Hölder--Zygmund regularity in algebras of generalized functions (Q1885059)

The author introduces a new intrinsic regularity notion in the special algebra \(\mathcal G\) of generalized functions. His ``generalized Hölder--Zygmund regularity'' incorporates the usual notion in the space of tempered distributions \(\mathcal S'\) consistently. That is, an embedded distribution of Hölder--Zygmund regularity is also of generalized Hölder--Zygmund regularity of the same order in \(\mathcal G\). As an application, a linear hyperbolic Cauchy problem of first order is studied. It is investigated how Hölder--Zygmund regularity of the PDE's coefficients and data translates to Hölder--Zygmund regularity of the waves (that is, the PDE's solution). This result requires a detailed asymptotic analysis of the generalized solution of the present PDE in \(\mathcal G\). As is explained in detail in the introduction, the choice of Hölder--Zygmund regularity is motivated by its value for a systematic qualitative analysis for seismological applications (in the distributional setting). The combination with generalized function algebras is quite fruitful, because such algebras are adapted to nonlinear problems, which in the context of distributions often cannot be rigorously formulated and solved. And such problems quite naturally appear in earth sciences. Finally, something should be said about the technicalities encountered in this paper. The mathematical motivation to develop the concept of Hölder--Zygmund regularity in generalized function algebras is motivated by the use of the continuous Littlewood--Paley decomposition for a characterization of this regularity notion inside \(\mathcal S'\). The distributional part of the paper is based on Theorem~1, which characterizes Hölder--Zygmund regularity by means of asymptotic growth properties of the wavelet transform (with respect to a scaled window, the scaling expressed by the smoothing parameter). The link to Colombeau theory is the fact that mollifying distributions with a Schwartz function \(g\) can be expressed in terms of the wavelet transform with respect to the window \(g\). And since mollifying is the key idea in the construction of the embedding of \(S'\) into \(\mathcal G\), it seems quite natural in view of Theorem~1 to formulate the notion of generalized Hölder--Zygmund regularity in terms of asymptotic growth properties of the representatives of generalized functions (and their derivatives) with respect to the smoothing parameter. A detailed proof of Theorem~1 can be found in the appendix of the paper, thus making the paper self-contained.