Commutators associated with Schrödinger operators on the nilpotent Lie group

Abstract

Abstract Assume that G is a nilpotent Lie group. Denote by L = − Δ + W $L=-\Delta +W $ the Schrödinger operator on G, where Δ is the sub-Laplacian, the nonnegative potential W belongs to the reverse Hölder class B q 1 $B_{q_{1}}$ for some q 1 ≥ D 2 $q_{1} \geq \frac{D}{2}$ and D is the dimension at infinity of G. Let R = ∇ ( − Δ + W ) − 1 2 $\mathcal{R}=\nabla (-\Delta +W)^{-\frac{1}{2}}$ be the Riesz transform associated with L. In this paper we obtain some estimates for the commutator [ h , R ] $[h,\mathcal{R}]$ for h ∈ Lip ν θ $h\in \operatorname{Lip}^{\theta }_{\nu }$ , where Lip ν θ $\operatorname{Lip}^{\theta }_{\nu }$ is a function space which is larger than the classical Lipschitz space.

Keywords

• commutator
• Lipschitz space
• nilpotent Lie groups
• reverse Hölder inequality
• Riesz transform
• Schrödinger operator