Widths of embeddings of 2-microlocal Besov spaces

We consider the asymptotic behaviour of the approximation, Gelfand and Kolmogorov numbers of compact embeddings between 2-microlocal Besov spaces with weights defined in terms of the distance to a $d$-set $U\subset \mathbb{R}^n$. The sharp estimates are shown in most cases, where the quasi-Banach setting is included.Comment: 28 page

Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type

Coorbit space theory is an abstract approach to function spaces and their atomic decompositions. The original theory developed by Feichtinger and Gr{\"o}chenig in the late 1980ies heavily uses integrable representations of locally compact groups. Their theory covers, in particular, homogeneous Besov-Lizorkin-Triebel spaces, modulation spaces, Bergman spaces, and the recent shearlet spaces. However, inhomogeneous Besov-Lizorkin-Triebel spaces cannot be covered by their group theoretical approach. Later it was recognized by Fornasier and the first named author that one may replace coherent states related to the group representation by more general abstract continuous frames. In the first part of the present paper we significantly extend this abstract generalized coorbit space theory to treat a wider variety of coorbit spaces. A unified approach towards atomic decompositions and Banach frames with new results for general coorbit spaces is presented. In the second part we apply the abstract setting to a specific framework and study coorbits of what we call Peetre spaces. They allow to recover inhomogeneous Besov-Lizorkin-Triebel spaces of various types of interest as coorbits. We obtain several old and new wavelet characterizations based on precise smoothness, decay, and vanishing moment assumptions of the respective wavelet. As main examples we obtain results for weighted spaces (Muckenhoupt, doubling), general 2-microlocal spaces, Besov-Lizorkin-Triebel-Morrey spaces, spaces of dominating mixed smoothness, and even mixtures of the mentioned ones. Due to the generality of our approach, there are many more examples of interest where the abstract coorbit space theory is applicable.Comment: to appear in Journal of Functional Analysi

Global well-posedness for the Yang-Mills equation in 4+14+1 dimensions. Small energy

We consider the hyperbolic Yang-Mills equation on the Minkowski space $\R^{4+1}$. Our main result asserts that this problem is globally well-posed for all initial data whose energy is sufficiently small. This solves a longstanding open problem.Comment: 53 page

Nonlinear Approximation and Function Space of Dominating Mixed Smoothness

As the main object of this thesis we study a generalization of function spaces with dominating smoothness. The basic idea for this generalization is based on a splitting of the d variables into N groups of possibly varying length. In this way the usual spaces of dominating mixed smoothness as well as the classical isotropic spaces occur as special cases (for d groups of length one each or for 1 group of length d, respectively). More precisely, we will study first Sobolev-type spaces, and afterwards Besov- and Triebel-Lizorkin-type spaces are introduced. For these spaces several basic properties such as Fourier multipliers and duality are discussed. As the main tool for further studies characterizations by local means are proved. The main result of the first part of the thesis consists in a characterization by tensor product Daubechies wavelets. One immediate corollary of this characterization is the identification of certain Besov spaces of dominating mixed smoothness as tensor products of isotropic ones, establishing a connection to many recent discussions in high-dimensional approximation. The second part of this thesis is devoted to the study of one particular method of nonlinear approximation, m-term approximation with respect to the mentioned tensor product spaces in the framework of the spaces of dominating mixed smoothness. Here the wavelet characterization comes into play, allowing a reformulation of this problem using associated sequence spaces and the canonical bases. After some preparatory considerations, including the investigation of continuous and compact embeddings and duality, some explicit constructions for m-term approximation in several different settings are studied. Our main attention is turned on the asymptotic behaviour of certain worst case errors of this method. After reformulating the results from the explicit constructions in this sense, these results are extended using assertions about real interpolation and reiteration. Finally, the results on these aysmptotic rates are transferred back to function spaces, using once more the wavelet characterization. The results obtained in this way improve earlier ones by Dinh Dung and Temlyakov