It is the purpose of this talk to discuss a variety of situations where invariance properties of function spaces under a certain group of operators, specifically time-frequency shifts or dilations, help to derive atomic characterizations, find minimal or maximal spaces, or prove boundedness properties of certain operators.

Aside from the well-known characterization of real Hardy spaces via
atomic decompositions (Coifman-Weiss, 1977) we can mention the work on the Segal algebra $\SORdN$ in the context of Gabor analysis , but also the proof of Wiener's Third Tauberian Theorem (see \cite{fe88}) for functions of bounded $p$-means on $\Rst^d$ (Wiener did the case $d=1, p=2$ in his book of 1933.

We will also present some known results concerning the {\it Fofana spaces} $(\Lqsp,\lpsp)^{\alpha}$. These spaces are defined as subspaces of Wiener Amalgam spaces $\Wsp(\Lqsp,\lpsp)(\Rdst)$, for $1 \leq p < \alpha < q \leq \infty$ (otherwise they are trivial). In particular we are able to describe them as dual Banach spaces and provide atomic characterizations of the predual.