Zygmund dilations: bilinear analysis and commutator estimates

TL;DR

This paper develops bilinear analysis and commutator estimates for Zygmund dilations, leading to a new $T1$ theorem for bilinear singular integrals and novel linear commutator bounds without weighted estimates.

Contribution

It introduces bilinear multiresolution methods and proves a $T1$ theorem for Zygmund dilation-invariant bilinear singular integrals, along with new linear commutator estimates.

Findings

Established a paraproduct free $T1$ theorem for bilinear singular integrals.

Constructed bilinear dyadic multiresolution frameworks for Zygmund dilations.

Proved linear commutator estimates without relying on weighted inequalities.

Abstract

We develop both bilinear theory and commutator estimates in the context of entangled dilations, specifically Zygmund dilations $(x_1, x_2, x_3) \mapsto (\delta_1 x_1, \delta_2 x_2, \delta_1 \delta_2 x_3)$ 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.