Mixed means over balls and annuli and lower bounds for operator norms of maximal functions

In this paper we prove mixed-means inequalities for integral power means of an arbitrary real order, where one of the means is taken over the ball B(x, δ|x|), centered at x ∈ Rn and of radius δ|x|, δ > 0. Therefrom we deduce the corresponding Hardy-type inequality, that is, the operator norm of the operator Sδ which averages |f| ∈ Lp(Rn) over B(x, δ|x|), introduced by Christ and Grafakos in Proc. Amer. Math. Soc. 123 (1995) 1687–1693. We also obtain the operator norm of the related limiting geometric mean operator, that is, Carleman or Levin–Cochran–Lee-type inequality. Moreover, we indicate analogous results for annuli and discuss estimations related to the Hardy– Littlewood and spherical maximal functions.  2003 Elsevier Inc. All rights reserved