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$L^p$ boundedness of commutators of Riesz transforms associated to Schrödinger operator. (English) Zbl 1140.47035
The authors consider the Schrödinger operator on $\Bbb{R}^{n}$, $n \geq 3$, $$P=-\Delta +V(x),$$ with a non-negative potential $V (x)$ for which the reverse Hölder inequality $$\biggl(\frac{1}{\vert B\vert} \int_{B} V^{q}\,dx\biggl)^{\frac{1}{q}} < c \biggl(\frac{1}{\vert B \vert}\int_{B} V\,dx\biggl)$$ holds with $q>\frac{n}{2}$ for every ball $B$ in $\Bbb{R}^{n}$. 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 $b\in\text{BMO}(\Bbb{R}^{n})$.

47F05Partial differential operators
35J10Schrödinger operator
Full Text: DOI arXiv
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