Superreflexive Banach Spaces Research Articles

On Whitney-type extension theorems on Banach spaces for C1,ω, C1,+, [formula omitted], and [formula omitted]-smooth functions

Our paper is a complement to a recent article by D. Azagra and C. Mudarra (2021, [2]). We show how older results on semiconvex functions with modulus ω easily imply extension theorems for C1,ω-smooth functions on super-reflexive Banach spaces which are versions of some theorems of Azagra and Mudarra. We present also some new interesting consequences which are not mentioned in their article, in particular extensions of C1,ω-smooth functions from open quasiconvex sets. They proved also an extension theorem for CB1,+-smooth functions (i.e., functions with uniformly continuous derivative on each bounded set) on Hilbert spaces. Our version of this theorem and new extension results for C1,+ and Cloc1,+-smooth functions (i.e., functions with uniformly, resp. locally uniformly continuous derivative), all of which are proved on arbitrary super-reflexive Banach spaces, are further main contributions of our paper. Some of our proofs use main ideas of the article by D. Azagra and C. Mudarra, but all are formally completely independent on their article.

Lorentz spaces and embeddings induced by almost greedy bases in superreflexive Banach spaces

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.

Nonlinear aspects of super weakly compact sets

The notion of super weak compactness for subsets of Banach spaces is a strengthening of the weak compactness that can be described as a local version of super-reflexivity. A recent result of K. Tu [32] which establishes that the closed convex hull of a super weakly compact set is super weakly compact has removed the main obstacle to further development of the theory. In this paper we provide a variety of results around super weak compactness in order to show the great scope of this notion. We also give non linear characterizations of super weak compactness in terms of the (non) embeddability of special trees and graphs. We conclude with a few relevant examples of super weakly compact sets in non super-reflexive Banach spaces.

Asymptotic geometry and Delta-points

We study Daugavet- and Delta-points in Banach spaces. A norm one element x is a Daugavet-point (respectively, a Delta-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 Delta-point. We show that asymptotically uniformly smooth spaces and reflexive asymptotically uniformly convex spaces do not contain Delta-points. We also show that the same conclusion holds true for the James tree space as well as for its predual. Finally, we prove that there exists a superreflexive Banach space with a Daugavet- or Delta-point provided there exists such a space satisfying a weaker condition.

RETRACTED - Compact reduction in Lipschitz-free spaces

We prove a general principle satisfied by weakly precompact sets of Lipschitz-free spaces. By this principle, certain infinite dimensional phenomena in Lipschitz-free spaces over general metric spaces may be reduced to the same phenomena in free spaces over their compact subsets. As easy consequences we derive several new and some known results. The main new results are: $\mathcal {F}(X)$ is weakly sequentially complete for every superreflexive Banach space $X$, and $\mathcal {F}(M)$ has the Schur property and the approximation property for every scattered complete metric space $M$.

C1,ω extension formulas for 1-jets on Hilbert spaces

We provide necessary and sufficient conditions for a 1-jet (f,G):E→R×X to admit an extension (F,∇F) for some F∈C1,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=Rn, we obtain an extension (nonlinear) operator whose norm does not depend on the dimension n. Furthermore, we construct extensions (F,∇F) in such a way that: (1) the (nonlinear) operator (f,G)↦(F,∇F) is bounded with respect to a natural seminorm arising from the constants in the given condition for extension (and the bounds we obtain are almost sharp); (2) F is given by an explicit formula; (3) (F,∇F) depend continuously on the given data (f,G); (4) if f is bounded (resp. if G is bounded) then so is F (resp. F is Lipschitz). We also provide similar results on superreflexive Banach spaces.

On uniformly convex functions

Non-convex functions that yet satisfy a condition of uniform convexity for non-close points can arise in discrete constructions. We prove that this sort of discrete uniform convexity is inherited by the convex envelope, which is the key to obtain other remarkable properties such as the coercivity. Our techniques allow to retrieve Enflo's uniformly convex renorming of super-reflexive Banach spaces as the regularization of a raw function built from trees. Among other applications, we provide a sharp estimation of the distance of a given function to the set of differences of Lipschitz convex functions. Finally, we prove the equivalence of several possible ways to quantify the super weakly noncompactness of a convex subset of a Banach space.

