Mixed weak type estimates: Examples and counterexamples related to a problem of E. Sawyer

TL;DR

This paper investigates mixed weighted weak-type inequalities for functions, extending key results in harmonic analysis and providing new insights into classical operators through counterexamples and applications.

Contribution

It extends a fundamental theorem on weighted weak-type inequalities, addressing a conjecture of Sawyer and broadening understanding in harmonic analysis.

Findings

Extended the main theorem on weighted weak-type inequalities.

Provided examples and counterexamples related to Sawyer's problem.

Applied results to classical harmonic analysis operators.

Abstract

We study mixed weighted weak-type inequalities for families of functions, which can be applied to study classical operators in harmonic analysis. Our main theorem extends the key result from D. Cruz-Uribe, J.M. Martell and C. Perez, Weighted weak-type inequalities and a conjecture of Sawyer, Int. Math. Res. Not., V. 30, 2005, 1849-1871.