Hardy inequalities on metric measure spaces

Abstract

In this note we give several characterisations of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardy's original inequality.We give examples obtaining new weighted Hardy inequalities on R n , on homogeneous groups, on hyperbolic spaces, and on Cartan-Hadamard manifolds.

Hardy inequalities on metric measure spaces, II: the case p > q

“…This is why we consider different powers α 1 , α 2 in this example. This is different from the case p ≤ q which was considered as an application in [21].…”

“…In our previous paper [21], we gave several important examples of polarizable metric spaces. Let us briefly recapture them here: On the Euclidean space double-struckRn, we can take λ ( r , ω ) = r n −1 , and more generally, we have (1.2) on all homogeneous groups with λ ( r , ω ) = r Q −1 , where Q is the homogeneous dimension of the group.…”

“…In our previous paper [21], for the case 1 < p ≤ q < ∞, we characterized the weights u and v for the Hardy inequalities (1.1) to hold on general metric measure spaces with polar decompositions. In this paper, complementary to [21], we consider the weight characterizations for the case right left right left right left right left right left right left3pt0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em0<q<p,1<p<∞. …”

“…In this case, λ ( ρ , ω ) depends on both variables ρ and ω . We refer to §3c for this example, and to [21] for more details on λ ( ρ , ω ) in this case.Arbitrary complete Riemannian manifolds M : let C ( p ) denote the cut locus of a point p ∈ M , which we may fix. Let us denote by M p the tangent space to M at p , and by | · | the Riemannian length.…”

“…In this case, λ ( ρ , ω ) depends on both variables ρ and ω . We refer to §3c for this example, and to [21] for more details on λ ( ρ , ω ) in this case.…”