  In this note we report on some recent progress in proving the AdS/CFT
correspondence for quantum fields using rigorously defined Euclidean path
integrals. We also comment on the infra-red problem in the AdS/CFT
correspondence and argue that it is different from the usual IR problem in
constructive quantum field theory. To illustrate this, a triviality proof based
on hypercontractivity estimates is given for the case of an ultra-violet
regularized potential of type $:\phi^4:$. We also give a brief discussion on
possible renormalization strategies and the specific problems that arise in
this context.