On the vanishing of reduced 1-cohomology for Banach representations

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 Pr\"ufer rank. Moreover all our results apply to isometric representations in a large class of Banach spaces, including reflexive Banach spaces. As applications, we obtain an ergodic theorem in for integrable cocycles, as well as a new proof of Bourgain's Theorem that the 3-regular tree does not embed quasi-isometrically into any superreflexive Banach space.

Uniformly convex renormings and generalized cotypes

We are concerned about improvements of the modulus of convexity by renormings of a super-reflexive Banach space. Typically optimal results are beyond Pisier's power functions bounds tp, with p≥2, and they are related to the notion of generalized cotype. We obtain an explicit upper bound for all the moduli of convexity of equivalent renormings and we show that if this bound is equivalent to t2, the best possible, then the space admits a renorming with modulus of power type 2. We show that a UMD space admits a renormings with modulus of convexity bigger, up to a multiplicative constant, than its cotype. We also prove the super-multiplicativity of the supremum of the set of cotypes.

Super weakly compact subsets of Banach spaces

The notion of super weakly compact sets is a localized setting of super reflexive Banach spaces. In this paper, we present a brief review of the research area of super weakly compact sets of Banach spaces and some new results about this topic are included.

A Proof of Komlós Theorem for Super-Reflexive Valued Random Variables

We give a geometrical proof of Komlós’ theorem for sequences of random variables with values in super-reflexive Banach space. Our approach is inspired by the elementary proof given by Guessous in 1996 for the Hilbert case and uses some geometric properties of smooth spaces.

A toolkit for constructing dilations on Banach spaces

We present a completely new structure theoretic approach to the dilation theory of linear operators. Our main result is the following theorem: if $X$ is a super-reflexive Banach space and $T$ is contained in the weakly closed convex hull of all invertible isometries on $X$, then $T$ admits a dilation to an invertible isometry on a Banach space $Y$ with the same regularity as $X$. The classical dilation theorems of Sz.-Nagy and Akcoglu-Sucheston are easy consequences of our general theory.

Lipschitz-free spaces over compact subsets of superreflexive spaces are weakly sequentially complete

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 Pe{\l}czy\'nski's property ($V^\ast$). As a consequence, the Lipschitz-free space $\mathcal{F}(M)$ is weakly sequentially complete.

Explicit formulas for C1,1 and [formula omitted] extensions of 1-jets in Hilbert and superreflexive spaces

Given X a Hilbert space, ω a modulus of continuity, E an arbitrary subset of X, and functions f:E→R, G:E→X, we provide necessary and sufficient conditions for the jet (f,G) to admit an extension (F,∇F) with F:X→R convex and of class C1,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 C1,1 extensions of general (not necessarily convex) 1-jets satisfying the usual Whitney extension condition, with best possible Lipschitz constants of the gradients of the extensions. Finally, if X is a superreflexive Banach space, we establish similar results for the classes Cconv1,α(X).

A remark on smooth images of Banach spaces

Let X be a non-separable super-reflexive Banach space. Then for any separable Banach space Y of dimension at least two there exists a C∞-smooth surjective mapping f:X→Y such that the restriction of f onto any separable subspace of X fails to be surjective. This solves a problem posed by Aron, Jaramillo, and Ransford (Problem 186 in the book [5]).

$$A_{p}$$ A p weights and strong convergence of operator-valued Fourier series for Stieltjes convolutions of Marcinkiewicz functions

