Analysis Seminar: "Multiresolution analysis, Zygmund dilations and weights"

Abstract: Zygmund dilations  $(x_1, x_2, x_3) \mapsto (\delta_1 x_1, \delta_2 x_2, \delta_1 \delta_2 x_3)$ on $\R^3$ are a group of dilations lying in between the standard product theory and the one-parameter setting. The dyadic multiresolution analysis and the related dyadic-probabilistic methods have been very impactful in the modern product singular integral theory. However, the multiresolution analysis has not been understood in the Zygmund dilation setting or in other intermediate product space settings.

We show how to develop this missing multiresolution analysis adapted to Zygmund dilations, and justify its usefulness by bounding, on weighted spaces, a general class of singular integrals that are invariant under Zygmund dilations. New examples of Zygmund $A_p$ weights and Zygmund kernels showcase the optimality of our kernel assumptions for weighted estimates.

Host: Christopher Felder