Zygmund regularity of Colombeau generalized functions and applications to differential equations with nonsmooth coefficients

We introduce an intrinsic notion of Zygmund regularity for Colombeau algebras of generalized functions. In case of embedded distributions belonging to some Zygmund-Hölder space this is shown to be consistent. The definition is motivated by the well-known use of the wavelet transform as a tool in studying Hölder regularity. It is based on a simple mollifier-wavelet interplay which translates wavelet estimates into properties of regularizations. We investigate basic properties of the newly defined subspaces as well as their application to differential equations whose coefficients and initial data are generalized functions in some Zygmund class. Problems of this kind occur, for example, in global seismology where Earth’s properties of fractal nature have to be taken into account.