Variable Calderón-Hardy spaces on the Heisenberg group
Pablo Rocha
TL;DR
The paper extends Calderón-Hardy theory to variable exponent Hardy spaces on the Heisenberg group by introducing variable Calderón-Hardy spaces $\mathcal{H}^{p(\cdot)}_{q,\gamma}(\mathbb{H}^{n})$ and proving that the sublaplacian $\mathcal{L}$ provides a bijective correspondence between $\mathcal{H}^{p(\cdot)}_{q,2}(\mathbb{H}^{n})$ and $H^{p(\cdot)}(\mathbb{H}^{n})$ under suitable log-Hölder and p-range conditions $1<q<\frac{n+1}{n}$ and $\underline{p} > \frac{Q}{2+Q/q}$. The work builds on classical Calderón-Hardy results and recent variable-exponent Hardy space theory, employing atomic decompositions, a potential representation via the fundamental solution of $\mathcal{L}$, and maximal-function controls to establish existence, uniqueness, and norm equivalences. It also identifies a zero-space regime when $p_{+} \le \frac{Q}{2+Q/q}$, highlighting the sharp dependence on the exponent function. Overall, the results generalize prior fixed-exponent and Orlicz settings to a flexible variable-exponent framework on a noncommutative geometric context with practical consequences for PDEs on the Heisenberg group.
Abstract
Let $\mathbb{H}^{n}$ be the Heisenberg group and $Q = 2n+2$. For $1 < q < \infty$, $γ> 0$ and an exponent function $p(\cdot)$ on $\mathbb{H}^n$, which satisfy log-Hölder conditions, with $0 < p_{-} \leq p_{+} < \infty$, we introduce the variable Calderón-Hardy spaces $\mathcal{H}^{p(\cdot)}_{q, γ}(\mathbb{H}^{n})$, and show for every $f \in H^{p(\cdot)}(\mathbb{H}^{n})$ that the equation \[ \mathcal{L} F = f \] has a unique solution $F$ in $\mathcal{H}^{p(\cdot)}_{q, 2}(\mathbb{H}^{n})$, where $\mathcal{L}$ is the sublaplacian on $\mathbb{H}^{n}$, $1 < q < \frac{n+1}{n}$ and $Q (2 + \frac{Q}{q})^{-1} < \underline{p}$.