  In this work, we have considered a $n-$dimensional Schwarzschild-Tangherlini
black hole spacetime with massless minimally coupled free scalar fields in its
bulk and $3-$brane. The bulk scalar field equation is separable using the
higher dimensional spherical harmonics on $(n-2)-$sphere. First, using the
Hamiltonian formulation with the help of the recently introduced near-null
coordinates we have obtained the expected temperature of the Hawking effect,
identical for both bulk and brane localized scalar fields. Second, it is known
that the spectrum of the Hawking effect as seen at asymptotic future does not
correspond to a perfect black body and it is properly represented by a greybody
distribution. We have calculated the bounds on this greybody factor for the
scalar field in both bulk and $3-$brane. Furthermore, we have shown that these
bounds predict a decrease in the greybody factor as the spacetime
dimensionality $n$ increases and suggest that for a large number of extra
dimensions the Hawking quanta is mostly emitted in the brane.
