Parabolic Singular Integrals with Nonhomogeneous Kernels

TL;DR

This paper proves $L^2$ boundedness for a class of parabolic singular integrals with nonhomogeneous kernels on regular Lip(1,1/2) graphs, advancing the understanding of parabolic uniform rectifiability.

Contribution

It extends previous results by including nonhomogeneous kernels, combining methods from earlier homogeneous cases with Coifman-David-Meyer techniques.

Findings

Established $L^2$ boundedness for nonhomogeneous kernels

Extended parabolic singular integral theory to more general kernels

Connected to parabolic uniform rectifiability

Abstract

We establish $L^2$ boundedness of all "nice" parabolic singular integrals on "Good Parabolic Graphs", aka {\em regular} Lip(1,1/2) graphs. The novelty here is that we include non-homogeneous kernels, which are relevant to the theory of parabolic uniform rectifiability. Previously, the third named author had treated the case of homogeneous kernels. The present proof combines the methods of that work (which in turn was based on methods described in Christ's CBMS lecture notes), with the techniques of Coifman-David-Meyer.