论文标题
Lipschitz扩展和近似
Lipschitz Extensions and Approximations
论文作者
论文摘要
经典的Hahn-Banach定理基于扩展有界线性函数的连续逐点过程。在通用度量域的设置中，条件的限制性较小，并且仅需要具有相同Lipschitz常数的Lipschitz。在这种情况下，连续的过程可以用麦克沙恩（McShane）和惠特尼（Whitney）在1930年代完成的更简单的过程取代。使用几乎相同的结构，Czipszer和Gehér在Lipschitz函数上显示了相似的扩展特性。在本文中，我们将这种结构与Hausdorff先前获得的另一个经典结果联系起来，并处理了半连续函数的Lipschitz近似值。此外，我们为本地Lipschitz函数提供了互补的扩展结果，这些函数自然而然地适合此框架。
The classical Hahn-Banach theorem is based on a successive point-by-point procedure of extending bounded linear functionals. In the setting of a general metric domain, the conditions are less restrictive and the extension is only required to be Lipschitz with the same Lipschitz constant. In this case, the successive procedure can be replaced by a much simpler one which was done by McShane and Whitney in the 1930s. Using virtually the same construction, Czipszer and Gehér showed a similar extension property for pointwise Lipschitz functions. In the present paper, we relate this construction to another classical result obtained previously by Hausdorff and dealing with pointwise Lipschitz approximations of semi-continuous functions. Moreover, we furnish complementary extension-approximation results for locally Lipschitz functions which fit naturally in this framework.
