Riesz transforms related to Schrödinger operators acting on BMO type spaces (Q1025818)

The authors study the operators \({\mathcal R_i}=\frac{\partial}{\partial x_i}(-\delta +V)^{-1/2}\) acting on functions defined on \(\mathbb R^d\), where \(\mathcal L=-\Delta + V\) is the Schrödinger operator, i.e a perturbation of the Laplace operator by a non-negative weight. For \(0\leq \beta< 1\) the space \(BMO^\beta_{\mathcal L}(w)\) is defined as the set of locally integrable functions such that \(\int_B |f-f_B|\leq Cw(B)|B|^{\beta/d}\) for any ball \(B\) and \(\int_B|f| \leq Cw(B)|B|^{\beta/d}\) for all \(B=B(x,R)\) with \(R\geq \rho(x)=\sup\{r>0 :\int_{B(x,r)}V\leq r^{d-2}\}\). The paper is devoted to get some conditions on the weight function \(w\) and the potential \(V\) for the operators \({ \mathcal R}_i\) and \({ \mathcal R}^*_i\) to be bounded on \(BMO^\beta_{\mathcal L}(w)\). The authors restrict themselves to weights such that \(w(tB) \leq t^{d\nu}w(B)\) sor some \(\nu\geq 1\) and the type of conditions which allow to show the boundedness on \(BMO^\beta_{\mathcal L}(w)\) are reverse-Hölder inequalities on the potential and \(A_p\)-conditions on the weight together with certain restrictions on the parameters \(\beta\) and \(\nu\).