Higher-order Parabolic Schrodinger Operators on Lebesgue Spaces

Abstract

Let $k \in \{1, 2, 3\, \ldots \}$, $d \in \{2k+1, 2k+2, \ldots \}$ and $V \in RH_\sigma \cap G_{k,d}^{(-\Delta )^k}$ with $\sigma \ge d/2 + k$. Here $RH_\sigma$ is a reverse Holder class and $G_{k,d}^{(-\Delta )^k}$ is a Gaussian class associated with $(-\Delta )^k$. Consider the parabolic Schrodinger operator $L = \partial _t + (-\Delta )^k + V^k$. We show that for certain k, $\alpha$ and V the Riesz transforms $\nabla _x^{2k}L^{-1}$ and $V^{k\alpha }L^{-\alpha }$ are bounded on $L^p(\mathbb {R}^{d+1}_+)$ for suitable values of $p \in (1,\infty )$.