Hardy inequalities on metric measure spaces, III: the case q ≤ p ≤ 0 and applications

In this paper, we obtain a reverse version of the integral Hardy inequality on metric measure space with two negative exponents. For applications we show the reverse Hardy–Littlewood–Sobolev and the Stein–Weiss inequalities with two negative exponents on homogeneous Lie groups and with arbitrary quasi-norm, the result of which appears to be new in the Euclidean space. This work further complements the ranges of p and q (namely, q≤p<0) considered in the work of Ruzhansky & Verma (Ruzhansky & Verma 2019 Proc. R. Soc. A 475, 20180310 (doi:10.1098/rspa.2018.0310); Ruzhansky & Verma. 2021 Proc. R. Soc. A 477, 20210136 (doi:10.1098/rspa.2021.0136)), which treated the cases 1q, respectively.


Introduction
In the famous work [1], G. H. Hardy showed the following (direct) integral inequality: where f ≥ 0, p > 1 and a > 0. The subject of the Hardy inequalities has been extensively investigated and we refer to the book [2].
The main goal of this paper is to extend the reverse Hardy inequalities to general metric measure space with two negative exponents. More specifically, we consider metric spaces X 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 for some set Σ r = {x ∈ X : d(x, a) = r} ⊂ X with a measure on it denoted by dω, and (r, ω) → a as r → 0. The condition (1.2) is rather general (see [15]) since we allow the function λ to depend on the whole variable x = (r, ω). Since X does not necessarily have a differentiable structure, the function λ(r, ω) cannot be in general obtained as the Jacobian of the polar change of coordinates. However, if such a differentiable structure exists on X, the condition (1.2) can be obtained as the standard polar decomposition formula. In particular, let us give several examples of X for which the condition (1.2) is satisfied with different expressions for λ(r, ω): (I) Euclidean space R n : λ(r, ω) = r n−1 .
(II) Homogeneous groups: λ(r, ω) = r Q−1 , where Q is the homogeneous dimension of the group. Such groups have been consistently developed by Folland & Stein [16] (see also an up-to-date exposition in [17,18]).  [19]). Let J(ρ, ω) be the density function on M (see e.g. [20]). Then we have the following polar decomposition: In [15,21], the (direct) integral Hardy inequality on metric measure spaces was established with applications to homogeneous Lie groups, hyperbolic spaces, Cartan-Hadamard manifolds with negative curvature and on general Lie groups with Riemannian distance for 1 < p ≤ q < ∞ and p > q, respectively. Also, in [22], the authors showed the integral Hardy inequality for p ∈ (0, 1) and q < 0 on metric measure space. In this paper, we continue the investigation of the integral Hardy inequality on a metric measure space, i.e. we show the reverse integral Hardy inequality with negative exponents. In [23], Hardy and Littlewood considered the one-dimensional fractional integral operator on (0, ∞) given by where they also showed the following L q − L p boundedness of this operator T λ : (1.5) In [24], Sobolev generalized theorem 1.1 for the multi-dimensional case in the following form: where C is a positive constant independent of u.
In [25], Stein and Weiss obtained the following radially weighted Hardy-Littlewood-Sobolev inequality, which is known as the Stein-Weiss inequality.
where C is a positive constant independent of u.
To the best of our knowledge, the Hardy-Littlewood-Sobolev inequality on the Heisenberg group was proved by Folland & Stein [26] and the best constants of the Hardy-Littlewood-Sobolev inequality, in the Euclidean space and Heisenberg group were obtained in [27,28], respectively. Also, in [18,29,30], the authors studied the Hardy-Littlewood-Sobolev and the Stein-Weiss inequalities on Heisenberg and homogeneous Lie groups. Note that systematic studies of different functional inequalities on general homogeneous (Lie) groups were initiated by the papers [31][32][33][34].
The reverse Stein-Weiss inequality in Euclidean setting has the following form:
Note, we obtain the reverse Hardy-Littlewood-Sobolev inequality if α = β = 0. Improved Stein-Weiss inequality was obtained in [36] on the Euclidean upper half-space and in [37] on homogeneous Lie groups. For more results about the reverse Hardy-Littlewood-Sobolev inequality in Euclidean space, we refer the reader to Beckner [38], Carrillo et al. [39], Dou & Zhu [40], Ngô & Nguyen [41] and the references therein. Note that the reverse Hardy-Littlewood-Sobolev and Stein-Weiss inequalities were shown in [37] for the case p ∈ (0, 1) and q < 0. In this paper, we show the reverse Hardy-Littlewood-Sobolev and Stein-Weiss inequalities with two negative exponents i.e. q < p < 0, which is also new in the Euclidean space.

Main result
Firstly, let us denote by B(a, r) a ball in X with centre a and radius r, i.e.,

B(a, r)
where d is the metric on X. Once and for all let us fix some point a ∈ X, and denote . Let p < 0, so that p = p/(p − 1) > 0. If non-negative functions satisfy 0 < X f p (x) dx < +∞ and 0 < X g p (x) dx < +∞, we have As the main results of this section, we show the reverse integral Hardy inequality as well as its conjugate.
Theorem 2.2. Assume that p, q < 0 are such that q ≤ p < 0. Let X be a metric measure space with a polar decomposition at a ∈ X. Suppose that u, v ≥ 0 are locally integrable functions on X. Then the inequality

