Let
L
L
be a left invariant sub-Laplacian on a connected Lie group
G
G
of polynomial volume growth, and let
{
E
λ
,
λ
⩾
0
}
\{ {E_\lambda },\lambda \geqslant 0\}
be the spectral resolution of
L
L
and
m
m
a bounded Borel measurable function on
[
0
,
∞
)
[0,\infty )
. In this article we give a sufficient condition on
m
m
for the operator
m
(
L
)
=
∫
0
∞
m
(
λ
)
d
E
λ
m(L) = \smallint _0^\infty m(\lambda )d{E_\lambda }
to extend to an operator bounded on
L
p
(
G
)
,
1
>
p
>
∞
{L^p}(G),\;1 > p > \infty
, and also from
L
1
(
G
)
{L^1}(G)
to weak-
L
1
(
G
)
{L^1}(G)
.