多复变几何函数论和函数空间的若干问题研究

Geometric function theory and function spaces in several complex variables is an important and active area of research in fundamental mathmatics, and have obtained substantial results, but there are many problems needed to be further investigated.Basing on the recent years' works and using mathmatical tools such as Lie group、Lie algebra and differential geometry etc., we intend to establish the determinantal distortion theorem for starlike mappings; obtain estimates for the k-th(k>3 ) order of homogeneous expansions for starlike mappings on the unit polydisk; extend close-to-convex functions to higher dimensions and study its growth、covering properties and the sharp estimations of homogeneous expansion systematically; generalize the famous Fekete-Szego inequalities to higher dimensional spaces; further study the conditions of boundness and compactness for Toeplitz operators on the Besov spaces, and the sufficient and necessary conditions for boundedness of Toeplitz products on Bergman spaces. Basing on preliminary research of previous years, we expect to make breakthroughs or significant progress on these issues.

多复变几何函数论和函数空间是基础数学函数论方向一个重要和活跃的研究领域，已取得十分丰硕的研究成果，但仍有很多问题有待进一步深入研究。本项目在近年来工作的基础上，拟用李群、李代数的表示理论以及微分几何等数学工具，建立多复变数星形映射行列式型偏差定理；给出单位多圆柱上星形映射齐次展开式的第 k(k>3)项系数估计；系统研究单复变近于凸函数在高维对应物的增长、掩盖性质和其齐次展开式系数的精确估计；把单复变著名的Fekete-Szego不等式推广到多复变数空间；进一步深入研究Besov空间上Toeplitz算子有界和紧的条件以及Bergman空间上Toeplitz乘积算子有界的充分必要条件。项目组对这些问题已有较充分的前期研究基础，有望取得突破或显著进展。

该项目基本上按预定计划进行，未作大的调整。 部分预定研究目标已取得有特色的创新性成果，譬如,在建立多复变Fekete-Szego不等式的研究中获得突破，取得系列研究成果；成功将单复变近于凸函数族及其子族推广到多复变数空间；部分解决了多复变Bieberbach 猜想；成功将单位圆盘上凸函数的第四种等价刻画推广到多复变数空间，并建立了欧氏单位上星形映照行列式型偏差定理。本项目的完成对促进多复变几何函数论的发展有着重要意义。