论文标题
在Orlicz--sobolev Maps下的Hausdorff措施的扭曲
Distortion of Hausdorff measures under Orlicz--Sobolev maps
论文作者
论文摘要
提供了关于Orlicz-Sobolev地图，欧几里得空间之间对具有零或有限Hausdorff措施的子集的影响的综合理论。我们的讨论中包括任意嵌入连续功能空间的任意Orlicz-Sobolev空间，而Hausdorff措施构建了。根据定义Orlicz-Sobolev空间的年轻函数，显示了在这些图的作用下对相关规函数失真的明确公式。检测到与地图弱衍生物的整合性程度以及集合度量概念的柔韧性的灵活性有关的新现象和特征。恢复了涉及标准的Sobolev空间和Hausdorff措施的经典结果，其最优性显示在更强的意义上。文献中可用的特殊实例，涉及非功率类型的年轻功能和量规函数，并且在不敏感的情况下也得到了改善。
A comprehensive theory of the effect of Orlicz-Sobolev maps, between Euclidean spaces, on subsets with zero or finite Hausdorff measure is offered. Arbitrary Orlicz-Sobolev spaces embedded into the space of continuous function and Hausdorff measures built upon general gauge functions are included in our discussion. An explicit formula for the distortion of the relevant gauge function under the action of these maps is exhibited in terms of the Young function defining the Orlicz-Sobolev space. New phenomena and features, related to the flexibility in the definition of the degree of integrability of weak derivatives of maps and in the notion of measure of sets, are detected. Classical results, dealing with standard Sobolev spaces and Hausdorff measures, are recovered, and their optimality is shown to hold in a refined stronger sense. Special instances available in the literature, concerning Young functions and gauge functions of non-power type, are also reproduced and, when not sharp, improved.
