  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.
