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Boundedness Estimates for Commutators of Riesz Transforms Related to Schrödinger Operators

Boundedness Estimates for Commutators of Riesz Transforms Related to Schrödinger Operators

Year:    2018

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

Abstract

Let L=+V be a Schrödinger operator on Rn(n3), where the nonnegative potential V belongs to reverse Hölder class  RHq1 for q1>n2. Let HpL(Rn) be the Hardy space associated with 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

Copyright:    COPYRIGHT: © Global Science Press

Pages:    17

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