An FIO-based approach to L^p-bounds for the wave equation on 2-step Carnot groups: the case of Métivier groups

Let L be a homogeneous left-invariant sub-Laplacian on a 2-step Carnot group. We devise a new geometric approach to sharp fixed-time Lpbounds with loss of derivatives for the wave equation driven by L, based on microlocal analysis and highlighting the role of the underlying sub-Riemannian geodesic flow. A major challenge here stems from the fact that, differently from the Riemannian setting, the conjugate locus of a point on a sub-Riemannian manifold may cluster at the point itself, thus making it indispensable to deal with caustics even when studying small-time wave propagation. Our analysis is based on the costruction of a parametrix by means of FIOs with complex phase, by suitably adapting a construction from the elliptic setting due to Laptev, Safarov and Vassiliev, which remains valid beyond caustics. A substantial problem arising here is that, after a natural decomposition and subsequent scalings, one effectively needs to deal with the long-time behaviour and according control of L1 -norms of the corresponding contributions to the wave propagator, a new phenomenon that is specific to sub-elliptic settings. A further challenge arises from the fact that the parametrix construction only converges in a certain conic region in frequency space, which we refer to as the FIO region; we thus need to develop a different method to tackle the remaining frequency region, referred to as the anti-FIO region, which seems not amenable to FIO techniques. For the class of M´etivier groups, we show how our approach, in combination with a variation of the key method of Seeger, Sogge and Stein for proving Lebesgue-space estimates for FIOs, yields Lp-bounds for the wave equation, which are sharp up to the endpoint regularity. In particular, we extend previously known results for distinguished sub-Laplacians on groups of Heisenberg type, by means of a more general and robust approach. The study of the wave equation on wider classes of 2-step Carnot groups via this approach will pose further challenges that we plan to address in subsequent works.