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Weighted Endpoint Estimates for Commutators of Riesz Transforms Associated with Schrödinger Operators

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  • Yu Liu
  • Jielai Sheng
  • Lijuan Wang

Abstract

Let L = −Δ + V be a Schrödinger operator, where Δ is the laplacian on ℝn and the nonnegative potential V belongs to the reverse Hölder class Bs1 for some s1 ≥ (n/2). Assume that ω ∈ A1(ℝn). Denote by HL1(ω) the weighted Hardy space related to the Schrödinger operator L = −Δ + V. Let ℛb = [b, ℛ] be the commutator generated by a function b ∈ BMOθ(ℝn) and the Riesz transform ℛ = ∇(−Δ + V) −(1/2). Firstly, we show that the operator ℛ is bounded from L1(ω) into Lweak1(ω). Secondly, we obtain the endpoint estimates for the commutator [b, ℛ]. Namely, it is bounded from the weighted Hardy space HL1(ω) into Lweak1(ω).

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Handle: RePEc:wly:jnlaaa:v:2013:y:2013:i:1:n:281562
DOI: 10.1155/2013/281562
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