Weak Type Estimates for Singular Integrals Related to a Dual Problem of Muckenhoupt–Wheeden

Abstract

A well-known open problem of Muckenhoupt–Wheeden says that any Calderón–Zygmund singular integral operator T is of weak type (1,1) with respect to a couple of weights (w,Mw). In this paper, we consider a somewhat “dual” problem: $$\sup_{\lambda >0}\lambda w\biggl\{x\in {\mathbb{R}}^{n}:\frac{|Tf(x)|}{Mw}>\lambda \biggr\}\le c\int_{{\mathbb{R}}^{n}}|f|\,dx.$$ We prove a weaker version of this inequality with M 3 w instead of Mw. Also we study a related question about the behavior of the constant in terms of the A 1 characteristic of w.