Abstract. Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Let also A be a smooth symmetrical positive (0, 2)-tensor field in M . By the Sobolev embedding theorem, we can write that there exist K, B > 0 such that for any u ∈ H 2 1 (M ),where H 2 1 (M ) is the standard Sobolev space of functions in L 2 with one derivative in L 2 . We investigate in this paper the value of the sharp K in the equation above, the validity of the corresponding sharp inequality, and the existence of extremal functions for the saturated version of the sharp inequality.Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Also let A be a smooth symmetrical (0, 2)-tensor field in M . In a local chart, A = (A ij ), i, j = 1, . . . , n. We assume that A is positive when acting on 1-forms in the sense that for any x ∈ M , and any η in the cotangent spaceThen, by the Sobolev embedding theorem, we can write that there exist K, B > 0 such that for any u ∈ H 2 1 (M ),where ∇u = (∂ i u) is the 1-form consisting (in local charts) of the first derivatives of u, dv g is the Riemannian volume element of g, and H 2 1 (M ) is the standard Sobolev space consisting of functions in L 2 with one derivative in L 2 . Clearly, the sharp constant B in (0.1) is V −2/n g , where V g is the volume of (M, g), and the corresponding sharp inequality holds true since it holds true for the classical Sobolev inequality [and |∇u| 2 is controled by A x (∇u, ∇u)]. On the other hand, as is easily understood by the fact that A charges some parts of the space M more than others, it is expected that A will affect the sharp constant K in (0.1). Note (0.1) is associated to the operator ∆ g A u = −div g (A x ∇u) which appears in several places in mathematical and physics literature.The questions we ask in this note are: what is the value K s = K s (g) of the sharp K in (0.1), does the corresponding sharp inequality hold true, and if yes, does its saturated version (where B is lowered to its minimum value under the constraint K = K s ) possess extremal functions? When A = g −1 , we are back to the classical problem (dealing with the classical Sobolev inequality). Possible references in book
