Function spaces with dominating mixed smoothness (Q1755284)

This monograph is about fundamental questions regarding anisotropic function spaces. Starting from spaces of univariate functions, there are several ways how to extend smoothness and integrability properties to the multivariate case, i.e., spaces of functions on $\mathbb{R}^{n}$. There are two special problems to be highlighted. The first one concerns the relation between function spaces on domains $\Omega\subset \mathbb{R}^{n}$ and on all of $\mathbb{R}^{n}$, and this is connected with the extension problem. Another problem is how to integrate anisotropy. There is a rich and highly developed theory of isotropic function spaces, such as, e.g., Besov and Sobolev type spaces. Also, the relation between function spaces on domains and on all of $\mathbb{R}^{n}$ is well understood. The present monograph is, among many other details, a systematic treatment of both the above problems. \par In Chapter 1 (Spaces on $\mathbb{R}^{n}$), the author complements the existing theory for both isotropic spaces and spaces of dominating mixed smoothness. The most prominent example for this class of spaces is $S_{1}^{1} W((0,1)^{n})$. It consists of all multivariate functions $f$ whose highest (weak) derivatives $\frac{\partial^{n}}{\partial x_{1}^{1}\dots \partial x_{n}^{1}}f$ are absolutely integrable. A relevant feature of this is the following: Let $\varphi$ be a (smooth) diffeomorphism on $(0,1)^{n}$. If a function $f\in S_{1}^{1} W((0,1)^{n}) $ has dominating mixed smoothness, then the transformed $f\circ \varphi$ will (in general) not have this property. Especially, such anisotropic spaces are aligned to fixed coordinate systems, in contrast to the isotropic spaces. It is clear that this fact has significant impact when treating the extension problem from arbitrary domains $\Omega$ to all of $\mathbb{R}^{n}$. For domains aligned to the axes, such as, e.g., $(0,1)^{n}$, things behave as expected, whereas for other domains a special calculus is required, and the extension problem does not have an affirmative answer in general. In order to thoroughly discuss this type of questions, the author collects material, standard and nonstandard, in this first chapter. It is good to know the corresponding discussion as provided in the earlier monographs [\textit{H. Triebel}, Function spaces and wavelets on domains. Zürich: European Mathematical Society (EMS) (2008; Zbl 1158.46002)] and [\textit{H. Triebel}, Bases in function spaces, sampling, discrepancy, numerical integration. Zürich: European Mathematical Society (EMS) (2010; Zbl 1202.46002)]. There is a special `intermezzo' (\S 1.3) which discusses certain types of problems which arise when comparing properties of isotropic and anisotropic spaces. \par Chapter 2 (Spaces on domains) deals with incorporating dominant mixed smoothness for functions on (arbitrary bounded) domains. In order to overcome the difficulties arising from the alignment to coordinate systems, the author pursues a way relying on the well-known Whitney decomposition of the given domain. By taking into account the distance of a given point to the boundary of the domain, weighted localization spaces are introduced. Properties of these (and refined spaces) are discussed, in particular, relating these spaces to Besov and Sobolev type spaces. \par This monograph presents an illuminating and concise view on the construction of spaces of dominating mixed smoothness. It is addressed to graduate students, but a profound knowledge of the classical theory of Besov and Sobolev type spaces seems indispensable.