Lipschitz扩张与逼近

摘要：

经典的Hahn-Banach定理是建立在对有界线性泛函逐点连续扩张的基础上的。在一般的度量域上，条件限制较少，仅要求扩张是具有相同Lipschitz常数的Lipschitz扩张。在这种情况下，可以用一个更简单的程序来代替连续程序，该程序由McShane和Whitney在20世纪30年代完成。使用几乎相同的结构，Czipszer和Gehér显示了逐点Lipschitz函数的类似扩展性质。在本文中，我们将这种结构与Hausdorff以前获得的另一个经典结果相联系，并处理半连续函数的逐点Lipschitz逼近。此外，我们提供了互补的局部Lipschitz函数，自然适合在这个框架中的扩展近似结果。

英文摘要：

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.

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