Multiparameter singular integrals on the Heisenberg group: uniform estimates

TL;DR

This paper studies multiparameter singular Radon integral operators on the Heisenberg group, revealing that unlike the Euclidean case, these operators are always $L^2$ bounded but with non-uniform bounds depending on polynomial coefficients.

Contribution

It uncovers the behavior of $L^2$ bounds for these operators on the Heisenberg group and compares it to the Euclidean case, highlighting new phenomena.

Findings

Operators are always $L^2$ bounded on the Heisenberg group.

Bounds are not uniform in polynomial coefficients.

The class of polynomials with uniform bounds matches the Euclidean case.

Abstract

We consider a class of multiparameter singular Radon integral operators on the Heisenberg group ${\mathbb H}^1$ where the underlying variety is the graph of a polynomial. A remarkable difference with the euclidean case, where Heisenberg convolution is replaced by euclidean convolution, is that the operators on the Heisenberg group are always $L^2$ bounded. This is not the case in the euclidean setting where $L^2$ boundedness depends on the polynomial defining the underlying surface. Here we uncover some new, interesting phenomena. For example, although the Heisenberg group operators are always $L^2$ bounded, the bounds are {\it not} uniform in the coefficients of polynomials with fixed degree. When we ask for which polynoimals uniform $L^2$ bounds hold, we arrive at the {\it same} class where uniform bounds hold in the euclidean case.