Strichartz estimates for higher order Schr\"odinger equations with Partial regular initial data

TL;DR

This paper develops refined Strichartz estimates for higher-order Schrödinger equations with initial data of partial regularity, enabling new well-posedness results and extending analysis to Dunkl Schrödinger equations with novel stationary phase techniques.

Contribution

It introduces refined Strichartz estimates for equations with partial regularity and extends the analysis to Dunkl Schrödinger equations using a new stationary phase adaptation.

Findings

Established refined Strichartz estimates for partial regularity data.

Proved well-posedness results for nonlinear Schrödinger equations.

Extended analysis to Dunkl Schrödinger equations with a new stationary phase method.

Abstract

In this paper, we establish refined Strichartz estimates for higher-order Schr\"odinger equations with initial data exhibiting partial regularity. By partial regularity, we mean that the initial data are not required to have full Sobolev regularity but only regularity with respect to a subset of the spatial variables. As an application of these estimates, we investigate the well-posedness of nonlinear Schr\"odinger equations with power-type nonlinearities. In addition, we extend our analysis to the Dunkl Schr\"odinger equations under partial regularity, defined with respect to two distinct root systems. This extension poses significant challenges, mainly due to the lack of a suitable stationary phase method in the Dunkl setting. To overcome this difficulty, we develop a new result that provides an adaptation of the stationary phase method to the framework of Dunkl analysis.