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$$L^ p$$ estimates for Schrödinger operators with certain potentials. (English) Zbl 0818.35021
Summary: We consider the Schrödinger operators $$-\Delta +V(x)$$ in $${\mathbb{R}}^ n$$ where the nonnegative potential $$V(x)$$ belongs to the reverse Hölder class $$B_ q$$ for some $$q\geq n/ 2$$. We obtain the optimal $$L^ p$$ estimates for the operators $$(- \Delta +V)^{i\gamma},\nabla^ 2 (- \Delta +V)^{-1}, \nabla (- \Delta +V)^{-1/ 2}$$ and $$\nabla(- \Delta +V)^{-1}$$ where $$\gamma\in{\mathbb{R}}$$. In particular we show that $$(- \Delta +V)^{i\gamma}$$ is a Calderón-Zygmund operator if $$V\in B_{n/ 2}$$ and $$\nabla (- \Delta +V)^{-1/ 2}, \nabla (- \Delta +V)^{-1}\nabla$$ are Calderón-Zygmund operators if $$V\in B_ n$$.

MSC:
 35J10 Schrödinger operator, Schrödinger equation 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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References:
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