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Factorizing the Classical Inequalities
Grahame Bennett, Indiana University, Bloomington, IN
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Memoirs of the American Mathematical Society
1996; 130 pp; softcover
Volume: 120
ISBN-10: 0-8218-0436-7
ISBN-13: 978-0-8218-0436-0
List Price: US$43
Individual Members: US$25.80
Institutional Members: US$34.40
Order Code: MEMO/120/576
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This volume describes a new way of looking at the classical inequalities. The most famous such results (Hilbert, Hardy, and Copson) may be interpreted as inclusion relationships, \(l^p\subseteq Y\), between certain (Banach) sequence spaces, the norm of the injection being the best constant of the particular inequality.

The authors' approach is to replace \(l^p\) by a larger space, \(X\), with the properties: \(\Vert l^p\subseteq X\Vert =1\) and \(\Vert X\subseteq Y\Vert =\Vert l^p\subseteq Y\Vert\), the norm on \(X\) being so designed that the former property is intuitive. Any such result constitutes an enhancement of the original inequality, because you now have the classical estimate, \(\Vert l^p\subseteq Y\Vert\), holding for a larger collection, \(X=Y\).

The authors' analysis has some noteworthy features: The inequalities of Hilbert, Hardy, and Copson (and others) all share the same space \(Y\). That space-alias ces(\(p\) )-being central to so many celebrated inequalities, the authors conclude, must surely be important. It is studied here in considerable detail. The renorming of \(Y\) is based upon a simple factorization, \(Y= l^p\cdot Z\) (coordinatewise products), wherein \(Z\) is described explicitly. That there is indeed a renorming, however, is not so simple. It is proved only after much preparation when duality theory is considered.

Readership

Graduate students and research mathematicians interested in real functions, functional analysis, and operator theory.

Table of Contents

  • Introduction
  • Outline
  • The spaces \(d({\mathbf a}, p)\) and \(g({\mathbf a},p)\)
  • Hardy
  • Hölder
  • Copson
  • Two techniques
  • Examples
  • The meaning of \(\ell ^p\)
  • \(ces(p)\) versus \(cop\) \((p)\)
  • Hilbert
  • Köthe-Toeplitz duality
  • The spaces \(\ell ^p\cdot d({\mathbf a},q)\)
  • Multipliers
  • Some non-factorizations
  • Examples
  • Other matrices
  • Summability matrices
  • Hausdorff matrices
  • Cesàro matrices
  • Integral analogues
  • References
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