Weighted weak-type inequalities for maximal operators and singular integrals

Abstract

We prove quantitative, one-weight, weak-type estimates for maximal operators, singular integrals, fractional maximal operators and fractional integral operators. We consider a kind of weak-type inequality that was first studied by Muckenhoupt and Wheeden (Indiana Univ. Math. J. 26(5):801–816, 1977) and later in Cruz-Uribe et al. (Int. Math. Res. Not. 30:1849–1871, 2005). We obtain quantitative estimates for these operators in both the scalar and matrix weighted setting using sparse domination techniques. Our results extend those obtained by Cruz-Uribe et al. (Rev. Mat. Iberoam. 37(4):1513–1538, 2021) for singular integrals and maximal operators when \(p=1\).