Boundedness Estimates for Commutators of Riesz Transforms Related to Schrödinger Operators

doi:10.4208/ata.OA-2017-0071

Boundedness Estimates for Commutators of Riesz Transforms Related to Schrödinger Operators

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Year: 2018

Analysis in Theory and Applications, Vol. 34 (2018), Iss. 4 : pp. 306–322

Abstract

Let $\mathcal{L} = −∆+V$ be a Schrödinger operator on $\mathbb{R}^n(n ≥ 3)$ , where the nonnegative potential $V$ belongs to reverse Hölder class $RH_{q_1}$ for $q_1 > \frac{n}{2}$ . Let $H^p_{\mathcal{L}}(\mathbb{R}^n)$ be the Hardy space associated with $\mathcal{L}$ . In this paper, we consider the commutator $[b,T_α]$ , which associated with the Riesz transform $T_α = V^α(−∆+V)^{-\alpha}$ with $0<α≤ 1$ , and a locally integrable function $b$ belongs to the new Campanato space $Λ^θ_β(ρ)$ . We establish the boundedness of $[b,T_α]$ from $L^p(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n)$ for $1<p<q_1/α$ with $1/q=1/p−β/n$ . We also show that $[b,T_α]$ is bounded from $H^p_{\mathcal{L}}(R^n)$ to $L^q(\mathbb{R}^n)$ when $n/ (n+β) < p ≤ 1$ , $1/q=1/p−β/n$ . Moreover, we prove that $[b,T_α]$ maps $H^{\frac{n}{n+\beta}}_{\mathcal{L}}(\mathbb{R}^n)$ continuously into weak $L^1(\mathbb{R}^n)$ .

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Journal Article Details

Publisher Name: Global Science Press

Language: English

DOI: https://doi.org/10.4208/ata.OA-2017-0071

Analysis in Theory and Applications, Vol. 34 (2018), Iss. 4 : pp. 306–322

Published online: 2018-01

AMS Subject Headings: Global Science Press

Pages: 17

Keywords: Riesz transform Schrödinger operator commutator Campanato space Hardy space.

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