On local-in-time Strichartz estimates for the Schr\"odinger equation with singular potentials

TL;DR

This paper establishes local-in-time Strichartz estimates for the Schrödinger equation with singular potentials without relying on resolvent estimates, allowing for broader potential classes like Morrey-Campanato.

Contribution

It introduces a new approach to derive Strichartz estimates directly, expanding applicability to more singular potentials beyond traditional resolvent-based methods.

Findings

Strichartz estimates obtained without resolvent estimates

Applicable to Morrey-Campanato class potentials

Broader potential classes covered

Abstract

There have been a lot of works concerning the Strichartz estimates for the perturbed Schr\"odinger equation by potential. These can be basically carried out adopting the well-known procedure for obtaining the Strichartz estimates from the weighted $L^2$ resolvent estimates for the Laplacian. In this paper we handle the Strichartz estimates without relying on the resolvent estimates. This enables us to consider various potential classes such as the Morrey-Campanato classes.