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Keywords = Besov, Lizorkin-Triebel and Sobolev spaces

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46 pages, 576 KiB  
Article
Lebesgue Points of Besov and Triebel–Lizorkin Spaces with Generalized Smoothness
by Ziwei Li, Dachun Yang and Wen Yuan
Mathematics 2021, 9(21), 2724; https://doi.org/10.3390/math9212724 - 27 Oct 2021
Cited by 10 | Viewed by 2242
Abstract
In this article, the authors study the Lebesgue point of functions from Hajłasz–Sobolev, Besov, and Triebel–Lizorkin spaces with generalized smoothness on doubling metric measure spaces and prove that the exceptional sets of their Lebesgue points have zero capacity via the capacities related to [...] Read more.
In this article, the authors study the Lebesgue point of functions from Hajłasz–Sobolev, Besov, and Triebel–Lizorkin spaces with generalized smoothness on doubling metric measure spaces and prove that the exceptional sets of their Lebesgue points have zero capacity via the capacities related to these spaces. In case these functions are not locally integrable, the authors also consider their generalized Lebesgue points defined via the γ-medians instead of the classical ball integral averages and establish the corresponding zero-capacity property of the exceptional sets. Full article
(This article belongs to the Special Issue Recent Developments of Function Spaces and Their Applications I)
47 pages, 431 KiB  
Article
Quantitative Hahn-Banach Theorems and Isometric Extensions for Wavelet and Other Banach Spaces
by Sergey Ajiev
Axioms 2013, 2(2), 224-270; https://doi.org/10.3390/axioms2020224 - 23 May 2013
Cited by 5 | Viewed by 5288
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Wavelets and Applications)
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