Best constants in second-order Sobolev inequalities on compact Riemannian manifolds in the presence of symmetries

Let (M,g) be a smooth compact 3 ≤ n-dimensional Riemannian manifold, and G a subgroup of the isometry group of (M,g). We establish the best constants in second-order for a Sobolev inequality when the functions are G-invariant. | Turk J Math 36 (2012) , 601 – 612. ¨ ITAK ˙ c TUB doi: Best constants in second-order Sobolev inequalities on compact Riemannian manifolds in the presence of symmetries Mohammed Ali Abstract Let (M, g) be a smooth compact 3 ≤ n -dimensional Riemannian manifold, and G a subgroup of the isometry group of (M, g) . We establish the best constants in second-order for a Sobolev inequality when the functions are G -invariant. Key Words: Best constants, compact Riemannian manifolds, Sobolev inequalities, isometries 1. Introduction Let (M, g) be a compact 3 ≤ n-dimensional Riemannian manifold, and G a subgroup of the isometry group Is(M, g). Assume that l is the minimum orbit dimension of G , and V is the minimum of the volume of the l -dimensional orbits. If 1 0 , (Ω, Ψ) can be chosen such that: ij l (g ) − (δ ij ) ≤ ε, Γ ≤ ε , (i) ij and 1−ε≤ det(gij ) ≤ 1 + ε on Ω, for 1 ≤ i, j ≤ n, l Γij ≤ ε , ij (˜ g ) − (δ ij ) ≤ ε, (ii) and 1−ε ≤ det(˜ gij ) ≤ 1 + ε on Vx , for 1 ≤ i, j ≤ N, x where g˜ is the metric induced by g on OG . Furthermore, (1 − ε)(δij ) ≤ (gij ) ≤ (1 + ε)(δij ) as bilinear forms. (4) For any f ∈ FGi,p , f ◦ Ψ−1 depends only on U2 variables. In order to prove Theorem , it suffices to prove the following lemmas. Lemma Let (M, g) be a compact 3 ≤ n-dimensional Riemannian manifold, and G a subgroup of the isometry group Is(M, g). Suppose that l is the minimum orbit dimension of G , and V is the minimum of the volume of the l -dimensional orbits. Assume that for any 1 0 , let B(x0 , δ) be a geodesic ball of radius δ and center x0 such that in normal coordinators of B(x0 , δ), the properties of Lemma are verified. p For any f ∈ Cc∞ (Bδ ) and ε small enough, there exist two real numbers A, B with A 0 and Cα,δ = O (δ −p 2 p ∂ f dx + C α,δ Bδ p |f| dx () Bδ ), plus the inequalities Bδ 2 p ∂ f dx ≤ Cn,p | f|p .