Caustics of weakly Lagrangian distributions

We study semiclassical sequences of distributions $u_h$ associated to a Lagrangian submanifold of phase space $\text{lag}\subset T^{\ast }X$. If $u_h$ is a semiclassical Lagrangian distribution, which concentrates at a maximal rate on $\text{lag},$ then the asymptotics of $u_h$ are well-understood by work of Arnol’d, provided $\text{lag}$ projects to $X$ with a stable Lagrangian singularity. We establish sup-norm estimates on $u_h$ under much more general hypotheses on the rate at which it is concentrating on $\text{lag}$ (again assuming a stable projection). These estimates apply to sequences of eigenfunctions of integrable and KAM Hamiltonians.