Discrete Hilbert Transform And Discrete Mikhlin Multiplier On Discrete Variable Lebesgue Space

In this paper, by using continuous Hilbert transform and maximal operator boundedness property in the variable Lebesgue space $L^{p(\cdot )}(\mathbb{R})$ we show that the discrete Hilbert transform is bounded in the variable discrete Lebesgue space ${\mathrm{\ell }}^{p_n}(\mathbb{Z})$. We show that the discrete Mikhlin multiplier ${\mathcal{T}}_m$ is a bounded operator on ${\mathrm{\ell }}^{p_n}(\mathbb{Z})$ when $1<{\underset{‾}{p}}_n<{\bar{p}}_n<\mathrm{\infty }$.