Lp boundedness of commutators of Riesz transforms associated to Schrödinger operator

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Abstract

In this paper we consider Lp boundedness of some commutators of Riesz transforms associated to Schrödinger operator P=Δ+V(x) on Rn, n3. We assume that V(x) is non-zero, non-negative, and belongs to Bq for some qn/2. Let T1=(Δ+V)−1V, T2=(Δ+V)1/2V1/2 and T3=(Δ+V)1/2. We obtain that [b,Tj] (j=1,2,3) are bounded operators on Lp(Rn) when p ranges in a interval, where bBMO(Rn). Note that the kernel of Tj (j=1,2,3) has no smoothness.

Keywords

Commutator
BMO
Smoothness
Boundedness
Riesz transforms associated to Schrödinger operators

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Research supported by NNSF of China No. 10471002, RFDP of China No. 20060001010.