  We show local smoothing estimates in $L^p$-spaces for solutions to the
Hermite wave equation. For this purpose, we obtain a parametrix given by a
Fourier Integral Operator, which we linearize. This leads us to analyze local
smoothing estimates for solutions to Klein-Gordon equations. We show
$\ell^2$-decoupling estimates adapted to the mass parameter to obtain local
smoothing with essentially sharp derivative loss. In one dimension as
consequence of square function estimates, we obtain estimates sharp up to
endpoints. Finally, we elaborate on the implications of local smoothing
estimates for Hermite Bochner--Riesz means.
