方向可微函数的可微性及其应用
On the differentiability of directionally differentiable functions and applications
摘要：
本文的第一部分利用k-切集建立了具有单侧方向导数的Lipshitz函数的不可微集的大小和结构的一种最优估计。这些结果可应用于分析中许多重要的特殊函数，如距离函数或不同的极大函数。在第二部分中，利用第一部分的结果，我们更仔细地研究了经典Hardy-Littlewood极大函数的可微性。例如，我们将证明，如果f是连续的，并且在切集的可数并之外是可微的，那么这同样适用于极大函数Mf（如果Mf不是恒无穷大）。作为另一个例子，我们的结果还意味着如果f几乎处处可微，则Mf a.e.可微.
英文摘要：
In the first part of this paper we establish, in terms of so called k-tangential sets, a kind of optimal estimate for the size and structure of the set of non-differentiability of Lipshitz functions with one-sided directional derivatives. These results can be applied to many important special functions in analysis, like distance functions or different maximal functions. In the second part, having the results from the first part in our use, we focus more carefully on the differentiability properties of the classical Hardy-Littlewood maximal function. For example, we will show that if f is continuous and differentiable outside a countable union of tangential sets, then the same holds to the maximal function Mf as well (if Mf is not identically infinity). As an another example, our results also imply that if f is differentiable almost everywhere, then Mf is differentiable a.e.
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