We present here a new method for approximating functions defined on superreflexive Banach spaces by differentiable functions with $α$-Hölder derivatives (for some $0<α\leq 1$). The smooth approximation is given by means of an explicit formula enjoying good properties from the minimization point of view. For instance, for any function $f$ which is bounded below and uniformly continuous on bounded sets this formula gives a sequence of $Δ$-convex ${\Cal{C}}^{1,α}$ functions converging uniformly on bounded sets to $f$ and preserving the infimum and the set of minimizers of $f$. The techniques we develop are based on the use of {\sl extended inf-convolution} formulas and convexity properties such as the preservation of smoothness for the convex envelope of certain differentiable functions.
The aim of this paper is to show that almost greedy bases induce tighter embeddings in superreflexive Banach spaces than in general Banach spaces. More specifically, we show that an almost greedy basis in a superreflexive Banach space $\mathbb{X}$ induces embeddings that allow squeezing $\mathbb{X}$ between two superreflexive Lorentz sequence spaces that are close to each other in the sense that they have the same fundamental function.
The class J of simultaneously almost transitive, uniformly convex and uniformly smooth Banach spaces is characterized in terms of convex-transitivity and weak geometry of the norm.
For a conditional quasi-greedy basis $\mathcal{B}$ in a Banach space the associated conditionality constants $k_{m}[\mathcal{B}]$ verify the estimate $k_{m}[\mathcal{B}]=\mathcal{O}(\log m)$. Answering a question raised by Temlyakov, Yang, and Ye, several authors have studied whether this bound can be improved when we consider quasi-greedy bases in some special class of spaces. It is known that every quasi-greedy basis in a superreflexive Banach space verifies $k_{m}[\mathcal{B}]=(\log m)^{1-ε}$ for some $0<ε<1$, and this is optimal. Our first goal in this paper will be to fill the gap in betw…
Let $M$ be a compact subset of a superreflexive Banach space. We prove that the Lipschitz-free space $\mathcal{F}(M)$, the predual of the Banach space of Lipschitz functions on $M$, has the Pełczyński's property ($V^\ast$). As a consequence, the Lipschitz-free space $\mathcal{F}(M)$ is weakly sequentially complete.
Let $X$ denote $\mathbb{R}^n$ or, more generally, a Hilbert space. Given an arbitrary subset $C$ of $X$ and a collection $\mathcal{H}$ of affine hyperplanes of $X$ such that every $H\in\mathcal{H}$ passes through some point $x_{H}\in C$, and $C=\{x_H : H\in\mathcal{H}\}$, what conditions are necessary and sufficient for the existence of a $C^{1,1}$ convex hypersurface $S$ in $X$ such that $H$ is tangent to $S$ at $x_H$ for every $H\in\mathcal{H}$? In this paper we give an answer to this question. We also provide solutions to similar problems for convex hypersurfaces of class $C^{1, ω}$ in Hilb…
Banach Spaces Generated by Finite-Valued Functions: Superreflexive Rigidity, Hankel Operators, and Universality
Konstantin Preslavsky University of Shumen（舒门康斯坦丁·普拉斯夫斯基大学）；University of Plovdiv Paisii Hilendarski（普罗夫迪夫帕伊西·希伦达斯基大学）；Technical University of Varna（瓦尔纳技术大学）
Is shown that any separable superreflexive Banach space X may be isometrically embedded in a separable superreflexive Banach space Z=Z(X) (which, in addition, is of the same type and cotype as X) such that its conjugate admits a continuous surjection on each its subspace. This gives an affirmative answer on S. Banach problem: Whether there exists a Banach space X, non isomorphic to a Hilbert space, which admits a continuous linear surjection on each its subspace and is essentially different from l_1?
