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A complete solution to the quaternionic contact Yamabe problem on the sevendimensional sphere is given. Extremals for the Sobolev inequality on the sevendimensional Hesenberg group are explicitly described and the best constant inthe $L^2$ Folland-Stein embedding theorem is determined.

Extremals for the Sobolev inequality on the seven-dimensional quaternionic Heisenberg group and the quaternionic contact Yamabe problem