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We study the problem of $L^p$-boundedness ($1\<p\<\infty$) of operators of the form $m(L_1,\cdots ,L_n)$ for a commuting system of self-adjoint left-invariant differential operators $L_1,\cdots ,L_n$ 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 $L_1,\cdots ,L_n$ are homogeneous, we prove analogues of the Mihlin-Hörmander and Marcinkiewicz multiplier theorems.

Let L be a homogeneous sublaplacian on the 6-dimensional free 2-step nilpotent Lie group $N_{3,2}$ on three generators. We prove a theorem of Mikhlin-Hörmander type for the functional calculus of L, where the order of differentiability s > 6/2 is required on the multiplier.