Spectral multipliers on exponential growth solvable Lie groups

Abstract

Let M be a measure space and let L be a positive definite operator on L2(M). By the spectral theorem, for any bounded Borel measurable function F : [0, ∞) 7→ C the operator F (L)f = ∫ ∞ 0 F (λ)dE(λ)f is bounded on L2(M). We are interested in sufficient conditions on F for F (L) to be bounded on Lp(M), p 6= 2. We direct the reader to [1], [3], [4], [8], [9], [10], [12] and [13] for more background on various multiplier theorems. In this paper we assume F is compactly supported and have some smoothness (finite number of derivatives) and we consider only the case p = 1. Our measure space G is semidirect product of stratified nilpotent Lie group N and the real line. The operator L is (minus) sublaplacian on G. Our group has exponential volume growth. The earlier theory suggested that one needs holomorphic F for F (L) to be bounded on L1, however the recent results [5], [6], [7] showed that estimates on only a finite number of derivatives of F imply boundedness of F (L) on L1 on some solvable G of exponential growth. In this case we say that G (more precisely L) has Ckfunctional calculus. On the other hand, Christ and Muller give an example of a solvable Lie group on which F must be holomorphic. The problem is to find the condition on G (and possibly L) which decides whether G has a Ck-functional calculus or not. Here, our condition is in terms of roots of adjoint representation of the Lie algebra of G. Our groups are of “rank one”,