A reduction theorem for supremum operators

✍ Scribed by Amiran Gogatishvili; Luboš Pick

• Book ID: 104005305
• Publisher: Elsevier Science
• Year: 2007
• Tongue: English
• Weight: 168 KB
• Volume: 208
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/j.cam.2006.10.048

✦ Synopsis

We show that the two-weight Hardy inequality restricted to nonincreasing functions, namely

, where 0 < p 1 and 0 < q < ∞, is equivalent to slightly different inequalities. Consequently, we can reduce this inequality to a pair of unrestricted inequalities (a reduction theorem). As an application, we prove an analogous assertion for a three-weight inequality involving a supremum operator, namely

, in which the weight u is assumed to be continuous on (0, ∞). This result in turn enables us to establish necessary and sufficient conditions on the weights (u, v, w) for which this inequality holds.