  By methods based on elementary Linear Algebra we obtain sharp constants in
cases of the Caffarelli-Kohn-Nirenberg inequality via quasi-conformal changes
of variables. Some of our results were obtained earlier by Lam and Lu. Our
proofs are radical simplifications of earlier proofs. In the case $\alpha>0$ we
establish that we have symmetry breaking and optimizers to the CKN inequalities
do not exist. This case was not treated by Lam and Lu. Our results thus are in
the full parameter range of \alpha.
