We provide estimates for the exponential sum
F
(
x
,
α
)
=
∑
n
≤
x
f
(
n
)
e
2
π
i
α
n
,
\begin{equation*}F(x,\alpha )=\sum _{n\le x} f(n)e^{2\pi i\alpha n}, \end{equation*}
where
x
x
and
α
\alpha
are real numbers and
f
f
is a multiplicative function satisfying
|
f
|
≤
1
|f|\le 1
. Our main focus is the class of functions
f
f
which are supported on the positive proportion of primes up to
x
x
in the sense that
∑
p
≤
x
|
f
(
p
)
|
/
p
≫
log
log
x
\sum _{p\le x}|f(p)|/p\gg \log \log x
. For such
f
f
we obtain rather sharp estimates for
F
(
x
,
α
)
F(x,\alpha )
by extending earlier results of H. L. Montgomery and R. C. Vaughan. Our results provide a partial answer to a question posed by G. Tenenbaum concerning such estimates.