$L^{p}$ estimates for commutators of Riesz transforms associated with Schrödinger operators on stratified groups

Abstract

We consider the Schrödinger operator $L = -\Delta _{G}+V$ on the stratified Lie group G, where $\Delta _{G}$ is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class $B_{q_{1}}$ for $q_1\ge \frac{Q}{2}$, where Q is the homogeneous dimension of G. Let $q_2 = 1$ when $q_1\ge Q$ and $\frac{1}{q_2}=1-\frac{1}{q_1}+\frac{1}{Q}$ when $\frac{Q}{2}<q_1<Q$. The commutator $[b,\mathcal {R}]$ is generated by a function $b\in \varLambda ^{\theta }_{\nu }(G)$ for $\theta >0,0<\nu <1$, where $\varLambda ^{\theta }_{\nu }(G)$ is a new function space on the stratified Lie group which is larger than the classical Companato space, and the Riesz transform $\mathcal {R}=\nabla _{G}(-\Delta _{G}+V)^{-\frac{1}{2}}$. We prove that the commutator $[b,\mathcal {R}]$ is bounded from $L^{p}(G)$ into $L^{q}(G)$ for $1<p<q^{'}_{2}$, where $\frac{1}{q}=\frac{1}{p}-\frac{\nu }{Q}$.