Directional derivatives of Lipschitz functions
Directional derivatives of Lipschitz functions
复制标题
Lipschitz 函数的方向导数
DOI：
10.1007/bf02773371
复制
发表时间：
2001
影响因子：
1
通讯作者：
L. Zajícek
中科院分区：
文献类型：
--
作者：
D. Preiss;L. Zajícek
设f是可分Banach空间X到Banach空间Y的Lipschitz映射。我们观察到f在一个方向生成集上可微但Gâteaux不可微的点的集合是σ-方向多孔的。由于Borelσ-定向多孔集除了是第一类集之外，在Aronszajn意义下（或等价地，在高斯意义下）是空的，我们得到了Rademacher定理（由于Aronszajn）关于Lipschitz映射的Gâteaux可微性的无限维推广的另一个证明。更好地了解σ-定向多孔集导致我们的一个新版本的Rademacher定理在无限维空间中，我们证明是强于Aronszajn得到的。更详细的分析表明，我们的观察（更强的版本）来自一个有点技术性的结果，表明斜率（f（x+t（u+v））-f（x+tv））/t ast → 0+的行为在某种意义上与v无关。特别地，这意味着在Lipschitz真实的值函数的情况下，上单侧导数与Michel和Penot定义的导数一致，除了σ-定向多孔集的点。这对上下方向导数有许多有趣的结果。例如，对于所有x ∈ X，除了那些属于σ-方向多孔集的，函数v →
$$\bar f$$
（x，v）（x在v方向上的右上导数）是凸的。
AbstractLetf be a Lipschitz mapping of a separable Banach spaceX to a Banach spaceY. We observe that the set of points at whichf is differentiable in a spanning set of directions but not Gâteaux differentiable isσ-directionally porous. Since Borelσ-directionally porous sets, in addition to being first category sets, are null in Aronszajn’s (or, equivalently, in Gaussian) sense, we obtain an alternative proof of the infinite-dimensional generalisation of Rademacher’s Theorem (due to Aronszajn) on Gâteaux differentiability of Lipschitz mappings. Better understanding ofσ-directionally porous sets leads us to a new version of Rademacher’s theorem in infinite dimensional spaces which we show to be stronger then the one obtained by Aronszajn. A more detailed analysis shows that (a stronger version of) our observation follows from a somewhat technical result showing that the behaviour of the slopes (f(x+t (u+v))−f(x+tv))/t ast → 0+is in some sense independent ofv. In particular, this implies that in the case of Lipschitz real valued functions the upper one-sided derivatives coincide with the derivatives defined by Michel and Penot, except for points of aσ-directionally porous set. This has a number of interesting consequences for upper and lower directional derivatives. For example, for allx ∈ X, except those which belong to aσ-directionally porous set, the functionv →
$$\bar f$$
(x, υ) (the upper right derivative off atx in the directionv) is convex.