  The paper studies short exact sequences of Banach modules over the
convolution algebra $L_1=L_1(G)$, where $G$ is a compact abelian group. The
main tool is the notion of a nonlinear $L_1$-centralizer, which in combination
with the Fourier transform, is used to produce sequences of $L_1$-modules
$0\rightarrow L_q \rightarrow Z \rightarrow L_p \rightarrow 0$ that are
nontrivial as long as the general theory allows it, namely for $p\in
(1,\infty], q\in[1,\infty)$. Concrete examples are worked in detail for the
circle group, with applications to the Hardy classes, and the Cantor group.
