Operator Theory On Weighted Bergman Spaces

Operator theory on function spaces is an important research branch of functional analysis,the study of classical operators on function spaces has always been a focus topic concerned by a great number of experts and scholars from home and abroad.Bergman space is a classical research object in the field of operator theory on function spaces,and the study on Bergman spaces is quite broad.Weighted function spaces is the product of the intersection and combination of harmonic analysis,complex analysis,functional analysis and many other fields.Different weights might make the properties of the operators on weighted function spaces have essential changes.In recent years,a class of doubling weights,called the two-sided doubling weights in this thesis,has gradually attracted a lot of interest from the scholars of operator theory,this class of weight is now shown to be able to equivalently characterize some important functional properties of radially weighted function spaces,and also some properties of the operators on such spaces.In this thesis,we study the boundedness and compactness of several classes of classical integral operators on the weighted Bergman spaces induced by two-sided doubling weights.We divide this thesis into six chapters,the structure and contents are as follows.Chapter 1 is the introduction part,we introduce the background and motivations of this thesis,followed by the research contents,main difficulties and the innovation points.In chapter 2,we first introduce three primary classes of analytic function spaces related to this thesis,then we introduce the definitions and some frequently used properties of several classes of weights,later we introduce the characterizations of the Carleson measures on the three classes of function spaces previously mentioned,finally we recall the atomic decomposition theorem for two-sided doubling weighted Bergman spaces.In chapter 3,we study the boundedness and compactness of the Toeplitz operators acting on weighted analytic Bergman spaces induced by two-sided doubling weights Lap(ω).First we use Carleson measure conditions and T1-type conditions together to equivalently characterize the boundedness and compactness of the Toeplitz operators with positive measures as symbols Tμ,ω:Lao(ω)→Laq(ω)(0<p≤1,q=1).Next we use the"logarithmic" condition and "bump" condition as sufficient conditions to give descriptions for the boundedness of the Toeplitz operators with integrable functions as symbols in the case p=q=1,respectively.Finally we give the sufficient and necessary conditions to characterize the boundedness and compactness of the Toeplitz operators acting between two Bergman spaces induced by standard weights Tμ,β:Lap(dAα)→Laq(dAβ)in two cases 0<p≤1,q=1,-1<α,β<∞ and 0<p≤1<q<∞,-1<β≤α<∞.In chapter 4,we study the boundedness and compactness of the Toeplitz operators acting on weighted harmonic Bergman spaces induced by two-sided doubling weights Lhp(ω).We first give the estimations for the reproducing kernels and descriptions for the Carleson measures on Lhp(ω),then equivalently characterize the boundedness and compactness of the Toeplitz operators with positive measures as symbols Tμ,ωh:Lhp(ω)→Lhq(ω)(1<p,q<∞).In chapter 5,we study the boundedness of the Small Hankel operators acting on weighted analytic Bergman spaces induced by two-sided doubling weights Lap(ω).First we introduce a new weight,which yields a novel representation of the Small Hankel operator.This makes us obtain the complete characterization of the boundedness of the Small Hankel operators with analytic functions as symbols,acting between two arbitrary weighted Bergman spaces induced by two-sided doubling weights in the range of 0<p,q<∞,here the analytic symbol is under a mild restriction.Finally as an application,we obtain the weak factorization theorem for Lap(ω)(1<p<∞).In chapter 6,we study the boundedness and compactness of the Volterra-type integration operators acting between weighted Bergman spaces induced by two-sided doubling weights Lap(ω)and Hardy spaces Hq,and give the sufficient and necessary conditions to characterize the boundedness and compactness of the Volterra-type integration operators with analytic symbols Jg:Lap(ω)→Hq and Jg:Hp→Laq(ω)when 0<p,q<∞.