Hardy’s inequality for 𝑊^{1,𝑝}₀-functions on Riemannian manifolds

Abstract

Abstract. We prove that for every Riemannian manifold X with the isoperimetric profile of particular type there holds an inequality of Hardy type for functions of the class W 1,p 0 (X ). We also study manifolds satisfying Hardy's inequality and, in particular, we establish an estimate for the rate of growth of the weighted volume of the noncompact part of such a manifold. Main resultsLet X be a connected noncompact Riemannian C 2 -manifold without boundary. We denote by ρ(x , x ) the distance between two points x , x ∈ X . For an arbitrary point x ∈ X we set ε(x) = infwhere the infimum is taken over all sequences of points (y k ) ⊂ X , which do not have points of accumulation in X . Let α(t), β(t) : [0, ∞) → [0, ∞) be positive continuous functions. We fix constants p, q such that 1 < p ≤ q < ∞. Let Ω ⊂ X be an arbitrary domain, and let ∂Ω be its boundary. We consider a weighted volumeHere, dv is the element of volume on the manifold X . Below we shall assume thatWe denote a weighted areawhere dH n−1 is the element of the (n − 1)-dimensional Hausdorff measure on ∂Ω. We consider isoperimetric profiles of the Riemannian manifold X with weight functions α and β. An isoperimetric profile of the manifold X is the function θ X : [0, I X ) → R + , θ(0) = 0, (1.1)

Optimal Hardy inequalities in cones

“…A sufficient condition for the validity of such a Hardy inequality is that Σ is boundary distance regular, and this condition holds true if Σ satisfies either the uniform interior cone condition or the uniform exterior ball condition (see the definitions in [37]). For other sufficient conditions for the validity of the Hardy inequality on Riemannian manifolds, see, for example, [28].…”

Extremal problems in geometric function theory

“…Note that Miklyukov and Vuorinen [111] (1999) considered generalized Hardytype inequalities on Riemannian manifolds of dimension n ⩾ 2 and proved an analogue of Theorem 4.1 for these inequalities. In place of p-capacity, they used certain geometric quantities related to the isoperimetric profile of the Riemannian manifold.…”

Some remarks on the Sobolev inequality in Riemannian manifolds

“…The connections between the validity of Sobolev-like inequalities and isoperimetric profiles (defined as the optimal h in (1.3) below) is well known; let us briefly recall that it appeared in [14,20]. The method of [20] allows one to reduce the proof of multidimensional Sobolev inequalities to one-dimensional Hardy type inequalities, and was also applied to derive Hardy inequalities in Riemannian manifolds by [21]. The symmetrization approach has also been used extensively in this field; we refer for example to [4,11,19,30,32]; the optimality of the constant in Sobolev-like inequalities has been analyzed also, with alternative approaches, in [6,12,16,17,23].…”

“…A property which certainly is necessary to us is p-hyperbolicity. This essentially amounts to the existence of a symmetric positive Green function G x for the p-Laplacian with pole at x, for every x ∈ M. For other definitions of p-hyperbolicity and comments on its necessity we refer to [13] (see also [21]). Here we need…”