99-2   Berichtsreihe des Mathematischen Seminars der Universität Kiel

Jean Ludwig, Detlef Müller:

Sub-Laplacians of holomorphic Lp-type on rank one AN-groups and related solvable groups

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 Lp-functional calculi, or may be of holomorphic Lp-type for a given $p\ne 2$, as recent studies of specific classes of groups G and sub-Laplacians L have revealed. By ''holomorphic Lp-type'' we mean that every Lp-spectral multiplier for L is necessarily holomorphic in a complex neighborhood of some point in the $L^2$-spectrum of L. This can only arize 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 $L^1(G)$ \cite{Poguntke}, also suffices for L to be of holomorphic $L^1$-type. Moreover, if this condition, which was first introduced by J.~Boidol \cite{Boidol} in a different context, holds for generic points in the dual $\g^*$ of the Lie algebra of G, then L is of holomorphic Lp-type for every $p\ne 2.$ 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 $\Om\subset\g^*$ such that the points of $\Om$ satisfy Boidol's condition.

In the course of the proof of our main results, whose principal strategy is similar as in \cite{ChristMueller}, 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.

Mathematics Subject Classification (1991): 22E30, 22E27, 43A20

Bibliographical note: J. Funct. Anal. 170 (2000), 366-427

Keywords: exponential solvable Lie group, sub-Laplacian, functional calculus, Lp-spectral multiplier, symmetry, unitary representation, holomorphic Lp-type, subelliptic estimate, heat kernel, perturbation

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[Thu Feb 19 18:56:34 2009]
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