丛测度的维数与正则性
On dimension and regularity of bundle measures
摘要：
本文对某些向量值测度的傅里叶变换的反对称性进行了量化。引入的尺度与内山定理中出现的条件有关，并用于给出这些测度的可求长维数的下界。此外，我们得到的估计较低的豪斯多夫维数假设某些更严格的版本的$2$波锥条件。我们考虑的结果可以被看作是一个不确定性类型的原则，在以下方式：这是不可能的同时本地化（束）措施和方向的傅里叶变换的小集合。所研究的类以BV函数的梯度为例进行建模。文章还包含一个定理关于正则性：我们证明了元素考虑类消失1-纯粹不可求长集。我们的结果可用于研究偏微分方程约束测度的性质。
英文摘要：
In this paper we quantify the notion of antisymmetry of the Fourier transform of certain vector valued measures. The introduced scale is related to the condition appearing in Uchiyama's theorem and is used to give a lower bound for the rectifiable dimension of those measures. Moreover, we obtain an estimate of the lower Hausdorff dimension assuming certain more restrictive version of the $2$-wave cone condition. Results of our considerations can be viewed as an uncertainty-type principle in the following way: it is impossible to simultaneously localize a (bundle) measure and a direction of its Fourier transform on small sets. The investigated class is modeled on the example of gradients of BV functions. The article contains also a theorem concerning regularity: we prove that elements of considered class vanish on 1-purely unrectifiable sets. Our results can be applied to studying the properties of PDE-constrainted measures.
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。
