This paper is dedicated to the study of derivative nonlinear Schrödinger equation on real line. The local well-posedness this in Sobolev spaces $$H^s({\mathop {{\mathbb {R}}}\nolimits })$$ well understood since a couple decades, while global not completely settled. For latter issue, best known results up-to-date concern either Cauchy data $$H^{\frac{1}{2}}({\mathop with mass strictly less than ...

We prove low regularity a priori estimates for the derivative nonlinear Schrödinger equation in Besov spaces with positive index conditional upon small L2-norm. This covers full subcritical range. use power series expansion of perturbation determinant introduced by Killip-Vi?an-Zhang completely integrable PDE. makes it possible to derive conservation laws from determinant.