Strichartz estimates for non-degenerate Schr\"odinger equations

TL;DR

This paper establishes local in time Strichartz estimates for Schr"odinger equations with non-degenerate metrics, including endpoint cases, under specific additional assumptions, advancing understanding of dispersive PDEs with non-standard geometries.

Contribution

It provides the first Strichartz estimates without loss for Schr"odinger equations with non-degenerate metrics, including endpoint cases, under new assumptions.

Findings

Strichartz estimates without loss are proved for non-degenerate metrics.

Additional assumptions are necessary for well-posedness and estimates.

The results include endpoint cases previously unresolved.

Abstract

We consider Schr\"odinger equation with a non-degenerate metric on the Euclidean space. We study local in time Strichartz estimates for the Schr\"odinger equation without loss of derivatives including the endpoint case. In contrast to the Riemannian metric case, we need the additional assumptions for the well-posedness of our Schr\"odinger equation and for proving Strichartz estimates without loss.