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Keywords:
Heisenberg group; singular Borel measure; $L^{p}$-improving property
Heisenberg group; singular Borel measure; $L^{p}$-improving property
Summary:
Let $\mu _A$ be the singular measure on the Heisenberg group $\mathbb {H}^{n}$ supported on the graph of the quadratic function $\varphi (y) = y^{t}Ay$, where $A$ is a $2n \times 2n$ real symmetric matrix. If $\det (2A \pm J) \neq 0$, we prove that the operator of convolution by $\mu _A$ on the right is bounded from $L^{\frac {(2n+2)}{(2n+1)}}(\mathbb {H}^{n})$ to $L^{2n+2}(\mathbb {H}^{n})$. We also study the type set of the measures ${\rm d}\nu _{\gamma }(y,s) = \eta (y) |y|^{-\gamma } {\rm d}\mu _{A}(y,s)$, for $0 \leq \gamma < 2n$, where $\eta $ is a cut-off function around the origin on $\mathbb {R}^{2n}$. Moreover, for $\gamma =0$ we characterize the type set of $\nu _{0}$.
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