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

Abstract

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