Commutators of higher order Riesz transform associated with Schrödinger operators
Let \(L = -\Delta + V\) be a Schrödinger operator on \(\mathbb R^n\) (\(n \geq 3\)), where \(V \not\equiv 0\) is a nonnegative potential belonging to a certain reverse Hölder class \(B_s\) for \(s \geq n\), i.e., \[ \left({1\over |B|}\int_B V(y)^s \, dy \right)^{1/q} \leq {C\over |B|} \int_B V(y)\, dy. \] The auxiliary function \(\rho(x)\) related to \(V(x)\) is defined as \[ \rho(x)=\sup_{r>0}\, \bigg \{ r:\; \frac{1}{r^{n-2}}\int_{B(x, r)}V(y)\, dy \leq 1 \bigg \}, \qquad x\in \mathbb{R}^n. \] Denote by \(\rho_1(x)\) the auxiliary function related to \(|\nabla V|\). For \(\theta>0\), a locally integrable function \(b\) is said to be in the class \(BMO_{\theta}(\rho) \) if \[ \frac{1}{|B(x,r)|}\int_{B(x,r)}|b(y)-b_B|\, dy\leq C\left(1+\frac{r}{\rho(x)}\right)^{\theta} \] for all \(x\in\mathbb{R}^d\) and \(r>0\), where \(b_B=\frac{1}{|B|}\int_B b(y)\, dy\). Define \(BMO_{\infty}(\rho)=\bigcup_{\theta>0}BMO_{\theta}(\rho) \). A norm for \(b\in BMO_\theta(\rho)\), denoted by \([b]_\theta\), is given by the infimum of the constants satisfying the above inequality. Clearly, \(BMO\subseteq BMO_{\infty}(\rho)\). The authors assume also that \(|\nabla V|\in B_{s_1}, \, s_1\geq n/2\); \(\rho(x)\leq 1\), \(\rho_1(x) \leq C\rho(x)\). The main result of the paper under review is the \(L^p\)-boundedness of the commutators \({\mathcal R}^H_b f = b{\mathcal R}^H f - {\mathcal R}^H(bf)\) generated by the higher order Riesz transform \({\mathcal R}^H = \nabla^2(-\Delta + V)^{-1}\), where \(b \in \text{BMO}_\infty(\rho)\), \(s_1^\prime < p < \infty, (1/s_1)+(1/s_1^\prime)=1\). The following endpoint inequality is also proved: \[ \left|\left\{x\in{\mathbb R}^n:\left|{\mathcal R}^H_b(f)(x) \right|>\lambda \right\} \right| \leq \frac{[b]_\theta}{\lambda}\|f\|_{H^1_L({\mathbb R}^n)}, \] where \(H^1_L({\mathbb R}^n)\) is the Hardy space associated with the Schrödinger operator \(L\).
scientific article; zbMATH DE number 6167919
scientific article