Best constants for two nonconvolution inequalities

Abstract

The norm of the operator which averages | f | |f| in L p ( R n ) {L^p}({\mathbb {R}^n}) over balls of radius δ | x | \delta |x| centered at either 0 or x is obtained as a function of n , p and δ \delta . Both inequalities proved are n-dimensional analogues of a classical inequality of Hardy in R 1 {\mathbb {R}^1} . Finally, a lower bound for the operator norm of the Hardy-Littlewood maximal function on L p ( R n ) {L^p}({\mathbb {R}^n}) is given.