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This paper provides a characterization of when two expansive matrices yield the same anisotropic local Hardy and inhomogeneous Triebel–Lizorkin spaces. The characterization is in terms of the coarse equivalence of certain quasi-norms associated to the matrices. For nondiagonal matrices, these conditions are strictly weaker than those classifying the coincidence of the corresponding homogeneous function spaces. The obtained results complete the classification of anisotropic Besov and Triebel–Lizorkin spaces associated to general expansive matrices. [ABSTRACT FROM AUTHOR]

In this paper, we consider traces at initial times for functions with mixed time‐space smoothness. Such results are often needed in the theory of evolution equations. Our result extends and unifies many previous results. Our main improvement is that we can allow general interpolation couples. The abstract results are applied to regularity problems for fractional evolution equations and stochastic evolution equations, where uniform trace estimates on the half‐line are shown. [ABSTRACT FROM AUTHOR]

The primitive equations are fundamental models in geophysical fluid dynamics and derived from the scaled Navier-Stokes equations. In the primitive equations, the evolution equation to the vertical velocity is replaced by the so-called hydrostatic approximation. In this paper, we give a justification of the hydrostatic approximation by the scaled Navier-Stokes equations in anisotropic spaces LH∞Lx3q(T³) for q ≥ 1. [ABSTRACT FROM AUTHOR]

We introduce and study Clarkson, Dol’nikov-Pichugov, Jacobi and mutual diameter constants reflecting the geometry of a Banach space and Clarkson, Jacobi and Pichugov classes of Banach spaces and their relations with James, self-Jung, Kottman and Schäffer constants in order to establish quantitative versions of Hahn-Banach separability theorem and to characterise the isometric extendability of Hölder-Lipschitz mappings. Abstract results are further applied to the spaces and pairs from the wide classes IG and IG+ and non-commutative Lp-spaces. The intimate relation between the subspaces and quotients of the IG-spaces on one side and various types of anisotropic Besov, Lizorkin-Triebel and Sobolev spaces of functions on open subsets of an Euclidean space defined in terms of differences, local polynomial approximations, wavelet decompositions and other means (as well as the duals and the lp-sums of all these spaces) on the other side, allows us to present the algorithm of extending the main results of the article to the latter spaces and pairs. Special attention is paid to the matter of sharpness. Our approach is quasi-Euclidean in its nature because it relies on the extrapolation of properties of Hilbert spaces and the study of 1-complemented subspaces of the spaces under consideration.

We introduce and study Clarkson, Dol'nikov-Pichugov, Jacobi and mutual diameter constants reflecting the geometry of a Banach space and Clarkson, Jacobi and Pichugov classes of Banach spaces and their relations with James, self-Jung, Kottman and Schäffer constants in order to establish quantitative versions of Hahn-Banach separability theorem and to characterise the isometric extendability of Hölder-Lipschitz mappings. Abstract results are further applied to the spaces and pairs from the wide classes IG and IG+ and non-commutative Lp-spaces. The intimate relation between the subspaces and quotients of the IG-spaces on one side and various types of anisotropic Besov, Lizorkin-Triebel and Sobolev spaces of functions on open subsets of an Euclidean space defined in terms of differences, local polynomial approximations, wavelet decompositions and other means (as well as the duals and the lp-sums of all these spaces) on the other side, allows us to present the algorithm of extending the main results of the article to the latter spaces and pairs. Special attention is paid to the matter of sharpness. Our approach is quasi-Euclidean in its nature because it relies on the extrapolation of properties of Hilbert spaces and the study of 1-complemented subspaces of the spaces under consideration. [ABSTRACT FROM AUTHOR]

Abstract: In this article we construct affine systems that provide a simultaneous atomic decomposition for a wide class of functional spaces including the Lebesgue spaces , . The novelty and difficulty of this construction is that we allow for non-lattice translations. We prove that for an arbitrary expansive matrix and any set —satisfying a certain spreadness condition but otherwise irregular—there exists a smooth window whose translations along the elements of and dilations by powers of provide an atomic decomposition for the whole range of the anisotropic Triebel–Lizorkin spaces. The generating window can be either chosen to be bandlimited or to have compact support. To derive these results we start with a known general “painless” construction that has recently appeared in the literature. We show that this construction extends to Besov and Triebel–Lizorkin spaces by providing adequate dual systems. [Copyright &y& Elsevier]

In this paper we generalize the analysis of the Campanato-type dual spaces 1, 3, for the highly anisotropic Hardy spaces on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^n$\end{document} introduced in 10. These Hardy spaces are constructed over multi-level ellipsoid covers of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^n$\end{document}, where the ellipsoids can change shape rapidly from point to point and from level to level. [ABSTRACT FROM AUTHOR]

Abstract: We investigate anisotropic function spaces defined over the multi-level ellipsoid covers of , where the ellipsoids can quickly change shape from point to point and from level to level. We explicitly define an anisotropic modulus of smoothness (already used implicitly in Dahmen et al. (2010) ) and investigate its properties. We show anisotropic variants of classic inequalities such as the Marchaud, Nikolskii and Ul’yanov, relationships with isotropic smoothness and applications to anisotropic Besov space embedding. [Copyright &y& Elsevier]

We present a general method for extending decomposition systems of L2 (Rd) to decomposition systems for the anisotropic Triebel--Lizorkin and Besov spaces, Fα,s,p,q and Bα,a p,q, respectively, for the full range of the indexes. Our approach is based on techniques from harmonic analysis and relies on the boundedness of almost diagonal operators on appropriate sequence spaces. Typical examples of such decomposition systems are the various wavelet-type unconditional bases for L2 (Rd). [ABSTRACT FROM AUTHOR]

The goal of this paper is to provide wavelet characterizations for anisotropic Besov spaces. Depending on the anisotropy, appropriate biorthogonal tensor product bases are introduced and Jackson and Bernstein estimates are proved for two-parameter families of finite-dimensional spaces. These estimates lead to characterizations for anisotropic Besov spaces by anisotropy-dependent linear approximation spaces and lead further on to interpolation and embedding results. Finally, wavelet characterizations for anisotropic Besov spaces with respect to Lp-spaces with 0