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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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