Abstract

Let D be a nonempty open set in a metric space (X, d) with ∂D ≠ Ø. Define $$h_{D,c}(x,y)=\log\left(1+c{{{d(x,y)}}\over{{\sqrt{d_{D}(x)d_{D}(y)}}}}\right).$$ where d_D(x) = d(x, ∂D) is the distance from x to the boundary of D. For every c ⩾ 2, h_D,c is a metric. We study the sharp Lipschitz constants for the metric h_D,c under Möbius transformations of the unit ball, the upper half space, and the punctured unit ball.

中文翻译：

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莫比乌斯变换下双曲型度量的 Lipschitz 常数

摘要

设D为度量空间 ( X, d )中的非空开集，且∂D ≠ Ø。定义$$h_{D,c}(x,y)=\log\left(1+c{{{d(x,y)}}\over{{\sqrt{d_{D}(x)d_{ D}(y)}}}}\right).$$其中d _D ( x ) = d ( x, ∂D ) 是从x到D边界的距离。对于每个c ⩾ 2，h _D,c是一个度量。我们研究了单位球、上半空间和穿孔单位球的莫比乌斯变换下度量h _{D,c 的尖锐 Lipschitz 常数。}