Endpoint estimates for commutators of Riesz transforms related to Schrödinger operators

Yanhui Wang () and Yueshan Wang ()
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Yanhui Wang: Jiaozuo University
Yueshan Wang: Jiaozuo University

Indian Journal of Pure and Applied Mathematics, 2021, vol. 52, issue 2, 449-458

Abstract: Abstract We consider the Schrödinger operator $${\mathcal {L}}=-\Delta +V$$ L = - Δ + V on $${\mathbb {R}}^n,$$ R n , where $$n\ge 3$$ n ≥ 3 and the nonnegative potential V belongs to reverse Hölder class $$RH_{s}$$ R H s for $$s>\frac{n}{2}.$$ s > n 2 . In this paper, we discuss the boundedness of Riesz transform $$T_{\alpha ,\beta }=V^\alpha {\mathcal {L}}^{-\beta }$$ T α , β = V α L - β and its commutator at the endpoint. We show that $$T_{\alpha ,\beta }$$ T α , β is bounded from $$L^1({\mathbb {R}}^n)$$ L 1 ( R n ) into $$L^{p_0}({\mathbb {R}}^n),$$ L p 0 ( R n ) , and prove that $$[b,T_{\alpha ,\beta }]$$ [ b , T α , β ] is bounded from $$H^{1}_{{\mathcal {L}}}({\mathbb {R}}^n)$$ H L 1 ( R n ) (Hardy space related to $${\mathcal {L}}$$ L ) into $$L^{p_0}({\mathbb {R}}^n),$$ L p 0 ( R n ) , where $$p_0=\frac{n}{n-2(\beta -\alpha )}$$ p 0 = n n - 2 ( β - α ) and b belongs to the BMO type space introduced by Bongioanni, Harboure and Salinas.

Keywords: Schrödinger operator; Riesz transform; Commutator; Hardy space; BMO; 42B35; 35J10 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s13226-021-00081-0

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