  We continue the analysis of the elliptic part of the trace formula initiated
in \cite{Altug:2015aa}. In that reference Poisson summation was applied to the
elliptic part and the dominant term was analyzed. The main aim of this paper is
to study the remaining terms after Poisson summation. We analyze the the
Fourier transforms of (smoothed) orbital integrals and obtain exact asymptotic
expansions. As an application we recover, using the Arthur-Selberg trace
formula, Kuznetsov's result (cf. \cite{Kuznetsov:1980aa}) that the trace of the
$p$th Hecke operator on cuspidal automorphic representations is bounded by
$p^{\frac14}$.
