### The class of convolution operators on the Marcinkiewicz spaces

Let ${\mathcal{T}}_{X}$ denote the operator-norm closure of the class of convolution operators ${\Phi}_{\mu}:X\to X$ where $X$ is a suitable function space on $R$. Let ${\mathcal{M}}_{r}^{p}$ be the closed subspace of regular functions in the Marinkiewicz space ${\mathcal{M}}^{p}$, $1\le p\<\infty $. We show that the space ${\mathcal{T}}_{{\mathcal{M}}_{r}^{p}}$ is isometrically isomorphic to ${\mathcal{T}}_{{L}^{p}}$ and that strong operator sequential convergence and norm convergence in ${\mathcal{T}}_{{\mathcal{M}}_{r}^{p}}$ coincide. We also obtain some results concerning convolution operators under the Wiener transformation. These are to improve a Tauberian theorem of Wiener on ${\mathcal{M}}^{2}$.