Asymptotic Constants in Averaged Hölder Inequalities

Abstract

For \(0 \lt q \le p\) we take \(L^\beta\)-averages over the Hölder inequality between the \(l^q\)-norm and the \(l^p\)-norm in \(\mathbb{R}^n\). We obtain precise limits as \(n\to\infty\) for the \(l^p\)-unit ball and, in case \(p\ge1\), also for the \(l^p\)-unit sphere, which coincide and are independent of \(\beta>0\). These are consequences of more general results on the asymptotic behavior of corresponding integrals over balls and spheres of certain bounded measurable functions.