Mathematical Sciences: Function and Operator Theory on Holomorphic Spaces

DMS-9622890 PI：阿拉巴马大学吴志坚@塔斯卡卢萨在过去的几年里，他的主要研究人员主要集中在解析(一个或多个变量)或单基因函数空间上的函数和算子理论的研究。这些算子包括Hankel、Toeplitz、交换子、乘子、双线性形式、逼近和原子分解。空间(解析的或调和的)是Bergman，Hardy，Dirichlet和容量相关的尺度空间，包括Bloch空间和有界平均振荡函数空间。这项研究还涉及d-bar、Laplace ian、Div、Curl和Dirac等微分算子。首席研究员建议继续他正在进行的研究，探索函数和算子理论与其他领域的更多联系和应用。虽然在自然界中出现了许多与研究有关的问题，但主要研究者计划集中于上述与势相关的函数空间上的算子、这些空间的函数理论、Dirichlet空间中的Nehari型问题、某些微分系统的最小解算子以及补偿紧性理论中的Clifford分析。他将在调和分析中使用函数和算符理论工具和技术，并在他的研究中引入新的思想和方法。除了它们在纯数学中的魅力之外，这个项目中的问题还与应用科学的许多领域有着丰富的联系和应用。例如，Hankel算子(或者更广泛地说是交换子和双线性形式)在工程设计和系统科学的各个方面非常流行。它们的行为为自动控制、导航系统和鲁棒稳定提供了重要信息。对Hankel及其相关算子在不同底层空间中的研究将为解决不同情况下的控制问题提供有效的方法。另一个很好的例子是最近发现的Clifford分析与多变量空间的非线性偏微分方程组的补偿紧性现象之间的联系。经验和以往的成就表明，首席调查员有潜力实现他的目标。

DMS-9622890 PI: Zhijian Wu University of Alabama @ Tuscaloosa In the past several years, the principal investigator has mainly focused his research on function and operator theory on analytic (one or several variables) or monogenic function spaces. The operators include Hankel, Toeplitz, commutators, multipliers, bilinear forms, approximation and atomic decomposition. The spaces (analytic or harmonic) are Bergman, Hardy, Dirichlet and the scale of capacity related spaces which includes the Bloch space and the space of bounded mean oscillation functions. The study also involves differential operators such as d-bar, Laplacian, Div, Curl and Dirac operators. The principal investigator proposes to continue his ongoing study, to explore more connections and applications of function and operator theory to other areas. Although there are many problems related to the study which arise in nature, the principal investigator plans to focus specifically on the aforementioned operators on potential related function spaces, the function theory of these spaces, Nehari type problems in Dirichlet spaces, minimum solution operators to some differential systems and Clifford analysis in the theory of compensated compactness. He will use function and operator theoretical tools and techniques in harmonic analysis, together with new ideas and methods in his investigation. Beside their own charm in pure mathematics, problems in this project have rich connections and applications to many fields of applied sciences. For example Hankel operators (or more generally commutators and bilinear forms) are very popular in various aspects of engineering design and system science. Their behavior provides important information to automatic control, navigation systems and robust stabilization. The study of Hankel and related operators in various underlying spaces will provide effective ways to deal with control problems in different situations. Another good example is the recent discover o f the connection between Clifford analysis and the compensated compactness phenomenon in non-linear partial differential equations of multi-variable spaces. Experience and previous achievement indicate that the principal investigator has the potential to accomplish his goal.