Best constants for two families of higher order critical Sobolev embeddings. (December 2018) - Shafrir, Itai; Spector, Daniel
Abstract: In this paper we obtain the best constants in some higher order Sobolev inequalities in the critical exponent. These inequalities can be separated into two types: those that embed into L ∞ ( R N ) and those that embed into slightly larger target spaces. Concerning the former, we show that for k ∈ 1, …, N − 1 , N − k even, one has an optimal constant c k > 0 such that ‖ u ‖ L ∞ ≤ c k ∫ | ∇ k ( − Δ ) ( N − k ) ∕ 2 u | for all u ∈ C c ∞ ( R N ) (the case k = N was handled in Shafrir, 2018). Meanwhile the most significant of the latter is a variation of D. Adams' higher order inequality of J. Moser: For Ω ⊂ R N, m ∈ N and p = N m, there exists A > 0 and optimal constant β 0 > 0 such that ∫ Ω exp ( β 0 | u | p ′ ) ≤ A | Ω | for all u such that ‖ ∇ m u ‖ L p ( Ω ) ≤ 1, where ‖ ∇ m u ‖ L p ( Ω ) is the traditional semi-norm on the space W m, p ( Ω ) .