Characterizations of infinitesimal relative boundedness for higher order Schrödinger operators

Abstract

Let $m\in N$ and $H={(-\Delta )}^{m/2}+V$ be a higher order Schrödinger operators in the Euclidean space $R^n$ with $V\in L_{\mathrm{loc}}^1\left(R^n\right)$. In this paper, the authors characterize the infinitesimal relative boundedness and Trudinger's subordination for H on $L^p\left(R^n\right)$ with $p\in [1,\infty )$, via the limit behavior of the family of operators ${\left\{V{({\lambda }^2-\Delta )}^{-m/2}\right\}}_{\lambda \in (0,\infty )}$ and a generalized Kato-type class condition. The latter is weaker than the classical Kato class condition corresponding to the case $p=1$. All these characterizations are new even when $H=-\Delta +V$ is a second order Schrödinger operator.