Multiresolution analysis and Zygmund dilations

Zygmund dilations are a group of dilations lying in between the standard product theory and the one-parameter setting -- in R³=R they are the dilations (x₁,x₂,x₃)↦ (δ₁ x₁,δ₂ x₂,δ₁δ₂ x₃). The dyadic multiresolution analysis and the related dyadic-probabilistic methods have been very impactful in the modern product singular integral theory. However, multiresolution analysis has not been understood in the Zygmund dilation setting or in other modified product space settings. In this paper we develop this missing dyadic multiresolution analysis of Zygmund type, and justify its usefulness by bounding, on weighted spaces, a general class of singular integrals that are invariant under Zygmund dilations. We provide novel examples of Zygmund Aₚ weights and Zygmund kernels showcasing the optimality of our kernel assumptions for weighted estimates.