$L^{p}$-$L^{q}$ estimates for some convolution operators with singular measures on the Heisenberg group

T. Godoy ¹   ; P. Rocha ¹

¹ Facultad de Matemática, Astronomía y Física – Ciem Universidad Nacional de Córdoba – Conicet Ciudad Universitaria, 5000 Córdoba, Argentina

Colloquium Mathematicum, Tome 132 (2013) no. 1, pp. 101-111

Résumé

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 $\varphi (w)=|w|^{2m}$.

Détail

DOI : 10.4064/cm132-1-8

Keywords: consider heisenberg group mathbb mathbb times mathbb borel measure mathbb defined int mathbb chi varphi eta where varphi sum vert vert dots mathbb mathbb eta eta vert vert eta infty mathbb characterize set pairs convolution operator mathbb l mathbb bounded obtain improving properties measures supported graph function varphi