Hölder analysis and geometry on Banach spaces: homogeneous homeomorphisms and commutative group structures, approximation and Tzar’kov’s phenomenon. Part I

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 non-commutative $L_p$-spaces, the classes $IG$ and $IG_+$ of independently generated spaces (for example, non-commutative-valued 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 re-homogenisation 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 three-space problem for the Hölder classification of infinite-dimensional 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.