  In this paper we provide necessary and sufficient conditions for the
$\textnormal{weak}(1,p)$ boundedness, $1< p<\infty,$ of convolution operators
on locally compact (Hausdorff) topological groups. So, we generalize a
classical result due to Sobolev-Hardy-Littlewood and Stepanov. Applications to
Fourier multipliers on Lie groups also are given.
