Gradient estimates for Schrödinger operators with characterizations of $BMO_{\mathcal {L}}$ on Heisenberg groups

Abstract:

Let $\mathcal {L}=-\Delta _{\mathbb {H}^n}+V$ be a Schrödinger operator with the nonnegative potential $V$ belonging to the reverse Hölder class $B_{Q}$, where $Q$ is the homogeneous dimension of the Heisenberg group $\mathbb {H}^n$. In this paper, we obtain pointwise bounds for the spatial derivatives of the heat and Poisson kernels related to $\mathcal {L}$. As an application, we characterize the space $BMO_{\mathcal {L}}(\mathbb {H}^n)$, associated to the Schrödinger operator $\mathcal {L}$, in terms of two Carleson type measures involving the spatial derivatives of the heat kernel of the semigroup $\{e^{-s\mathcal {L}}\}_{s>0}$ and the Poisson kernel of the semigroup $\{e^{-s\sqrt {\mathcal {L}}}\}_{s>0}$, respectively. At last, we pose a conjecture about the converse characterization of $BMO_{\mathcal {L}}(\mathbb {H}^n)$.