H\"older-Zygmund regularity in algebras of generalized functions

TL;DR

This paper develops an intrinsic notion of Hölder-Zygmund regularity within Colombeau algebras, linking wavelet analysis with generalized functions to handle non-smooth coefficients in differential equations, with applications in geophysics.

Contribution

It introduces a new scale of subspaces in Colombeau algebras based on Hölder-Zygmund regularity, connecting wavelet estimates with regularization properties.

Findings

Defined Hölder-Zygmund regularity for Colombeau functions

Established consistency with classical distribution spaces

Applied to differential equations with fractal-like coefficients

Abstract

We introduce an intrinsic notion of Hoelder-Zygmund regularity for Colombeau generalized functions. In case of embedded distributions belonging to some Zygmund-Hoelder space this is shown to be consistent. The definition is motivated by the well-known use of Littlewood-Paley decomposition in characterizing Hoelder-Zygmund regularity for distributions. It is based on a simple interplay of differentiated convolution-mollification with wavelet transforms, which directly translates wavelet estimates into properties of the regularizations. Thus we obtain a scale of new subspaces of the Colombeau algebra. We investigate their basic properties and indicate first applications to differential equations whose coefficients are non-smooth but belong to some Hoelder-Zygmund class (distributional or generalized). In applications problems of this kind occur, for example, in seismology when Earth's geological properties of fractal nature have to be taken into account while the initial data typically involve strong singularities.