On horizontal Hardy, Rellich, CaffarelliKohnNirenberg and subLaplacian inequalities on stratified groups
Abstract.
In this paper, we present a version of horizontal weighted HardyRellich type and CaffarelliKohnNirenberg type inequalities on stratified groups and study some of their consequences. Our results reflect on many results previously known in special cases. Moreover, a new simple proof of the BadialeTarantello conjecture [2] on the best constant of a Hardy type inequality is provided. We also show a family of Poincaré inequalities as well as inequalities involving the weighted and unweighted subLaplacians.
Key words and phrases:
Hardy inequality, Rellich inequality, CaffarelliKohnNirenberg inequality, subLaplacian, horizontal estimate, stratified group2010 Mathematics Subject Classification:
22E30, 43A801. Introduction
Consider the following inequality
(1.1) 
where is the standard gradient in , , and the constant is known to be sharp. The onedimensional version of (1.1) for was first discovered by Hardy in [28], and then for other in [29], see also [29] for the story behind these inequalities. Since then the inequality (1.1) has been widely analysed in many different settings (see e.g. [1][10], [12], [13], [16], [17], [20], [30], [31]). Nowadays there is vast literature on this subject, for example, the MathSciNet search shows about 5000 research works related to this topic. On homogeneous Carnot groups (or stratified groups) inequalities of this type have been also intensively investigated (see e.g. [14], [25], [26], [27], [33], [34], [35], [36], [38]). In this case inequality (1.1) takes the form
(1.2) 
where is the homogeneous dimension of the stratified group , is the horizontal gradient, and is the socalled gauge, which is a particular homogeneous quasinorm obtained from the fundamental solution of the subLaplacian, that is, is a constant multiple of Folland’s [22] (see also [23]) fundamental solution of the subLaplacian on . For a short review in this direction and some further discussions we refer to our recent papers [40, 41, 42, 43, 44] and [39] as well as to references therein.
The main aim of this paper is to give analogues of Hardy type inequalities on stratified groups with horizontal gradients and weights. Actually we obtain more than that, i.e., we prove general (horizontal) weighted Hardy, Rellich and CaffarelliKohnNirenberg type inequalities on stratified groups. Our results extend known Hardy type inequalities on abelian and Heisenberg groups, for example (see e.g. [2] and [11]). For the convenience of the reader let us now briefly recapture the main results of this paper. Let be a homogeneous stratified group of homogeneous dimension , and let be leftinvariant vector fields giving the first stratum of the Lie algebra of , , with the subLaplacian
Denote the variables on by where corresponds to the first stratum. For precise definitions we refer to Section 2.
Thus, to summarise briefly, in this paper we establish the following results:

(Hardy inequalities) Let be a stratified group with being the dimension of the first stratum, and let . Then for all complexvalued functions and we have the following CaffarelliKohnNirenberg type inequality
(1.3) where and is the Euclidean norm on . If then the constant is sharp. In the special case of , and , inequality (1.3) implies
(1.4) where the constant is sharp for . One novelty of this is that we do not require that . In turn, for , the inequality (1.4) gives a stratified group version of Hardy inequality
(1.5) again with being the best constant.

(BadialeTarantello conjecture) Let In [2] Badiale and Tarantello proved that for and there exists a constant such that
(1.6) where is the standard Euclidean gradient. Clearly, for this gives the classical Hardy’s inequality with the best constant
It was conjectured in [2, Remark 2.3] that the best constant in (1.6) is given by
(1.7) This conjecture was proved in [45]. As a consequence of our techniques, we give a new proof of the BadialeTarantello conjecture.

(Critical Hardy inequality) For , the inequality (1.5) fails. In this case the Hardy inequality (1.1) is replaced by a logarithmic version, an analogue of which we establish on stratified groups as well. For a bounded domain with and we have
(1.8) where In the abelian case of being the Euclidean space, inequality (1.8) reduces to the logarithmic Hardy inequality of Edmunds and Triebel [19].

(subLaplacian) Let be a stratified group with being the dimension of the first stratum, and let with be such that . Then for all we have and
(1.9) where is the Euclidean norm on and is the subLaplacian operator defined in (2.3).

(Higher order HardyRellich inequalities) Let For any we have
for any realvalued function , , and such that and
as well as such that and

