  We prove low regularity a priori estimates for the derivative nonlinear
Schr\"{o}dinger equation in Besov spaces with positive regularity index
conditional upon small $L^2$-norm. This covers the full subcritical range. We
use the power series expansion of the perturbation determinant introduced by
Killip--Vi\c{s}an--Zhang for completely integrable PDE. This makes it possible
to derive low regularity conservation laws from the perturbation determinant.
