On reverse Hardy's inequality (Q1332994)

The author, following \textit{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. \textit{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)].