Pointwise variable anisotropic function spaces on \(\mathbb{R}^n\)
This monograph is dedicated to an exhaustive analysis of pointwise variable anisotropic function spaces on \(\mathbb R^n\), including Hardy spaces, Campanato spaces, Besov spacs and Sobolev spaces. In [\textit{M. Bownik}, Anisotropic Hardy spaces and wavelets. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1036.42020)], Bownik first extended the theory of Hardy spaces to the anisotropic setting, where the spaces were defined via the maximal function \(M^0_\varphi f(x)=\sup_{k\in\mathbb Z}|f\ast \varphi_k(x)|\) with \(\varphi_k\) generated by the dilation structure \(\varphi_k(x)=|{\det A}|^{-k}\varphi(A^{-k}x)\) for an appropriate matrix \(A\) and given function \(\varphi\). In this monograph, the author further considers pointwise variable anisotropic function spaces related to affine transformations of the form \(A_{x,t}(y)=M_{x, t}y+x\) for all \(x,y\in \mathbb R^n\) and \(t\in \mathbb R\) such that \(M_{x,t}\) is a positive definite \(n\times n\) matrix (or \(A_{x,m}(y)=M_{x,m}y+x\) for all \(x,y\in\mathbb R^n\) and \(m\in\mathbb Z\)). Here are the topics covered in the book. Chapter 1 contains basic information of local approximation over domains in \(\mathbb R^n\), including \(K\)-functionals, Marchaud inequalities, the Bramble-Hilbert lemma and Whitney theorem for convex domains. Chapter 2 is devoted to the anisotropic multilevel ellipsoid covers of \(\mathbb R^n\), where ellipsoids are defined via affine transformations. In particular, construction and properties of continuous (and discrete) ellipsoid covers are presented. The purpose of Chapter 3 is to present multiresolution analysis constructions that are subordinate to the anisotropic quasi-distance induced by the ellipsoid covers. The multiresolution analysis construction on \(\mathbb R^n\) provides polynomial reproduction of arbitrary order and has arbitrarily high regularization. The author also provides anisotropic locally stable basis consisting of bumps supported over the ellipsoids of the covers, or supported over interactions of ellipsoids from adjacent levels. Chapter 4 contains a detailed study of anisotropic wavelets. The wavelet operators are defined via the differences of two-level adjacent quasi-projection kernels, and useful localized representations of the difference between two adjacent scales are constructed. It is then shown that two wavelet operators from different scales are almost orthogonal, which generalizes known results for the isotropic case. Using a discrete Calderón reproducing formula, the author further constructs discrete wavelet frames of \(L^2(\mathbb R^n)\). Chapter 5 is devoted to smoothness spaces. The author first generalizes the moduli of smoothness over \(\mathbb R^n\) to moduli that are subordinate to anisotropic ellipsoid covers. Then the author discusses the anisotropic Besov spaces and anisotropic Campanato spaces, and provides the Marchaud inequality and Ul'yanov inequality. In Chapter 6, the author deals with the anisotropic Hardy spaces for \(p\in(0, 1]\). The author introduces the spaces via the grand maximal functions defined by continuous ellipsoid covers \(\Theta\), and presents radial and nontangential maximal functions and atomic characterizations. The duals of the anisotropic spaces \(H^p(\Theta)\) are also identified with Campanato spaces. The final part of the monograph, Chapter 7, is devoted to the boundedness of anisotropic singular operators on \(H^p(\Theta)\) for \(p\in(0, 1]\). The author shows that a variable anisotropic singular integral operator \(T\) with respect to a continuous ellipsoid cover \(\Theta\) is bounded on \(H^p(\Theta)\) and from \(H^p(\Theta)\) to \(L^p(\mathbb R^n)\). The author also shows that a smooth variable anisotropic singular integral operator maps smooth atoms to smooth molecules with some quantifiable reduced regularity.
scientific article; zbMATH DE number 7438179