调和映照和拟共形调和映照的若干极值问题

In this project we study some hot extremal problems on harmonic mappings, quasiconformal harmonic mappings and their generalized mappings. We will use the geometric characterizations of the ranges of harmonic mappings, properties of singular integral operators, the generalized extremal criterions given by Reich and Strebel, modular inequalities and the new or refined inequalities on harmonic mappings to study the following problems: (1) Sharp Schwarz-Pick inequalities for weighted harmonic mappings or generalized quasiconformal harmonic mappings, norm esitmate of Beurling transformation, distortion theorems of Hausdorff dimensions and Hausdorff measures on generalized quasicircles; (2) Optimal continuous modulus of K-quasiconformal weighted harmonic mappings, the Landau theorem and the Bloch constant for generalized quasiregular mappings, sharp coefficient estimates for subclasses of harmonic mappings; (3) Extremality of quasiconformal harmonic mappings, quasiconformality and boundary characterization of univalent harmonic mappings, quasiconformal extension theorms for harmonic mappings; (4) Generalization of planar results on harmonic mappings or quasiconformal harmonic mappings to the ones in higher dimensional spaces such as the hypercomplex space. The results obtained in this project will contribute to the development of theories on quasiconformal mappings, Teichmüller spaces and modular spaces. 8 to 10 papers are expected in the SCI journals, and 6 to 8 articles are expected in the domestic journals.

研究调和映照、拟共形调和映照及其推广映照倍受关注的一些极值问题。利用调和映照像区域的几何特征、奇异积分算子的性质、推广的Reich-Strebel极值判别方法和模不等式，推广或建立涉及调和映照的不等式来研究如下问题: 1)带势调和映照与广义拟共形调和映照的精确Schwarz-Pick型不等式，Beurling变换的范数估计，广义拟圆周的维数和测度偏差；2）K-拟共形带势调和映照的最优连续模，广义拟正则调和映照的Landau和 Bloch型定理，调和映照类的精确系数估计；3）判别K-拟共形调和映照的极值性问题，单叶调和映照的拟共形性与边界特征，调和映照的拟共形延拓问题；4）调和映照与拟共形调和映照的平面结果向超复空间高维推广问题。项目结论对拟共形映照理论、Teichmüller 空间理论、模空间理论的发展有实际的推动作用。计划在SCI刊物发表8-10篇论文、国内期刊6-8篇。

与调和映照、拟共形调和映照及其推广映照相关的极值问题是本项目的研究重点。本项目证明了Iwaniec猜想(即Beurling奇异积分算子精确范数估计)在对数调和映照类等几类非拉伸映照类成立；建立有界域上Burkholder泛函精确的下界估计，并应用于估计拟共形基本解和具有恒等边界拟共形映照的Burkholder精确积分估计；利用双曲度量构造单连通区域的微分不等式，并用于获得ρ调和拟共形的双曲Lipschitz连续性及用拟共形偏差系数表示的Lipschitz常数，得到ρ调和映照的精确双曲梯度下界估计，利用边界条件和几何特征给出欧氏调和映照多种欧氏双Lipschitz判别法则；给出α-调和映照的积分表示、级数表示和解析函数表示，借此得到α-调和映照的Rado-Kneser-Choquet反例定理，推广了欧氏调和映照得精确Heinz不等式，给出α-调和映照的(K,K^')-拟共形性等价判别条件，对给定指数α得到T_α-调和函数的精确Schwarz引理；通过引入复参数，统一单叶函数与调和映照的拟共形延拓理论；建立单位圆周上一个积分恒等式，借此获得有界调和映照的Landau型定理，获得调和映照在Nagpal和Rovichandran积分算子下可继承精确凸半径；简化C^n空间单位球上多调和映照Harnack不等式的证明；获得流形之间拟共形映照的角偏差定理；给出超复空间上超复Beltrami方程的基本解的存在性定理。