Calder\'on-Hardy type spaces and the Heisenberg sub-Laplacian

TL;DR

This paper introduces Calderón-Hardy spaces on the Heisenberg group and proves the existence and uniqueness of solutions to a sublaplacian equation within these spaces.

Contribution

It defines new Calderón-Hardy spaces on the Heisenberg group and establishes solvability of a sublaplacian equation in these spaces.

Findings

Unique solutions to  = f exist in Calderf3n-Hardy spaces for given f.

The spaces are characterized for specific ranges of p, q, and b3.

The results connect harmonic analysis on the Heisenberg group with PDE solutions.

Abstract

For $0 < p \leq 1 < q < \infty$ and $\gamma > 0$, we introduce the Calder\'on-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 sublaplacian on $\mathbb{H}^{n}$, $1 < q < \frac{n+1}{n}$ and $(2n+2) \, (2 + \frac{2n+2}{q})^{-1} < p \leq 1$.