Besov regularity in non-linear generalized functions (Q6102534)

This paper is a contribution to the theory of Colombeau type algebras. The regularity theory in this context was initiated in [\textit{M. Oberguggenberger}, Multiplication of distributions and applications to partial differential equations. Harlow: Longman Scientific \& Technical; New York: Wiley (1992; Zbl 0818.46036)], Hölder type regularity in the Colombeau setting is studied in [\textit{G. Hörmann}, Z. Anal. Anwend. 23, No. 1, 139--165 (2004; Zbl 1111.46023)] (from the list of references), and precise characterizations of Hölder-Zygmund classes inside Colombeau algebras can be found in previous work of the authors, [\textit{S.~Pilipović} et al., Monatsh. Math. 170, No.~2, 227--237 (2013; Zbl 1271.26005)] and [\textit{S.~Pilipović} et al., Oper. Theory: Adv. Appl. 231, 307--322 (2013; Zbl 1275.46026)]. In this paper, the authors are interested in describing regularity theory of Colombeau type algebras in the context of Besov type spaces. To that end, a new scale of spaces of generalized functions, denoted by \( \mathcal{G}_q (\Omega)\), is introduced and studied. The basic notions of moderateness and negligibility are given in terms of \(L^q\)-integrability bounds with respect to the net parameter. The obtained spaces turn out to be modules, but not algebras (except when \(q= \infty\)). The authors first characterize local Besov spaces as subspaces of \( \mathcal{G}_q (\Omega)\), and proceed with a characterization of local Besov regularity for distributions. This work is then complemented with global versions of the obtained results. In addition, the Schwartz \(L^q\)-based distribution spaces are embedded in global versions of \( \mathcal{G}_q (\Omega)\).