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

In this note we continue giving the characterisation 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. This is a continuation of our paper [M. Ruzhansky and D. Verma. Hardy inequalities on metric measure spaces, Proc. R. Soc. A., 475(2223):20180310, 2018] where we treated the case $p\leq q$. Here the remaining range $p>q$ is considered, namely, $0<q<p$, $1<p<\infty.$ We give examples obtaining new weighted Hardy inequalities on $\mathbb R^n$, on homogeneous groups, on hyperbolic spaces, and on Cartan-Hadamard manifolds. We note that doubling conditions are not required for our analysis.


Introduction
After the Hardy inequality was proved by G. H. Hardy in [Har20], a large amount of literature is available on this inequality. The integral inequality of the type b a x a f (t)dt q u(x)dx is well-known with a and b real numbers satisfying −∞ ≤ a < b ≤ ∞, and p, q real parameters satisfying 0 < q ≤ ∞, 1 ≤ p ≤ ∞. The problem of characterising the weights u and v in this inequality has been investigated by many authors. There are too many references to give an entire overview here, so we refer to only a few: Hardy, Littlewood and Polya [HLP52], Mitrinovic, Pecaric and Fink [MPF91], Heinig and Sinnamon [HS98], Kokitishvili [Kok79], Opic and Kufner [OK90], Davies [Dav99], Kufner, Persson and Samko [KP03,KPS17], Edmunds and Evans [EE04], Mazya [Maz85,Maz11], Ghoussoub and Moradifam [GM13], Balinsky, Evans and Lewis [BEL15], and references therein. We can also refer to the recent open access book [RS19] devoted to Hardy, Rellich and other inequalities in the setting of nilpotent Lie groups.
In our previous paper [RV18], for the case 1 < p ≤ q < ∞, we characterised 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 [RV18], we consider the weight characterisations for the case 0 < q < p, 1 < p < ∞.
The setting of these papers is rather general, and we consider polarisable metric measure spaces. These are metric spaces (X, d) with a Borel measure dx allowing for the following polar decomposition at a ∈ X: we assume that there is a locally integrable function λ ∈ L 1 loc such that for all f ∈ L 1 (X) we have where (r, ω) → a as r → 0. Here the sets Σ r = {x ∈ G : d(x, a) = r} ⊂ X are equipped with measures, which we denote by dω r . The polar decomposition (1.2) is a rather general condition in the sense that we allow the function λ to be dependent on the full variable x = (r, ω). In the examples described below, in the presence of the differential structure, the function λ(r, ω) can appear naturally as the Jacobian of the polar change of variables. However, since we do not assume that X must have a differentiable structure, we impose (1.2) as a condition on metric and measure.
In our previous paper [RV18] we gave several important examples of polarisable metric spaces, let us briefly recapture them here: (I) On the Euclidean space R n , 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. We can also refer to Folland and Stein [FS82] and to [FR16] for details of such groups. (II) Hyperbolic spaces H n with λ(r, ω) = (sinh r) n−1 , or more general symmetric spaces of noncompact type. (III) Cartan-Hadamard manifolds, that is, complete, simply connected Riemannian manifolds with non-positive sectional curvature. In this case λ(ρ, ω) depends on both variables ρ and ω. We refer to Section 3.3 for this example, and to [RV18] for more details on λ(ρ, ω) in this case. (IV) 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. We also denote D p := M\C(p) and S(p; r) := {x ∈ M p : |x| = r}. In this paper, as usual, we will write A ≈ B to indicate that the expressions A and B are equivalent.
The authors would like to thank Aidyn Kassymov and Bolys Sabitbek for checking some calculations in this paper.

