Covering spheres of Banach spaces by balls

If the unit sphere of a Banach space X can be covered by countably many balls no one of which contains the origin, then, as an easy consequence of the separation theorem, X* is w*-separable. We prove the converse under suitable renorming. Moreover, the balls of the countable covering can be chosen as translates of the same ball. Research of V. P. Fonf was supported in part by Israel Science Foundation, Grant # 139/02 and by the Istituto Nazionale di Alta Matematica of Italy. Research of C. Zanco was supported in part by the Ministero dell’Università e della Ricerca Scientifica e Tecnologica of Italy and by the Center for Advanced Studies in Mathematics at the Ben-Gurion University of the Negev, Beer-Sheva, Israel.     

Density of Lipschitz mappings in the class of Sobolev mappings between metric spaces

We prove that Lipschitz mappings are dense in the Newtonian–Sobolev classes N ^1,p (X, Y) of mappings from spaces X supporting p-Poincaré inequalities into a finite Lipschitz polyhedron Y if and only if Y is [p]-connected, π ₁(Y) = π ₂(Y) = · · · = π _[p](Y) = 0, where [p] is the largest integer less than or equal to p. This work was supported by the NSF grant DMS-0500966.     

Unconditionality in spaces of smooth functions

Our main result shows that subspaces of L¹([0, 1]) on which the blow-up operators act compactly are isometric to dual spaces, and their natural predual belongs to the Banach-Mazur closure of quotient spaces of . Related general results are shown for subspaces X of or of reflexive K?the function spaces, which imply that when X consists of smooth functions it embeds into a Banach space with an unconditional basis. Received: 25 September 2008     

Characterization of nearly Schroeder-Bernstein quadruples for Banach spaces

Let X and Y be Banach spaces such that each of them is isomorphic to a complemented subspace of the other. In 1996, W. T. Gowers solved the Schroeder-Bernstein problem for Banach spaces by showing that X is not necessarily isomorphic to Y . Let (p, q, r, s) be a quadruple in with p + q  ≥  2 and r + s ≥  2. Suppose that for every pair of Banach spaces X and Y isomorphic to complemented subspaces of each other and satisfying the following Decomposition Scheme

Application of the Trace Inequality to the Poisson Equation

The purpose of this paper is to show that solutions of the Poisson equation

Blow up of balls and coverings in metric spaces

We obtain some refinements and extensions of the Basic Covering Theorem in a metric space (X, ρ). The properties of the metric ρ are used to define an inclusion coefficient k in this theorem, and this is related to the supremum of numbers t such that ρ ^t is a metric in X. The inclusion coefficient k characterizes ultrametric spaces.     

Odd Degree Polynomials on Real Banach Spaces

A classical result of Birch claims that for given k, n integers, n-odd there exists some N = N(k, n) such that for an arbitrary n-homogeneous polynomial P on , there exists a linear subspace of dimension at least k, where the restriction of P is identically zero (we say that Y is a null space for P). Given n > 1 odd, and arbitrary real separable Banach space X (or more generally a space with w*-separable dual X*), we construct an n-homogeneous polynomial P with the property that for every point 0 ≠ x ∈ X there exists some k ∈ such that every null space containing x has dimension at most k. In particular, P has no infinite dimensional null space. For a given n odd and a cardinal τ , we obtain a cardinal N = N(τ, n) = expⁿ⁺¹τ such that every n-homogeneous polynomial on a real Banach space X of density N has a null space of density τ . Some of the work on this paper was done while the first author was a visitor to the Departamento de Análisis Matemático of the Universidad Complutense de Madrid, to which great thanks are given. The research of the second author was supported by grants: Institutional Research Plan AV0Z10190503, A100190502, GA ČR 201/04/0090.     

The level sets of the resolvent norm and convexity properties of Banach spaces

We construct a bounded linear operator on a separable, reflexive and strictly convex Banach space with the resolvent norm that is constant in a neighbourhood of zero.      

Relatively weakly open sets in closed balls of Banach spaces,and the centralizer

Let f be a transcendental entire function and let I(f) denote the set of points that escape to infinity under iteration. We give conditions which ensure that, for certain functions, I(f) is connected. In particular, we show that I(f) is connected if f has order zero and sufficiently small growth or has order less than 1/2 and regular growth. This shows that, for these functions, Eremenko’s conjecture that I(f) has no bounded components is true. We also give a new criterion related to I(f) which is sufficient to ensure that f has no unbounded Fatou components.     

