Sharp weighted isoperimetric and Caffarelli–Kohn–Nirenberg inequalities

Abstract Using the optimal mass transport method and a suitable quasi-conformal mapping, we study the sharp weighted isoperimetric, Sobolev, Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities. The class of weight functions under consideration includes all nonnegative homogeneous weights satisfying a concavity condition that is equivalent to a usual curvature-dimension bound and the nonnegativity of a Bakry–Émery Ricci tensor. Though our densities are not radial in general, the optimizers are radially symmetric.