Sharp quantitative stability of Poincaré-Sobolev inequality in the hyperbolic space and applications to fast diffusion flows

Consider the Poincaré-Sobolev inequality on the hyperbolic space: for every n ≥ 3 n ≥ 3 and 1 < p ≤ n+2/n-2, 1 < p ≤ n + 2 n - 2 , there exists a best constant Sn,p, λ(Bⁿ)>0 S n , p , λ ( B n ) > 0 such that $${aligned} Sn, p, λ({B}ⁿ)( ~∫ _{{B}ⁿ}|u|^{{p+1}} \, dv_{{B}^n} ) 2/p+1 ≤ ∫ _{{B}ⁿ}( |∇ _{{B}ⁿ}u|²-λ u²) \, dv_{{B}^n}, {aligned}$$ S n , p , λ ( B n ) ∫ B n | u | p + 1 d v B n 2 p + 1 ≤ ∫ B n | ∇ B n u | 2 - λ u 2 d v B n , holds for all $$u∈ C_c∞({B}^n),$$ u ∈ C c ∞ ( B n )</m