83% of cell towers in Puerto Rico are still not functioning. Details are sparse, but logical reasons for lack of service are destruction of the tower, no electricity, no backhaul (the way your phone and cell data reach the Internet and POTS (Plain Old Telephine System), or the central office is non-functional for the same reasons.
The FCC just enthusiastically gave Alphabet (Google) permission to have an airborn LTE transmitter. Google had been working on this technology to provide service in isolated parts of the world or in emergencies.
It is important to note this is LTE data services, not GSM or CDMA phone services. A basic cell phone will not be helped. To place a phone call over the Google Balloon, you need a smart phone with “Voice over LTE” or an app that supports Voice over IP (VoIP).
The carriers in Puerto Rico are signing over some of their LTE spectrum to Google for up to six months while they reconstruct their networks. LTE would also allow sending emails and some text messages for people with smart phones.