The Lipschitz free Banach spaces of -spaces

The aim of this note is to prove that if is any infinite metric compact space, then the Lipschitz free spaces of and are isomorphic. This gives an example of non-Lipschitz-homeomorphic Banach spaces whose free Lipschitz spaces are isomorphic. We also derive some results about Lipschitz homogeneity for Banach spaces, from the results of G. Godefroy and N. J. Kalton on Lipschitz free Banach spaces.

     

A note on non-existence results for semi-linear cooperative elliptic systems via moving spheres

In this note, we extend some earlier non-existence, monotonicity and one-dimensionality results of W. Reichel and the author, by replacing the local Lipschitz continuity hypothesis on the non-linearities by a so-called boundedly uniform Lipschitz condition in the magnitude of .

     

Non-commutative metric topology on matrix state space

We present an operator space version of Rieffel's theorem on the agreement of the metric topology, on a subset of the Banach space dual of a normed space, from a seminorm with the weak*-topology. As an application we obtain a necessary and sufficient condition for the matrix metric from an unbounded Fredholm module to give the BW-topology on the matrix state space of the -algebra. Motivated by recent results we formulate a non-commutative Lipschitz seminorm on a matrix order unit space and characterize those matrix Lipschitz seminorms whose matrix metric topology coincides with the BW-topology on the matrix state space.

     

Semigroups of mappings with rigid Lipschitz constant

It is shown that two well-known uniformly fixed point free lipschitzian semigroups of mappings have minimal Lipschitz constant on the positive part of the unit ball of . This implies that a question raised by T. Kuczumow has a negative answer.

     

A brief remark on van der Waerden spaces

We demonstrate that Martin's axiom for -centered notions of forcing implies the existence of a van der Waerden space that is not a Hindman space. Our proof is an adaptation of the one given by M. Kojman and S. Shelah that such a space exists if one assumes the continuum hypothesis to be true.

     

Reflexivity and sets of Fréchet subdifferentiability

We show that the sets of Fréchet subdifferentiability of Lipschitz functions on a Banach space are Borel if and only if is reflexive. This answers a question of L. Zajíček.

     

Characterizations of regular almost periodicity in compact minimal abelian flows

Regular almost periodicity in compact minimal abelian flows was characterized for the case of discrete acting group by W. Gottschalk and G. Hedlund and for the case of -dimensional phase space by W. Gottschalk a few decades ago. In 1995 J. Egawa gave characterizations for the case when the acting group is . We extend Egawa's results to the case of an arbitrary abelian acting group and a not necessarily metrizable phase space. We then show how our statements imply previously known characterizations in each of the three special cases and give various other applications (characterization of regularly almost periodic functions on arbitrary abelian topological groups, classification of uniformly regularly almost periodic compact minimal - and -flows, conditions equivalent with uniform regular almost periodicity, etc.).

     

Generalized subdifferentials: a Baire categorical approach

We use Baire categorical arguments to construct pathological locally Lipschitz functions. The origins of this approach can be traced back to Banach and Mazurkiewicz (1931) who independently used similar categorical arguments to show that ``almost every continuous real-valued function defined on [0,1] is nowhere differentiable". As with the results of Banach and Mazurkiewicz, it appears that it is easier to show that almost every function possesses a certain property than to construct a single concrete example. Among the most striking results contained in this paper are: Almost every 1-Lipschitz function defined on a Banach space has a Clarke subdifferential mapping that is identically equal to the dual ball; if is a family of maximal cyclically monotone operators defined on a Banach space then there exists a real-valued locally Lipschitz function such that for each ; in a separable Banach space each non-empty weakcompact convex subset in the dual space is identically equal to the approximate subdifferential mapping of some Lipschitz function and for locally Lipschitz functions defined on separable spaces the notions of strong and weak integrability coincide.

     

Memorandum on multiplicative bijections and order

Let denote the semigroup of continuous functions from the topological space X to , equipped with the pointwise multiplication. The paper studies semigroup homomorphisms , with emphasis on isomorphisms. The crucial observation is that, in this setting, homomorphisms preserve order, so isomorphisms preserve order in both directions and they are automatically lattice isomorphisms. Applications to uniformly continuous and Lipschitz functions on metric spaces are given. Sample result: if Y and X are complete metric spaces of finite diameter without isolated points, every multiplicative bijection has the form Tf=f○τ, where τ:X→Y is a Lipschitz homeomorphism. F. Cabello Sánchez and J. Cabello Sánchez are supported in part by DGICYT projects MTM2004-02635 and MTM2007-6994-C02-02. J. Cabello Sánchez is supported in part by a grant of the UEx (Programa Propio–Acción 2).     

