TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 351, Number 9, Pages 3743–3755 S 0002-9947(99)02501-5 Article electronically published on April 27, 1999 ON THE L 2 → L ∞ NORMS OF SPECTRAL MULTIPLIERS OF “QUASI-HOMOGENEOUS” OPERATORS ON HOMOGENEOUS GROUPS ADAM SIKORA Abstract. We study the L 2 → L ∞ norms of spectral projectors and spec- tral multipliers of left-invariant elliptic and subelliptic second-order differential operators on homogeneous Lie groups. We obtain a precise description of the L 2 → L ∞ norms of spectral multipliers for some class of operators which we call quasi-homogeneous. As an application we prove a stronger version of Alexopoulos’ spectral multiplier theorem for this class of operators. 1. Introduction Let G be a nilpotent Lie group with Lie algebra g. For a system X 1 ,...,X k ∈ g of left-invariant vector fields on G satisfying H¨ormander’s condition we define an operator L by the formula L = − k  i=1 X 2 i . (1) It is well known that the closure of the operator L in L 2 (G) is self-adjoint. Thus it admits a spectral resolution L =  ∞ 0 λdE L (λ). For any bounded Borel function F we define an operator F (L) by the formula F (L)=  ∞ 0 F (λ)dE L (λ). By K F (L) we denote the kernel of the operator F (L), i.e. a distribution such that F (L)ψ = ψ ∗ K F (L) . (2) In the present paper we investigate the L 2 (G) → L ∞ (G) norms of spectral multipliers of the operator L: ‖F (L)‖ 2 L 1 →L 2 = ‖F (L)‖ 2 L 2 →L ∞ = ‖K F (L) ‖ 2 L 2 . In the case of elliptic differential and pseudo-differential self-adjoint operators on a compact manifold M the L 2 → L ∞ norms of spectral multipliers were investigated by H¨ ormander [10], [9], by Sogge [16], [17], [18] and by Christ and Sogge in [3]. In [10] H¨ ormander proved that if χ [a,b) denotes the characteristic function of an Received by the editors November 10, 1996. 1991 Mathematics Subject Classification. Primary 42B15; Secondary 43A22, 35P99. c 1999 American Mathematical Society 3743 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use