Several Classes Of Operators On Compact Manifolds

This dissertation focuses on the boundedness and convergence rate of several classes of operators on compact manifolds.The following four issues should be considered:The convergence rate of the fractional power dissipative operator on compact manifolds;The boundedness of wave operators on compact manifolds;The Hp estimation of wave operators on compact manifolds and the convergence rate of fractional Schrodinger operators on noncompact manifolds.It mainly uses the existing multipliers theory on compact manifolds,embedding theorem,atomic decomposition theorem,Hadamard parametrix construction and interpolation theorem of analytic operator family to solve relevant problems through careful and complicated calculation.The key problem is to solve the estimation of the corresponding operator kernel.The way to solve this problem is to decompose the frequency space and physical space and divide the kernel into several parts to discuss separately.The results of several classes of operators involved in this paper are very rich in Euclidean spaces.The work of this paper is to generalize the corresponding results to manifolds.The thesis is divided into five chapters:The first chapter introduces the research background of several problems and the main conclusions.At the same time,the required preparatory knowledge and lemmas are given.In Chapter 2,we investigate the convergence rate of the fractional power dissipative operator e-t|L|α,which is also called fractional heat operator,on compact manifolds Mn.Here we first extend the conclusion on the Euclidean space Rn to the n dimensional torus Tn by a transference method,and then to the n dimension connected compact manifolds Mn.In Chapter 3,the Lp boundedness of wave operators on compact manifolds is studied.We give sufficient and necessary conditions for the LP boundedness of wave operator except the endpoint case,and at the endpoint we give the result for n=3.In Chapter 4,we continue to consider the boundedness of the wave operator.Here we give the Hp estimate of the wave operator.This result not only generalizes the corresponding conclusion on Euclidean space,but also generalizes the conclusion on compact Lie group.The fifth chapter proves the almost everywhere convergence of the fractional Schrodinger operator eit|L|α/2 on noncompact manifolds.In addition,when a noncompact manifold is a Lie group,we give the convergence rate of the operator.