Abstract
Let \({A=-(\nabla-i{\vec a})\cdot (\nabla-i{\vec a}) +V}\) be a magnetic Schrödinger operator acting on \({L^2({\mathbb R}^n)}\), n ≥ 1, where \({{\vec a}=(a_1, \ldots, a_n)\in L^2_{\rm loc}({\mathbb R}^n, {\mathbb R}^n)}\) and \({0\leq V\in L^1_{\rm loc}({\mathbb R}^n)}\). In this paper, we show that when a function \({b\in {\rm BMO}({\mathbb R}^n)}\), the commutators [b, T k ]f = T k (b f) − b T k f, k = 1, . . . , n, are bounded on \({L^p({\mathbb R}^n)}\) for all 1 < p < 2, where the operators T k are Riesz transforms (∂/∂x k − i a k )A −1/2 associated with A.
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X. T. Duong is supported by a grant from Australia Research Council. L. X. Yan is supported by NCET of Ministry of Education of China and NNSF of China (Grant No. 10571182/10771221).
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Duong, X.T., Yan, L. Commutators of Riesz transforms of magnetic Schrödinger operators. manuscripta math. 127, 219–234 (2008). https://doi.org/10.1007/s00229-008-0202-y
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DOI: https://doi.org/10.1007/s00229-008-0202-y