The pilot must follow the lost comm procedure of the US, since he/she is over the US.
The pilot must follow the lost comm procedure of Italy, since the aircraft is registered in Italy.
The pilot must follow the lost comm procedure of France, since the operator is in France.
Italy and USA rules apply, however Italy has priority because the aircraft is registered in Italy.
ICAO Annex 2
Chapter 2. Applicability of The Rules of the Air
2.1 Territorial application of the rules of the air
2.1.1 The rules of the air shall apply to aircraft bearing the nationality and registration marks of a Contracting State, wherever they may be, to the extent that they do not conflict with the rules published by the State having jurisdiction over the territory overflown.
When an aircraft is flying within the airspace of the state of registration, the rules of the air of that state are applicable. Indeed, for an Italian registered aeroplane, the Italian rules apply wherever the aeroplane is flown providing there is no confliction with local rules. Where an Italian registered aircraft is flying over a foreign state, the rules of the air of that state apply. Do not confuse Rules with Law! The application of the rules can be summarized thus:
Italian registered aircraft over Italy - Italian rules apply
Italian registered aircraft over USA - Italian and USA rules apply (USA have priority)
Italian registered aircraft over the high seas - ICAO rules apply without exception
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