Extremals for Sharp Trace Gagliardo--Nirenberg--Sobolev Inequalities on Convex Cones

Zugmeyer~\cite{Zug} established sharp trace Gagliardo–Nirenberg–Sobolev inequalities on convex cones for $a\ge n>p>1$. We classify all nonzero real-valued extremals of these inequalities. Let the cone be the epigraph of a finite convex function that is positively homogeneous of degree one, and let $H$ be any norm. We prove that equality holds precisely for functions of the form $AH(x+te+\mathrm{\ell }{)}^{-(a-p)\mathrm{/}(p-1)}$, where $A\in \mathbb{R}\mathrm{\backslash }\{0\}$, $t>0$, $e=(0,\dots ,0,1)$, and $\mathrm{\ell }$ belongs to the largest vector subspace contained in the cone. The result includes the critical Sobolev trace inequality at $a=n$, as well as nonsmooth norms and cones, half-spaces, and cones containing lines.