  We present conjectures for the space-time form and leading large N dependence
of extremal and near-extremal correlation functions in the \N=8 superconformal
Yang-Mills theory in d=3 as well as in the (0,2) superconformal theory in d=6,
using their gravity duals with M-theory on $AdS_4\times \S^7$ and $AdS_7 \times
\S^4$ respectively. As a key part of the conjectures, we argue that the bulk
couplings associated with extremal and near-extremal field configurations in
the corresponding $AdS_4$ and $AdS_7$ gauged supergravities vanish. The
vanishing of these couplings constitutes a generalization of the property of
consistent truncation of the Kaluza-Klein modes.
