Factorizing the classical inequalities. (English) Zbl 0857.26009

Bennett, Grahame

Mem. Am. Math. Soc. 576, 130 p. (1996).

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.
To illustrate results, consider Hardy’s inequality \[ \sum^\infty_{n=1} \Biggl({1\over n} \sum^n_{k=1} |x_k|\Biggr)^p\leq (p')^p\sum^\infty_{k=1} |x_k|^p\tag{1} \] (here \(p'={p\over p-1}\) and \(p>1\)). This inequality can be interpreted as an inclusion theorem between sequence spaces: \[ l^p\subseteq\text{ces}(p),\tag{2} \] where \[ \text{ces}(p):=\Biggl\{{\mathbf x}=(x_1,x_2,\dots); |{\mathbf x}|_{\text{ces}(p)}=\Biggl(\sum^\infty_{n=1} \Biggl({1\over n} \sum^n_{k=1} |x_k|\Biggr)^p\Biggr)^{1/p}<\infty\Biggr\}. \] In this context natural questions arise:
(i) Is it possible to replace \(\text{ces}(p)\) in (2) by a smaller space?
(ii) Can one replace \(l^p\) by something larger?
The author is concerned almost exclusively with the second question and he proves the following Theorem:
Let \(p>1\). A sequence belongs to \(\text{ces}(p)\) if and only if it admits a factorization \[ {\mathbf x}= {\mathbf y}\cdot{\mathbf z}:=(y_1z_1,y_2z_2,\dots)\tag{3} \] with \[ {\mathbf y}\in l^p\quad\text{and}\quad {\mathbf z}\in g(p'):=\{v; |v_1|^{p'}+\cdots+ |v_n|^{p'}=O(n)\}.\tag{4} \] Moreover, \[ (p-1)^{-1/p}|{\mathbf x}|_p\leq |{\mathbf x}|_{\text{ces}(p)}\leq p'|{\mathbf x}|_p,\tag{5} \] where \(|{\mathbf x}|_p:=\inf\{|{\mathbf y}|_p\cdot|{\mathbf z}|_{g(p')}\}\), the infimum being extended over all factorizations (4), (5).
(Note that (1) follows from (4), (5) by taking \({\mathbf y}={\mathbf x}\) and \({\mathbf z}=(1,1,\dots)\)).

Reviewer: B.Opic (Praha)

MSC:

26D15	Inequalities for sums, series and integrals
46B45	Banach sequence spaces
47A30	Norms (inequalities, more than one norm, etc.) of linear operators
47B37	Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
40G05	Cesàro, Euler, Nörlund and Hausdorff methods

Keywords:

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