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 λ>0 λw x ∈ R n : |T f(x)| Mw > λ ≤ c Z Rn |f| dx. We prove a weaker version of this inequality with M3w instead of Mw. Also we study a related question about the behavior of the constant in terms of the A1 characteristic of w.