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Zbl 0857.26009
Bennett, Grahame
Factorizing the classical inequalities. (English)
[J] Mem. Am. Math. Soc. 576, 130 p. (1996). ISSN 0065-9266

This memoir describes a new way of looking at the classical inequalities. The classical inequalities (those of Hardy, Hölder, Hilbert and Copson) are factorized and analogous results for integral transforms of the type $g(x)=\int^\infty_0 K(x,y)f(y)dy$ are briefly mentioned in the final section.\par To illustrate results, consider Hardy's inequality $$\sum^\infty_{n=1} \Biggl({1\over n} \sum^n_{k=1} |x_k|\Biggr)^p\le (p')^p\sum^\infty_{k=1} |x_k|^p\tag1$$ (here $p'={p\over p-1}$ and $p>1$). This inequality can be interpreted as an inclusion theorem between sequence spaces: $$l^p\subseteq\text{ces}(p),\tag2$$ where $$\text{ces}(p):=\Biggl\{{\bold x}=(x_1,x_2,\dots); |{\bold x}|_{\text{ces}(p)}=\Biggl(\sum^\infty_{n=1} \Biggl({1\over n} \sum^n_{k=1} |x_k|\Biggr)^p\Biggr)^{1/p}<\infty\Biggr\}.$$ In this context natural questions arise:\par (i) Is it possible to replace $\text{ces}(p)$ in (2) by a smaller space?\par (ii) Can one replace $l^p$ by something larger?\par The author is concerned almost exclusively with the second question and he proves the following Theorem:\par Let $p>1$. A sequence belongs to $\text{ces}(p)$ if and only if it admits a factorization $${\bold x}= {\bold y}\cdot{\bold z}:=(y_1z_1,y_2z_2,\dots)\tag3$$ with $${\bold y}\in l^p\quad\text{and}\quad {\bold z}\in g(p'):=\{v; |v_1|^{p'}+\cdots+ |v_n|^{p'}=O(n)\}.\tag4$$ Moreover, $$(p-1)^{-1/p}|{\bold x}|_p\le |{\bold x}|_{\text{ces}(p)}\le p'|{\bold x}|_p,\tag5$$ where $|{\bold x}|_p:=\inf\{|{\bold y}|_p\cdot|{\bold z}|_{g(p')}\}$, the infimum being extended over all factorizations (4), (5).\par (Note that (1) follows from (4), (5) by taking ${\bold y}={\bold x}$ and ${\bold z}=(1,1,\dots)$).
[B.Opic (Praha)]

MSC 2000:: *26D15 Inequalities for sums, series and integrals of real functions
46B45 Banach sequence spaces
47A30 Operator norms and inequalities
47B37 Operators on sequence spaces, etc.
40G05 Traditional summability methods

Keywords: norms; operators on sequence spaces; Hardy inequality; Hölder inequality; Hilbert inequality; Copson inequality; factorization

Cited in: Zbl 1220.26014 Zbl 1068.26017 Zbl 1071.42013 Zbl 0973.26016 Zbl 0963.26007 Zbl 0990.26015 Zbl 0933.26009 Zbl 0916.26005 Zbl 0926.26011 Zbl 0908.26012 Zbl 0894.26008

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