  We prove an upper bound for the exponential sum associated to a localized
$k-$divisor function, i.e., the counting function of the number of ways to
write a positive integer $n$ as a product of $k\ge 2$ positive integers, each
of them belonging to a specified interval. In particular, this gives an
estimate for the exponential sum for the $k-$divisor function, $d_k(n)$.
