  In this paper we study the complex representations of reductive groups over
local non-Archimedean fields. We use the building of the reductive group to
give upper-bounds for the absolute value of the character of an admissible
representation and for the Weyl integration formula for certain regular
elements. The upper-bound for the character of a representation is based on the
alternative description, depending on the building, of the character as given
by R. Meyer and M. Solleveld [MS12]. Once the character and the Weyl
integration formula are related to the building, the upper-bounds will follow
from a similar argument. Both upper-bounds generalize the upper-bounds given by
Harish-Chandra [HC70] to groups defined over fields of positive characteristic.
At last following Harish-Chandra's method we combine both upper-bounds to show
that for a maximal torus $T$ containing a maximal split torus the character is
locally summable on $\{ gtg^{-1} : g\in G,t\in T\}$.
