The Real-Variable Theory of Function Spaces and its Applications

函数空间实变量理论及其应用

• 批准号: 392255916
• 负责人: Professorin Dr. Dorothee Haroske
• 金额: --
• 依托单位: National Natural Science Foundation of China
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2018
• 资助国家: 德国
• 起止时间: 2017-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/392255916?language=en
• 关键词: Real; Variable; Theory; Function; Spaces

The topic of this project is related to the real-variable theory of function spaces on Euclidean spaces, domains and metric measure spaces (including graphs) as well as some problems in partial differential equations, numerical analysis and geometric analysis. The theory of function spaces is one of the central topics in modern harmonic analysis and has found wide applications. Smoothness function spaces, especially Sobolev spaces, are widely used in calculus of variations and PDE. As more general scale of function spaces, Besov spaces and Triebel-Lizorkin spaces, are connected with the study of traces and interpolation of Sobolev spaces. They have also been used in various PDEs, studying the well-posedness of solutions and longtime behaviour for Euler equations, Hydrodynamic equations such as Navier-Stokes equations and some nonlinear partial differential equations and nonlinear dispersion equations. Moreover, the theory of function spaces has implications on some areas like signal analysis, data interpolation and applied wavelet theory, potential analysis and approximation theory. Recently the theory of function spaces with variable exponents has attracted a lot of attention due to its special and rich structures and applications in calculus of variations and fluid mechanics. Another topic concerns high-dimensional approximation which has become a very active field of research. This was motivated by needs of numerical mathematics and applications to financial mathematics, chemistry and other areas, where the dimension of the underlying domain could be very large. Though the asymptotic behaviour of certain characteristic quantities of related embeddings is well known, in most cases this means up to multiplicative constants. For practical purposes such estimates are useless, unless one has additional information on the hidden constants, in particular their dependence on the dimension. There are first results in some special cases which indicate that the situation can be completely different from what was known before. This is an interesting effect that is of great importance for practical problems.Altogether we want to study the following problems: characterize the class of pointwise multipliers on Besov-type and Triebel-Lizorkin-type spaces on Euclidean spaces and domains; develop a theory of Besov-type and Triebel-Lizorkin-type spaces on some general domains; find the interpolation spaces of variable Besov(-type) and Triebel-Lizorkin(-type) spaces via different methods; find sharp asymptotic and pre-asymptotic estimates for Sobolev spaces defined on cubes; find the conditions which imply that certain semilinear equations on graphs have a (unique) solution; find the critical index on the uniqueness of the non-negative solution for some differential inequalities related to elliptic operators on Riemannian manifolds.The Chinese and German teams have sufficient expertise and strength to cope with these challenging and up-to-date questions.

该项目的主题涉及欧几里得空间、域和度量测度空间（包括图）上函数空间的实变量理论以及偏微分方程、数值分析和几何分析中的一些问题。函数空间理论是现代调和分析的中心课题之一，并得到了广泛的应用。 光滑函数空间，特别是Sobolev空间，在变分法和偏微分方程中有着广泛的应用。Besov空间和Triebel-Lizorkin空间作为函数空间的更一般的标度，与Sobolev空间的迹和插值的研究有关。它们也被用于各种偏微分方程，研究Euler方程，流体动力学方程，如Navier-Stokes方程和一些非线性偏微分方程和非线性色散方程的解的适定性和长期行为。此外，函数空间理论在信号分析、数据插值和应用小波理论、势分析和逼近理论等领域都有应用。 近年来，变指数函数空间理论由于其特殊而丰富的结构以及在变分法和流体力学中的应用而引起了人们的广泛关注。另一个主题涉及高维近似，这已成为一个非常活跃的研究领域。这是出于数值数学和金融数学，化学和其他领域的应用的需要，其中底层域的维度可能非常大。虽然相关嵌入的某些特征量的渐近行为是众所周知的，但在大多数情况下，这意味着乘法常数。为了实际的目的，这样的估计是无用的，除非一个人有额外的信息隐藏的常数，特别是他们的依赖于尺寸。在某些特殊情况下，有第一个结果表明，情况可能与以前所知道的完全不同。这是一个有趣的效应，在实际问题中具有重要的意义，我们主要研究以下几个问题：刻画欧氏空间和域上的Besov型和Triebel-Lizorkin型空间上的点态乘子类，发展一些一般域上的Besov型和Triebel-Lizorkin型空间的理论;用不同的方法找到变Besov（-型）空间和Triebel-Lizorkin（-型）空间的插值空间，找到定义在立方体上的Sobolev空间的精确渐近估计和准渐近估计;找到暗示某些图上的半线性方程有（唯一）解的条件;找到非唯一性的临界指数，黎曼流形上与椭圆算子相关的一些微分不等式的负解。中德团队有足够的专业知识和实力来科普这些具有挑战性和最新的问题。