Boundedness of second-order Riesz transforms on weighted Hardy and \(BMO\) spaces associated with Schrödinger operators. (English) Zbl 1475.35129
Summary: Let \(d\in\{3,4,5,\dots\}\) and a weight \(w\in A^\rho_\infty\). We consider the second-order Riesz transform \(T=\nabla^2\) \(L^{-1}\) associated with the Schrödinger operator \(L=-\Delta+V\), where \(V\in RH_\sigma\) with \(\sigma>\frac{d}{2}\). We present three main results. First \(T\) is bounded on the weighted Hardy space \(H^1_{w,L}(\mathbb{R}^d)\) associated with \(L\) if \(w\) enjoys a certain stable property. Secondly \(T\) is bounded on the weighted \(BMO\) space \(BMO_{w,\rho}(\mathbb{R}^d)\) associated with \(L\) if \(w\) also belongs to an appropriate doubling class. Thirdly \(BMO_{w,\rho}(\mathbb{R}^d)\) is the dual of \(H^1_{w,L}(\mathbb{R}^d)\) when \(w\in A^\rho_1\).
MSC:
| 35J10 | Schrödinger operator, Schrödinger equation |
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 42B30 | \(H^p\)-spaces |
| 42B37 | Harmonic analysis and PDEs |
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