Generalizations Of The Boundary Schwarz Lemma And Norm Estimates Of The Bergman Type Operators

Boundary Schwarz lemma and Bergman projection are two important research objects in analytic function theory.This thesis mainly considers the extension of the boundary Schwarz lemma for analytic functions to more general classes of functions,such as harmonic mappings,bi-harmonic mappings and so on;another purpose of this thesis is to study the norm estimates of generalized Bergman type integral operators and their operator theory in function space.Firstly,we generalize the classical Schwarz lemma to harmonic mappings having zero of order p,and then establish the boundary Schwarz lemma for this class of mappings by using the series expansion of boundary fixed points and the existence of radial limit and angular limit.Furthermore,we establish the boundary Schwarz lemma for the solutions to inhomogeneous bi-harmonic equations by using the above research methods and combining with the kernel function characteristics of Poisson integral and Green integral.Secondly,we study the norm estimates of generalized Bergman type integral operators and their operator theory by using Fourier series theory,special function theory and the projection theorem of Hilbert space.This thesis is divided into four chapters,and the contents of each chapter are as follows:In chapter one,we introduce the background of the research questions,the related concepts and the main results.In chapter two,by improving the Schwarz lemma for harmonic mappings,we establish the boundary Schwarz lemma for harmonic mappings having zero of order p.In chapter three,we first study the forms of solutions to inhomogeneous bi-harmonic equations satisfying some Dirichlet boundary conditions,and then establish the boundary Schwarz lemma for the solutions.Our results generalize the relative results of planar harmonic mappings and are sharp under certain boundary conditions.In chapter four,we study the generalized Bergman type integral operators,and obtain the asymptotically sharp norm estimates of the integral operators.In addition,we also discuss the compactness of the operators and their relation to Schatten p-class.