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Hardy’s inequality in Orlicz-type sequence spaces, for operators related to generalized Hausdorff matrices. (English) Zbl 0584.40001

This paper contains inequalities of the form \(\| Ax\| \leq C\| x\|\), in which \(A=(a_{n,r})\) is an infinite ”square” matrix and \(x=(x_ n)\) is a column sequence. A recent inequality of this kind, due to D. Borwein [Math. Z. 183, 483-487 (1983; Zbl 0508.40013)], has A a generalized Hausdorff matrix and \(\|.\|\) the \(\ell^ p\)-norm. It generalizes an earlier inequality of A. Jakimovski, B. E. Rhoades and J. Tzimbalario, which was itself a very substantial generalization of the original Hardy’s inequality.
The inequalities of this paper are for Orlicz-Luxemburg norms. They revolve around a theorem which, stated approximately (for brevity) is as follows. Theorem 1: Let \(k_ n>0\). For \(t\in I\) let \(\alpha\) (t) be non- decreasing, \(\gamma_{n,r}(t)\geq 0\), \(\sum^{\infty}_{r=0}\gamma_{n,r}(t)\leq 1\) and \(\sum^{\infty}_{r=0}(k_ r/k_ n)\gamma_{r,n}(t)\leq \beta (t)\). If \(\|.\|\) is an Orlicz-Luxemburg norm made from \(k_ n\) and \(\phi\) (t), \(0\leq a_{n,r}\leq \int_{I}\gamma_{n,r}(t)d\alpha (t)\) and \(C=\int_{I}\phi^{-1}(\beta (t))d\alpha (t),\) then \(\| Ax\| \leq C\| x\|\). Extensions of the inequality of Jakimovski, Rhoades and Tzimbalario, and of that of Borwein, are made using Theorem 1. Theorem 2 contains the former inequality as a special case; and Theorem 3 the latter, that is, Borwein’s. Theorem 3 is the specialization of Theorem 1 in which \(k_ n\) is non-decreasing, \(I=[0,1]\), \(\beta (t)=\nu t^{-c}\) for suitable constants \(\nu\) and c, and \(a_{n,r}\leq (k_ r/k_ n)h_{n,r}\) where \((h_{n,r})\) is a generalized Hausdorff matrix.

MSC:

40C05 Matrix methods for summability
40J05 Summability in abstract structures

Citations:

Zbl 0508.40013
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References:

[1] Borwein, D.: Generalized Hausdorff matrices as bounded operators onl p. Math. Z.183, 483-487 (1983) · Zbl 0508.40013 · doi:10.1007/BF01173925
[2] Hardy, G.H.: An inequality for Hausdorff means. J. Lond. Math. Soc.18, 46-50 (1943) · Zbl 0061.12704 · doi:10.1112/jlms/s1-18.1.46
[3] Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge: Cambridge Univ. Press 1934
[4] Jakimovski, A., Rhoades, B.E., Tzimbalario, J.: Hausdorff matrices as bounded operators overl p . Math. Z.138, 173-181 (1974) · Zbl 0282.47003 · doi:10.1007/BF01214233
[5] Krasnoselski, M.A., Ruticki, Y.B.: Convex functions and Orlicz spaces. Leiden: Noordhoff 1961
[6] Lorentz, G.G.: Bernstein polynomials. Toronto: Univ. of Toronto Press 1953
[7] Love, E.R.: Hardy’s inequality in Orlicz-type sequence spaces (unpublished talk given at GTE Conference on Sequence Spaces at St. Laurence University, Canton, N.Y., U.S.A. July, 1985)
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