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

摘要

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, deltax), centered at X is an element of R-n and of radius deltax, delta > 0. Therefrom we deduce the corresponding Hardy-type inequality, that is, the operator norm of the operator S-delta which averages if is an element of L-p (R-n) over B(x, deltax), 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. (C) 2003 Elsevier Inc. All rights reserved. [References: 15]