Calderón-Hardy type spaces and the Heisenberg sub-Laplacian
Pablo Rocha
TL;DR
This work extends Calderón-Hardy spaces from the Euclidean setting to the noncommutative Heisenberg group by solving the sublaplacian equation $\mathcal{L}F=f$ for $f \in H^{p}(\mathbb{H}^{n})$. It introduces Calderón-Hardy spaces $\mathcal{H}^{p}_{q,2}(\mathbb{H}^{n})$ based on a quotient-type construction and a maximal operator $N_{q,\gamma}$, then uses the fundamental solution $c_n\rho^{-2n}$ to build inverse images via atomic decompositions of $f$ and convolution. The main theorem establishes bijectivity of $\mathcal{L}$ from $\mathcal{H}^{p}_{q,2}(\mathbb{H}^{n})$ onto $H^{p}(\mathbb{H}^{n})$ for $Q(2+Q/q)^{-1}<p\le 1$, with precise norm equivalences, and shows the space vanishes when $0<p\le Q(2+Q/q)^{-1}$. This provides a sharp, non-Euclidean PDE framework for inverting the sublaplacian on $\mathbb{H}^{n}$ and connects Euclidean Calderón-Hardy theory with subelliptic harmonic analysis.
Abstract
For $0 < p \leq 1 < q < \infty$ and $γ> 0$, we introduce the Calderón-Hardy spaces $\mathcal{H}^{p}_{q, γ}(\mathbb{H}^{n})$ 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 $(2n+2) \, (2 + \frac{2n+2}{q})^{-1} < p \leq 1$.