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 $.

Categories: math.FA