$L^{\infty} $ norms of Husimi distributions of eigenfunctions

Husimi distributions of Laplace eigenfunctions are special types of `microlocal lifts' of eigenfunctions to phase space. Their weak * limits are the well-known quantum limits or microlocal defect measures of an orthonormal basis $\{ \phi_j\}$ of eigenfunctions on a Riemannian manifold $(M,g)$ . Husimi distributions are normalized mod squares of analytic continuations of eigenfunctions to the complexification of $M$, which may be identified with an open subset of the cotangent bundle $T^*M$. Husimi distributions are probability measures whose density at $\zeta$ is the probability density of a quantum particle at the phase space point $\zeta$. We given universal upper bounds on the sup norms of the Husimi distributions. We also give necessary conditions to obtain the upper bounds in terms of the type of the geodesic through $\zeta$. The bounds are sharp and are achieved by complexified Gaussian beams. These results open the question of relating sup norms (or other natural norms) of Husimi distributions to properties of the weak * limits.