Hasse principle and weak approximation for multinorm equations

In this note, we are interested in local-global principles for multinorm equations where k is a global field, L (i) /k are finite separable field extensions and a a k*. In particular, we prove a result relating the Hasse principle and weak approximation for this equation to the Hasse principle and weak approximation for some classical norm equation N (F/k) (w) = a where . It provides a proof of a "weak approximation" analogue of a recent conjecture by Pollio and Rapinchuk about the multinorm principle. We also provide a counterexample to the original conjecture concerning the Hasse principle.