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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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