Weighted Hardy and Rellich Types Inequalities on the Heisenberg Group with Sharp Constants

This paper aims at deriving some weighted Hardy type and Rellich type inequalities with sharp constants on the Heisenberg group. The improved versions of these inequalities are established as well. The technique adopted involve the application of some elementary vectorial inequalities and some properties of Heisenberg group.

where ∇ is the Euclidean gradient, for 1 ≤ p < n , which can also be extended to the whole of ℝ n ⧵ {0} for n < p < ∞ . It is well known that the constant n−p p p is sharp but never achieved by a nontrivial function . On the other hand, generalization of (1.1) to higher order derivative is the following Rellich inequality where Δ is the Euclidean Laplacian, with suitable positive weight functions Φ, Ψ and positive constant C. This was originally obtained by Rellich [30] for p = 2 and n ≥ 5 in the sharp form A generalization of (1.3) was obtained by Davies and Hinz [9] as follows where n > 2p and the constant is sharp. Recently, there has been considerable development of the concepts of Hardy and Rellich type inequalities in sub-elliptic context. For instance, Garofalo and Lanconelli [16], Niu et al. [27], Xi and Dou [33], and D'Ambrosio [8] have obtained Hardy inequalities on the Heisenberg group with sharp constants (see also the book [31] for several related results).
Let ℍ be the Heisenberg group of homogeneous dimension Q (detail description is given below) and cc be the Carnot-Carathéodory norm on ℍ . Motivated by the above quoted works, Yang [34] proved analogous Hardy type and Rellich type inequalities on the Heisenberg group with cc being the weight function. Precisely, they proved via a representation formula associated with cc that for 1 < p < Q , then and for Q ≥ 4 , then The motivation for studying this class of inequalities on sub-Riemannian setting with noncommutative vector fields satisfying Hörmander's condition is due to the link between Hardy and Rellich inequalities on one hand and their numerous applications as highlighted above on the other hand. From mathematical point of view, the Heisenberg group is the simplest and most important model in this setting. It is (1.4) an important model to study many significant problems such as existence, continuation, eigenvalue problems, therefore, obtaining vital integral inequalities and their improvements such as those discussed in this paper become essential, and hence the theme of this paper. Our results generalize and improve the above quoted results with weight involving Koranyi-Folland homogeneous norm. In fact this paper complements some other existing results [1,3,4,11,12,18,19,32].

Preliminaries
Basic facts on the Heisenberg group are summarily recalled here. Detail description can be found in Refs. [15,31]. Heisenberg group arises from many fields such as quantum physics, differential geometry, harmonic analysis. It is a unimodular, connected and simply connected step two (Carnot) Lie group. Let = (x, y, t) ∈ ℝ n × ℝ n × ℝ , n ≥ 1 . The Heisenberg group ℍ is the set ℍ n = ℝ 2n × ℝ equipped with the non-commutative group law, • , having the structure where x ⋅ y is the usual Euclidean inner product in ℝ n . The distance between points and ′ in ℍ is defined by where �−1 is the inverse of ′ given by �−1 = − � with respect to the group law • . Let z = (x, y) , = (z, t) and � = (z � , t) . By (1.7) and (1.8), one can obtain the explicit expression for d ℍ n ( , � ) . Therefore the Heisenberg distance function (homogeneous norm) is given by where |z| 2 = x 2 + y 2 and 0 is the centre. A basis for the Lie algebra of the left invariant vector fields on ℍ n is given by ( By the Lie bracket definition one can easily check that and which constitute Heisenberg canonical commutation relation of quantum mechanics for position and momentum, whence the name Heisenberg group.

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A family of dilation on ℍ is defined by (z, t) = ( z, 2 t), > 0 . It is also easy to check that | ⋅ | ℍ , X j , Y j , (j = 1, 2 ⋯ n) are homogeneous of degree 1 with respect to dilations (where | ⋅ | ℍ is as defined by (1.8)). The anisotropic dilation on ℍ induces a homogeneous norm also known as Korányi-Folland non-isotropic gauge, while the homogeneous dimension with respect to dilations is Q = 2n + 2.
The horizontal gradient and divergence on ℍ are respectively defined by and Hence, sub-Laplacian on ℍ (Kohn Laplacian) is defined by The Heisenberg group belongs to a class of homogeneous Carnot groups. It was shown by Folland [14] (see also Ref. [15]) that the sub-Laplacian on a general stratified Lie group has a unique fundamental solution u in distributional sense, which in the Heisenberg setting means where 0 is the Dirac delta-distribution at the neutral element 0 of ℍ n . The Δ ℍ -gauge d( ) (i.e. the homogeneous quasi-norm on ℍ ) can be written in term of the fundamental solution as as a continuous function smooth away from the origin. The action of Δ ℍ on d is given as A simple computation therefore gives the following identities ( [27,31]) , Similarly, the following identities can be established by direct computation.

