Boundedness Of Commutators Generalized By Riesz Potential Related To Schrodinger Operators

The boundedness of the singular integral operator and its commutator in some function space is an important topic in the research of harmonic analysis. Riesz potential is a classic singular integral operator in harmonical analysis, which not only plays an impotant role in the harmonic analysis, but also has extensive application in PDEs. Since Schodinger found that the Schrodinger equation could be used to describe the state of microscopic particle in quantum mechanics, then the boundedness of some Riesz potential and their commuta-tors associated with the Schrodinger differential operator which has nonnega-tive potential have drawn widespread attention.This paper aims to explore the the boundedness of Toeplitz operators gen-erated by the Riesz potential related to Schrodinger operator and BMO or Lip-schitz type functions. And, we also studied the boundedness of commutators generated by the Riesz potential related to Schrodinger operator and general-ized Lipschitz type function Lipβ,∞(ρ) on Lebesgue space.In chapter1, we introduce the background and some related theory of the paper. These involved in the following aspects:the theory of the singular inte-gral operator and it’s commutator; Schrodinger operator and the Riesz transfor-m associated with the Schrodinger operator; the main research content of this paper.In chapter2, using the pointwise estimate of Sharp function, we studied the boundedness of Toeplitz operators generalized by Riesz transform T=▽V(一△+V)-1/2which related to Schrodinger operator and BMOL function on Lp(Rn) space.In chapter3, we discuss the boundedness of the Toeplitz operator defined by chapter2, furtherly. By the pointwise estimate of Sharp function, we studied the the boundedness of Toeplitz operators generalized by Riesz transform T=▽V(一△+V)-1/2which related to Schrodinger operator and∧βε(Rn) function on Lp(Rn) space.In chapter4, by intensive pointwise estimate of Sharp function, we got the boundedness of the commutators generalized by Riesz transforms T=T=▽V(一△+V)-1/2and Lipβ,∞(ρ) function on Lp(Rn) space.