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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$46 Individual Members: US$27.60
Institutional Members: US\$36.80
Order Code: MEMO/120/576

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.

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

• 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