided into three chapters.The first chapter introduces maximal operators, the Calderón-Zygmund decomposition, BMO functions, commutators, weight functions, extrapolation theorems and other related concepts.Chapter two deals with the boundedness for commutators generated by BMO functions and singular integral operators with kernels of some weak regularity conditions. By a new interpolation theorem and some precise estimates from using John-Str(?)mberg sharp maximal operator, the boundedness for commutators can be proved as follows: by the boundedness of the singular integral operators, we can obtain two endpoint estimates for first order commutators, which by our new interpolation theorem gives the boundedness for those commutators; the boundedness for singular integral operators and first order commutators imply two endpoint estimates for second order commutators, and the boundedness for second order commutators then follows; repeating the argument will lead to the boundedness and two endpoint estimates for higher order commutators. It should be pointed out that the regularity conditions what we consider in this chapter does not lead to any weighted estimate with weights for singular integral operators, so the boundedness for commutators can not be deduced from the result of Alvarez, Bagby, Kurtz and Peréz.The third chapter establishes an extrapolation theorem for sublinear operators with general weights. The new extrapolation theorem is very useful to study weighted estimates for commutators. Using this extrapolation theorem, we point out the fact that the weighted estimate with general weights for the commutators of standard Calderón-Zygmund operators can be obtained by weak type endpoint estimates.