, Volume 19, Issue 1, pp 115-139
Date: 17 Oct 2012

Sharp Weak Type Inequalities for the Dyadic Maximal Operator

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Abstract

We obtain sharp estimates for the localized distribution function of \(\mathcal{M}\phi \) , when ϕ belongs to L p,∞ where \(\mathcal{M}\) is the dyadic maximal operator. We obtain these estimates given the L 1 and L q norm, q<p and certain weak-L p conditions.In this way we refine the known weak (1,1) type inequality for the dyadic maximal operator. As a consequence we prove that the inequality 0.1 is sharp allowing every possible value for the L 1 and the L q norm for a fixed q such that 1<q<p, where ∥⋅∥ p,∞ is the usual quasi norm on L p,∞.

Communicated by Loukas Grafakos.