The complexity of the Specht modules corresponding to hook partitions

We show that the complexity of the Specht module corresponding to any hook partition is the p-weight of the partition. We calculate the variety and the complexity of the signed permutation modules. Let E _s be a representative of the conjugacy class containing an elementary abelian p-subgroup of a symmetric group generated by s disjoint p-cycles. We give formulae for the generic Jordan types of signed permutation modules restricted to E _s and of Specht modules corresponding to hook partitions μ restricted to E _s where s is the p-weight of μ.      

Compact multipliers on spaces of analytic functions

In the paper compact multiplier operators from Banach spaces of analytic functions on the unit disk into Banach sequence lattices are studied. If , then the characterization of compact multipliers is obtained through calculating the Hausdorff measure of noncompactness of diagonal operators between Banach sequence lattices. Furthermore, in the general case , necessary and sufficient conditions for compactness are presented. Received: 12 August 2008, Revised: 11 January 2009     

Constant mean curvature spheres in Riemannian manifolds

We prove the existence of embedded spheres with large constant mean curvature in any compact Riemannian manifold (M, g). This result partially generalizes a result of R. Ye which handles the case where the scalar curvature function of the ambient manifold (M, g) has non-degenerate critical points.     

Concept of convergence in Banach algebras related to a spectral property

The aim of this note is a discussion about a concept of convergence related to a spectral property in unital complex Banach algebras.     

Linear fractional mappings: invariant sets, semigroups and commutativity

We study commutativity and embeddability (into continuous semi-groups) properties of linear fractional self-mappings of the open unit disk in the complex plane. The common thread in our approach is the classical notion of the Kœnigs function which we use in each of the three possible cases (dilation, hyperbolic and parabolic). Since we are interested in a classical subject, the paper is written in the style of a survey, in order to make it accessible to a wider audience. Therefore it contains, in addition to our new results, an exposition of most relevant facts. Dedicated to Professor Felix E. Browder with admiration and respect     

On surfaces of prescribed weighted mean curvature

Utilising a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and von der Mosel. Next we study graphs of prescribed weighted mean curvature, for which a quasilinear elliptic equation is proved. Using this equation, we can show height and boundary gradient estimates. Finally, we solve the Dirichlet problem for graphs of prescribed weighted mean curvature.     

On the Structure of L 1 of a Vector Measure via its Integration Operator

Geometric and summability properties of the integration operator associated to a vector measure m can be translated in terms of structure properties of the space L¹(m). In this paper we study the cases of the integration operator being: (i) p-concave on L^p(m), or (ii) positive p-summing on L¹(m) (where ). We prove that (i) is equivalent to saying that L¹(m) contains continuously the L^p space of a (non-negative scalar) control measure for m. On the other hand, we show that (ii) holds if and only if L¹(m) is order isomorphic to the L¹ space of a non-negative scalar measure. J.M. Calabuig was supported by MEC and FEDER (MTM2005-08350-C03-03) and Generalitat Valenciana (GV/2007/191). J. Rodríguez was supported by MEC and FEDER (MTM2005-08379) and Generalitat Valenciana (GVPRE/2008/312). E.A. Sánchez-Pérez was supported by MEC and FEDER (MTM2006-11690-C02-01).     

PAC fields over finitely generated fields

We prove the following theorem for a finitely generated field K: Let M be a Galois extension of K which is not separably closed. Then M is not PAC over K. Research supported by the Minkowski Center for Geometry at Tel Aviv University, established by the Minerva Foundation. This work constitutes a part of the Ph.D. dissertation of L. Bary-Soroker done at Tel Aviv University under the supervision of Prof. Dan Haran.     

On statistical approximation in spaces of continuous functions

Our goal is to present approximation theorems for sequences of positive linear operators defined on C(X), where X is a compact metric space. Instead of the uniform convergence we use the statistical convergence. Examples and special cases are also provided.      

Unique continuation of semi-conformality for a harmonic mapping onto a surface

We show that a harmonic mapping ϕ from either a three-manifold (with a condition on its Ricci curvature) or from a surface with values in a surface which has rank 2 somewhere, satisfies the following unique continuation property: if ϕ is semi-conformal on an open set, then it is semi-conformal everywhere.