Multilinear operators with non-smooth kernels and commutators of singular integrals

Abstract

We obtain endpoint estimates for multilinear singular integral operators whose kernels satisfy regularity conditions significantly weaker than those of the standard Calderón-Zygmund kernels. As a consequence, we deduce endpoint L 1 × ⋯ × L 1 L^1 \times \dots \times L^1 to weak L 1 / m L^{1/m} estimates for the m m th-order commutator of Calderón. Our results reproduce known estimates for m = 1 , 2 m = 1, 2 but are new for m ≥ 3 m \ge 3 . We also explore connections between the 2 2 nd-order higher-dimensional commutator and the bilinear Hilbert transform and deduce some new off-diagonal estimates for the former.