Sharp Weighted Sobolev Trace Inequalities on Convex Cones and Rigidity of Extremals

We study critical weighted Sobolev trace inequalities on convex cones endowed with positively homogeneous weights. Using mass transportation, we first establish the continuity of the critical trace embedding under general homogeneity and concavity assumptions on the weight. We then identify a compatibility condition between the weight, the cone, and a distinguished direction that makes the transport argument sharp. In this regime, we determine the optimal constant and prove the existence of extremals. Finally, we obtain a complete classification of all nontrivial extremals modulo translation in the lineality space.