Suppose that \(1<p<\infty \) and let w be a bilateral weight sequence satisfying the discrete Muckenhoupt \(A_{p}\) weight condition (in symbols, \(w\in A_{p}\left( \mathbb {Z}\right) \)). In this setting, the left bilateral shift \(\mathcal {L}\) acting on the sequence space \(\ell ^{p}\left( w\right) \) is known to be a trigonometrically well-bounded operator (that is, \(\mathcal {L}\) has a “unitary-like” spectral decomposition \(E:\mathbb {R}\rightarrow \mathfrak {B}\left( \ell ^{p}\left( w\right) \right) \) that consists of idempotent operators and has additional properties reminiscent of, but weaker than, those that would be inherited from a countably additive Borel spectral measure on \(\mathbb {R}\)). In this framework we show that for any Marcinkiewicz multiplier \(\psi :\mathbb {T} \rightarrow \mathbb {C}\), the operator-valued Fourier series of the Stieltjes convolution \(\psi \,\,\mathfrak {\circledast }\)dE (which can be regarded as an operator-theoretic analogue of the Fourier series of \(\psi \)) converges to \(\psi \,\,\mathfrak {\circledast }\)dE, relative to the strong operator topology of \(\mathfrak {B}\left( \ell ^{p}\left( w\right) \right) \), at each point of \(\mathbb {T}\). In particular, the partial sums of the Fourier series of \(\psi \) are uniformly bounded in the Banach algebra of Fourier multipliers for \(\ell ^{p}\left( w\right) \). This outcome is made possible by the mutually advantageous interplay between spectral theory and real variable methods in harmonic analysis—the fundamental links being the operator ergodic discrete Hilbert transform and exrapolation of weights. These results are transferred to the framework of invertible, disjoint, modulus mean-bounded operators acting on \(L^{p}\) spaces of sigma-finite measures. We note that in a still wider framework than that dealt with here—specifically the class of all super-reflexive Banach spaces—trigonometrically well-bounded operators and their spectral families of projections have recently been shown (Berkson, Studia Math 200:221–246, 2010) to constitute a disguised generalization of the dual conjugates of the operator ergodic discrete Hilbert transform generated by not necessarily power-bounded operators, in particular providing new transference vistas.

Finite slicing in superreflexive Banach spaces

For every superreflexive Banach space X there exists a supermultiplicative function which is the supremum, in a very natural ordering, of the set of all the moduli of convexity of equivalent norms. If this supremum is actually a maximum achieved under some equivalent renorming of X, then its modulus of convexity is the best possible in asymptotic sense. Otherwise, we can give an almost optimal uniformly convex renorming of X beyond the classical power type bound obtained by Pisier [21].

Coarse differentiation and quantitative nonembeddability for Carnot groups

We give lower bound estimates for the macroscopic scale of coarse differentiability of Lipschitz maps from a Carnot group with the Carnot–Carathéodory metric (G,dcc) to a few different classes of metric spaces. Using this result, we derive lower bound estimates for quantitative nonembeddability of Lipschitz embeddings of G into a metric space (X,dX) if X is either an Alexandrov space with nonpositive or nonnegative curvature, a superreflexive Banach space, or another Carnot group that does not admit a biLipschitz homomorphic embedding of G. For the same targets, we can further give lower bound estimates for the biLipschitz distortion of every embedding f:B(n)→X, where B(n) is the ball of radius n of a finitely generated nonabelian torsion-free nilpotent group G. We also prove an analogue of Bourgain's discretization theorem for Carnot groups and show that Carnot groups have nontrivial Markov convexity. These give the first examples of metric spaces that have nontrivial Markov convexity but cannot biLipschitzly embed into Banach spaces of nontrivial Markov convexity.

Operator ergodic theory for one-parameter decomposable groups

After initial treatment of the Fourier analysis and operator ergodic theory of strongly continuous decomposable one-parameter groups of operators in the Banach space setting, we show that in the setting of a super-reflexive Banach space X these groups automatically transfer from the setting of R to X the behavior of the Hilbert kernel, as well as the Fourier multiplier actions of functions of higher variation on R . These considerations furnish one-parameter groups with counterparts for the single operator theory in Berkson (2010) [4]. Since no uniform boundedness of one-parameter groups of operators is generally assumed in the present article, its results for the super-reflexive space setting go well beyond the theory of uniformly bounded one-parameter groups on UMD spaces (which was developed in Berkson et al., 1986 [13]), and in the process they expand the scope of vector-valued transference to encompass a genre of representations of R that are not uniformly bounded.

Spectral theory and operator ergodic theory on super-reflexive Banach spaces

Spectral theory and operator ergodic theory on super-reflexive Banach spaces