Zygmund dilations: bilinear analysis and commutator estimates

Abstract

We develop both bilinear theory and commutator estimates in the context of entangled dilations, specifically Zygmund dilations $(x_1,x_2,x_3)\mapsto (\hat{I}{´}_1{x}_1,\hat{I}{´}_2{x}_2,\hat{I}{´}_1\hat{I}{´}_2{x}_3)$ in ${\mathbb{R}}^3$. We construct bilinear versions of recent dyadic multiresolution methods for Zygmund dilations and apply them to prove a paraproduct free $T1$ theorem for bilinear singular integrals invariant under Zygmund dilations. Independently, we prove linear commutator estimates even when the underlying singular integrals do not satisfy weighted estimates with Zygmund weights. This requires new paraproduct estimates.

To appear in Trans. Amer. Math. Soc