Sub-Laplacians of Holomorphic Lp-type on Rank One AN-Groups and Related Solvable Groups
✍ Scribed by J. Ludwig; D. Müller
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 547 KB
- Volume
- 170
- Category
- Article
- ISSN
- 0022-1236
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✦ Synopsis
Consider a right-invariant sub-Laplacian L on an exponential solvable Lie group G, endowed with a left-invariant Haar measure. Depending on the structure of G and possibly also that of L, L may admit differentiable L p -functional calculi, or may be of holomorphic L p -type for a given p{2, as recent studies of specific classes of groups G and sub-Laplacians L have revealed. By holomorphic L p -type'' we mean that every L p -spectral multiplier for L is necessarily holomorphic in a complex neighborhood of some point in the L 2 -spectrum of L. This can only arise if the group algebra L 1 (G) is non-symmetric. In this article we prove that, for large classes of exponential groups, including all rank one AN-groups, a certain Lie algebraic condition, which characterizes the non-symmetry of L 1 (G) [37], also suffices for L to be of holomorphic L 1 -type. Moreover, if this condition, which was first introduced by J. Boidol [6] in a different context, holds for generic points in the dual g\* of the Lie algebra of G, then L is of holomorphic L p -type for every p{2. Besides the non-symmetry of L 1 (G), also the closedness of coadjoint orbits plays a crucial role. We also discuss an example of a higher rank AN-group. This example and our results in the rank one case suggest that sub-Laplacians on exponential Lie groups may be of holomorphic L 1 -type if and only if there exists a closed coadjoint orbit 0/g\* such that the points of 0 satisfy Boidol's condition. In the course of the proof of our main results, whose principal strategy is similar as in [8], we develop various tools which may be of independent interest and largely apply to more general Lie groups. Some of them are certainly known as folklore'' results. For instance, we study subelliptic estimates on representation spaces, the relation between spectral multipliers and unitary representations, and develop some holomorphic'' and continuous'' perturbation theory for images of sub-Laplacians under ``smoothly varying'' families of irreducible unitary representations.