Maximizers for the Strichartz inequalities and the Sobolev-Strichartz inequalities for the Schr\"odinger equation

TL;DR

This paper proves the existence of maximizers for both the non-endpoint Strichartz inequalities and the Sobolev-Strichartz inequalities for the Schrödinger equation across all dimensions, using profile decomposition techniques.

Contribution

It introduces a new proof of the linear profile decomposition for the Schrödinger equation with Sobolev initial data, establishing the existence of maximizers.

Findings

Existence of maximizers for non-endpoint Strichartz inequalities in all dimensions.

New proof of linear profile decomposition for Schrödinger equation with Sobolev data.

Existence of maximizers for Sobolev-Strichartz inequalities.

Abstract

In this paper, we first show that there exists a maximizer for the non-endpoint Strichartz inequalities for the Schr\"odinger equation in all dimensions based on the recent linear profile decomposition results. We then present a new proof of the linear profile decomposition for the Schr\"oindger equation with initial data in the homogeneous Sobolev space; as a consequence, there exists a maximizer for the Sobolev-Strichartz inequality.