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

TL;DR

This paper demonstrates the boundedness of the discrete Hilbert transform and Mikhlin multipliers on variable discrete Lebesgue spaces, extending continuous space results to discrete settings using properties of the continuous Hilbert transform.

Contribution

It establishes the boundedness of discrete Hilbert transform and Mikhlin multipliers on variable discrete Lebesgue spaces, bridging continuous and discrete harmonic analysis.

Findings

Discrete Hilbert transform is bounded on variable discrete Lebesgue spaces.

Discrete Mikhlin multipliers are bounded under certain p-norm conditions.

Results extend continuous space boundedness to discrete variable Lebesgue spaces.

Abstract

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 $.