Get Sub-spaces Of Homogeneous Type Boundedness And Only Extension,

In 1971, Coifman and Weiss introduced the definition of homogeneous type spaces. A typical case is the Euclidean space, and the other example is the Heisenberg group, a class of non-Able group which exists substaintial distinguish from the Euclidean space. The commutative harmonic analysis has been studied very well in the Euclidean space, and then the harmonic analysis for non-Able groups naturally becomes extensively attention. The paper is concerned with properties of several class of operators in the homogeneous type space, which includes boundedness of integral operator and unique continuation proerties of a differential operator. The following four parts consist in the mian results of this paper.In the first part, by generalizing classical Morrey spaces in the Euclidean space, Boundedness of the weighted maximal operator,the Hardy-Littlewood maximal operator, the singular integral operator and the commutator of the singular integral operator is showed in the generalized weighted Morrey space, respectively.In the second part, we are interested in the vanishing generalized Morrey space, which is a subspace of the generalized Morrey space in homogeneous type spaces. Moreover, a sufficient condition of boundedness for the commutator of the fractional integral operator is established on the vanishing generalized Morrey space, based on estensions concerning the sharp maximal function and the fractional maximal function on classical Morrey spaces in the Euclidean space.The third part focus on the generalized weighted Campanato space in homogeneous type spaces. An estimate which plays a crucial role in investigating the boundedness of the Littlewood-paley operator in the Campanato space is established.In the last part, we pay attention to unique continuation properties for a class of differential operator in Heisenberg type group. The results obtained involve those in Heisenberg groups and quaternionic Heisenberg groups.