Weighted endpoint estimates for commutators of Riesz transforms associated with Schrödinger operators (Q2015311)

Summary: Let \(L=-\Delta+V\) be a Schrödinger operator, where \(\Delta\) is the Laplacian on \(\mathbb R^n\) and the nonnegative potential \(V\) belongs to the reverse Hölder class \(B_{s_1}\) for some \(s_1\geq(n/2)\). Assume that \(\omega\in A_1(\mathbb R^n)\). Denote by \(H^1_L(\omega)\) the weighted Hardy space related to the Schrödinger operator \(L=-\Delta+V\). Let \(\mathcal R_b=[b,\mathcal R]\) be the commutator generated by a function \(b\in\mathrm{BMO}_\theta(\mathbb R^n)\) and the Riesz transform \(\mathcal R=\nabla(-\Delta+V)^{-(1/2)}\). Firstly, we show that the operator \(\mathcal R\) is bounded from \(L^1(\omega)\) into \(L^1_{\mathrm{weak}}(\omega)\). Secondly, we obtain the endpoint estimates for the commutator \([b,\mathcal R]\). Namely, it is bounded from the weighted Hardy space \(H^1_L(\omega)\) into \(L^1_{\mathrm{weak}}(\omega)\).