Cocompact embeddings of function spaces

A sequence  in a Banach space $E$ is called $G$-weak convergent, relative to a set $G$
of homeomorphisms of $E$  if $\forall  g_k \in G$, $g_k(u_k-u)\rightharpoonup 0$. An
embedding of two Banach spaces is called $G$-cocompact if every $G$-weakly convergent
sequence in $E$ is convergent in $F$. Cocompact embeddings allow to improve convergence
of bounded sequences beyond weak convergence when there is no pertinent compactness.
On a series of examples in Sobolev, Besov, Lorenz-Zygmund, and Stricharz spaces, we show
how cocompactness follows from compactness via a suitable decomposition, for example the
Littlewood-Paley decomposition or the wavelet expansion.