Guo, Zihua; Li, Pengtao; Peng, Lizhong \(L^p\) boundedness of commutators of Riesz transforms associated to Schrödinger operator. (English) Zbl 1140.47035 J. Math. Anal. Appl. 341, No. 1, 421-432 (2008). The authors consider the Schrödinger operator on \(\mathbb{R}^{n}\), \(n \geq 3\), \[ P=-\Delta +V(x), \] with a non-negative potential \(V (x)\) for which the reverse Hölder inequality \[ \biggl(\frac{1}{| B|} \int_{B} V^{q}\,dx\biggl)^{\frac{1}{q}} < c \biggl(\frac{1}{| B |}\int_{B} V\,dx\biggl) \] holds with \(q>\frac{n}{2}\) for every ball \(B\) in \(\mathbb{R}^{n}\). Denote \(T_{1}=(-\Delta+V)^{-1}V\), \(T_{2}=(-\Delta+V)^{-\frac{1}{2}}V^{\frac{1}{2}}\) and \(T_{3}=(-\Delta+V)^{-\frac{1}{2}}\nabla\). The authors study the \(L_{p}\)-boundedness of the commutator operators \([b,T_{j}]=bT_{j}-T_{j}b\) (\(j = 1,2,3\)), where \(b\in\text{BMO}(\mathbb{R}^{n})\). Reviewer: Petru A. Cojuhari (Kraków) Cited in 1 ReviewCited in 53 Documents MSC: 47F05 General theory of partial differential operators 35J10 Schrödinger operator, Schrödinger equation Keywords:commutator; BMO; smoothness; boundedness; Riesz transforms associated to Schrödinger operators PDFBibTeX XMLCite \textit{Z. Guo} et al., J. Math. Anal. Appl. 341, No. 1, 421--432 (2008; Zbl 1140.47035) Full Text: DOI arXiv References: [1] Coifman, R.; Rochberg, R.; Weiss, G., Factorization theorems for Hardy spaces in several variables, Ann. of Math., 103, 611-635 (1976) · Zbl 0326.32011 [2] Gehring, F., The \(L^p\)-integrability of the partial derivatives of a quasi-conformal mapping, Acta Math., 130, 265-277 (1973) · Zbl 0258.30021 [3] Gilberg, D.; Trudinger, N., Elliptic Partial Differential Equations of Second Order (1983), Springer-Verlag [4] Janson, S., Mean oscillation and commutators of singular integral operators, Ark. Mat., 16, 263-270 (1978) · Zbl 0404.42013 [5] Meyer, Y., La minimalité de le espace de Besov \(B_1^{0, 1}\) et la continuité des opérateurs définis par des intégrales singulières, Monografias de Matematicas, vol. 4 (1985), Univ. Autonoma de Madrid [6] Torchinsky, A., Real-Variable Methods in Harmonic Analysis (1986), Academic Press · Zbl 0621.42001 [7] Shen, Z., \(L^p\) estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier, 45, 2, 513-546 (1995) · Zbl 0818.35021 [8] J. Zhong, Harmonic analysis for some Schrödinger type operators, PhD thesis, Princeton University, 1993; J. Zhong, Harmonic analysis for some Schrödinger type operators, PhD thesis, Princeton University, 1993 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.