Remark 2.3.
In (2.5), by simple calculation, we have that for the case q ≤ p < 0 Proof. Let us divide a proof of this theorem to two steps.
Step 1. Firstly, let us denote By using the reverse Hölder inequality (2.2) with the polar decomposition, we compute (2.14) Multiplying by u, integrating over X with q < 0 and by using (direct) Minkowski's inequality with q/p ≥ 1 (see [42], theorem 2.9, p. 26), we compute Finally, we obtain Step 2. Now it remains to show that (2.3) yields (2.4). Let us fix t > 0 and denote the following function: if |x| a > t, (2.19) where f 1 is any function satisfying B(a,|x| a ) f 1 (y) dy < ∞ and |x| a ≥t v(x)f p 1 (x) dx < ∞, and α > 0. Then we compute Summarizing the above facts with q ≤ p < 0 and taking the limit as α → 0, we obtain Theorem 2.4. Assume that p, q < 0 such that q ≤ p < 0. Let X be a metric measure space with a polar decomposition at a ∈ X. Suppose that u, v ≥ 0 are locally integrable functions on X. Then the inequality holds for all non-negative real-valued measurable functions f , if and only if and D 2 (|x| a ) is non-increasing. Moreover, the largest constant C 2 (p, q) satisfies where 1/p + 1/p = 1.
Proof. The main idea of the proof of this theorem is similar to that of theorem 2.2 with the only difference that D 2 (|x| a ) is non-increasing, so we omit the details.

Consequences on homogeneous groups
In this section, we consider several consequences of the main results for the reverse integral Hardy, Hardy-Littlewood-Sobolev and Stein-Weiss inequalities on homogeneous groups.
Let us recall that a Lie group (on R n ) G with the dilation D λ (x) := (λ ν 1 x 1 , . . . , λ ν n x n ), ν 1 , . . . , ν n > 0, D λ : R n → R n , which is an automorphism of the group G for each λ > 0, is called a homogeneous (Lie) group. For simplicity, throughout this paper we use the notation λx for the dilation D λ (x). The homogeneous dimension of the homogeneous group G is denoted by Q := ν 1 + . . . + ν n . Also, in this paper we denote a homogeneous quasi-norm on G by |x|, which is a continuous non-negative function Let us also recall the following well-known fact about quasi-norms.

Proposition 3.1 (e.g. [17], proposition 3.1.38 and [43], proposition 1.2.4).
If | · | is a homogeneous quasi-norm on G, there exists C > 0 such that for every x, y ∈ G, we have |xy| ≤ C(|x| + |y|). (3. 2) The following polarization formula on homogeneous Lie groups will be used in our proofs: there is a (unique) positive Borel measure σ on the unit quasi-sphere S := {x ∈ G : |x| = 1}, so that for every f ∈ L 1 (G) we have We refer to Folland & Stein [16] for the original appearance of such groups, to Fischer & Ruzhansky [17] and to Ruzhansky & Suragan [43] for a recent comprehensive treatment. Let us define quasi-ball centred at x with radius r in the following form: B(x, r) := {y ∈ G : |x −1 y| < r}.

(a) Reverse integral Hardy inequality
In this subsection, we show the reverse integral Hardy inequality on homogeneous Lie groups.

Theorem 3.2. Let G be a homogeneous Lie group of homogeneous dimension Q with a quasi-norm | · |.
Assume that q ≤ p < 0 and α, β ∈ R. Then the reverse integral Hardy inequality

Theorem 3.3. Let G be a homogeneous Lie group of homogeneous dimension Q with a quasi-norm | · |.
Assume that q ≤ p < 0 and α, β ∈ R. Then the reverse conjugate integral Hardy inequality holds for some C 2 > 0 and for all non-negative measurable functions f , if α + Q < 0, β(1 − p ) + Q < 0 and (Q + α)/q + (Q + β(1 − p ))/p = 0. Moreover, the biggest constant C 2 for (3.6) satisfies Proof. Proof of this theorem is similar to the previous case, where we use theorem 2.4 instead of theorem 2.2.

(b) The reverse Hardy-Littlewood-Sobolev inequality and Stein-Weiss inequality
In this subsection, we obtain the reverse Hardy-Littlewood-Sobolev inequality and Stein-Weiss inequality on Euclidean space and homogeneous Lie groups. Let us introduce the Riesz operator on homogeneous Lie groups in the following form: where * is the convolution. Hence, by taking G = (R n , +), Q = n and | · | = | · | E (| · | E is the Euclidean distance), we get the Riesz operator on Euclidean space: Firstly, let us present the Hardy-Littlewood-Sobolev inequality on Euclidean space.
Proof. By using the reverse Hölder inequality with 1/q + 1/q = 1, we calculate Thus for (3.9), it is enough to show that where B E (0, |x| E ) is the Euclidean ball centred at 0 with radius |x| E . By using |y| E ≤ |x| E , we get Then for any λ < 0, we have Let us show that the condition (2.4) is satisfied. From the assumption, we have which means n + λq > 0. By using this fact, we obtain Finally, by using the assumption 1/p + 1/q + λ/n = 0, (3.17) which implies D 1 (|x| E ) is a non-decreasing function. Thus, completing the proof. Also, let us now present the reverse Hardy-Littlewood-Sobolev inequality on G.

where C is a positive constant independent of f and h.
Proof. The proof of this theorem is similar to theorem 3.4, but here we use proposition 3.1 and the polar decomposition formula (3.3).

Theorem 3.9 (The reverse Stein-Weiss inequality on G)
. Let G be a homogeneous group of homogeneous dimension Q ≥ 1 and let | · | be an arbitrary homogeneous quasi-norm on G. Assume that q ≤ p < 0, λ < 0 and 1/p + 1/q + (α + β + λ)/Q = 0, where 1/p + 1/p = 1 and 1/q + 1/q = 1. Then for all non-negative functions f ∈ L q (G) and 0 < G h p (x)dx < ∞, we have (3.31) if one of the following conditions is satisfied: Proof. The proof is similar to the previous theorem, but here we use proposition 3.1 and the polar decomposition formula (3.3).
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