Regularity of fractional heat semigroups associated with Schrödinger operators on Heisenberg groups

Abstract

Let L = - Δ ℍ n + V {L=-{\Delta}_{\mathbb{H}^{n}}+V} be a Schrödinger operator on Heisenberg groups ℍ n {\mathbb{H}^{n}} , where Δ ℍ n {{\Delta}_{\mathbb{H}^{n}}} is the sub-Laplacian, the nonnegative potential V belongs to the reverse Hölder class B 𝒬 / 2 {B_{\mathcal{Q}/2}} . Here 𝒬 {\mathcal{Q}} is the homogeneous dimension of ℍ n {\mathbb{H}^{n}} . In this article, we introduce the fractional heat semigroups { e - t ⁢ L α } t > 0 {\{e^{-tL^{\alpha}}\}_{t>0}} , α > 0 {\alpha>0} , associated with L. By the fundamental solution of the heat equation, we estimate the gradient and the time-fractional derivatives of the fractional heat kernel K α , t L ⁢ ( ⋅ , ⋅ ) {K_{\alpha,t}^{L}(\,\cdot\,,\cdot\,)} , respectively. As an application, we characterize the space BMO L γ ⁢ ( ℍ n ) {\mathrm{BMO}_{L}^{\gamma}(\mathbb{H}^{n})} via { e - t ⁢ L α } t > 0 {\{e^{-tL^{\alpha}}\}_{t>0}} .