Boundedness Of Commutators Associated With Schr(?)dinger Operator And Fourier Multiplier

Commutators theory is more and more thorough in the past several dozens year in research and the development, more and more widespread, Specially the singular integral and the BMO function generated commutators has provided the powerful tool for the research variable coefficient differential equation. Found under the quantum system from Austrian physicist Schrodinger matter satisfied equation of motion -Schrodinger equation. The people discovered has the non- negative potential Schrodinger -â–³+V (x) to study certain ellipses operator to be extremely useful, recent year, but discusses the the nature of Schrodinger operator regarding the potential function different selection ,more and more to be valued.In this paper, we study the boundedness of commutators generated by Riesz transform associated with Schrodinger operator and BMO type functions , and by Schrodinger operator associated semigroup and Lipschitz functions. We also concerned with commutators generated by linear operators and Fourier multiplier operators. This paper is organized as follows.In chapter 1, we simply introduce the development and background in the study of the boundedness of Schrodinger operators, and of Herz type spaces and Besov spaces.In chapter 2, we discuss the boundedness of commutators of Riesz transform on the L~p(R~n) space which is generated by operator associated with Schrodinger and BMO functions associated with Schrodinger operator. In this chaper, we prove that if p and potential V(x) satisfy certain condition then the commutators are bounded on L~P(R~n) spaces.In chaper 3, we study the boundedness of commutators generated by Schrodinger operator associated semigroup and Lipschitz functions. We prove that it is a bounded operator from L~P(R~n) to Triebel-Lizorkin spaces, and L~p(R~n) to L~q(R~n).In chapter 4, we discuss the boundedness of commutators generated by Linear operators and Fourier multipllier on Herz type Besov spaces.