Bounded Toeplitz products on the Bergman space of the polydisk

We consider the question for which square integrable analytic functions f and g on the polydisk the densely defined products are bounded on the Bergman space. We prove results analogous to those we obtained in the setting of the unit disk [K. Stroethoff, D. Zheng, J. Funct. Anal. 169 (1999) 289-313].     

Elliptic equations with BMO coefficients in Lipschitz domains

In this paper, we study inhomogeneous Dirichlet problems for elliptic equations in divergence form. Optimal regularity requirements on the coefficients and domains for the estimates are obtained. The principal coefficients are supposed to be in the John-Nirenberg space with small BMO semi-norms. The domain is supposed to have Lipschitz boundary with small Lipschitz constant. These conditions for the theory do not just weaken the requirements on the coefficients; they also lead to a more general geometric condition on the domain.

     

Strong Convergence of Iterative Algorithms for Variational Inequalities in Banach Spaces

Let C be a nonempty closed convex subset of a Banach space E with the dual E ^*, let T:C→E ^* be a Lipschitz continuous mapping and let S:C→C be a relatively nonexpansive mapping. In this paper, by employing the notion of generalized projection operator, we study the following variational inequality (for short, VI(T−f,C)): find x∈C such that

Lipschitz representations of subsets of the cube

We show that for any class of uniformly bounded functions with a reasonable combinatorial dimension, the vast majority of small subsets of the -dimensional combinatorial cube cannot be represented as a Lipschitz image of a subset of , unless the Lipschitz constant is very large. We apply this result to the case when consists of linear functionals of norm at most one on a Hilbert space.

     

Nonlinear structure of some classical quasi-Banach spaces and -spaces

We investigate the Lipschitz structure of ℓ_p and L_p for 0<p<1 as quasi-Banach spaces and as metric spaces (with the metric induced by the p-norm) and show that they are not Lipschitz isomorphic. We prove that the -space L₀ is not uniformly homeomorphic to any other L_p space, that the L_p spaces for 0<p<1 embed isometrically into one another, and reduce the problem of the uniform equivalence amongst L_p spaces to their Lipschitz equivalence as metric spaces.     

On the derivatives of the Berezin transform

Improving upon a recent result of L. Coburn and J. Xia, we show that for any bounded linear operator on the Segal-Bargmann space, the Berezin transform of is a function whose partial derivatives of all orders are bounded. Similarly, if is a bounded operator on any one of the usual weighted Bergman spaces on a bounded symmetric domain, then the appropriately defined ``invariant derivatives' of any order of the Berezin transform of are bounded. Further generalizations are also discussed.

     

On the regularity of products and intersections of complete intersections

This paper proves the formulae

for arbitrary monomial complete intersections and , and provides examples showing that these inequalities do not hold for general complete intersections.

     

Fourier multipliers of Lipschitz spaces

Let G denote an infinite, compact, metrizable, 0-dimensional, Abelian group. The following are characterized: (i) the multipliers from one Lipschitz space Lip(α, p; G) to another Lipschitz space Lip(β, q; G) for 0 < α < β < ∞ and 1 ? p, q ? ∞; and (ii) the multipliers from Lip(α, p; G) to Lip(β, q; G) for 0 < β ? α < ∞ and 1 < q ? 2 ? p < ∞. Two special cases of (i), namely the case q = ∞ and the case p = 1, were obtained by the authors in an earlier publication (1981). A. Zygmund (J. Math. Mech.8 (1959), 889–895) and T. Mizuhara (Tôhoku Math. J.24 (1972), 263–268) have characterized the multipliers of certain Lipschitz spaces defined on the circle group.     

Minimal displacement and retraction problems in infinite-dimensional Hilbert spaces

We give the first constructive example of a Lipschitz mapping with positive minimal displacement in an infinite-dimensional Hilbert space We use this construction to obtain an evaluation from below of the minimal displacement characteristic in the space In the second part we present a simple and constructive proof of existence of a Lipschitz retraction from a unit ball onto a unit sphere in the space , and we improve an evaluation from above of a retraction constant

     

Flow invariance for nonlinear accretive evolutions under range conditions

Let A be a nonlinear accretive operator in a real Banach space X and f : J × X  X be continuous or of Carathéodory type, where We investigate the existence of mild solutions of the evolution system