论文标题
Orlicz空间中的最佳问题
Optimality problems in Orlicz spaces
论文作者
论文摘要
在数学建模中，数据和解决方案表示为可测量的功能，其质量经常被成员捕获到某个功能空间。分析模型的核心问题之一是数据与解决方案质量之间的相互关系。获得的结果的最佳性值得特别重点。它需要仔细选择功能空间的家庭在表达性之间平衡的家庭，即捕获模型的精细特性及其可访问性的能力，即其在实际使用方面的技术困难。本文提出了在Orlicz空间中解决最佳问题的统一和一般方法。 Orlicz空间通过单个凸功能进行了参数化，并整齐地平衡了表现力和可访问性。我们证明了一个一般原则，该原则可为各种任务中最佳的Orlicz空间的存在或不存在易于验证的必要条件和足够的条件。我们证明了它在特定问题中的使用，包括Sobolev嵌入的连续性以及整体操作员的界限，例如Hardy-Little Wood-Liles Wood Maximal Operator和Laplace Transform。
In mathematical modelling, the data and solutions are represented as measurable functions and their quality is oftentimes captured by the membership to a certain function space. One of the core questions for an analysis of a model is the mutual relationship between the data and solution quality. The optimality of the obtained results deserves a special focus. It requires a careful choice of families of function spaces balancing between their expressivity, i.e. the ability to capture fine properties of the model, and their accessibility, i.e. its technical difficulty for practical use. This paper presents a unified and general approach to optimality problems in Orlicz spaces. Orlicz spaces are parametrized by a single convex function and neatly balance the expressivity and accessibility. We prove a general principle that yields an easily verifiable necessary and sufficient condition for the existence or the non-existence of an optimal Orlicz space in various tasks. We demonstrate its use in specific problems, including the continuity of Sobolev embeddings and boundedness of integral operators such as the Hardy--Littlewood maximal operator and the Laplace transform.
