Optimal Sobolev inequalities of arbitrary order on compact Riemannian manifolds
Let \(M\) denote a smooth compact \(n\)-dimensional Riemannian manifold, \(n \geq 2\). For \(p \in (1,n)\) one knows an explicit value \(K(n,p)\) such that for every \(\varepsilon > 0\) there exists a constant \(B = B(M,p,\varepsilon)\) such that \[ \left( \int_M | u| ^{p^\ast} \right)^{1/p^\ast} \leq (K(n,p)+\varepsilon)\left( \int_M | \nabla u| ^{p} \right)^{1/p} + B \left( \int_M | u| ^{p} \right)^{1/p} \] for all \(u \in H_1^p(M)\) where \(p^\ast = {np \over n-p}\). The question arises if one can also put \(\varepsilon = 0\). We say that the optimal Sobolev inequality (\(I_{p,\text{opt}}^1\)) holds for \(M\) and \(p\) if there exists a constant \(B = B(M,p)\) such that \[ \left( \int_M | u| ^{p^\ast} \right)^{1/p^\ast} \leq K(n,p)\left( \int_M | \nabla u| ^{p} \right)^{1/p} + B \left( \int_M | u| ^{p} \right)^{1/p} \] for all \(u \in H_1^p(M)\). Similarly, we say (\(I_{p,\text{opt}}^p\)) holds for \(M\) and \(p\) if there exists a constant \(C = C(M,p)\) such that \[ \left( \int_M | u| ^{p^\ast} \right)^{p/p^\ast} \leq K(n,p)^p \int_M | \nabla u| ^{p} + C \int_M | u| ^{p} \] for all \(u \in H_1^p(M)\). One checks that (\(I_{p,\text{opt}}^p\)) implies (\(I_{p,\text{opt}}^1\)). The following results were known before: (1) (\(I_{p,\text{opt}}^1\)) holds for \(n=2\) and \(p\in (1,2)\). (2) (\(I_{p,\text{opt}}^1\)) holds for \(n\geq 3\) and \(p\in (1,n)\) if \(M\) has constant sectional curvature. (3) (\(I_{p,\text{opt}}^p\)) holds for \(n\geq 3\) and \(p\in (1,2]\) if \(M\) is the standard sphere. (4) (\(I_{p,\text{opt}}^p\)) holds for \(n\geq 3\) and \(p=2\). See \textit{T. Aubin} [J. Differ. Geom. 11, 573-598 (1976; Zbl 0371.46011)] for (1)--(3) and \textit{E. Hebey} and \textit{M. Vaugon} [Ann. Inst. H. Poincaré Anal. Non Linéaire 13, No. 1, 57-93 (1996; Zbl 0849.53035)] for (4). In the present paper the following results are shown: (5) (\(I_{p,\text{opt}}^p\)) is false for \(p \in (2,\sqrt{n})\) if the scalar curvature is positive somewhere on \(M\). (6) (\(I_{p,\text{opt}}^p\)) holds for \(n\geq 2\) and \(p\in (1,n)\) if \(M\) is a flat torus or a hyperbolic manifold. (7) (\(I_{p,\text{opt}}^p\)) is false for \(p \in (4,\sqrt{n})\) if \(M\) is Ricci flat but not flat.
smooth compact \(n\)-dimensional Riemannian manifold
scientific article; zbMATH DE number 1236711