Improvements on Sawyer type estimates for generalized maximal functions

Fabio Berra, Marilina Carena and Gladis Pradolini

Mathematische Nachrichten, 2020, vol. 293, issue 10, 1911-1930

Abstract: In this paper we prove mixed inequalities for the maximal operator MΦ, for general Young functions Φ with certain additional properties, improving and generalizing some previous estimates for the Hardy–Littlewood maximal operator proved by E. Sawyer. We show that given r≥1, if u,vr are weights belonging to the A1‐Muckenhoupt class and Φ is a Young function as above, then the inequality uvrx∈Rn:MΦ(fv)(x)v(x)>t≤C∫RnΦ|f(x)|tu(x)vr(x)dxholds for every positive t. A motivation for studying these type of estimates is to find an alternative way to prove the boundedness properties of MΦ. Moreover, it is well‐known that for the particular case Φ(t)=t(1+log+t)m with m∈N these maximal functions control, in some sense, certain operators in harmonic analysis.

Date: 2020
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