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\(H^1\) boundedness for Riesz transform related to Schrödinger operator on nilpotent groups. (English) Zbl 1215.35053
Summary: Let \(\mathbb G\) be a nilpotent Lie group equipped with a Hörmander system of vector fields \(X=(X_1,\dots,X_m)\) and \(\Delta= \sum_{i=1}^m X_i^2\) be the sub-Laplacians associated with \(X\). Let \(A=-\Delta+W\) be the Schrödinger operator with the potential function \(W\) belonging to the reverse Hölder class \(B_q\) for some \(q\geq D/2\), where \(D\) denotes the dimension at infinity. In this paper, we prove that the Riesz transform \(\nabla A^{-1/2}\) related to the Schrödinger operator \(A\) is bounded from the Hardy space \(H^1(\mathbb G)\) to itself.
MSC:
35J10 Schrödinger operator, Schrödinger equation
42B35 Function spaces arising in harmonic analysis
42B30 \(H^p\)-spaces
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