Classical discrete operators on variable ${\mathrm{\ell }}^{p(\cdot )}(\mathbb{Z})$ spaces

We show, by applying discrete weighted norm inequalities and the Rubio de Francia algorithm, that the discrete Hilbert transform and discrete Riesz potential are bounded on variable ${\mathrm{\ell }}^{p(\cdot )}(\mathbb{Z})$ spaces whenever the discrete Hardy-Littlewood maximal is bounded on ${\mathrm{\ell }}^{p^{\mathrm{\prime }}(\cdot )}(\mathbb{Z})$. We also obtain vector-valued inequalities for the discrete fractional maximal operator.