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

Abstract

We consider the Schrödinger operator $$L = -\\Delta _{G}+V$$L=-ΔG+V on the stratified Lie group G, where $$\\Delta _{G}$$ΔG is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class $$ B_{q_{1}}$$Bq1 for $$q_1\\ge \\frac{Q}{2}$$q1≥Q2, where Q is the homogeneous dimension of G. Let $$q_2 = 1$$q2=1 when $$q_1\\ge Q$$q1≥Q and $$\\frac{1}{q_2}=1-\\frac{1}{q_1}+\\frac{1}{Q}$$1q2=1-1q1+1Q when $$\\frac{Q}{2}<q_1<Q$$Q2<q1<Q. The commutator $$[b,\\mathcal {R}]$$[b,R] is generated by a function $$b\\in \\varLambda ^{\\theta }_{\\nu }(G)$$b∈Λνθ(G) for $$\\theta >0,0<\\nu <1$$θ>0,0<ν<1, where $$\\varLambda ^{\\theta }_{\\nu }(G)$$Λνθ(G) is a new function space on the stratified Lie group which is larger than the classical Companato space, and the Riesz transform $$\\mathcal {R}=\\nabla _{G}(-\\Delta _{G}+V)^{-\\frac{1}{2}}$$R=∇G(-ΔG+V)-12. We prove that the commutator $$[b,\\mathcal {R}]$$[b,R] is bounded from $$L^{p}(G)$$Lp(G) into $$L^{q}(G)$$Lq(G) for $$1<p<q^{ }_{2}$$1<p<q2′, where $$\\frac{1}{q}=\\frac{1}{p}-\\frac{\\nu }{Q}$$1q=1p-νQ.