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We propose a priori estimates for a weak solution to the derivative nonlinearSchr\"odinger equation (DNLS) on torus with small $L^2$-norm datum in lowregularity Sobolev spaces. These estimates allow us to show the existence ofsolutions 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 finitedimensional approximation solutions of DNLS using the Fourier-Lesbesgue type asauxiliary spaces, despite the fact that Nahmod, Oh, Rey-Bullet and Staffilani\cite{nors} have already seen them.

A priori estimates and weak solutions for the derivative nonlinear Schr\"{o}dinger equation on torus below $H^{1/2}$