A restriction theorem for the Heisenberg group
If p is close enough to one, then the Fourier transform of an \(L^ p\)- function on \({\mathbb{R}}^ n\) has a restriction to the unit sphere which is square integrable. It follows that the convolution operator \({\mathcal P}f=f*K\), where K denotes the Fourier transform of the surface measure of the unit sphere, is bounded from \(L^ p\) to \(L^{p'}\). In this paper one proves an analogue for the Heisenberg group \(H_ n\). One considers the spectral resolution E(\(\alpha\)) of the Kohn-Laplacian L on \(H_ n\), \(L=\int^{\infty}_{0}\lambda dE(\lambda)\), and one defines \[ {\mathcal P}_{\alpha}=\lim_{\beta \to \alpha}(E(\beta)-E(\alpha))/(\beta - \alpha). \] Then \({\mathcal P}_{\alpha}\) is a convolution operator, \({\mathcal P}_{\alpha}f=f*G_{\alpha}\), where \(G_{\alpha}\) is computed explicitly. The main result is: if \(1\leq p<2\), then \(\| {\mathcal P}_ 1f\|_{(p',\infty)}\leq C_ p\| f\|_{(p,1)}\). One proves also estimates for the operators \(E_{\alpha,\beta}=E(\beta)-E(\alpha)\). The proofs use the Gelfand transform for the commutative algebra \(L^ 1_ r\) of polyradial integrable functions on \(H_ n\).
scientific article; zbMATH DE number 4208790