Critical-exponent limitation of the analytic method

Determine whether the degeneration of the analytic Beckner–Sobolev constants to \(1/(n\kappa)\) at the critical exponent \(q=2n/(n-2)\) is merely a limitation of the methods or reflects an underlying mathematical reason.

Background

The paper contrasts geometric and analytic approaches to improving Beckner–Sobolev constants under positive Ricci curvature. The geometric approach yields an improvement involving the diameter, whereas the analytic approach based on positive solutions improves the constant only below the critical exponent.

At the critical exponent, the constants from the cited Riemannian results and the paper’s theorem revert to the non-improved value. The authors explicitly leave unresolved whether this is methodological or structural.