Hölder-Zygmund Regularity in Algebras of Generalized Functions

Abstract

We introduce an intrinsic notion of Hölder-Zygmund regularity for Colombeau 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 Littlewood-Paley decomposition in characterizing Hölder-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 Hölder-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.

Classes of Generalized Functions with Finite Type Regularities

“…The following remarks make some partial comparisons between our definition and Hörmann's definition [9]. We also formulate an open question.…”

“…The global classes that we introduce are capable of recovering the embedded image of the classical Zygmund and Hölder spaces of functions. We also compare our global Zygmund type generalized function spaces with the one proposed in [7,9].…”

“…Originally [9], the "tilde" did not appear in the notation but since we will introduce a new definition, which is intrinsically related to the classical definition of Zygmund spaces, we leave the notation G r * (R d ) for our space. …”

“…One can find many articles in the literature where local and microlocal properties of generalized functions in generalized function algebras have been considered; besides the quoted monographs, we refer to the papers [5,7,9,10,14,17,21]. The motivation of this article came partly from the papers [7,9], where Zygmund type algebras of generalized functions were studied and used in the qualitative analysis of certain hyperbolic problems.…”

“…The motivation of this article came partly from the papers [7,9], where Zygmund type algebras of generalized functions were studied and used in the qualitative analysis of certain hyperbolic problems. We shall define new classes of generalized functions that are also intrinsically connected with the classical Zygmund spaces.…”