From the L¹ norms of the complex heat kernels to a Hörmander multiplier theorem for sub-Laplacians on nilpotent Lie groups

Xuan Thinh Duong

Vol. 173 (1996), No. 2, 413–424

DOI: 10.2140/pjm.1996.173.413

Abstract

This paper aims to prove a Hörmander multiplier theorem for sub-Laplacians on nilpotent Lie groups. We investigate the holomorphic functional calculus of the sub-Laplacians, then we link the L¹ norm of the complex time heat kernels with the order of differentiability needed in the Hörmander multiplier theorem. As applications, we show that order d∕2 + 1 suffices for homogeneous nilpotent groups of homogeneous dimension d, while for generalised Heisenberg groups with underlying space R^2n+k and homogeneous dimension 2n + 2k, we show that order n + (k + 5)∕2 for k odd and n + 3 + k∕2 for k even is enough; this is strictly less than half of the homogeneous dimension when k is sufficiently large.

Mathematical Subject Classification 2000

Primary: 43A22

Secondary: 22E30, 42B15, 47F05

Milestones

Received: 10 May 1993

Revised: 4 August 1993

Published: 1 April 1996