$L^p$ estimates for Schrödinger operators with certain potentials

Shen, Zhongwei

Annales de l'Institut Fourier, Tome 45 (1995) no. 2, pp. 513-546.

Nous considérons des opérateurs de Schrödinger $-\Delta +V\left(x\right)$ dans ${ℝ}^n$ où le facteur non négatif $V\left(x\right)$ appartient à la classe de Hölder inversée $B_q$ pour tout $q\ge n/2$. Nous obtenons les estimations optimales $L^p$ pour les opérateurs $(-\Delta +V{)}^{i\gamma },{\nabla }^2(-\Delta +V{)}^{-1},\nabla (-\Delta +V{)}^{-1/2}$ et $\nabla (-\Delta +V{)}^{-1}$ où $\gamma \in ℝ$. En particulier nous montrons que $(-\Delta +V{)}^{i\gamma }$ est un opérateur de Calderón-Zygmund si $V\in B_{n/2}$ and $\nabla (-\Delta +V{)}^{-1/2},\nabla (-\Delta +V{)}^{-1}\nabla$ sont des opérateurs de Calderón-Zygmund si $V\in B_n$.

We consider the Schrödinger operators $-\Delta +V\left(x\right)$ in ${ℝ}^n$ where the nonnegative potential $V\left(x\right)$ belongs to the reverse Hölder class $B_q$ for some $q\ge n/2$. 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 $\gamma \in ℝ$. 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 $V\in B_n$.

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