$L^p$ Estimates for Some Schrödinger Type Operators and a Calderóon-Zygmund Operator of Schrödinger Type

Abstract

We consider the Schrödinger and Schrödinger type operators $H_{1}=-\Delta+V$ and $H_2=(-\Delta)^2+V^2$ with non-negative potentials $V$ on $\mathbf{R}^n$. We assume that the potential $V$ belongs to the reverse Hölder class which includes non-negative polynomials. We establish estimates of the fundamental solution for $H_{2}$ and show some $L^p$ estimates for Schrödinger type operators. Moreover, we show that the operator $\nabla^4H_{2}^{-1}$ is a Calderón-Zygmund operator.