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

Abstract EN

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).