On the best Sobolev inequality
Sobolev embeddings \(W^{1,p}\subset L^{p^*}\), \(1\leq p<n\) and \(p^*= {np\over n-p}\), hold for the first-order Sobolev spaces \(W^{1,p}\) defined on a complete \(n\)-dimensional Riemannian manifold with positive injectivity radius and the Riccati curvature bounded from below. Let \(K= K(n,p)\) be the norm of the inclusion \(W^{1,p}\subset L^{p^*}\) on \(\mathbb{R}^n\). Then for any \(\varepsilon> 0\) there exists a constant \(A_P(\varepsilon)\) such that every \(\varphi\in W^{1,p}\) satisfies \[ \|\varphi\|_{L^{p^*}}\leq(K+ \varepsilon)\|\nabla\varphi\|_{L^p}+ A_p(\varepsilon)\|\varphi\|_{L^p},\tag{1} \] and \(K\) is the smallest constant having this property. The best constant is achieved if there exist \(A_p(0)\). One can expect that the sharp constant exists. In an earlier work T. Aubin made a conjecture that there exist \(A(p)\) such that any \(\varphi\in W^{1,p}\) satisfies \[ \|\varphi\|^p_{L^{p^*}}\leq K^p\|\nabla\varphi\|_{L^p}+ A(p)\|\varphi\|^p_{L^p}\quad\text{if }1\leq p\leq 2,\tag{2} \] and \[ \|\varphi\|^2_{L^{p^*}}\leq K^2\|\nabla\varphi\|^2_{L^p}+ A(p)\|\varphi\|^2_{L^p}\quad\text{if }2< p<n.\tag{3} \] The main theorem of the paper asserts that if \(M\) is a complete \(n\)-dimensional Riemannian manifolds with positive injectivity radius and if the Riemann curvature tensor and its covariant derivative are bounded on \(M\) then the inequality (2) holds for \(1<p\leq 2\) and (3) holds with \(2<p<n\). In particular, the theorem holds if \(M\) is compact. It is proved even more. The exponent in the inequality (3) can be improved if the dimension \(n\geq 3\). The improved version of the inequality (2) is also proved.
Riemannian manifolds with positive injectivity radius
scientific article; zbMATH DE number 1344437
scientific article