arXiv math/04021602004-02-10math.FA

关于巴拿赫空间间Lipschitz映射的Fréchet可微性

On Fréchet differentiability of Lipschitz maps between Banach spaces

Joram Lindenstrauss, David Preiss

备注：: 32 pages, published version
发表信息：: Ann. of Math. (2), Vol. 157 (2003), no. 1, 257--288

摘要：

一个著名的公开问题是：在具有可分对偶的巴拿赫空间X上，Lipschitz函数的可数集合是否有一个公共的Frechet可微点。我们证明了对于某些无限维X，答案是肯定的。以前，即使是由两个函数组成的集合，这也只对有限维X是已知的（尽管对于一个函数，答案在完全一般性上是肯定的）。通过在巴拿赫空间中引入一类新的空集（称为$Γ$-null sets），我们的目的得以实现，它的定义包含了范畴和测度的概念，并证明了所需的可微性关于它几乎处处成立，我们甚至得到了某些无限维巴拿赫空间之间Lipschitz函数的Fréchet导数的存在性;以前没有这样的结果。我们的主要结果表明，可分巴拿赫空间之间的Lipschitz映射是Fréchet可微的$Γ$-几乎处处，只要它是正则的Gateaux可微的$Γ$-几乎处处，并且Gateaux导数保持在一个范数可分空间内.很容易看出，X到具有Radon-Nikodym性质的空间的Lipschitz映射$Γ$-几乎处处是Gateaux可微的。此外，除了另一类可忽略集（即所谓的$σ$-多孔集）之外，Gateaux可微性还包含正则Gateaux可微性.因此，在每个$σ$-多孔集为$Γ$-空的空间中，问题的答案是肯定的。我们证明了这对具有$K$可数紧的$C(K)$、Tsirelson空间和$c_0$的所有子空间都成立，但对希尔伯特空间不成立.

英文摘要：

A well-known open question is whether every countable collection of Lipschitz functions on a Banach space X with separable dual has a common point of Frechet differentiability. We show that the answer is positive for some infinite-dimensional X. Previously, even for collections consisting of two functions this has been known for finite-dimensional X only (although for one function the answer is known to be affirmative in full generality). Our aims are achieved by introducing a new class of null sets in Banach spaces (called $Γ$-null sets), whose definition involves both the notions of category and measure, and showing that the required differentiability holds almost everywhere with respect to it. We even obtain existence of Fréchet derivatives of Lipschitz functions between certain infinite-dimensional Banach spaces;no such results have been known previously. Our main result states that a Lipschitz map between separable Banach spaces is Fréchet differentiable $Γ$-almost everywhere provided that it is regularly Gateaux differentiable $Γ$-almost everywhere and the Gateaux derivatives stay within a norm separable space of operators. It is easy to see that Lipschitz maps of X to spaces with the Radon-Nikodym property are Gateaux differentiable $Γ$-almost everywhere. Moreover, Gateaux differentiability implies regular Gateaux differentiability with exception of another kind of negligible sets, so-called $σ$-porous sets. The answer to the question is therefore positive in every space in which every $σ$-porous set is $Γ$-null. We show that this holds for $C(K)$ with $K$ countable compact, the Tsirelson space and for all subspaces of $c_0$, but that it fails for Hilbert spaces.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文