Reverse conformally invariant Sobolev inequalities on the sphere

We consider the optimization problem corresponding to the sharp constant in a conformally invariant Sobolev inequality on the $n$-sphere involving an operator of order $2s> n$. In this case the Sobolev exponent is negative. Our results extend existing ones to noninteger values of $s$ and settle the question of validity of a corresponding inequality in all dimensions $n\geq 2$.