
This article is cited in 1 scientific paper (total in 1 paper)
Hölder analysis and geometry on Banach spaces: homogeneous homeomorphisms and commutative group structures, approximation and Tzar’kov’s phenomenon. Part I
S. S. Ajiev^{} ^{} School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW, 2522, Australia
Abstract:
In an explicit quantitative and often precise manner, we construct the homogeneous Hölder homeomorphisms and study the approximation of uniformly continuous mappings by the Hölder–Lipschitz ones between the pairs of abstract and concrete metric and (quasi) Banach spaces including, in particular, Banach lattices, general noncommutative $L_p$spaces, the classes $IG$ and $IG_+$ of independently generated spaces (for example, noncommutativevalued Bochner–Lebesgue spaces) and anisotropic Sobolev, Nikol’skii–Besov and Lizorkin–Triebel spaces of functions on an open subset or a class of domains of an Euclidean space defined with underlying mixed $L_p$norms in terms of differences, local approximations by polynomials, wavelet decompositions and systems of closed operators, such as holomorphic functional calculus and Fourier multipliers of smooth Littlewood–Paley decompositions. Our approach also allows to treat both the finite (as in the initial and/or boundary value problems in PDE) and infinite $l_p$sums of these spaces, their duals and “Bochnerizations”. Many results are automatically extended to the setting of the function spaces with variable smoothness, including the weighted ones. The sharpness of the approximation results, shown for the majority of the pairs under some mild conditions and underpinning the corresponding sharpness of the Hölder continuity exponents of the homogeneous homeomorphisms, indicates that the range of the exponents is often a proper subset of $(0,1]$, that is the presence of Tsar’kov’s phenomenon. We also consider the approximation by the mappings taking the values in the convex envelope of the range of the original approximated mapping. The negative results on the absence of uniform embeddings of the balls of some function spaces, particularly including $BMO$, $VMO$, Nikol’skii–Besov and Lizorkin–Triebel spaces with $q=\infty$ and their $VMO$like separable subspaces, into any Hilbert space are established. Relying on the solution to the problem of the global Hölder continuity of metric projections and the existence of the Hölder continuous homogeneous right inverses of closed surjective operators and retractions onto closed convex subsets, as well as our results on the bounded extendability of the Hölder–Lipschitz mappings and rehomogenisation technique, we develop and employ our key explicit quantitative tools, such as the global (on arbitrary bounded subsets) Hölder continuity of the duality mapping and Lozanovskii factorisation, the answer to the threespace problem for the Hölder classification of infinitedimensional spheres, the Hölder continuous counterpart of the Kalton–Pełczyńki decomposition method, the Hölder continuity of the homogeneous homeomorphism induced by the complex interpolation method and such counterparts of the classical Mazur mappings as the abstract and simple Mazur ascent and complex Mazur descent. Important role is also played by the study of the local unconditional structure and other complementability results, as well as the existence of equivalent geometrically friendly norms.
Keywords and phrases:
Hölder classification of spheres, Hölder–Lipschitz mappings, approximation of uniformly continuous mappings, Tsar’kov’s phenomenon, Mazur mappings, Lozanovskii factorisation, homogeneous right inverses, metric projection, asymmetric uniform convexity and smoothness, dimensionfree estimates, $\lambda$horn condition, local unconditional structure, Nikol’skii–Besov, Lizorkin–Triebel, Sobolev, noncommutative $L_p$ and $IG$spaces, Banach lattices, wavelets, threespace problem, Kalton–Pełczyńki decomposition, bounded extension of Hölder mappings, Markov type and cotype, complemented subspaces, UMD.
Full text:
PDF file (663 kB)
References:
PDF file
HTML file
MSC: 46Txx, 46Exx, 47Jxx, 47Lxx, 58Dxx, 46T20, 46E35, 47L05, 15A60, 47J07, 46L52 Received: 17.03.2011
Language:
Citation:
S. S. Ajiev, “Hölder analysis and geometry on Banach spaces: homogeneous homeomorphisms and commutative group structures, approximation and Tzar’kov’s phenomenon. Part I”, Eurasian Math. J., 5:1 (2014), 7–60
Citation in format AMSBIB
\Bibitem{Aji14}
\by S.~S.~Ajiev
\paper H\"older analysis and geometry on Banach spaces: homogeneous homeomorphisms and commutative group structures, approximation and Tzar’kov’s phenomenon. Part I
\jour Eurasian Math. J.
\yr 2014
\vol 5
\issue 1
\pages 760
\mathnet{http://mi.mathnet.ru/emj148}
Linking options:
http://mi.mathnet.ru/eng/emj148 http://mi.mathnet.ru/eng/emj/v5/i1/p7
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles
Cycle of papers
This publication is cited in the following articles:

S. S. Ajiev, “Hölder analysis and geometry on Banach spaces: homogeneous homeomorphisms and commutative group structures, approximation and Tzar'kov's phenomenon. Part II”, Eurasian Math. J., 5:2 (2014), 7–51

Number of views: 
This page:  205  Full text:  108  References:  53 
