Weak-type inequalities for maximal operators acting on Lorentz spaces

Abstract

We prove sharp a priori estimates for the distribution function of the dyadic maximal function Mφ, when φ belongs to the Lorentz space L, 1 < p < ∞, 1 ≤ q < ∞. The approach rests on a precise evaluation of the Bellman function corresponding to the problem. As an application, we establish refined weak-type estimates for the dyadic maximal operator: for p, q as above and r ∈ [1, p], we determine the best constant Cp,q,r such that for any φ ∈ L p,q , ||Mφ||r,∞ ≤ Cp,q,r||φ||p,q .