Sharp Sobolev inequalities of second order

Abstract

Let (M, g) be a smooth compact Riemannian manifold of dimension n≥5, and 2 2 (M) be the Sobolev space consisting of functions in L2(M) whose derivatives up to the order two are also in L2(M). Thanks to the Sobolev embedding theorem, there exist positive constants A and B such that for any U ∈ H 2 2 (M), $$\left\| u \right\|_{2^\sharp }^2 \leqslant A\left\| {\Delta _g u} \right\|_2^2 + B\left\| u \right\|_{H_1^2 }^2 $$ where 2#=2n/(n−4) is critical, and\(\left\| \cdot \right\|_{H_1^2 } \) is the usual norm on the Sobolev space H 1 2 (M) consisting of functions in L2(M) whose derivatives of order one are also in L2(M). The sharp constant A in this inequality is K 0 2 where K0, an explicit constant depending only on n, is the sharp constant for the Euclidean Sobolev inequality\(\left\| u \right\|_{2^\sharp } \leqslant K\left\| {\nabla u} \right\|_2 \). We prove in this article that for any compact Riemannian manifold, A=K 0 2 is attained in the above inequality.