Let $k$ be a local field of characteristic not $2$, and let
$G$ be the group of $k$-rational points of a connected
reductive linear algebraic group defined over $k$ with a
simple derived group of $k$-rank at least $2$. We construct
new uniform pointwise bounds for the matrix coefficients of
all infinite-dimensional irreducible unitary representations
of $G$. These bounds turn out to be optimal for ${\rm SL}\sb
n(k), n\geq 3$, and ${\rm Sp}\sb {2n}(k),n\geq 2$. As an
application, we discuss a simple method of calculating Kazhdan
constants for various compact subsets of semisimple $G$.