Abstract

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 a consequence of square function estimates, we obtain estimates with essentially sharp derivative loss in $L^p$-spaces for $p\ge 2$. Finally, we elaborate on the implications of local smoothing estimates for Hermite Bochner–Riesz means.

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