Regularity of vector fields with piecewise regular curl and divergence

We consider a bounded Lipschitz domain [Formula: see text] with sufficiently smooth boundary and prove piecewise Sobolev regularity of vector fields that have piecewise regular curl and divergence, but may be discontinuous across mutually disjoint and sufficiently smooth surfaces inside of [Formula: see text]. The main idea behind our approach is to employ recently developed parametrices for the curl-operator and the regularity theory of Poisson transmission problems. We conclude our work by applying our findings to the heterogeneous time-harmonic Maxwell equations with either a) impedance, b) natural or c) essential boundary conditions and providing wavenumber-explicit piecewise regularity estimates for these equations.