Main results
Let d is the metric on X. We denote by B(a, r) the corresponding balls with respect to d, centred at a ∈ X and having radius r, namely, B(a, r) := {x ∈ X : d(x, a) < r}.
To simplify the notation, for all arguments, we fix some point a ∈ X, and then we will denote |x| a := d(a, x).
The main result of this paper is to characterise the weights u and v for which the corresponding Hardy inequality holds on X. For X=R, such result has been proved by Sinnamon and Stepanov [SS96]. For an alternative approach to these estimates in the case 1 < q < p < ∞ we refer to [RY18b,Theorem 1.13]. Now, we formulate one of our main results: Theorem 2.1. Suppose 0 < q < p, 1 < p < ∞ and 1/r = 1/q − 1/p. Let X be a metric measure space with a polar decomposition at a. Let u, v > 0 be measurable and positive a.e in X such that u ∈ L 1 (X\{a}) and v 1−p ′ ∈ L 1 loc (X). Then the inequality holds for all measurable functions f : X → C if and only if Moreover, the smallest constant C for which (2.1) holds satisfies Before proving the above theorem, we will need to prove several auxiliary facts. Throughout this paper, we will use the following notations: Lemma 2.2. Let us denote

Then
A r 2 = (q/p ′ )A r 1 . (2.8) Proof. Using integration by parts, we have Lemma 2.3. Suppose that α, β and γ are non-negative functions and γ is a radial non-decreasing function of | · | a . If X\B(a,|x|a) α(y)dy ≤ X\B(a,|x|a) β(y)dy for all x, then X γα ≤ X γβ.
Proof. Let us denote for |x| a = r. Given that, X\B(a,|x|a) α(y)dy ≤ X\B(a,|x|a) β(y)dy, changing to polar coordinates, we get Using [SS96, Lemma 2.1] which says if α, β, γ are non-negative functions and γ is non-decreasing, and if completing the proof.
Proposition 2.4. Suppose that u, b and F are non-negative functions with F nondecreasing such that X\B(a,|x|a) b(y)dy < ∞ for all x = a and X b(x)dx = ∞. If 0 < q < p < ∞, F is radial in |x| a , and 1/r = 1/q − 1/p, then Applying Hölder's inequality with indices q/r and q/p, we get On interchanging the order of integration and using r/p + 1 = r/q, the first factor becomes To complete the proof we apply Lemma 2.3 to the second factor. We take in Lemma 2.3. As γ is non-decreasing by assumption, it remains to check that Finally, by using Lemma 2.3 we get which completes the proof.
Proposition 2.5. Suppose 1 < p < ∞ and w is a non-negative function satisfying for all measurable functions f ≥ 0.
Proof. Let us denote: Consider the left hand side of (2.10) and change it into polar coordinates, to get By using Hölder's inequality to the indices 1/p and 1/p ′ , the LHS of (2.10) can be estimated by completing the proof. Now, we prove our Theorem 2.1 : Proof. Set w = v 1−p ′ . Suppose that inequality (2.1) holds for all f ≥ 0 and let u 0 and w 0 be L 1 functions such that 0 < u 0 ≤ u and 0 < w 0 ≤ w. We denotẽ Let us apply inequality (2.1) to the function After changing to polar coordinates and using We then have where the second last equality is integration by parts. Since u 0 and w 0 are in L 1 and are positive, the integral on the right hand side is finite. Therefore, we have (p ′ q/r)(q/p ′ ) 1/p X X\B(a,|x|a) u 0 (y)dy Approximating u and w by increasing sequence of L 1 functions, using (p ′ q/r)(q/p ′ ) 1/p = (p ′ )(p ′ ) −(1/p) q 1/p (q/r) = (p ′ ) 1/p ′ q 1/p (q(1/q−1/p)) = (p ′ ) 1/p ′ q 1/p (1−q/p) and applying the Monotone Convergence Theorem, we conclude that Suppose now that A 2 < ∞ and, for the moment, that (2.9) holds for w. Set in the first factor and applying Proposition 2.5 to the second factor, we reach the inequality In the fourth last equality, we used r(p−1) and in the second last equality, we used Lemma 2.2.
To establish sufficiency for general w, we fix positive functions u and w. If w = 0 almost everywhere on some ball B(a, |x| a ) then translating u, w on the left will reduce the problem to the one in which this does not occur. (If w = 0 almost everywhere on X, sufficiency holds trivially). We therefore assume that 0 < B(a,|x|a) w(y)dy, for all x = a. For each n > 0, set u n = uχ (B(a,n)) and w n = min(w, n) + χ (X\B(a,n)) . Then w n clearly satisfies (2.9), so from previous arguments we have , for all f ≥ 0. Here c = (r/q) 1/r (p ′ ) 1/p ′ p 1/p . If we take f = g min(w, n) 1/p ′ χ (B(a,n)) and use the definitions of u n and w n , the inequality becomes , for all non-negative g. We let n → ∞, apply the Monotone Convergence Theorem and substitute f w −1/p ′ for g to get the desired inequality and complete the proof.