(Weighted subLaplacian) Let and be such that
(1.11) where is a weighted subLaplacian defined in (5.8). Then we have
(1.12) for all realvalued functions and . Here is a positive constant. For the inequality (1.12) is replaced by an analogous one while for it becomes an identity, see Remark 5.6 and Remark 5.5, respectively.
In Section 2 we very briefly recall the main concepts of stratified groups and fix the notation. In Section 3 we derive versions of CaffarelliKohnNirenberg type inequalities on stratified groups and discuss their consequences including higher order cases as well as a new proof of the BadialeTarantello conjecture. An analogue of the critical Hardy inequality is proved in Section 4. HardyRellich type inequalities and their weighted versions on stratified groups are presented and analysed in Section 5.
2. Preliminaries
A Lie group is called a stratified group (or a homogeneous Carnot group) if it satisfies the following conditions:
(a) For some natural numbers , that is the decomposition is valid, and for every the dilation given by
is an automorphism of the group Here and for
(b) Let be as in (a) and let be the left invariant vector fields on such that for Then
for every i.e. the iterated commutators of span the Lie algebra of
That is, we say that the triple is a stratified group. See also e.g. [21] for discussions from the Lie algebra point of view. Here is called a step of and the left invariant vector fields are called the (Jacobian) generators of . The number
is called the homogeneous dimension of . The second order differential operator
(2.1) 
is called the (canonical) subLaplacian on . The subLaplacian is a left invariant homogeneous hypoelliptic differential operator and it is known that is elliptic if and only if the step of is equal to 1. We also recall that the standard Lebesque measure on is the Haar measure for (see, e.g. [21, Proposition 1.6.6]). The left invariant vector field has an explicit form and satisfies the divergence theorem, see e.g. [40] for the derivation of the exact formula: more precisely, we can write
(2.2) 
see also [21, Section 3.1.5] for a general presentation. We will also use the following notations
for the horizontal gradient,
for the horizontal divergence,
(2.3) 
for the horizontal Laplacian (or subLaplacian), and
for the Euclidean norm on
3. Horizontal CaffarelliKohnNirenberg type inequalities and consequences
In this section and in the sequel we adopt all the notation introduced in Section 2 concerning stratified groups and the horizontal operators.
3.1. CaffarelliKohnNirenberg inequalities
In this section we establish the following horizontal CaffarelliKohnNirenberg type inequalities on the stratified group and then discuss their consequences and proofs. The proof is analogous to [39] in the case of homogeneous groups, but here we rely on the divergence theorem rather on the polar decomposition which is less suitable for the stratified setting. We refer e.g. to [6] and [7] for Euclidean settings of CaffarelliKohnNirenberg inequalities.
Theorem 3.1.
Let be a homogeneous stratified group with being the dimension of the first stratum, and let . Then for any and all we have
(3.1) 
where and is the Euclidean norm on . If then the constant is sharp.
In the abelian case , we have , , so (3.1) implies the CaffarelliKohnNirenberg type inequality (see e.g. [10] and [18]) for with the sharp constant:
(3.2) 
for all and In the case
that is, taking and , the inequality (3.1) implies that
(3.3) 
for any and all .
When and , the inequality (3.3) gives the following stratified group version of Hardy inequality
(3.4) 
again with being the best constant (see [11] and [47] for the version on the Heisenberg group). In the abelian case , , (3.4) implies the classical Hardy inequality for :
for all and
The inequality (3.4) implies the following HeisenbergPauliWeyl type uncertainly principle on stratified groups (see e.g. [9], [42], [40] and [39] for different settings): For each , using Hölder’s inequality and (3.4), we have
(3.5) 
that is,
(3.6) 
In the abelian case , taking , we obtain that (3.6) with implies the classical uncertainty principle for : for all we have
which is the HeisenbergPauliWeyl uncertainly principle on .
On the other hand, directly from the inequality (3.1), using the Hölder inequality, we can obtain a number of HeisenbergPauliWeyl type uncertainly inequalities which have various consequences and applications. For instance, when , we get
(3.7) 
and if and , then
(3.8) 
both with sharp constants.
Proof of Theorem 3.1.
We may assume that since for the inequality (3.1) is trivial. By using the identity (2.5), the divergence theorem and Schwarz’s inequality one calculates
Here we have used Hölder’s inequality in the last line. Thus, we arrive at
(3.9) 
This proves (3.1). Now it remains to show the sharpness of the constant. Let us examine the equality condition in above Hölder’s inequality as in the abelian case (see [18]). For this we consider the function
(3.10) 
where and Then it can be checked that
(3.11) 
which satisfies the equality condition in Hölder’s inequality. It shows that the constant is sharp. ∎
3.2. BadialeTarantello conjecture.
The proof of Theorem 3.1 gives the following similar statement in
Proposition 3.2.
Let and . Then for any and all we have
(3.12) 
where and is the Euclidean norm on . If then the constant is sharp.
The proof is similar to the proof of Theorem 3.1. However, for the sake of completeness here we give the details.
Proof of Proposition 3.2.
We may assume that since for the inequality (3.12) is trivial. By using the identity
for all and with , where is the standard divergence on , the divergence theorem and Schwarz’s inequality one calculates
where , that is is the standard gradient on , as well as is the gradient on Here we have used Hölder’s inequality in the last line. Thus, we arrive at
(3.13) 
This proves (3.12). Now it remains to show the sharpness of the constant. Let us examine the equality condition in above Hölder’s inequality. Consider
(3.14) 
where and Then it can be checked that
(3.15) 
which satisfies the equality condition in Hölder’s inequality. It shows that the constant is sharp. ∎
As above, taking and the inequality (3.12) implies that
(3.16) 
for any and for all with the sharp constant. When and the inequality (3.16) implies that
(3.17) 
again with being the best constant. This proves the BadialeTarantello conjecture, which is stated in the introduction (see also [2, Remark 2.3] for the original statement).
3.3. Horizontal higher order versions
In this subsection we show how by iterating the established CaffarelliKohnNirenberg type inequalities one can get inequalities of higher order. Putting instead of and instead of in (3.3) we consequently have
for Here and after we understand , that is, Combining it with (3.3) we get
(3.18) 
for each such that and This iteration process gives