For satellite communication, the downlink frequency is lower

Heres a guess, power transmission, I think that if you transmit a lower freq it carries longer distance with less power.

Does this sound correct, if so then a sattelite will have limited power compared to a ground station.

I would like to know if this is correct?

It's correct. The resources on the comm. satellite are scarce so saving power by transmitting at the lower freq is the standard choice.
Good guess :D

Just another guess..similar size antenna with lower frequency means lower gain and so wide coverage.

as we know the radiated power from the antenna at a distance r is inversely proportional to f2 (f=frequency) and this is in any medium.
so as f increases the attenuation of the power will increase. so we need to increase the power from the source.
as there is a frequence for uplink and one for downlink so we choose the high f for the uplink because we can increase the sending power of the earth station which is difficult to make with the satellite because it has a certain amount of power to use it in the communications and to be in its orbit.