Oscillating convolution operators on the Heisenberg group

TL;DR

This paper studies oscillating convolution operators on the Heisenberg group, establishing $L^2$ boundedness through oscillatory integral estimates and improving previous results for certain parameter ranges.

Contribution

It extends $L^2$ boundedness results for oscillating convolution operators on the Heisenberg group, using degenerate phase estimates and improving prior work in specific cases.

Findings

Established $L^2$ boundedness for the operators.

Improved previous bounds for the ratio $rac{a^2}{b^2} \

Enhanced understanding of oscillatory integrals on the Heisenberg group.

Abstract

In this paper, we consider oscillating convolution operotors on the Heisenberg group $H^n_a$ with respect to the norm $\rho(x,t) = \rho_1(b x, b t)$ with $\rho_1(x,t)= (|x|^4 + t^2)^{1/4}$. We obtain $L^2$ boundedness properties using the oscillatory integral estimates for degenerate phases in the Euclidean setting. Our result contains an improvement of the Lyall's result for the cases $\frac{a^2}{b^2} \geq C_{\beta}$.