Generalized Schrödinger operators on the Heisenberg group and Hardy spaces. (English) Zbl 1540.35135
Summary: Let \(L = - \Delta_{\mathbb{H}^n} + \mu\) be a generalized Schrödinger operator on the Heisenberg group \(\mathbb{H}^n\), where \(\Delta_{\mathbb{H}^n}\) is the sub-Laplacian, and \(\mu\) is a nonnegative Radon measure satisfying certain conditions. In this paper, we first establish some estimates of the fundamental solution and the heat kernel of \(L\). Applying these estimates, we then study the Hardy spaces \(H_L^1( \mathbb{H}^n)\) introduced in terms of the maximal function associated with the heat semigroup \(e^{- t L} \); in particular, we obtain an atomic decomposition of \(H_L^1( \mathbb{H}^n)\), and prove the Riesz transform characterization of \(H_L^1( \mathbb{H}^n)\). The dual space of \(H_L^1( \mathbb{H}^n)\) is also studied.
MSC:
| 35J10 | Schrödinger operator, Schrödinger equation |
| 35R03 | PDEs on Heisenberg groups, Lie groups, Carnot groups, etc. |
| 42B30 | \(H^p\)-spaces |
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