Carlen–Jauslin–Lieb–Loss Carlen et al. (Int Math Res Not 24:18604–18612, 2021) recently observed a remarkable property of a convolution inequality, namely the convolution inequality \(f\ge f*f\) for real-valued \(f \in L^1(\mathbb {R}^n)\) implies that f is non-negative and has a non-trivial \(L^1\) -bound. In this note, we improve their result by proving that these two consequences are still valid under the weaker assumption in terms of the N times iterated convolution \(f \ge f*f*\cdots *f\) . Moreover, we extend these results to the convolution on some class of Lie groups especially including the Heisenberg group \({\mathbb H}^n\) . We then apply our result on the iterated convolution on \(\mathbb {R}^n\) to the study of some integro–differential equations which generalise the equation investigated in Carlen–Jauslin–Lieb Carlen et al. (Pure Appl Anal 2:659–984, 2020)
