Cutoff Resolvent Estimates and the Semilinear Schr\"odinger Equation

TL;DR

This paper links abstract resolvent estimates to local smoothing effects in the Schr"odinger equation, demonstrating how losses in resolvent bounds affect regularity and applying these insights to establish well-posedness of the semilinear Schr"odinger equation.

Contribution

It establishes a connection between resolvent estimates and local smoothing, and applies this to derive well-posedness results for the semilinear Schr"odinger equation.

Findings

Loss in resolvent estimates leads to loss in local smoothing regularity

Abstract resolvent bounds can predict regularity properties of solutions

Well-posedness results are obtained for the semilinear Schr"odinger equation

Abstract

This paper shows how abstract resolvent estimates imply local smoothing for solutions to the Schr\"odinger equation. If the resolvent estimate has a loss when compared to the optimal, non-trapping estimate, there is a corresponding loss in regularity in the local smoothing estimate. As an application, we apply well-known techniques to obtain well-posedness results for the semi-linear Schr\"odinger equation.