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The classical Sobolev and Escobar inequalities are embedded into the sameone-parameter family of sharp trace-Sobolev inequalities on half-spaces.Equality cases are characterized for each inequality in this family by tweakinga well-known mass transportation argument and lead to a new comparison theoremfor trace Sobolev inequalities. The case $p=2$ corresponds to a family ofvariational problems on conformally flat metrics which was previously settledby Carlen and Loss with their method of competing symmetries. In this caseminimizers interpolate between conformally flat spherical and hyperbolicgeometries, passing through the Euclidean geometry defined by the fundamentalsolution of the Laplacian.

A bridge between Sobolev and Escobar inequalities and beyond