Hardy spaces associated with Schrodinger operators on the Heisenberg group

Details Hardy spaces associated with Schrodinger operators on the Heisenberg group Hardy spaces associated with Schrodinger operators on the Heisenberg group

2011-06-24

arxiv.org/abs/1106.4960v1

Mathematics

Analysis of PDEs

42 pages

Scientific paper

Let $L= -\Delta_{\mathbb{H}^n}+V$ be a Schr\"odinger operator on the Heisenberg group $\mathbb{H}^n$, where $\Delta_{\mathbb{H}^n}$ is the sub-Laplacian and the nonnegative potential $V$ belongs to the reverse H\"older class $B_{\frac{Q}{2}}$ and $Q$ is the homogeneous dimension of $\mathbb{H}^n$. The Riesz transforms associated with the Schr\"odinger operator $L$ are bounded from $L^1(\mathbb{H}^n)$ to $L^{1,\infty}(\mathbb{H}^n)$. The $L^1$ integrability of the Riesz transforms associated with $L$ characterizes a certain Hardy type space denoted by $H^1_L(\mathbb{H}^n)$ which is larger than the usual Hardy space $H^1(\mathbb{H}^n)$. We define $H^1_L(\mathbb{H}^n)$ in terms of the maximal function with respect to the semigroup $\big \{e^{-s L}:\; s>0 \big\}$, and give the atomic decomposition of $H^1_L(\mathbb{H}^n)$. As an application of the atomic decomposition theorem, we prove that $H^1_L(\mathbb{H}^n)$ can be characterized by the Riesz transforms associated with $L$. All results hold for stratified groups as well.

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