Limits of higher-order Besov spaces and sharp reiteration theorems

We compute the limits of higher-order Besov norms and derive the sharp constants for certain forms of the Sobolev embedding theorem. Our results extend to higher-order spaces the recent work by Brézis-Bourgain-Mironescu and Maz'ya-Shaposhnikova. The interpolation methods we develop are of interest on their own and could have applications to related inequalities.