A weak form of beyond endoscopic decomposition for the stable trace formula of odd orthogonal groups

We show that the cuspidal component of the stable trace formula of a special odd orthogonal group over a number field, satisfies a weak form of beyond endoscopic decomposition. We also study the $r$-stable trace formula, when $r$ is the standard or the second fundamental representation of the dual group. The results are consequences of Arthur's works on endoscopic classification of automorphic representations, together with known results concerning a class of Langlands $L$-functions for special odd orthogonal groups.