Some estimates of Schrödinger type operators on the Heisenberg group

Abstract

In this paper, we consider the Schrödinger type operator ${H = (-\Delta _{\mathbb {H}}^n)^2 +V ^{2}}$, where the nonnegative potential V belongs to the reverse Hölder class ${B_{{q}_{1}}\, {\rm for}\, q_{1}\geq {\frac {Q}{ 2}},Q \geq 6}$, and ${\Delta_{\mathbb {H}^n}}$ is the sublaplacian on the Heisenberg group ${\mathbb {H}^n}$. An L ^p estimate and a weak type L ¹ estimate for the operator ${\nabla^4_{\mathbb {H}^n} H^{-1}}$ when ${V \in B_{{q}_{1}}}$ for ${1 < p \leq \frac{q_{1}}{2}}$ are obtained.