Tempered homogeneous function spaces, II

TL;DR

This paper develops a new framework for homogeneous Besov-Sobolev type function spaces using tempered distributions and Gauss-Weierstrass semi-groups, providing clearer Fourier-analytical descriptions relevant for PDEs like heat and Navier-Stokes.

Contribution

It introduces a novel approach to homogeneous function spaces that avoids polynomial ambiguity by using domestic norms within the tempered distribution framework.

Findings

Provides a Fourier-analytical description of homogeneous function spaces

Applies the framework to PDE models such as heat, Navier-Stokes, and chemotaxis systems

Clarifies the structure of homogeneous Besov-Sobolev spaces in tempered distributions

Abstract

This paper deals with homogeneous function spaces of Besov-Sobolev type within the framework of tempered distributions in Euclidean $n$-space based on Gauss-Weierstrass semi-groups. Related Fourier-analytical descriptions are incorporated afterwards as so-called domestic norms. This approach avoids the usual ambiguity modulo polynomials when homogeneous function spaces are considered in the context of homogeneous tempered distributions. The motivation to deal with these spaces comes from (nonlinear) heat and Navier-Stokes equations, but also from Keller-Segel sytems and other PDE models of chemotaxis.