$L^{p}$-$L^{q}$ estimates for some convolution operators with singular measures on the Heisenberg group
Volume 132 / 2013
Colloquium Mathematicum 132 (2013), 101-111
MSC: 43A80, 42A38.
DOI: 10.4064/cm132-1-8
Abstract
We consider the Heisenberg group $\mathbb{H}^{n}=\mathbb{C}^{n}\times \mathbb{R}$. Let $\nu $ be the Borel measure on $\mathbb{H}^{n}$ defined by $ \nu (E)=\int_{\mathbb{C}^{n}}\chi _{E}( w,\varphi (w)) \eta (w)\,dw$, where $\varphi (w)=\sum_{j=1}^{n}a_{j}\vert w_{j}\vert ^{2}$, $w=(w_{1},\dots,w_{n})\in \mathbb{C}^{n}$, $a_{j}\in \mathbb{R}$, and $\eta (w)=\eta _{0}( \vert w\vert ^{2}) $ with $\eta _{0}\in C_{c}^{\infty }(\mathbb{R})$. We characterize the set of pairs $(p,q)$ such that the convolution operator with $\nu $ is $L^{p}(\mathbb{H}^{n})$-$L^{q}(\mathbb{H}^{n})$ bounded. We also obtain $L^{p}$-improving properties of measures supported on the graph of the function $\varphi (w)=|w|^{2m}$.
