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

Chaichenets, Leonid; Hundertmark, Dirk; Kunstmann, Peer; Pattakos, Nikolaos

##### Abstract:
We introduce a Littlewood-Paley characterization of modulation spaces and use it to give an alternative proof of the algebra property, implicitly contained in [STW11], 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 $s\ge0$. 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 [BO09, Theorem 1.1] by Bényi and Okoudjou, where only the case $q=1$ is considered, and closes a gap in the literature. If $q>1$ and $s>d(1-\frac{1}{q})$ or if $q=1$ and $s\geq0$ 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 obtain a new Hölder-type inequality for modulation spaces.

 Zugehörige Institution(en) am KIT Institut für Analysis (IANA)Sonderforschungsbereich 1173 (SFB 1173) Publikationstyp Forschungsbericht/Preprint Publikationsjahr 2019 Sprache Englisch Identifikator ISSN: 2365-662X KITopen-ID: 1000105752 Verlag Karlsruher Institut für Technologie (KIT) Umfang 14 S. Serie CRC 1173 ; 2019/27 Projektinformation SFB 1173/2 (DFG, DFG KOORD, SFB 1173/2 2019) Externe Relationen Siehe auch Schlagwörter nonlinear Schrödinger equation, modulation spaces, local well-posedness, Littlewood-Paley characterization, Hölder-type inequality
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