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The Rockland condition for nondifferential convolution operators. (English) Zbl 0678.43002

A connected simply connected Lie group \({\mathcal N}\) is said to be homogeneous if its Lie algebra N admits a positive graduation. Then \({\mathcal N}\) is nilpotent; thus may be identified with N via the exponential mapping. Moreover, there exists a group \((\delta_ s)_{s>0}\) of dilations of N and \({\mathcal N}\) associated with the graduation. Let \((\mu_ t)_{t>0}\) be a continuous convolution semigroup of symmetric probability measures on \({\mathcal N}\) that is \((\delta_ s)_{s>0}\)-stable with exponent \(r>0\) i.e. \(\delta_ s(\mu_ t)=\mu_{s^ rt}\) for all \(s,t>0\). The generating functional P of \((\mu_ t)_{t>0}\) is defined at least on the space \({\mathcal D}({\mathcal N})\) of test functions on \({\mathcal N}\). With every unitary representation \(\pi\) of \({\mathcal N}\) there is canonically associated a closable operator \(\pi_ p\) on the space \(C^{\infty}(\pi)\) of smooth vectors for \(\pi\) [cf. M. Duflo, Ann. Inst. Fourier 28, No.3, 225-249 (1978; Zbl 0368.22006)].
Now the main result of the paper can be formulated: All the measures \(\mu_ t\) are absolutely continuous with square integrable densities (with respect to Haar measure) iff P satisfies the ‘Rockland condition’: For every nontrivial irreducible unitary representation \(\pi\) of \({\mathcal N}\) the closure \(\overline{\pi_ p}\) of \(\pi_ p\) is injective on its domain.
The difficult proof makes heavy use of the machinery of B. Helffer and J. Nourrigat developed for the investigation of hyperelliptic operators on graduated Lie groups [cf. Commun. Partial Differ. Equations 4, 899-958 (1979; Zbl 0423.35040)]. The proof is by induction on the dimension of \({\mathcal N}\); together with some desintegration techniques for representations. For the Heisenberg group the author has proved this theorem previously by another method [Stud. Math. 79, 105-138 (1984; Zbl 0563.43002)].
Reviewer’s remark: It is easy to see that the Rockland condition is equivalent to the condition that the measures \(\mu_ t\) are full i.e. are not supported by a proper closed connected (normal) subgroup of \({\mathcal N}\) [cf. W. Hazod, S. Nobel in: Probability Measures on Groups IX, Proc. Oberwolfach 1988, Lect. Notes Math. 1379, 90-106 (1989)].
Reviewer: E.Siebert

MSC:

43A05 Measures on groups and semigroups, etc.
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
47D03 Groups and semigroups of linear operators
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References:

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