Convolution operators and variable Hardy spaces on the Heisenberg group

Abstract

Let ${\mathbb{H}}^n$ be the Heisenberg group. For $0\le \hat{I}\pm <Q=2n+2$ and $N\in \mathbb{N}$ we consider exponent functions $p(\cdot ):{\mathbb{H}}^n\to (0,+\infty )$, which satisfies Hölder conditions, such that $\frac{Q}{Q+N}<p_{-}\le p(\cdot )\le p_{+}<\frac{Q}{\hat{I}}\pm$. In this article we prove the $H^{p(\cdot )}({\mathbb{H}}^n)\to L^{q(\cdot )}({\mathbb{H}}^n)$ and $H^{p(\cdot )}({\mathbb{H}}^n)\to H^{q(\cdot )}({\mathbb{H}}^n)$ boundedness of convolution operators with kernels of type $(\hat{I}\pm ,N)$ on ${\mathbb{H}}^n$, where $\frac{1}{q(\cdot )}=\frac{1}{p(\cdot )}-\frac{\hat{I}}{\pm }Q$. In particular, the Riesz potential on ${\mathbb{H}}^n$ satisfies such estimates.

23 pages