How is mobile communication(between pilot and ATC ) made possible from airplanes?
http://computer.howstuffworks.com/in-flight-mobile-phone-services1.htm

Basically an extra "cell" is created on the plane, and because the base station is moving at the same speed as the mobile phone, there is no doppler effect. Backhaul is done over satellite, which is perpendicular to the plane's direction of travel and therefore does not suffer from doppler either.