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On reverse Hardy’s inequality. (English) Zbl 0816.26006

The author, following C. J. Neugebauer [Publ. Mat., Barc. 35, No. 2, 429-447 (1991; Zbl 0746.42014)], studies a converse of Hardy’s inequality with weights, valid for non-increasing functions. The argument is based on an extension of the Riesz convexity theorem to operators that act on non-increasing functions.
{Reviewer’s remark: A general theory of interpolation with respect to cones was developed by Sagher, including interpolation of operators acting on \(L^ p\) spaces restricted to non-increasing functions [cf. Y. Sagher, Stud. Math. 44, 239-252 (1972; Zbl 0258.42005); ibid. 41, 169-181 (1972; Zbl 0258.42004); Proc. Conf. Oberwolfach 1974, 169-180 (1974; Zbl 0322.46034)].

MSC:

26D15 Inequalities for sums, series and integrals
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
42B25 Maximal functions, Littlewood-Paley theory
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