Sobolev inequalities with jointly concave weights on convex cones

Abstract

Using optimal mass transport arguments, we prove weighted Sobolev inequalities of the form \[\left(\int_E |u(x)|^q\,ω(x) \,dx\right)^{1/q}\leq K_0\,\left(\int_E |\nabla u(x)|^p\,σ(x)\,dx\right)^{1/p},\ \ u\in C_0^\infty(\mathbb R^n),\ \ \ \ \ \ {\rm (WSI)}\] where $p\ge 1$ and $q>0$ is the corresponding Sobolev critical exponent. Here $E\subseteq {\mathbb{R}}^n$ is an open convex cone, and $\omega ,\sigma :E\to (0,\infty )$ are two homogeneous weights verifying a general concavity-type structural condition. The constant $K_0=K_0(n,p,q,\omega ,\sigma )>0$ is given by an explicit formula. Under mild regularity assumptions on the weights, we also prove that $K_0$ is optimal in (WSI) if and only if $\omega$ and $\sigma$ are equal up to a multiplicative factor. Several previously known results, including the cases for monomials and radial weights, are covered by our statement. Further examples and applications to PDEs are also provided.

35 pages; some references are updated. To appear in the Proceedings of the London Mathematical Society