It is known that for a conditional quasi-greedy basis $\mathcal{B}$ in a Banach space $\mathbb{X}$, the associated sequence $(k_{m}[\mathcal{B}])_{m=1}^{\infty}$ of its conditionality constants verifies the estimate $k_{m}[\mathcal{B}]=\mathcal{O}(\log m)$ and that if the reverse inequality $\log m =\mathcal{O}(k_m[\mathcal{B}])$ holds then $\mathbb{X}$ is non-superreflexive. Indeed, it is known that a quasi-greedy basis in a superreflexive quasi-Banach space fulfils the estimate $k_{m}[\mathcal{B}]=\mathcal{O}(\log m)^{1-ε}$ for some $ε>0$. However, in the existing literature one finds very f…
The main goal of this paper is to develop a new embedding method which we use to show that some finite metric spaces admit low-distortion embeddings into all non-superreflexive spaces. This method is based on the theory of equal-signs-additive sequences developed by Brunel and Sucheston (1975-1976). We also show that some of the low-distortion embeddability results obtained using this method cannot be obtained using the method based on the factorization between the summing basis and the unit vector basis of $\ell_1$, which was used by Bourgain (1986) and Johnson and Schechtman (2009).
It is known that for a conditional quasi-greedy basis $\mathcal{B}$ in a Banach space $\mathbb{X}$, the associated sequence $(k_{m}[\mathcal{B}])_{m=1}^{\infty}$ of its conditionality constants verifies the estimate $k_{m}[\mathcal{B}]=\mathcal{O}(\log m)$ and that if the reverse inequality $\log m =\mathcal{O}(k_m[\mathcal{B}])$ holds then $\mathbb{X}$ is non-superreflexive. However, in the existing literature one finds very few instances of non-superreflexive spaces possessing quasi-greedy basis with conditionality constants as large as possible. Our goal in this article is to fill this gap.…
This paper is devoted to theoretical aspects on optimality of sparse approximation. We undertake a quantitative study of new types of greedy-like bases that have recently arisen in the context of nonlinear $m$-term approximation in Banach spaces as a generalization of the properties that characterize almost greedy bases, i.e., quasi-greediness and democracy. As a means to compare the efficiency of these new bases with already existing ones in regards to the implementation of the Thresholding Greedy Algorithm, we place emphasis on obtaining estimates for their sequence of unconditionality param…
It is proved that there exists a separable reflexive Banach space W that contains an isomorphic image of every separable superreflexive Banach space. This gives the affirmative answer on one J. Bourgain's question
In this paper we give some results about the approximation of a Lipschitz function on a Banach space by means of $Δ$-convex functions. In particular, we prove that the density of $Δ$-convex functions in the set of Lipschitz functions for the topology of uniform convergence on bounded sets characterizes the superreflexivity of the Banach space. We also show that Lipschitz functions on superreflexive Banach spaces are uniform limits on the whole space of $Δ$-convex functions.
Let $\H$ denote the discrete Heisenberg group, equipped with a word metric $d_W$ associated to some finite symmetric generating set. We show that if $(X,\|\cdot\|)$ is a $p$-convex Banach space then for any Lipschitz function $f:\H\to X$ there exist $x,y\in \H$ with $d_W(x,y)$ arbitrarily large and \begin{equation}\label{eq:comp abs} \frac{\|f(x)-f(y)\|}{d_W(x,y)}\lesssim \left(\frac{\log\log d_W(x,y)}{\log d_W(x,y)}\right)^{1/p}. \end{equation} We also show that any embedding into $X$ of a ball of radius $R\ge 4$ in $\H$ incurs bi-Lipschitz distortion that grows at least as a constant multipl…
A Banach space X is superreflexive if each Banach space Y that is finitely representable in X is reflexive. Superreflexivity is known to be equivalent to J-convexity and to the non-existence of uniformly bounded factorizations of the summation operators S_n through X. We give a quantitative formulation of this equivalence. This can in particular be used to find a factorization of S_n through X, given a factorization of S_N through [L_2,X], where N is `large' compared to n.
In article is constructed a wide couple of pairwice non-isomorphic separable superreflexive Banach spaces E that are subspace homogeneous. Their conjugates are quotient homogeneous. None of this couple neither isomorphic to its Cartesian square nor has the approximation property. At the same time, any such E is isomorphic to E+ E+ W for some Banach space W and, hence, solves the Schroeder - Bernstein problem.
