×

On operators satisfying the Rockland condition. (English) Zbl 0924.22007

Main theorem of this paper: Let \(G\) be a homogeneous Lie group with dilations such that \(\delta_t\delta_s=\delta_{ts}\) and \(\delta_t x\to e\) as \(t\to 0\) for all \(x\in G\). Let \(\Gamma\) be a closed subset of \(G^*\) such that \(Ad^*(G)\Gamma\subset\Gamma\) and \(\delta_t\Gamma\subset\Gamma\) for all \(t>0\). Then: (1) For any \(\alpha\geq 0\), there exists a regular kernel \(P\) of order \(\alpha\), (i.e., \(P\) coincides with a smooth function away from the origin, and \(\delta_tP=t^\alpha P\) for all \(t>0)\), such that (i) \(\pi_l(P)=0\) for all \(l\in\Gamma\); (ii) \(\overline{\pi_l(P)}\) is positive definite and injective on its domain for \(l\notin\Gamma\).
(2) For any \(0>\alpha>-Q\), there exists a kernel satisfying the conditions in (1) except for \(l=0\). – Moreover, there is a Schwartz class function \(H\) on \(G\), such that (i’) \(\pi_l(H)=0\) for all \(l\in\Gamma\); (ii’) the operator \(\pi_l(H)\) is positive definite and injective for \(l\notin\Gamma\). – An application is given.
Reviewer: Su Weiyi (Nanjing)

MSC:

22E30 Analysis on real and complex Lie groups
PDFBibTeX XMLCite
Full Text: DOI EuDML