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Reverse Stein-Weiss, Hardy-Littlewood-Sobolev, Hardy, Sobolev and Caffarelli-Kohn-Nirenberg inequalities on homogeneous groups

In this note we prove the reverse Stein-Weiss inequality on general homogeneous Lie groups. The obtained results extend previously known inequalities. Special properties of homogeneous norms and the reverse integral Hardy inequality play key roles in our proofs. Also, we show reverse Hardy, Hardy-Littlewood-Sobolev, Lp-Sobolev and Lp-Caffarelli-Kohn-Nirenberg inequalities on homogeneous groups.


INTRODUCTION
In one of their pioneering work [24], Hardy and Littlewood considered the one dimensional fractional integral operator on (0, ∞) given by and proved the following theorem: Theorem 1.1. Let 1 < < < ∞ and ∈ (0, ∞) with 1 = 1 + − 1. Then where is a positive constant independent of .
where is a positive constant independent of .
The Hardy-Littlewood-Sobolev inequality on Euclidean spaces and the regularity of fractional integrals was studied in [10], [12], [26] and [30]. On the Heisenberg group, Folland and Stein in [14] obtained the Hardy-Littlewood-Sobolev inequality and in [23] the authors also proved an analogue of the Stein-Weiss inequality. In [18] the authors studied the Stein-Weiss inequality on the Carnot groups. On homogeneous Lie groups, the Hardy-Littlewood-Sobolev and Stein-Weiss inequalities were obtained in [43] and [21]. In [8] the author proved the Stein-Weiss inequality on the Euclidean half-space.
The reverse Hardy-Littlewood-Sobolev inequality in the Euclidean space was obtained in the works [9], [11] and [27]. In [5] and [4], the authors obtained the reverse Stein-Weiss inequality on the Euclidean space and half-space, respectively. In this paper, we first show the reverse Stein-Weiss inequality on the homogeneous groups. In the proof we use special properties of homogeneous norms of the homogeneous Lie groups and reverse integral Hardy inequality, which are playing key roles in our calculations. Thus, in Theorem 2.5 we establish the reverse Stein-Weiss inequality on general homogeneous groups based on the reverse integral Hardy inequalities with one negative exponent. In particular, the obtained result recovers the previously known results of Abelian (Euclidean), Heisenberg, Carnot groups since the class of the homogeneous Lie groups contains those and since we can work with an arbitrary homogeneous quasi-norm. Note that in this direction systematic studies of different functional inequalities on general homogeneous (Lie) groups were initiated by the paper [37]. We refer to this and other papers by the authors (e.g. [39]) for further discussions.
We also note that the best constant in the Hardy-Littlewood-Sobolev inequality on the Heisenberg group is now known, see Frank and Lieb [16] (in the Euclidean case this was done earlier by Lieb in [25]). The expression for the best constant depends on the particular quasi-norm used and may change for a different choice of a quasi-norm.
A multidimensional version of one of the well-known inequalities of G.H. Hardy is where ∈ ∞ 0 (ℝ ), ∇ is the Euclidean gradient and constant − is sharp. The Hardy inequality was intensively studied. For example, the Hardy inequalities were considered on the Euclidean space in [19], [20], on the Heisenberg group in [6], [7], on stratified groups in [3], [31,32,33], [36], [42], for the vector fields in [34] and [38], on homogeneous groups in [37], [39]. In [37] (see e.g., [35]), authors showed the Hardy inequality with radial derivative on homogeneous groups in the following form: where ∇ is the standard Euclidean gradient. By using the Cauchy-Schwartz inequality from the inequality (1.8), we get (1.6). In this note, we show the reverse Hardy inequality on homogeneous general Lie groups.
We are also interested in Sobolev and Caffarelli-Kohn-Nirenberg inequalities, let us recall them briefly. In the classical work of Sobolev, he showed the following inequality: where * = − . The Sobolev inequalities on stratified groups were obtained in [39,40,41]. In [41], on homogeneous groups, the authors showed -Sobolev inequality in the following form: where ∇ is the standard Euclidean gradient. In the Euclidean case, this -Sobolev inequality has been considered in [28].
In this note, we show the reverse Hardy, Hardy-Littlewood-Sobolev, -Sobolev, -Caffarelli-Kohn-Nirenberg inequalities on homogeneous Lie groups. The main idea for proving such reverse inequalities was developed in [37]. In addition, to the best of our knowledge, such reverse inequalities above on a homogeneous group are new even in the Euclidean case.
Summarising our main results of the present note, we prove the following facts: • The reverse Stein-Weiss and Hardy-Littlewood-Sobolev inequalities on homogeneous groups; • The reverse Hardy inequality on homogeneous groups; • The reverse -Sobolev inequality on homogeneous groups; • The reverse -Caffarelli-Kohn-Nirenberg inequality on homogeneous groups.

MAIN RESULTS
Let us recall that a Lie group (on ℝ ) with the dilation which is an automorphism of the group for each > 0, is called a homogeneous (Lie) group. For simplicity, throughout this paper we use the notation for the dilation . The homogeneous dimension of the homogeneous group is denoted by ∶= 1 + … + . Also, in this note we denote a homogeneous quasi-norm on by | |, which is a continuous non-negative function with the properties i) | | = | −1 | for all ∈ , ii) | | = | | for all ∈ and > 0, iii) | | = 0 iff = 0. Moreover, the following polarisation formula on homogeneous Lie groups will be used in our proofs: there is a (unique) positive Borel measure on the unit quasi-sphere ∶= { ∈ ∶ | | = 1}, so that for every ∈ 1 ( ) we have The quasi-ball centred at ∈ with radius > 0 can be defined by We refer to [15] for the original appearance of such groups, and to [13,35] for a recent comprehensive treatment. Let us consider the integral operator where * is the convolution. Let us recall briefly the reverse Hölder's inequality.

Proposition 2.2 ([13], Theorem 3.1.39). Let be a homogeneous Lie group. Then there exists a homogeneous quasi-norm on which is a norm, that is, a homogeneous quasi-norm | ⋅ | which satisfies the triangle inequality
Furthermore, all homogeneous quasi-norms on are equivalent.
The next theorem is the reverse integral version of Hardy inequalities on general homogeneous groups that will be instrumental in our proof.

( , ) > 0 and all non-negative measurable functions , if and only if
Moreover, the biggest constant 1 ( , ) in (2.6) has the following relation to 1 : Also, inequality holds for some 2

( , ) > 0 and all non-negative measurable functions , if and only if
Moreover, the biggest constant 2 ( , ) in (2.9) has the following relation to 2 : Remark 2.4. In our sense, the negative exponent < 0 of 0, we understand in the following form: and ℎ ∈ ( ) we have where is a positive constant independent of and ℎ.
Proof of Theorem 2.5. By using reverse Hölder's inequality with 1 + 1 ′ = 1 (Theorem 2.1) in (2.12), we calculate, So for (2.12), it is enough to show that 6 and by changing ( ) = ℎ( )| | , this is equivalent to We have that Therefore, we obtain Similarly with (2.14), we have By summarising above facts, from (2.14)-(2.15), we have From now on, in view of Proposition 2.2 we can assume that our quasi-norm is actually a norm.