Local nonfactorization of functions on locally compact groups (Q1094620)

The well-known Cohen-Hewitt factorization theorem states that \(L^ p(G)=L^ 1(G)*L^ p(G)\) for \(1\leq p<\infty\) and any locally compact group G. But \(L^ p(G)=W*L^ p(G)\) is not always valid for an arbitrary subset W in \(L^ 1(G)\). \textit{J. Dieudonné} had shown that \(\{\) f*g \(|\) \(g\in L^ p(G)\}\) is strictly contained in \(L^ p(G)\) for certain \(f\in L^ 1(G)\), i.e. there exists a convolution operator \(T_ f: g\to f*g\) which is not surjective on \(L^ p(G)\). This paper shows that some subsets W actually do not suffice to obtain \(L^ p(G)\) through factorization.