Some Norm Bounds for the Spectral Projections of the Heisenberg Sublaplacian

This paper is a summary of the lectures delivered by the author at the Gent Summer School in Analysis and PDE in September 2023. It concerns the boundedness of the generalized spectral projections \(\mathcal P_\lambda \) associated to the sublaplacian in the Heisenberg group, seen as operators from \(L^1_t L^p_{z}\) to \(L^\infty _t L^q_{z}\) . A conjecture about the range of \(p,q\) for which the projections \(\mathcal P_\lambda \) are bounded is formulated and some motivations are provided.