Hardy spaces associated with Schrodinger operators on the Heisenberg group

Let $L= -Δ_{\mathbb{H}^n}+V$ be a Schrödinger operator on the Heisenberg group $\mathbb{H}^n$, where $Δ_{\mathbb{H}^n}$ is the sub-Laplacian and the nonnegative potential $V$ belongs to the reverse Hölder class $B_{\frac{Q}{2}}$ and $Q$ is the homogeneous dimension of $\mathbb{H}^n$. The Riesz transforms associated with the Schrödinger 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.

No reviews yet.

No comments yet.