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

Arash Ghorbanalizadeh, Reza Roohi Seraji

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 $.

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

Abstract

In this paper, by using continuous Hilbert transform and maximal operator boundedness property in the variable Lebesgue space we show that the discrete Hilbert transform is bounded in the variable discrete Lebesgue space . We show that the discrete Mikhlin multiplier is a bounded operator on when .
Paper Structure (5 sections, 7 theorems, 45 equations)

This paper contains 5 sections, 7 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Variable Lebesgue Spaces
  3. Variable Discrete Lebesgue Spaces
  4. Auxiliary lemmas and the proof of main result
  5. The Discrete Mikhlin Multiplier

Key Result

Theorem 1.3

Let $\bar{p}_n < \infty$. Then, the discrete Hilbert transform $\mathcal{H}$ is bounded on ${\ell}^{p_n}(\mathbb{Z})$, i.e.,

Theorems & Definitions (17)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 7 more