Local Well-Posedness for the Derivative Nonlinear Schr\"odinger Equation in Besov spaces

It is shown that the cubic derivative nonlinear Schr"odinger equation is locally well-posed in Besov spaces $B_{2,\mathrm{\infty }}^s(\mathbb{X})$, $s\ge \frac{1}{2}$, where we treat the non-periodic setting $\mathbb{X}=\mathbb{R}$ and the periodic setting $\mathbb{X}=\mathbb{T}$ simultaneously. The proof is based on the strategy of Herr for initial data in $H^s(\mathbb{T})$, $s\ge \frac{1}{2}$.