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 ...

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