A Frobenius integrability theorem for plane fields generated by quasiconformal deformations

Abstract

We generalize the classical Frobenius integrability theorem to plane fields of class $C^Q$, a regularity class introduced by Reimann [9] for vector fields in Euclidean spaces. Reimann showed that a $C^Q$ vector field is uniquely integrable and its flow is a quasiconformal deformation. We prove that an a.e. involutive $C^Q$ plane field (defined in a suitable way) in $R^n$ is integrable, with integral manifolds of class $C^{1,Q}$.