Convolution operators with singular measures of fractional type on the Heisenberg group
Tom 245 / 2019
Studia Mathematica 245 (2019), 213-228
MSC: Primary 43A80; Secondary 42A38.
DOI: 10.4064/sm8781-12-2017
Opublikowany online: 24 August 2018
Streszczenie
We consider the Heisenberg group $\mathbb{H}^{n}=\mathbb{C}^{n} \times \mathbb{R}$. Let $\mu_{\gamma}$ be the fractional Borel measure on $\mathbb{H}^{n}$ defined by $$ \mu_{\gamma}(E) = \int_{\mathbb{C}^{n}}\chi_{E}(w,\varphi(w)) \prod_{j=1}^{n} \eta_j ( |w_j|^{2}) | w_j |^{-{\gamma}/{n}}\,dw, $$ where $0 \lt \gamma \lt 2n$, $\varphi(w) = \sum_{j=1}^{n} a_{j} \vert w_{j}\vert^{2}$, $w=(w_{1},\ldots ,w_{n}) \in \mathbb{C}^{n}$, $a_{j} \in \mathbb{R}$, and $\eta_j \in C_{c}^{\infty}(\mathbb{R})$. In this paper we study the set of pairs $(p,q)$ such that right convolution with $\mu_{\gamma}$ is bounded from $L^{p}(\mathbb{H}^{n})$ into $L^{q}(\mathbb{H}^{n})$.
