Calderón-Hardy type spaces and the Heisenberg sub-Laplacian

Abstract

For (0 \lt p \leq 1 \lt q \lt \infty) and (\gamma \gt 0), we introduce the Calderón-Hardy spaces (\mathcal{H}^{p}{q,\gamma}(\mathbb{H}^{n})) on the Heisenberg group (\mathbb{H}^{n}), and show for every (f \in H^{p}(\mathbb{H}^{n})) that the equation [\mathcal{L}F=f] has a unique solution (F) in (\mathcal{H}^{p}{q,2}(\mathbb{H}^{n})), where (\mathcal{L}) is the sub-Laplacian on (\mathbb{H}^{n}), [1 \lt q \lt \frac{n+1}{n} \quad \text{and} \quad (2n+2)\left(2+\frac{2n+2}{q}\right)^{-1} \lt p \leq 1.]