An FIO-based approach to $L^p$-bounds for the wave equation on $2$-step Carnot groups: the case of M\'etivier groups

Let $\mathcal{L}$ be a homogeneous left-invariant sub-Laplacian on a $2$-step Carnot group. We devise a new geometric approach to sharp fixed-time $L^p$-bounds with loss of derivatives for the wave equation driven by $\mathcal{L}$, 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 case, 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 of the wave propagator on a $2$-step Carnot group allows us to reduce microlocally to two conic regions in frequency space: an anti-FIO region, which seems not amenable to FIO techniques, and an FIO region. For the latter, we construct a parametrix by means of FIOs with complex phase, by 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 scalings, one must deal with the long-time behaviour and control of $L^1$-norms of the corresponding contributions to the wave propagator, a new phenomenon that is specific to sub-elliptic settings. 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 $L^p$-estimates for FIOs, yields $L^p$-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.