Riesz transforms associated with higher-order Schrödinger type operators

Abstract

In this paper, let $L=L_0+V$ be a Schrödinger type operator where $L_0$ is higher order elliptic operator with complex coefficients in divergence form and $V$ is signed measurable function, under the strongly subcritical assumption on $V$, the authors study the $L^q$ boundedness of Riesz transforms ${\nabla }^m{L}^{-1/2}$ for $q\le 2$ and obtain a sharp result. Furthermore, the authors impose extra regularity assumptions on $V$ to obtain the $L^q$ boundedness of Riesz transforms ${\nabla }^m{L}^{-1/2}$ for $q>2$. As an application, the main results can be applied to the operator $L=(-\Delta {)}^m-\gamma |x{|}^{-2m}$ for suitable $\gamma$