Abstract
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.
Highlights
In a sense, the theory of function spaces has appeared as a result of “building a bridge” between abstract methods of functional analysis and mathematical physics (PDE, or harmonic/real analysis in a wider sense) by Sergey L
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 on the other side, allows us to present the algorithm of extending the main results of the article to the latter spaces and pairs
While our Dol’nikov-Pichugov constant serves as the quantitative reflection of the quantitative Hahn-Banach theorems, the introduction of Pichugov classes in Section 5.2 and the study of them and their relations with other constants and classes of Banach spaces is the most important tool for establishing the quantitative Hahn-Banach theorems for the subspaces and quotients of the spaces from very large auxiliary classes of independently generated spaces IG and IG+ that we introduce in Section 2.1 to cover, firstly, Lebesgue and sequence lp space with mixed norm, sequence spaces appearing as wavelet spaces
Summary
The theory of function spaces has appeared as a result of “building a bridge” between abstract methods of functional analysis and mathematical physics (PDE, or harmonic/real analysis in a wider sense) by Sergey L. The spaces ltp,q and lt∗p,q contain isometric and 1-complemented copies of lp and lq according to the lemma. Direct constructions of the isomorphic (and often complemented) copies of various sequence spaces (for example, mixed lp and Lorentz Lp,r ) in various anisotropic Besov, Sobolev and Lizorkin-Triebel spaces of functions defined on open subsets of Rn are presented in Section 3 in [2]. It is convenient to think about the parameters as parameter functions p = pX : P → [1, ∞] defined on one and the same parameter position set P = V(X) (here we slightly abuse the notation in the sense that the vertexes that are classes ltp,q or lt∗p,q are doubled to cover both their parameters, or that the value of p on them is 2-dimensional with the vector operations as in the beginning of this section) determined by T and the spaces at its vertexes. The resulting space Y ∈ IG0+ is what we are looking for
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