Local well-posedness for the derivative nonlinear Schrödinger equation with <inline-formula><tex-math id="M1">$ L^2 $</tex-math></inline-formula>-subcritical data

Considering the Cauchy problem of the derivative NLS

\begin{document}$ \begin{align} {\rm i} u_{t} + \partial_{xx} u = {\rm i} \mu \partial_x (|u|^2u),\quad u(0,x) = u_0(x), \end{align} $\end{document}

we will show its local well-posedness in modulation spaces \begin{document}$ M^{1/2}_{2,q}(\mathbb{R}) $\end{document} \begin{document}$ (4{\leqslant} q<\infty) $\end{document}. It is well-known that \begin{document}$ H^{1/2} $\end{document} is a critical Sobolev space of the derivative NLS. Noticing that \begin{document}$ H^{1/2} \subset M^{1/2}_{2,q} \subset B^{1/q}_{2,q} $\end{document} \begin{document}$ (q{\geqslant} 2) $\end{document} are sharp inclusions, our result contains a class of functions in \begin{document}$ L^2\setminus H^{1/2} $\end{document}.