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\(L^p\)-improving properties of certain singular measures on the Heisenberg group. (English) Zbl 1513.43012

Summary: Let \(\mu_A\) be the singular measure on the Heisenberg group \(\mathbb{H}^n\) supported on the graph of the quadratic function \(\varphi(y)=y^tAy\), 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 \(L^{(2n+2)/(2n+1)}(\mathbb{H}^{n})\) to \(L^{2n+2}(\mathbb{H}^{n})\). We also study the type set of the measures \(\mathrm{d}\nu_{\gamma}(y,s)=\eta(y)|y|^{-\gamma}\mathrm{d}\mu_{A}(y,s)\), 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 \(\nu_0\).

MSC:

43A80 Analysis on other specific Lie groups
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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