Lemma 1.1 Let d be as defined above. Then we have
Proof by applying (1.9) and (1.11).
by applying (1.12) and (1.13). ◻ Note that the standard Lebesgue measure on ℝ n is the Haar measure for the homogeneous group, so we write d = dzdt as the Lebesgue measure on ℍ . Throughout, we denote by This section is concluded with the following elementary vectorial inequalities that will be of great applications to the proof of some of the results in the next sections. Lemma 1.2 [26] There exists some constant c(p) > 0 such that for all u, v ∈ ℝ n , there holds the following inequalities
The rest of the paper is structured as follows: Sect. 2 is devoted to the proof of weighted L p -Hardy type inequalities, ( p ≠ Q + , Q ≥ 3 and ∈ ℝ ), and their improvements on the Heisenberg group. The claims for the sharp constant will also be established. While Sect. 3 is devoted to the proofs of sharp weighted Rellich type inequalities and thier improvements.

Weighted Hardy Type Inequalities
This section is concerned with sharp weighted Hardy type inequalities and their improvements.

Weighted L p -Hardy Inequalities
and define a function g = d − f , where ≠ 0 will be chosen later. Direct computation implies Clearly by Lemma 1.2 we have for u, v ∈ ℝ n and p ≥ 1 Thus, one can write Multiplying the last inequality by d and integrating by parts over ℍ gives Therefore the last integral in (2.2) vanishes and we thus arrive at which implies the required inequality (2.1).
In the next we prove the claim that the constant is sharp in the sense that must hold. In other words, we need to show that C ℍ = A ℍ . We use the idea from Ref. [8] (see also Refs. [22,23,35] for similar methods). Obviously, by the inequality (2.1) and by passing to the infimum in the last equation . Thus, we need to show that and for this purpose, we define the following radial function which can be approximated by some smooth function with compact support in ℍ for > 0 . Therefore Since the integrands on the right hand side (of the last equation) are integrable, both integrals on that side are finite and so also the integral on the left hand side. Also Thus we have This concludes the proof. ◻
, holds for all compactly supported smooth function The last integral in (2.7) vanishes due to integration by parts and identities in (1.13). This can be shown as follows: Therefore ( in (2.5), obiviously, the middle integral on the right hand side of the equation vanishes at once. Then one can conclude at that point using (2.8) without resulting to the maximization of coefficient.

Improved L p -Hardy Type Inequalities
Following the same steps as in the proof of Theorem 2.1 we arrive at (2.1) but with an extra term which will be analyzed further, that is Define a function ∶=  .10) gives (2.13)

Rellich Type Inequalities
In this section, we are concerned with Rellich type inequalities with weights given in terms of the homogeneous quasi norm (Heisenberg distance function). First, we start with an L 2 -Rellich inequalities and their improved versions and we conclude with L p -Rellich inequalities and their improved versions is sharp. , holds for all compactly support function f ∈ C ∞ 0 (Ω).

Proof A direct computation shows that
On the other hand, using integration by parts we get (f is assumed to be real-valued without loss of generality) Since Q + − 4 > 0 , we have Rearranging we have Applying the weighted Hardy inequality (2.1) replacing by − 2 and p = 2 gives The last inequality implies after rearranging and adding like terms Using Cauchy-Schwarz inequality

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Combining ( In the next we prove that the constant (Q+ −4) 2 (Q− ) 2 16 appearing in (3.1) is sharp in the sense that must hold.
Thus, we are required to prove that D Q, ∶= (Q+ −4) 2 (Q− ) 2 16 = D ℍ n . Clearly by (3.1) and passing to the infimum in (3.8) we have D Q, ≤ D ℍ . Hence, it only remains to prove that D Q, ≥ D ℍ .
For > 0 , we define the following radial function which can be approximated by smooth function with compact support in ℍ Sending → 0 and noting that ∫ ℍ⧵B 1 . The proof is therefore complete. ◻ The next theorem contains the L p -Rellich type inequalities and improvement.

Conclusion
In this paper we have derived some weighted Hardy type and Rellich type inequalities with sharp constants and their improved versions on the Heisenberg group. Improvement of this class of inequalities is being a hot research topic since few decades ago, see Refs. [2,5,8,12,13,19,24,25,31,34] for instance. The book [31] contains detail discussion on this class of inequalities on homogeneous groups. Some more general results and some improvement of Hardy and Rellich inequalities were established in Ref. [33] (this reference was pointed to us by one of the referees at the review stage). The present paper therefore provides simple and more direct proofs of improved Hardy type and Rellich type inequalities on the Heisenberg group with respect to homogeneous norm, horizontal gradient and Kohn sub-Laplace operators. The results obtained generalize and improved several results in literature as reported in the paper. From mathematical point of view, the Heisenberg group is the simplest and most important model in sub-Riemannian setting with noncommutative vector fields satisfying Hörmander's condition. Hence, it would be interesting to extend these results to some other examples of homogeneous groups.
Journal of Nonlinear Mathematical Physics (2023) 30:677-698 you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.