Injection theorem for local Ditkin sets (Q258999)

This paper is devoted to present an injection theorem for local Ditkin sets in locally compact groups. Let \(G\) be a locally compact group, \(H\) be an arbitrary closed subgroup of \(G\), and \(1<p<\infty\). The extended notion of locally \(p\)-Detkin sets has been introduced using the canonical action of the Figà-Talamanca-Herz algebras \(A^p(G)\), of the locally compact group \(G\) on the Banach space consists of all bounded linear operators of the Banach function space \(L^p(G)\). As the injection result, it is shown that a closed subset \(F\) of \(H\), is locally \(p\)-Ditkin in \(G\) if and only if it is locally \(p\)-Ditkin in \(H\). To this end, the author employed an interesting characterization of the notion of locally \(p\)-Ditkin sets involving not only convolution operators but also general bounded operators of \(L^p(G)\).