【作者】 张晗；

【导师】 仝策中；

【作者基本信息】 河北工业大学 ， 数学， 2020， 硕士

【摘要】 本文研究了单位球上的Békollé-Bonami型加权Bergman空间上的原子分解定理.首先着重从Bergman空间,加权Bergman空间理论的研究背景及现状,原子分解理论的研究背景及意义来介绍.在研究背景方面讲述了这些理论的发展进程;对于研究意义,目前也拓展到了理论与各个空间之间的联系.Bergman空间理论一直是函数论中的热点专题,随后标准的加权Bergman空间也得到了广泛的关注,所涉及的诸如原子分解、插值等问题长期以来被国内外广大数学学者所研究.原子分解理论在研究函数空间和算子理论时是一个非常重要的工具,在本文中先推广了原子分解定理,随后利用原子分解定理研究复合算子的紧差分问题.第三章利用Békollé-Bonami型加权Bergman空间中的再生核函数给出该加权Bergman空间的原子分解定理,将复n维空间上的单位球上利用加权Bergman空间A2(u)的再生核推广Luecking的原子分解定理.随后在第四章中我们刻画了当0<q<p时,由标准加权Bergman空间到Lebesgue空间复合算子的差分的有界性,类比于不加权的情形,利用不加权的Bergman空间的原子分解定理证明了紧差分与加权复合算子紧性的等价条件,我们推广到Békollé-Bonami加权Bergman空间的复合算子的差分的刻画.在这个过程中我们发现了算子的差分和某些加权复合算子之间的有趣联系,并因此得到了标准加权Bergman空间之间的有界性和紧差分的完整刻画,给出了复合算子的差分是有界的一个充分条件,加权复合算子有界的充分必要条件.最后利用原子分解定理给出了后续的研究思路和目标.更多还原

【Abstract】 On the unit ball in the n dimensional complex Euclidean space,we investigate the atomic decomposition of the weighted Bergman spaces with Békollé-Bonami weights.First of all,it focuses on the research background and current situation of Bergman space theory,weighted Bergman space theory,and the research background and significance of atomic decomposi-tion theory.In terms of research background,it describes the development process of these theories;for the research significance,it also extends to the connection between theory and each space.Bergman space theory has always been a hot topic in function theory,then the standard weighted Bergman space has been widely concerned,the problems involved such as atomic decomposition and interpolation have been studied by many mathematicians for a long time.Atomic decomposition theory is a very important tool in the study of function space and operator theory.In this paper,we first generalize the atomic decomposition theo-rem,then we use the atomic decomposition theorem to study the compact difference problem of composite operators.In chapter 3,we give the atomic decomposition theorem of this weighted Bergman space by using the reproducing kernel function of Békollé-Bonami weighted Bergman space,then we modify the Luecking’s atomic decomposition theorem by the reproducing kernel functions of A2(u)of the weighted Bergman space.Then in chapter 4,we characterize the bounded difference of composition operators from the standard weighted Bergman space to the Lebesgue space when 0<q<p.Compared with the case of unweighted,the equivalent condition of compact difference and the com-pactness of weighted composite operator is proved by using the atomic decomposition the-orem of unweighted Bergman space,we generalize to the characterization of difference of composite operators in Békollé-Bonami weighted Bergman spaces.In this process,we find the interesting relationship between the difference operators and some weighted composition operators,the boundedness between standard weighted Bergman Spaces and the complete characterization of compact difference are obtained.A sufficient condition of the difference of composite operator is bounded and the necessary and sufficient condition for the bound-edness of weighted composite operators are given.Finally,the following research ideas and objectives are given by using the atomic decomposition theorem.更多还原