Measurable differentiable structures and the Poincaré inequality
Measurable differentiable structures and the Poincaré inequality
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可测可微结构和庞加莱不等式
DOI：
10.1512/iumj.2004.53.2417
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发表时间：
2004
影响因子：
1.1
通讯作者：
S. Keith
中科院分区：
文献类型：
--
作者：
S. Keith
本文的主要结果是对Cheeger [2，定理4.38]中提出的可微结构的改进，其条件与[2]相同，即给定的度量测度空间允许一个具有加倍测度的Poincare不等式.更确切地说，本文证明了可微结构的坐标函数可以取为距离函数。在这个过程中，给出了包含在[2]中定义的Sobolev空间H1，p中的函数的微分的近似极限的表示，因此N1，p由Shanmugalingam [16]定义。这些结果的进一步应用包括确定任意Sobolev函数的最小广义上梯度[2，定义2.9]，以及对Franchi，Hajlasz和Kokorna [5]和Semmes [8，定理5.1]中定义在Lp中Lipschitz函数上的微分算子的可闭性的另一种证明.
The main result of this paper is an improvement for the differentiable structure presented in Cheeger [2, Theorem 4.38] under the same assumptions of [2] that the given metric measure space admits a Poincare inequality with a doubling measure. To be precise, it is shown in this paper that the coordinate functions of the differentiable structure can be taken to be distance functions. In the process, a representation is given in terms of approximate limits for the differential of functions contained in the Sobolev space H 1,p defined in [2], and therefore N 1,p defined by Shanmugalingam [16]. Further application of these results includes identifying the minimal generalized upper gradient [2, Definition 2.9] of an arbitrary Sobolev function, and an alternate proof to that of Franchi, Hajlasz and Koskela [5] and Semmes [8, Theorem 5.1] for the closability of the differential operator defined on Lipschitz functions in L p .