Variable Calderón-Hardy spaces on the Heisenberg group

Abstract

Let ${\mathbb{H}}^n$ be the Heisenberg group and $Q=2n+2$. For $1<q<\infty$, $\gamma >0$ and an exponent function $p(\cdot )$ on ${\mathbb{H}}^n$, which satisfy log-Hölder conditions, with $0<p_{-}\le p_{+}<\infty$, we introduce the variable Calderón-Hardy spaces ${\mathcal{H}}_{q,\gamma }^{p(\cdot )}({\mathbb{H}}^n)$, 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 ${\mathcal{H}}_{q,2}^{p(\cdot )}({\mathbb{H}}^n)$, where $\mathcal{L}$ is the sublaplacian on ${\mathbb{H}}^n$, $1<q<\frac{n+1}{n}$ and $Q(2+\frac{Q}{q}{)}^{-1}<\underline{p}$.

arXiv comment18 pages. arXiv admin note: substantial text overlap with arXiv:2505.12163