ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 26, 46, 47 Grosse-Erdmann, Karl-Goswin, Fernuniversität Hagen, Germany The blocking technique and the weighted Hardy inequality The classical Hardy inequality asserts that there is a constant M > 0 such that for all non-negative numbers x_{k}

∑_{n}(\frac{1}{n} ∑_{k=1}^{n} x_{k})^{p} ≤ M ∑_{n} x_{n}^{p},

where p > 1. More generally, one may ask for which weights a_{n} ≥ 0, b_{n} ≥ 0 the weighted Hardy inequality

(∑_{n} (a_{n} ∑_{k=1}^{n} b_{k}x_{k})^{q})^{1/q} ≤ M(∑_{n} x_{n}^{p})^{1/p}

holds, where 0 < p, q ≤ ∞. In [2] the so-called blocking-technique is developed in order to solve this problem completely, thus confirming earlier results of Bennett [1] and solving one of his open problems. The corresponding question for integrals is also discussed.

[1] G. Bennett: Some elementary inequalities. III, Quart. J. Math. Oxford 42 (1991), 149–174.

[2] K. -G. Grosse-Erdmann: The blocking technique, weighted mean operators and Hardy’s inequality, Lecture Notes Math. 1679, Springer, Berlin