Local well-posedness for the nonlinear Schr\"odinger equation in the intersection of modulation spaces $M_{p, q}^s(\mathbb{R}^d) \cap M_{\infty, 1}(\mathbb{R}^d)$

TL;DR

This paper establishes local well-posedness for the nonlinear Schrödinger equation in a specific intersection of modulation spaces, using a new Littlewood-Paley characterization to prove an algebra property and extend previous results.

Contribution

It introduces a Littlewood-Paley characterization of modulation spaces and applies it to prove local well-posedness in an intersection space, improving and extending prior work.

Findings

Proved algebra property of the intersection of modulation spaces.

Established local well-posedness for the cubic nonlinear Schrödinger equation in the intersection.

Reobtained a Hölder-type inequality for modulation spaces.

Abstract

We introduce a Littlewood-Paley characterization of modulation spaces and use it to give an alternative proof of the algebra property, somehow implicitly contained in Sugimoto (2011), of the intersection $M^s_{p,q}(\mathbb{R}^d) \cap M_{\infty, 1}(\mathbb{R}^d)$ for $d \in \mathbb{N}$, $p, q \in [1, \infty]$ and $s \geq 0$. We employ this algebra property to show the local well-posedness of the Cauchy problem for the cubic nonlinear Schr\"odinger equation in the above intersection. This improves Theorem 1.1 by B\'enyi and Okoudjou (2009), where only the case $q = 1$ is considered, and closes a gap in the literature. If $q > 1$ and $s > d \left(1 - \frac{1}{q}\right)$ or if $q = 1$ and $s \geq 0$ then $M^s_{p,q}(\mathbb{R}^d) \hookrightarrow M_{\infty, 1}(\mathbb{R}^d)$ and the above intersection is superfluous. For this case we also reobtain a H\"older-type inequality for modulation spaces.