Oscillating convolution operators on the Heisenberg group

Abstract

In this paper, we consider oscillating convolution operotors on the Heisenberg group $H_a^n$ with respect to the norm $\rho (x,t)={\rho }_1(bx,bt)$ 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 $\frac{a^2}{b^2}\ge C_{\beta }$.

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