Learning On Caffarelli-Kohn-Nirenberg Inequality And Conformal Invariant

This article focuses on the paper "A Conformal Geometric Point of View on the Caffarelli-Kohn-Nirenberg Inequality" written by Dupaigne Louis,Gentil,Ivan and Zugmeyer Simon,introducing the ideas and improving the proof process.Translating the Caffarelli-Kohn-Nirenberg inequality into Sobolev inequality by Γ-calculus introducd by Bakry and (?)mery(through the carre du champoperator):uses Markov Triple to define three n-conformal invariant weighted Riemannian manifolds(They can be shown to be conformal-invariant,furthermore,the corresponding Sobolev inequality on it is also equivalent),convert the solution to the problem to the Sobolev inequality on the spherical CKN space(weighted Riemannian manifold),and finally,using the equivalence of Poincar’e inequality and the curvature dimension condition CD(p,n)of integral forms further explains the optimality of the Felli-Schneider region.This paper first introduces the basic concepts,operators and properties in Riemannian manifolds and Markov Triple,then introduces conformal invariant,and operators in conformal invariant classes,the transformation formulas of operators in nconformal invariant classes,and finally,it is proved that the spherical CKN space satisfies the curvature dimension condition of integral form,and then the optimal constant and extremal function of the Sobolev inequality are given,and the corresponding Poincar’e inequality is derived.