Spectral multipliers for sub-Laplacians on amenable Lie groups with exponential volume growth

Abstract

Let G be a Lie group and ∆ a left invariant sub-Laplacian on G. If L(G) denotes the space of square integrable functions with respect to the right invariant Haar measure on G, then ∆ is a selfadjoint operator on L(G). Therefore every bounded Borel function f on R induces a continuous operator f(∆) on L(G). It is now natural to ask, under which additional conditions on f the operator f(∆) is necessarily bounded on L(G), p 6= 2. In this case we call f an L-multiplier for ∆. For more background information and various multiplier theorems we refer to [1], [3], [2], [5], [10], [8] and the literature mentioned therein. Here we focus our attention on amenable groups with exponential volume growth and continuous functions f with compact support. Our aim is to show for a reasonably large class of Lie groups and sub-Laplacians that a certain degree of differentiability of f is sufficient for f(∆) to extend to a bounded operator on L(G), i. e. that ∆ has differentiable L-functional calculus. That this is not true on any group with exponential growth (in contrast to the situation on Lie groups with polynomial growth, cf. [1]), was shown by M. Christ and D. Muller in [2]. They gave examples of sub-Laplacians ∆ on solvable Lie groups, which are for any p 6= 2 of holomorphic L-type, i. e., there exists some non-isolated point λ in the L-spectrum of ∆ and an open complex neighbourhood U of λ in C such that every continuous Lmultiplier, which vanishes at infinity, extends holomorphically to U . (More recent articles dealing with this topic are [8] and [7].)