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Commutators of BMO functions and degenerate Schrödinger operators with certain nonnegative potentials. (English) Zbl 1236.42016

Summary: 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 the reverse Hölder class with respect to the measure \(\omega(x)dx\), where \(\omega(x)\) satisfies the \(A_2\) condition of Muckenhoupt and let \(a_{i,j}(x)\) be a real symmetric matrix satisfying \[ \lambda^{-1}\omega(x)|\xi|^2 \leq \sum^n_{i,j=1} a_{i,j}(x) \xi_i \xi_j \leq \lambda\omega(x) |\xi|^2. \] 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 \(b \in \text{BMO}_\omega\) and \(0 < \alpha \leq 1\).

MSC:

42B30 \(H^p\)-spaces
35J10 Schrödinger operator, Schrödinger equation
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