  We propose a priori estimates for a weak solution to the derivative nonlinear
Schr\"odinger equation (DNLS) on torus with small $L^2$-norm datum in low
regularity Sobolev spaces. These estimates allow us to show the existence of
solutions in $H^s(\mathbb{T})$ with some $s<1/2$ in a relatively weak sense.
Furthermore we make some remarks on the error estimates arising from the finite
dimensional approximation solutions of DNLS using the Fourier-Lesbesgue type as
auxiliary spaces, despite the fact that Nahmod, Oh, Rey-Bullet and Staffilani
\cite{nors} have already seen them.
