Sharp weighted CR trace Sobolev inequalities

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