Applications and examples
In this section we present several examples of applications of our results to characterise the weights u and v in several settings: homogeneous groups, hyperbolic spaces, and more general Cartan-Hadamard manifolds.
3.1. Homogeneous groups. Let X = G be a homogeneous group in the sense of Folland and Stein [FS82], see also an up-to-date exposition in the open access books [FR16] and [RS19]. Here condition (1.2) is always satisfied with function λ(r, ω) = r Q−1 , with Q being the homogeneous dimension of the group.
Without loss of generality, let us fix a = 0 to be the identity element of the group G. To simplify the notation further, we denote |x| a by |x|. We note that this is consistent with the notation for the quasi-norm | · | on a homogeneous group G.
Let us consider an example of the power weights Then by Theorem 2.1 the inequality holds for 0 < q < p, 1 < p < ∞, if and only if Let us consider which is finite for which means since we have r/p + 1 = r(1/p + 1/r) = r(1/p + 1/q − 1/p) = r/q.

Now, consider the other part
which is finite for Summarising that we get Corollary 3.1. Let G be a homogeneous group of homogeneous dimension Q, an we equip it with a quasi-norm | · |. Let 0 < q < p, 1 < p < ∞, 1/r = 1/q − 1/p, and let α 1 , α 2 , β ∈ R. Assume that α 1 + Q = 0. Let holds for all measurable functions f : G → C if and only if the parameters satisfy the following conditions: It is interesting to note that in view of the last two conditions, it is not possible to have Hardy inequality (3.2) with weights u and v in (3.1) with α 1 = α 2 . 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 [RV18].
The case α 1 + Q = 0 can be treated in a similar way.
However, we can note that in (a), if α 1 + n ≥ 0, then the second condition is automatically satisfied under our assumption β(1 − p ′ ) + n > 0.
3.3. Cartan-Hadamard manifolds. Let (M, g) be a Cartan-Hadamard manifold. This means that M is a complete and simply connected Riemannian manifold with non-positive sectional curvature, that is, the sectional curvature of M satisfies K M ≤ 0 along each (plane) section at each point of M. Then condition (1.2) is automatically satisfied by taking λ(ρ, ω) = J(ρ, ω)ρ n−1 , where J(ρ, ω) is the density function on M, see e.g. [GHL04], Helgason [Hel01]. Let us fix a point a ∈ M and denote by ρ(x) = d(x, a) the geodesic distance from x to a on M. The exponential map exp a : T a M → M is a diffeomorphism, see e.g. Helgason [Hel01]. Let us assume that the sectional curvature K M is constant, in which case it is known that the function J(t, ω) depends only on t. More precisely, let us denote K M = −b for b ≥ 0. Then we have J(t, ω) = 1 if b = 0, and J(t, ω) = ( sinh √ bt √ bt ) n−1 for b > 0, see e.g. [Ngu17]. In the case b = 0, then let us take the weights u(x) = (sinh |x| a ) α 1 if |x| < 1, (sinh |x| a ) α 2 if |x| ≥ 1, v(x) = (sinh |x| a ) β , then inequality (2.1) holds for 0 < q < p, 1 < p < ∞, 1/r = 1/q − 1/p, if and only if which is finite if and only if conditions of Corollary 3.1 hold with Q = n (which is natural since the curvature is zero). When b > 0, let Then inequality (2.1) holds for 0 < q < p, 1 < p < ∞, 1/r = 1/q − 1/p, if and only if A 2 is finite. We have