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 $\ell^{p_n}(\mathbb{Z})$. We show that the discrete Mikhlin multiplier $\mathcal{T}_m$ is a bounded operator on $\ell^{p_n}(\mathbb{Z})$ when $1<\underline{p}_n<\bar{p}_n<\infty$.