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Axioms 2013, 2(2), 224-270; doi:10.3390/axioms2020224
Article

Quantitative Hahn-Banach Theorems and Isometric Extensions for Wavelet and Other Banach Spaces

Received: 4 March 2013; in revised form: 12 May 2013 / Accepted: 14 May 2013 / Published: 23 May 2013
(This article belongs to the Special Issue Wavelets and Applications)
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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.
Keywords: wavelet norms; quantitative Hahn-Banach theorem; isometric extension; Hölder-Lipschitz mapping; IG-spaces; non-commutative Lp-spaces; Clarkson, Jacobi and Pichugov classes; Besov, Lizorkin-Triebel and Sobolev spaces; anisotropic function spaces; Schatten-von Neumann classes wavelet norms; quantitative Hahn-Banach theorem; isometric extension; Hölder-Lipschitz mapping; IG-spaces; non-commutative Lp-spaces; Clarkson, Jacobi and Pichugov classes; Besov, Lizorkin-Triebel and Sobolev spaces; anisotropic function spaces; Schatten-von Neumann classes
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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MDPI and ACS Style

Ajiev, S. Quantitative Hahn-Banach Theorems and Isometric Extensions for Wavelet and Other Banach Spaces. Axioms 2013, 2, 224-270.

AMA Style

Ajiev S. Quantitative Hahn-Banach Theorems and Isometric Extensions for Wavelet and Other Banach Spaces. Axioms. 2013; 2(2):224-270.

Chicago/Turabian Style

Ajiev, Sergey. 2013. "Quantitative Hahn-Banach Theorems and Isometric Extensions for Wavelet and Other Banach Spaces." Axioms 2, no. 2: 224-270.

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