A Study Of The Boundedness Of Singular Integrals Related To Schr(?)dinger Operators On Heisenberg Groups

Some singular integral operators related to Schr(?)dinger operator have been concerned widely.It is one of the important contents in harmonic analysis to study the boundedness of singular integrals associated with Schr(?)dinger operator in the function spaces in recent years.Assume L=-?Hn+V be a Schr(?)dinger operator on Hn,where Hn is the Heisenberg group and ?Hn is the sub-Laplacian.V is a nonnegative potential belonging to the reverse Holder class BQ/2 and Q is the homogeneous dimension of Mn.The H_L^p(Hⁿ),Q/(Q+?)<p?1,for some ?>0,are defined Hardy spaces associte with L.In this paper,we review some basic knowledge of Heisenberg group,and give the definition of Hardy type spaces on Heisenberg group firstly.Then we give some basic results about auxiliary functions and we give the estimates of the kernels related to L on the Heisenberg group.After the estimates of the kernels related to L on the Heisenberg group are obtained,we establish a molecular characterization of H_L^p(Hⁿ).By the molecular decomposition theorem,we obtain the boundedness of Riesz transform VHn,gL-1/2 and L^i?(Hⁿ)on the Hardy spaces H_L^p(Hⁿ).