| The singular integral operator (SIO) plays an important role in harmonic analysis, and this theory has an important effect on the development of partial d-ifferential equation. The purpose of this paper is to extend the weak-type (1-1) inequality for Monge-Ampere singular integral operator from Rn to homoge-neous group. This paper mainly studies several questions about singular integral operator on the homogeneous group and promotes the related results. This pa-per consists of four chapters. In the first chapter, we state the background and the basic concepts about singular integral operator, and then give some conclu-sions that we need. In the second chapter, we firstly give the concepts of homo-geneous group and its related concepts, including section, pseudo-distance and quasi-metric, on the basis of which we give the doubling property and engulf-ing property on the homogeneous group. In the third chapter, after introducing Vitali type, Whitney type covering lemma and Calderon-Zygmund decomposi-tion theory on the homogeneous group, we prove the singular integral operator is bounded in Lμ2(G). In the fourth chapter, we study the SIO’s strong-type (p,q) and weak-type (p, q), finally we prove weak-type (1-1) inequality for singular integral operator (SIO) on the homogeneous group. |
