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

Abstract

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