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# Wydawnictwa / Czasopisma IMPAN / Colloquium Mathematicum / Wszystkie zeszyty

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

### Tom 132 / 2013

Colloquium Mathematicum 132 (2013), 101-111 MSC: 43A80, 42A38. DOI: 10.4064/cm132-1-8

#### Streszczenie

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

#### Autorzy

• T. GodoyFacultad de Matemática, Astronomía y Física – Ciem
Universidad Nacional de Córdoba – Conicet