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Commutators of BMO functions and degenerate Schrödinger operators with certain nonnegative potentials

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Let \({\mathcal{L}f(x)=-\frac{1}{\omega}\sum_{i,j} \partial_i(a_{i,j}(\cdot)\partial_jf)(x)+V(x)f(x)}\) with the non-negative potential V belonging to reverse Hölder class with respect to the measure ω(x)dx, where ω(x) satisfies the A 2 condition of Muckenhoupt and a i,j (x) is a real symmetric matrix satisfying \({\lambda^{-1}\omega(x)|\xi|^2\le \sum^n_{i,j=1}a_{i,j}(x)\xi_i\xi_j\le\lambda\omega(x)|\xi|^2. }\) We obtain some estimates for \({V^{\alpha}\mathcal{L}^{-\alpha}}\) on the weighted L p spaces and we study the weighted L p boundedness of the commutator \({[b, V^{\alpha} \mathcal{L}^{-\alpha}]}\) when \({b\in BMO_\omega}\) and 0 < α ≤ 1.

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Correspondence to Yu Liu.

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Supported by NNSF of China under Grant #10901018 and Theory Foundation of Metallurgy Research Institute of USTB under Grant No. 00009503.

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Liu, Y. Commutators of BMO functions and degenerate Schrödinger operators with certain nonnegative potentials. Monatsh Math 165, 41–56 (2012). https://doi.org/10.1007/s00605-010-0228-6

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  • DOI: https://doi.org/10.1007/s00605-010-0228-6

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