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

• Final Report Year: 2024

Final Report Abstract

The topic of this project was 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. The scales of Besov spaces and Triebel-Lizorkin spaces are of particular interest in view of the study of traces and interpolation. They have also been used in various PDEs, as well as in signal analysis, applied wavelet theory, (high-dimensional) approximation and various other areas. We concentrated in this project on the systematic study of the theory of function spaces of Besov and Triebel- Lizorkin type on Euclidean spaces and on bounded domains, in particular on Morrey smoothness spaces and spaces with variable exponents. Another topic concerned high-dimensional approximation with a variety of recent applications, e.g. in financial mathematics, chemistry and other areas, where the dimension of the underlying domain could be very large. In addition we studied existence and uniqueness of certain PDE in different settings like on graphs and on Riemannian manifolds. There was an obvious and major obstacle we (all) had to struggle with: the pandemic with all its constraints just during the main part of our project. So direct exchange and contact via conferences, workshops, visits etc. was very difficult or later impossible. We tried to cope with this as much as possible and managed to organise at least one big international conference in each of the countries. In addition we continued the joint work online and obtained a number of very interesting results. We also started new ideas for our joint cooperation in future.

Publications

• An asymptotic sharp Sobolev regularity for planar infinity harmonic functions. Journal de Mathématiques Pures et Appliquées, 132, 457-482.
  Koch, Herbert; Zhang, Yi Ru-Ya & Zhou, Yuan
  
• New molecular characterizations of anisotropic Musielak–Orlicz Hardy spaces and their applications. Journal of Mathematical Analysis and Applications, 475(2), 1341-1366.
  Liu, Jun; Haroske, Dorothee D. & Yang, Dachun
  
• Some sharp Sobolev regularity for inhomogeneous infinity Laplace equation in plane. Journal de Mathématiques Pures et Appliquées, 132, 483-521.
  Koch, Herbert; Zhang, Yi Ru-Ya & Zhou, Yuan
  
• Decompositions with Atoms and Molecules for Variable Exponent Triebel–Lizorkin–Morrey Spaces. Constructive Approximation, 53(1), 201-234.
  Caetano, António & Kempka, Henning
  
• Entropy numbers of compact embeddings of smoothness Morrey spaces on bounded domains. Journal of Approximation Theory, 256, 105424.
  Haroske, Dorothee D. & Skrzypczak, Leszek
  
• Hardy's inequality and Green function on metric measure spaces. Journal of Functional Analysis, 281(3), 109020.
  Cao, Jun; Grigor'yan, Alexander & Liu, Liguang
  
• How anisotropic mixed smoothness affects the decay of singular numbers for Sobolev embeddings. Journal of Complexity, 63, 101523.
  Kühn, Thomas; Sickel, Winfried & Ullrich, Tino
  
• Regularity for Robin boundary problems of Laplace equations and Hardy spaces on C1 and (semi-)convex domains. Journal of Differential Equations, 279, 198-244.
  Yang, Sibei; Sickel, Winfried; Yang, Dachun & Yuan, Wen
  
• Discrete Tori and Trigonometric Sums. The Journal of Geometric Analysis, 32(12).
  Grigor’yan, Alexander; Lin, Yong & Yau, Shing-Tung
  
• Morrey smoothness spaces: A new approach. Science China Mathematics, 66(6), 1301-1358.
  Haroske, Dorothee D. & Triebel, Hans