The Real-Variable Theory of Function Spaces and its Applications

这个项目的主题是关于欧氏空间、区域和度量空间(包括图)上的函数空间的实变量理论以及偏微分方程组、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，有着广泛的应用。光滑函数空间，特别是Soblev空间，在变分和偏微分方程组中有着广泛的应用。作为更一般尺度的函数空间，Besov空间和Triebel-Lizorkin空间与Sobolev空间的迹和内插的研究密切相关。它们也被用于各种偏微分方程组，研究了Euler方程、流体力学方程如Navier-Stokes方程以及一些非线性偏微分方程组和非线性色散方程解的适定性和长期行为。此外，函数空间理论还涉及信号分析、数据内插、应用小波理论、位势分析和逼近理论等领域。近年来，变指数函数空间理论因其独特而丰富的结构及其在变分和流体力学中的应用而备受关注。另一个话题是关于高维逼近，它已经成为一个非常活跃的研究领域。这是由于数值数学的需要以及金融数学、化学和其他领域的应用程序的需要，在这些领域，基础领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，这意味着直到乘法常数。出于实际目的，这样的估计是无用的，除非有关于隐藏常量的额外信息，特别是它们对维度的依赖。在一些特殊情况下的第一个结果表明，情况可能与以前所知的完全不同。我们主要研究以下问题：刻画欧氏空间和区域上的Besov型和Triebel-Lizorkin型空间上的逐点乘子类；发展一些一般区域上的Besov型和Triebel-Lizorkin型空间的理论；通过不同的方法找到变量Besov(-型)和Triebel-Lizorkin(-型)空间的内插空间；对定义在立方体上的Sobolev空间找到精确的渐近和预渐近估计；找到图上某些半线性方程有(唯一)解的条件；找出黎曼流形上一些与椭圆算子有关的微分不等式非负解唯一性的临界指标。中国和德国的团队有足够的专业知识和实力来应对这些具有挑战性的最新问题。

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.