We study biorthogonal sequences with special properties, such as weak or weak-star convergence to 0, and obtain an extension of the Josefson-Nissenzweig theorem. This result is applied to embed analytic disks in the fiber over 0 of the spectrum of H^infinity (B), the algebra of bounded analytic functions on the unit ball B of an arbitrary infinite dimensional Banach space. Various other embedding theorems are obtained. For instance, if the Banach space is superreflexive, then the unit ball of a Hilbert space of uncountable dimension can be embedded analytically in the fiber over 0 via an embed…
Given $X$ a Hilbert space, $ω$ a modulus of continuity, $E$ an arbitrary subset of $X$, and functions $f:E\to\mathbb{R}$, $G:E\to X$, we provide necessary and sufficient conditions for the jet $(f,G)$ to admit an extension $(F, \nabla F)$ with $F:X\to \mathbb{R}$ convex and of class $C^{1, ω}(X)$, by means of a simple explicit formula. As a consequence of this result, if $ω$ is linear, we show that a variant of this formula provides explicit $C^{1,1}$ extensions of general (not necessarily convex) $1$-jets satisfying the usual Whitney extension condition, with best possible Lipschitz constants…
We study Daugavet- and $Δ$-points in Banach spaces. A norm one element $x$ is a Daugavet-point (respectively a $Δ$-point) if in every slice of the unit ball (respectively in every slice of the unit ball containing $x$) you can find another element of distance as close to $2$ from $x$ as desired. In this paper we look for criteria and properties ensuring that a norm one element is not a Daugavet- or $Δ$-point. We show that asymptotically uniformly smooth spaces and reflexive asymptotically uniformly convex spaces do not contain $Δ$-points. We also show that the same conclusion holds true for th…
We study property (T) and the fixed point property for actions on $L^p$ and other Banach spaces. We show that property (T) holds when $L^2$ is replaced by $L^p$ (and even a subspace/quotient of $L^p$), and that in fact it is independent of $1\leq p<\infty$. We show that the fixed point property for $L^p$ follows from property (T) when $1<p< 2+\e$. For simple Lie groups and their lattices, we prove that the fixed point property for $L^p$ holds for any $1< p<\infty$ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products …
A theorem of Delorme states that every unitary representation of a connected Lie group with nontrivial reduced first cohomology has a finite-dimensional subrepresentation. More recently Shalom showed that such a property is inherited by cocompact lattices and stable under coarse equivalence among amenable countable discrete groups. We give a new geometric proof of Delorme's theorem which extends to a larger class of groups, including solvable $p$-adic algebraic groups, and finitely generated solvable groups with finite Prüfer rank. Moreover all our results apply to isometric representations in…
We prove that SL(3,R) has Strong Banach property (T) in Lafforgue's sense with respect to the Banach spaces that are $θ>0$ interpolation spaces (for the complex interpolation method) between an arbitrary Banach space and a Banach space with sufficiently good type and cotype. As a consequence, every action of SL(3,R) or its lattices by affine isometries on such a Banach space X has a fixed point, and the expanders contructed from SL(3,Z) do not admit a coarse embedding into X. We also prove a quantitative decay of matrix coefficients (Howe-Moore property) for representations with small exponent…
We provide necessary and sufficient conditions for a $1$-jet $(f, G):E\rightarrow \mathbb{R} \times X$ to admit an extension $(F, \nabla F)$ for some $F\in C^{1, ω}(X)$. Here $E$ stands for an arbitrary subset of a Hilbert space $X$ and $ω$ is a modulus of continuity. As a corollary, in the particular case $X=\mathbb{R}^n$, we obtain an extension (nonlinear) operator whose norm does not depend on the dimension $n$. Furthermore, we construct extensions $(F, \nabla F)$ in such a way that: (1) the (nonlinear) operator $(f, G)\mapsto (F, \nabla F)$ is bounded with respect to a natural seminorm ari…
