Best constants for two nonconvolution inequalities

Christ, Michael; Grafakos, Loukas

doi:10.1090/S0002-9939-1995-1239796-6

Best constants for two nonconvolution inequalities
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by Michael Christ and Loukas Grafakos PDF

Proc. Amer. Math. Soc. 123 (1995), 1687-1693 Request permission

Abstract:

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

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