拟共形映射下球面度量的偏差

Hausdorff测度的拟共形畸变

Quasiconformal distortion of Hausdorff measures

本文证明了：若f是平面K-拟共形映射，且0<t<2，t' = 2 t/（2K-Kt+t），则f将有限（t'）-Hausdorff测度集变换为有限t-Hausdorff测度集.我们还证明了以下更定量的陈述：如果E是平面集，则H^t（E）\leq C（K）H^{t '}（f（E））^{t/（t' K）}，其中H^s代表s-Hausdorff测度。

模方程与偏差函数

Modular Equations and Distortion Functions

模方程出现在数论中，但较少有人知道这类方程也出现在拟共形映射变形性质的研究中。研究了两个重要的平面拟共形偏差函数，得到了它们的单调性和凸性，并找到了它们的精确界。应用程序提供有关的拟共形Schwarz引理和肖特基定理。这些结果也给出了完全椭圆积分奇异值的新的界。

Bilipschitz和拟共形旋转、拉伸和多重分形谱

Bilipschitz and quasiconformal rotation, stretching and multifractal spectra

我们建立了平面拟共形映射的局部旋转和Hölder-畸变的精确界。此外，我们给出了相应的联合拟共形多重分形谱的尖锐估计，基于新的估计Burkholder泛函与复杂的参数。因此，我们得到最佳的旋转估计也双Lipschitz映射。

拟共形映射的改进 Hölder 连续性

Improved Hölder Continuity of Quasiconformal Maps

本文结合等周不等式与拟圆长度分析，改进复平面拟共形映射的 Hölder 连续性下界，刻画极值子并应用于椭圆偏微分方程。

Extension of boundary homeomorphisms to mappings of finite distortion

We provide sufficient conditions so that a homeomorphism of the real line or of the circle admits an extension to a mapping of finite distortion in the upper half-plane or the disk, respectively. Moreover, we can ensure that the quasiconformal dilatation of the extension satisfies certain integrability conditions, such as $p$-integrability or exponential integrability. Mappings satisfying the latter integrability condition are also known as David homeomorphisms. Our extension operator is the same as the one used by Beurling and Ahlfors in their celebrated work. We prove an optimal bound for th…

拟共形映射与 $f\in BV(\mathbb{R}^n; \mathbb{R}^n)$ 的 $\tfrac {dDf}{d|Df|}$ 的秩

Quasiconformal mappings and the rank of $\tfrac {dDf}{d|Df|}$ for $f\in BV(\mathbb{R}^n; \mathbb{R}^n)$

本文通过引入畸变数的精细松弛版本，刻画了 BV 函数在几乎处处意义下导数密度矩阵满秩与有限畸变之间的等价关系。