Convolution operators and variable Hardy spaces on the Heisenberg group
Pablo Rocha
TL;DR
This work extends the theory of Hardy spaces to variable exponent settings on the Heisenberg group $\mathbb{H}^{n}$ and studies convolution operators with kernels of type $(\alpha,N)$. Using an atomic decomposition for the variable Hardy space $H^{p(\cdot)}(\mathbb{H}^{n})$ and a refined vector-valued Fefferman–Stein framework for the fractional maximal operator in $L^{p(\cdot)}$ spaces, the authors prove boundedness of right convolution by kernels of type $(\alpha,N)$ from $H^{p(\cdot)}(\mathbb{H}^{n})$ to $L^{q(\cdot)}(\mathbb{H}^{n})$ and to $H^{q(\cdot)}(\mathbb{H}^{n})$ under the condition $\frac{1}{q(\cdot)} = \frac{1}{p(\cdot)} - \frac{\alpha}{Q}$ with $0 \leq \alpha < Q$ and $p(\cdot) \in \mathcal{P}^{\log}(\mathbb{H}^{n})$ satisfying $\frac{Q}{Q+N} < p_{-} \le p_{+} < \frac{Q}{\alpha}$. The Riesz potential $\mathcal{R}_{\alpha}$ is included as a primary example. The results hinge on the variable-exponent atomic decomposition, off-diagonal maximal inequalities, and kernel Taylor estimates, offering a robust generalization of Folland–Stein theory to variable exponent, noncommutative settings with potential applications to a broad class of singular integrals on $\mathbb{H}^{n}$.
Abstract
Let $\mathbb{H}^{n}$ be the Heisenberg group. For $0 \leq α< Q=2n+2$ and $N \in \mathbb{N}$ we consider exponent functions $p(\cdot) : \mathbb{H}^{n} \to (0, +\infty)$, which satisfies Hölder conditions, such that $\frac{Q}{Q+N} < p_{-} \leq p(\cdot) \leq p_{+} < \frac{Q}α$. 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 $(α, N)$ on $\mathbb{H}^{n}$, where $\frac{1}{q(\cdot)} = \frac{1}{p(\cdot)} - \fracα{Q}$. In particular, the Riesz potential on $\mathbb{H}^{n}$ satisfies such estimates.