Abstract. For a general subcritical second-order elliptic operator P in a domain Ω ⊂ R n (or noncompact manifold), we construct Hardyweight W which is optimal in the following sense. The operator P − λW is subcritical in Ω for all λ < 1, null-critical in Ω for λ = 1, and supercritical near any neighborhood of infinity in Ω for any λ > 1. Moreover, if P is symmetric and W > 0, then the spectrum and the essential spectrum of W −1 P are equal to [1, ∞), and the corresponding Agmon metric is complete.Our method is based on the theory of positive solutions and applies to both symmetric and nonsymmetric operators. The constructed Hardy-weight is given by an explicit simple formula involving two distinct positive solutions of the equation P u = 0, the existence of which depends on the subcriticality of P in Ω.

Let (M, g) be a complete Riemannian manifold which satisfies a Sobolev inequality of dimension n, and on which the volume growth is comparable to the one of R n for big balls; if there is no non-zero L 2 harmonic 1-form, and the Ricci tensor is in L n 2 −ε ∩ L ∞ for an ε > 0, then we prove a Gaussian estimate on the heat kernel of the Hodge Laplacian acting on 1-forms. This allows us to prove that, under the same hypotheses, the Riesz transform d∆ −1/2 is bounded on L p for all 1 < p < ∞. Then, in presence of non-zero L 2 harmonic 1-forms, we prove that the Riesz transform is still bounded on L p for all 1 < p < n, when n > 3.

Let Ω be a domain in R n or a noncompact Riemannian manifold of dimension n ≥ 2, and 1 < p < ∞. Consider the functional Q(ϕ) := Ω (|∇ϕ| p + V |ϕ| p ) dν defined on C ∞ 0 (Ω), and assume that Q ≥ 0. The aim of the paper is to generalize to the quasilinear case (p = 2) some of the results obtained in [6] for the linear case (p = 2), and in particular, to obtain "as large as possible" nonnegative (optimal) Hardy-type weight W satisfyingOur main results deal with the case where V = 0, and Ω is a general punctured domain (for V = 0 we obtain only some partial results). In the case 1 < p ≤ n, an optimal Hardy-weight is given bywhere G is the associated positive minimal Green function with a pole at 0. On the other hand, for p > n, several cases should be considered, depending on the behavior of G at infinity in Ω. The results are extended to annular and exterior domains.