Sharp Weak Type Inequalities for the Dyadic Maximal Operator

We obtain sharp estimates for the localized distribution function of $\mathcal{M}\phi $ , when ? 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,∞.