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

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 $M_{p,q}^s({\mathbb{R}}^d)\cap M_{\infty ,1}({\mathbb{R}}^d)$ for $d\in \mathbb{N}$, $p,q\in [1,\infty ]$ and $s\ge 0$. We employ this algebra property to show the local well-posedness of the Cauchy problem for the cubic nonlinear Schrödinger equation in the above intersection. This improves Theorem 1.1 by Bényi 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 $s\ge 0$ then $M_{p,q}^s({\mathbb{R}}^d)\hookrightarrow M_{\infty ,1}({\mathbb{R}}^d)$ and the above intersection is superfluous. For this case we also reobtain a Hölder-type inequality for modulation spaces.

14 pages

References in corpus (2)

• Nonlinear Schrödinger equation, differentiation by parts and modulation spaces
• NLS in the modulation space $M_{2,q}(\mathbb{R})$

Cited by in corpus (1)

• On the global wellposedness of the Klein-Gordon equation for initial data in modulation spaces