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Geodesic trapping is an obstruction to dispersive estimates for solutions tothe Schr\"odinger equation. Surprisingly little is known about solutions to theSchr\"odinger equation on manifolds with degenerate trapping, since theconditions for degenerate trapping are not stable under perturbations. In thispaper we extend some of the results of [CM14] on inflection-transmission typetrapping on warped product manifolds to the case of multi-warped products. Themain result is that the trapping on one cross section does not interact withthe trapping on other cross sections provided the manifold has only oneinfinite end and only inflection-transmission type trapping.

Local Smoothing for the Schr\"odinger Equation on a Multi-Warped Product Manifold with Inflection-Transmission Trapping