Best Constants for Two Non-convolution Inequalities

The norm of the operator which averages |f | in L p (R n) over balls of radius δ|x| centered at either 0 or x is obtained as a function of n, p and δ. Both inequalities proved are n-dimensional analogues of a classical inequality of Hardy in R 1. Finally, a lower bound for the operator norm of the Hardy-Littlewood maximal function on L p (R n) is given. A classical result of Hardy [HLP] states that if f is in L p (R 1) for p > 1, then (0.1) ∞ 0 1 x x 0 |f (t)| dt p dx 1/p ≤ p p − 1 ∞ 0 |f (t)| p dt 1/p and the constant p/(p − 1) is the best possible. By considering two-sided averages of f instead of one-sided, (0.1) can be equivalently formulated as: (0.2) ∞ −∞ 1 2|x| |x| −|x| |f (t)| dt p dx 1/p ≤ p p − 1 ∞ −∞ |f (t)| p dt 1/p. We denote by D(a, R) the ball of radius R in R n centered at a. Let (T f)(x) be the average of |f | ∈ L p (R n) over the ball D(0, |x|). The analogue of (0.2) for R