On Herz's extension theorem

Abstract

‎We present a self-contained proof of the following famous extension theorem due to Carl Herz‎. ‎A closed subgroup $H$ of a locally compact group $G$ is a set of $p$-synthesis in $G$ if and only if‎, ‎for every $u\in A_p(H)\cap C_{00}(H)$ and for every $\varepsilon > 0$‎, ‎there is $v\in A_p(G)\cap C_{00}(G)$, an extension of $u$, such that \[\|v\|_{A_p(G)} < \|u\|_{A_p(H)}+\varepsilon.\]