First of all, AirVPN would need a username to monitor a certain user. Which should be nearly impossible to know for the law enforcement. Atleast without doing illegal things. Then, if they would have the username and would force AirVPN to log traffic for user, then they could easily ban that person. You would think, why ban that person? Well, there have been no logs before, and now the person is banned, so there is nothing to keep logs of. Please correct me if im wrong
Given that AirVPN is not under U.S. Government Jurisdiction and have terminated Italian servers to prevent legal complications, they can easily refuse to log for them and all the government could likely do is say they can't host servers in the US anymore, and even that is a long stretch on which I doubt they can do and given what AirVPN stands for even IF the U.S. government got the Italian government to work with them AirVPN would likely try to find a way to still have their service and refuse to cooperate or in worst case shut down temporarily and reopen with different name and/or different owner. Now this is mostly just speculation from my part on what AirVPN would do, only they can know for certain.