Analysis of joint spectral multipliers on Lie groups of polynomial growth

We study the problem of L^p-boundedness (1 < p < \infty) 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-H"ormander and Marcinkiewicz multiplier theorems.