The Real-Variable Theory of Function Spaces and its Applications

• Applicant: Professorin Dr. Dorothee Haroske
• Subject Area: Mathematics
• Term: from 2018 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 392255916

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.

• DFG Programme: Research Grants
• International Connection: China
• Partner Organisation: National Natural Science Foundation of China
• Cooperation Partner: Professor Dr. Dachun Yang