Boundedness Of Related Operators Of Schr(?)dinger Operator Semigroup And Related Problem

Austrian theoretical physicist Schr(?)dinger proposed that the Schr(?)dinger equation is the basic equation of quantum mechanics,which is a partial differential equation describing how the quantum state of the physical system evolves with time.The study of the solution of the Schr(?)dinger equation has attracted the attention of many scholars in the field of partial differential equations and mathematical physics.Among them,the Schr(?)dinger operator is extended to obtain the Laplace operator.This paper is mainly divided into three parts:(i)Firstly,by the regularity estimation of the integral kernel Kγ2,tL,γ1(·,·)related to L and the weighted estimation of the maximal function Φγ1,γ2L,*,we prove the weighted Lp-boundedness of the maximal function Φγ1,γ2L,*and the related commutator[b,Φγ1,γ2L,*]We also prove the compactness of the commutator[b,Φγ1,γ2L,*]by introducing a family of truncated operators,and we obtain that the weighted Lp-compactness of the commutator[b,Φγ1,γ2L,*]covers many maximal functions related to the Schr(?)dinger operator.(ii)Secondly,by the subordinative formula,we obtain the regularity of the fractional integral kernel UiL(·,·),i=1,2 related to L,and then obtain the bounded operator of the fractional integral operator IiL,i=1,2 from Lp(Rn)to Lq(Rn)through its regularity.Then we prove the weighted(Lp,Lq)-boundedness and compactness of the fractional integral operator IiL,i=1,2 and its commutator[b,IiL],i=1,2.(iii)Finally,we briefly introduce the characterization of Hardy spaces by fractional heat kernels associated with degenerate Schr(?)dinger operators.Then we establish a T1 criterion for the boundedness of operators on Campanato type spaces related with Schr(?)dinger operators in the setting of Heisenberg groups.We apply the T1 criterion to the operators generated by fractional heat semigroups and obtain their boundedness on Campanato type spaces.