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 (x1, x2, x3) → (δ1x1, δ2x2, δ1δ2x3) in R 3 . We construct bilinear versions of recent dyadic multiresolution methods for Zygmund dilations and apply them to prove a paraproduct free T 1 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.2010 Mathematics Subject Classification. 42B20. Key words and phrases. singular integrals, multi-parameter analysis, Zygmund dilations, multiresolution analysis, weighted estimates. E.A. was supported by Academy of Finland through Grant No. 321896 "Incidences on Fractals" (PI = Orponen) and No. 314829 "Frontiers of singular integrals" (PI = Hytönen). K. L. was supported by the National Natural Science Foundation of China through project number 12222114 and 12001400.