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

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). [ABSTRACT FROM AUTHOR]