Growth Bound and Nonlinear Smoothing for the Periodic Derivative Nonlinear Schr\"odinger Equation

A polynomial-in-time growth bound is established for global Sobolev $H^s(\mathbb{T})$ solutions to the derivative nonlinear Schr"odinger equation on the circle with $s>1$. These bounds are derived as a consequence of a nonlinear smoothing effect for an appropriate gauge-transformed version of the periodic Cauchy problem, according to which a solution with its linear part removed possesses higher spatial regularity than the initial datum associated with that solution.