Air carrier Captain reported an inability to communicate with MEM Center on arrival into GTR.

Narrative

Descending into GTR; Memphis Center (127.1) cleared us to 3;000 and asked what runway we wanted. We requested Rwy 18. We reached 3;000 ft approximately 10 miles from the airport. We were nearing the airport so I started to reduce our speed. We weren't particularly concerned at this point because we could still hear the Controller talking to other aircraft. We queried the Controller about an approach clearance 3 times; if I recall correctly; but never received an answer. We surmised he couldn't hear us possibly due to our low altitude so we attempted to contact Columbus Approach a few times but got no reply there either. By this time we were almost over the top of GTR at 3000 ft. We called the Tower and he answered. We explained what happened and he said we were cleared to land Rwy 18. We overflew the field and entered a right traffic pattern for 18 and landed. After we landed and we're taxiing I asked the Tower Controller if he knew why we were unable to contact Memphis Center or Columbus Approach. He stated that Columbus Approach doesn't open until 9am and that this type of situation has happened before with Memphis Center. Lost contact with Memphis Center. Our company pages denote that Memphis Center provides approach services when Columbus AFB Approach Control is closed; however there are no operating hours listed for Columbus AFB Approach Control. Hours of operation for Columbus Approach Control would have been useful so we didn't waste time trying to contact them. Also; if ATC has a known area where radio coverage is lost at certain altitudes that would also be useful information for flight crews as well.