Symmetry for Extremal Functions in Subcritical and Fractional Caffarelli–Kohn–Nirenberg Inequalities

Abstract

We determine the best constants for Gagliardo–Nirenberg inequalities the applications to nonlinear diffusions, the first order interpolation inequalities with weights, the two subtle convex nonlocal approximations of the bounded variation norm, the limiting embedding theorems for Sobolev spaces and the BBM formula. We establish Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators with the nonlinear ground state representations and sharp Hardy inequalities with fractional Hardy-Sobolev-Maz'ya inequality for domains. The Caffarelli-Kohn-Nirenberg inequalities with remainder terms, sharp constants, existence, nonexistence and fractional order are investigated. The symmetry of optimizers of extremal functions in subcritical and fractional Caffarelli–Kohn–Nirenberg inequalities are obtained.