Weighted Convolution Operators on $\ell_p$ The main results deal with conditions for the validity of the weighted convolution inequality $\sum_{n\in\mathbb Z}\left|b_n\sum_{k\in\mathbb Z} a_{n-k}x_k\right|^p\le C^p\sum_{k\in\mathbb Z} |x_k|^p$ when $p\ge1$. Keywords:convolution operators on $\ell_p$Categories:40G10;, 40E05