Failure of the Hörmander kernel condition for multilinear Calderón–Zygmund operators

Abstract

It is well known that the Hörmander smoothness condition supy≠0⁡∫|x|≥2|y||K(x−y)−K(x)|dx<∞ implies weak-type (1,1) estimates for associated L2-bounded Calderón–Zygmund operators. It has been an open question to know whether Hörmander's condition also suffices to guarantee weak-type (1,1,1/2) estimates for bilinear Calderón–Zygmund operators that are bounded at one point. In this paper, we provide a negative answer to this question.