Multiparameter singular integrals on the Heisenberg group: uniform estimates

Abstract:

We consider a class of multiparameter singular Radon integral operators on the Heisenberg group ${\mathbb H}^1$ where the underlying submanifold 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 not uniform in the coefficients of polynomials with fixed degree. When we ask for which polynomials uniform $L^2$ bounds hold, we arrive at the same class where uniform bounds hold in the euclidean case.