On joint estimates for maximal functions and singular integrals on weighted spaces

Abstract

We consider a conjecture attributed to Muckenhoupt and Wheeden which suggests a positive relationship between the continuity of the Hardy-Littlewood maximal operator and the Hilbert transform in the weighted setting. Although continuity of the two operators is equivalent for A p A_p weights with 1 > p > ∞ 1 > p > \infty , through examples we illustrate this is not the case in more general contexts. In particular, we study weights for which the maximal operator is bounded on the corresponding L p L^p spaces while the Hilbert transform is not. We focus on weights which take the value zero on sets of nonzero measure and exploit this lack of strict positivity in our constructions. These types of weights and techniques have been explored previously by the first author and independently with C. Thiele.