It important to note that there must be a match between the UPNs and the verified domain names in Office365. For the sake of this discussion, let us assume that you are trying to synchronize the LUCERNE onpremise domain with Office365. In this scenario, the best approach is to have a UPN suffix of lucernepublishing.com set up in ActiveDirectoryDomains and Trusts, and ensure that all users have that UPN suffix applied. The users then have primary on-premise SMTP addresses that match their UPNs. In Office365, you register the lucernepublishing.com domain to Office365 and set it up for use with ExchangeOnline.