Sharp weighted CR trace Sobolev inequalities

Abstract

We establish a sharp Sobolev trace inequality on the Siegel domain $\Omega_{n+1}$ involving the weighted norm-$W^{2,2}(\Omega_{n+1}, \rho^{1-2[\gamma]})$. The inequality is closely related the realization of fractional powers of the sub-Laplacian on the Heisenberg group $H^n=\partial \Omega_{n+1}$ as generalized Dirichlet-to-Neumann operators associated to the weighted poly-sublaplacian, generalizing observations of Frank--Gonz\'alez--Monticelli--Tan.