Analysis of joint spectral multipliers on Lie groups of polynomial growth

We study the problem of L p-boundedness (1 < p < ∞) of operators of the form m(L 1,·,L n) for a commuting system of self-adjoint left-invariant differential operators L 1,·, L n on a Lie group G of polynomial growth, which generate an algebra containing a weighted subcoercive operator. In particular, when G is a homogeneous group and L 1,·,L n are homogeneous, we prove analogues of the Mihlin-Hörmander and Marcinkiewicz multiplier theorems.