Alexander Brudnyi,Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, T2N 1N4, Canada; e-mail: abrudnyi@ucalgary.ca

Almaz Buraev, Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, T2N 1N4, Canada; e-mail: butaev@ucalgary.ca

Abstract/Résumé:

Let \(C_b^{k,\omega}({\mathbb R}^n)\) be the Banach space of \(C^k\) functions on \({\mathbb R}^n\) bounded together with all derivatives of order \(\le k\) , where the derivatives of order \(k\) have moduli of continuity majorization by \(c\,\omega\) , \(c\in\mathbb R_+\) , for some \(\omega\in C(\mathbb R_+)\) . For a closed set \(S\subset{\mathbb R}^n\) the jet space \(J_b^{k,\omega}(S)\) is the Banach space of vector functions whose components are partial derivatives of functions in \(C_b^{k,\omega}({\mathbb R}^n)\) evaluated at points of \(S\) equipped with the corresponding quotient norm. The geometric predual \(G_J^{k,\omega}(S)\) of \(J_b^{k,\omega}(S)\) is the minimal closed subspace of the dual \(\bigl(C_b^{k,\omega}({\mathbb R}^n)\bigr)^*\) containing the evaluation functionals of all partial derivatives of order \(\le k\) at points in \(S\) . In the paper we study some geometric properties of spaces \(G_J^{k,\omega}(S)\) related to the classical Whitney problems.

Soit \(C_b^{k,\omega}({\mathbb R}^n)\) l’espace de Banach des fonctions \(C^k\) sur \({\mathbb R}^n\) bornées avec toutes les dérivées d’ordre \(k\) , où les dérivés d’ordre \(k\) ont des modules de continuités majorés par \(c\,\omega\) , \(c\in\mathbb R_+\) , pour quelques \(\omega\in C(\mathbb R_+)\) . Pour un ensemble fermé \(S\subset{\mathbb R}^n\) l’espace de jet \(J_b^{k,\omega}(S)\) est l’espace de Banach des fonctions vectorielles dont les composantes sont des dérivées partielles des fonctions en \(C_b^{k,\omega}({\mathbb R}^n)\) évaluées aux points de \(S\) équipés de la norme du quotient correspondante. Le prédual géométrique \(G_J^{k,\omega}(S)\) de \(J_b^{k,\omega}(S)\) est le sous-espace minimal fermé du dual \(\bigl(C_b^{k,\omega}({\mathbb R}^n)\bigr)^*\) contenant les fonctionnelles d’évaluation de toutes les dérivées partielles d’ordre \(\le k\) aux points de \(S\) . Dans cet article, nous étudions certaines propriétés géométriques des espaces \(G_J^{k,\omega}(S)\) liées aux problèmes classiques de Whitney.

Keywords: Predual space, Whitney problems, approximation property, second dual space, trace space

AMS Subject Classification: Geometry and structure of normed linear spaces, Banach spaces of continuous; differentiable or analytic functions 46B20, 46E15

PDF(click to download): On Geometric Preduals of Jet Spaces on Closed Subsets of ${mathbb R}^n$

Alexander Brudnyi,Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4; e-mail: abrudnyi@ucalgary.ca

Abstract/Résumé:

Let \(C_b^{k,\omega}({\mathbb R}^n)\) be the Banach space of \(C^k\) functions on \({\mathbb R}^n\) bounded together with all derivatives of order \(\le k\) and with derivatives of order \(k\) having moduli of continuity majorated by \(c\cdot\omega\), \(c\in{\mathbb R}_+\), for some \(\omega\in C({\mathbb R}_+)\). Let \(C_b^{k,\omega}(S):=C_b^{k,\omega}({\mathbb R}^n)|_S\) be the trace space to a closed subset \(S\subset{\mathbb R}^n\). The geometric predual \(G_b^{k,\omega}(S)\) of \(C_b^{k,\omega}(S)\) is the minimal closed subspace of the dual \(\bigl(C_b^{k,\omega}({\mathbb R}^n)\bigr)^*\) containing evaluation functionals of points in \(S\). We study geometric properties of spaces \(G_b^{k,\omega}(S)\) and their relations to the classical Whitney problems on the characterization of trace spaces of \(C^k\) functions on \({\mathbb R}^n\).

Soit \(C_b^{k, \omega} ({\mathbb R}^n)\) l’espace de Banach des fonctions \(C^k\) sur \({\mathbb R}^n\) bornées avec toutes leurs dérivées d’ordre jusqu’à \(k\) et avec les dérivées d’ordre \(k\) ayant des modules de continuité majorés par \(c \cdot \omega\), \(c \in {\mathbb R}_+\), pour quelque \(\omega \in C ({\mathbb R}_+)\). Soit \(C_b ^ {k, \omega} (S): = C_b^{k, \omega} ({\mathbb R}^n) |_S\) l’espace de trace à un fermé \(S\subset{\mathbb R} ^ n\). Le predual géométrique \(G_b^{k, \omega}(S)\) de \(C_b^{k, \omega} (S)\) est le sous-espace minimal fermé du dual \(\bigl (C_b^ {k, \omega} ({\mathbb R}^n) \bigr)^*\) contenant les fonctionnelles d’évaluation aux points de \(S\). Nous étudions les propriétés géométriques des espaces \(G_b^{k, \omega} (S)\) et leur relation avec les problèmes classiques de Whitney sur la caractérisation des espaces de trace des fonctions \(C^k\) sur \({\mathbb R}^n\).

Keywords: Finiteness Principle, Predual space, Weak Markov set, Whitney problems, approximation property, dual space, linear extension operator, weak$^*$ topology

AMS Subject Classification: Geometry and structure of normed linear spaces, Banach spaces of continuous; differentiable or analytic functions 46B20, 46E15

PDF(click to download): On Properties of Geometric Preduals of ${mathbf C^{k,omega}}$ Spaces

Z. Lou

Abstract/Résumé:

No abstract available but the full text pdf may be downloaded at the title link below.

Keywords: ?-Bloch function, Carleson measure, Predual space, continuity, little?-Bloch function

AMS Subject Classification: Bloch functions; normal functions; normal families, Blaschke products; bounded mean oscillation; bounded characteristic; bounded functions; functions with positive real part 30D45, 30D50

PDF(click to download): Some properties of ?-Bloch functions