The Regularity Of The Fractional Calculus Of The Schr(?)dinger Operator

The Heisenberg group is a non-exchange group,and the study of harmonic analysis problems on the Heisenberg group is an extension of Euclidean space.Heisenberg group is widely used in many fields,such as quantum mechanics,partial differential equations,number theory and so on.Many scholars have studied some related operators on the Heisenberg group,such as the Schr(?)dinger operator and Hardy type operator.In this paper,we study the common Schr(?)dinger operators.Let L=-?Hn+V be a Schr(?)dinger operator on the Heisenberg group Hn,where?Hn is the sub-Laplacian,the nonnegative potential V belongs to the reverse Holder class.The main contents of this paper include the following two aspects:(1)Using the subordination formula,to obtain the regularity estimate for the fractional derivatives of the Poisson semigroup associated with L;(2)To study the space associated with L on the Heisenberg group,for example,BMO type space.We get a descripation of the function space(BMOL?(Hn)space),and to study the relationship between the function spaces,such as,the dual of HLQ/LQ+?(Hn)is the space BMOL?(Hn).