Lp estimates for a singular integral operator motivated by Calderónʼs second commutator

We prove a wide range of Lp estimates for a trilinear singular integral operator motivated by dropping one average in Calderónʼs second commutator. For comparison by dropping two averages in Calderónʼs second commutator one faces the trilinear Hilbert transform. The novelty in this paper is that in order to avoid difficulty of the level of the trilinear Hilbert transform, we choose to view the symbol of the operator as a non-standard symbol. The methods used come from time-frequency analysis but must be adapted to the fact that our symbol is non-standard.