Some Estimates Of Multilinear Operators

Multi-linear operator theory has been developing in the last decade. In this dissertation, we focus on the boundedness of bilinear pseudo-differential operators, the boundedness of multilinear Calderon-Zygmund operators and their commuta-tors, and the bounded properties of the commutators generated by Marcinkiewicz integral.By researching the bilinear pseudo-differential operators with prohibited sym-bols, we show the bounded properties of the operators on Morrey type Sobolev spaces and Morrey type Besov spaces. Furthermore, the boundedness of these operators on Herz type Sobolev spaces and on Herz type Besov spaces are also obtained.We also study the the boundedness of the multilinear Calderon-Zygmund op-erators and their commutators when the underlying measure is substituted by a nonnegative Radon measure with some growth condition. When the correspond-ing measure only satisfies some growth conditions we present the properties of the multilinear Calderon-Zygmund operators and their commutators on Morrey spaces. In additional, we demonstrate the boundedness of commutators generated by the multilinear Calderon-Zygmund operators and BMO functions on Gener-alized Morrey Spaces and Hardy spaces, and the weighted boundedness of these commutators by using sharp maximal function.Finally, we focus on the behaviors of commutators associated to the Marcinki-ewicz integrals and the Lipschitz functions on Lebesgue spaces and Hardy spaces under the condition of non-doubling measure. Also we propose the boundedness of the higher order commutators on some spaces, respectively.The dissertation is supported by the National Natural Science Foundation of China (No.10771054and No.10861010) and Natural Science Fundation of Xinjiang University(YX080106).