The Real-Variable Theory of Function Spaces and its Applications

本课题涉及欧氏空间、定义域和度量空间（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谐波分析的核心问题之一，有着广泛的应用。光滑函数空间，尤其是Sobolev空间，在变分学和偏微分方程中有着广泛的应用。Besov空间和triiebel - lizorkin空间作为更一般的函数空间尺度，与Sobolev空间的迹线和插值研究联系在一起。它们也被用于各种偏微分方程，研究欧拉方程、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、势分析和近似理论等领域也具有重要意义。变指数函数空间理论由于其特殊而丰富的结构及其在变分学和流体力学中的应用，近年来引起了人们的广泛关注。另一个话题是高维近似，这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在金融数学、化学和其他领域的应用，在这些领域，基础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。对于实际目的，这种估计是无用的，除非有关于隐藏常数的附加信息，特别是它们对维度的依赖。在一些特殊情况下的初步结果表明，情况可能与以前所知道的完全不同。这是一个有趣的效应，对实际问题很重要。总之，我们想研究以下问题：欧几里得空间和域上besov型和triiebel - lizorkin型空间上的点乘子类的刻画；在一些一般域上发展besov型和triiebel - lizorkin型空间理论；通过不同的方法求出变量Besov（type）和triiebel - lizorkin （type）空间的插值空间；寻找定义在立方体上的Sobolev空间的渐近估计和前渐近估计；找出图上某些半线性方程有（唯一）解的条件；寻找黎曼流形上与椭圆算子有关的微分不等式非负解唯一性的临界指标。中德团队有足够的专业知识和实力应对这些具有挑战性和时代性的问题。

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.