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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 $$\lambda^{-1}\omega(x)|\xi|^2 \le \sum^n_{i,j=1} a_{i,j}(x) \xi_i \xi_j \le \lambda\omega(x) |\xi|^2.$$ 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 $b \in \text{BMO}_\omega$ and $0 < \alpha \leq 1$.

42B30$H^p$-spaces (Fourier analysis)
35J10Schrödinger operator
Full Text: DOI
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