Exact optimal constant in the differential Bakry–Émery inequality on projective space

Determine the exact optimal constant L+(RP^{d−1}) in the differential Bakry–Émery (log-Sobolev) inequality on the real projective space RP^{d−1} that controls ∫ T2(log F) by L+(RP^{d−1}) ∫ T1(log F) for even densities F, which governs the sharp value of K+ in the Fisher-information monotonicity criterion for the spatially homogeneous Boltzmann and Landau equations.

Background

The constants L, L+, and the spectral gap X1 quantify optimal Poincaré/log-Sobolev-type inequalities on the sphere and on the projective space RP^{d−1}. In Theorem 13.1, the decay of Fisher information is reduced to such inequalities on the sphere/projective space.

While lower bounds are known (e.g., L+(RP^{d−1}) ≥ 6d/(d+1) in these notes and improved bounds by Ji), the exact optimal value is not known and would directly sharpen the monotonicity criterion.