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

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.