Calderón-Hardy type spaces and the Heisenberg sub-Laplacian

Abstract

For $0<p\le 1<q<\infty$ and $\gamma >0$, we introduce the Calderón-Hardy spaces ${\mathcal{H}}_{q,\gamma }^p({\mathbb{H}}^n)$ 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 ${\mathcal{H}}_{q,2}^p({\mathbb{H}}^n)$, where $\mathcal{L}$ is the sublaplacian on ${\mathbb{H}}^n$, $1<q<\frac{n+1}{n}$ and $(2n+2)(2+\frac{2n+2}{q}{)}^{-1}<p\le 1$.

26 pages