Coercive Inequalities in Higher-Dimensional Anisotropic Heisenberg Group

In the setting of higher-dimensional anisotropic Heisenberg group, we compute the fundamental solution for the sub-Laplacian, and we prove Poincar\'e and $\beta-$Logarithmic Sobolev inequalities for measures as a function of this fundamental solution.


Introduction
In 1975, G.B. Folland showed in [15] that on a Carnot group G, the sub-Laplacian △ := n i=1 X 2 i admits a unique fundamental solution N 2−Q , i.e.
where δ is the delta-distribution at the unit element of G, X 1 , ..., X n are the Jacobian generators of G, Q is the homogeneous dimension, and N is a homogeneous norm on G.
In [2], Z. Balogh and J. Tyson introduced the concept of polarizable Carnot groups defined by the condition that N is ∞−harmonic in G\{0}, i.e. for ▽ :=(X i ) 1≤i≤n , They have shown that using the ∞−harmonicity of N one can provide a procedure to construct polar coordinates of special type where the curves passing through the points on the unit sphere {N = 1} are horizontal.
Moreover, they showed in [2] that the fundamental solution of the p−sub-Laplacian can be expressed as the fundamental solution N of the sub-Laplacian, proved capacity formulas, and Key words and phrases. Poincaré inequality, Logarithmic-Sobolev inequality, Anisotropic Heisenberg group, subgradient, fundamental solution, probability measures.
For the time being, there is no a classification of polarizable Carnot groups, and the only examples till now are Euclidean spaces and Heisenberg-type groups. In addition, the concept of a polarizable Carnot group is a delicate one in the sense that under a small pertubation of the Lie algebra, the group is no longer polarizable. Z. Balogh and J. Tyson provided in [2] the anisotropic Heisenberg group in R 4 as a counterexample with the following generators of the Lie algebra: (Note that if a = 1, we have the polarizable Heisenberg group.) To show (1.1) does not hold true for the anisotropic Heisenberg group, they computed explicitly the fundamental solution of the sub-Laplacian using R. Beals, B. Gaveau, and P. Greiner's [3] explicit intergal representation for the fundamental solution in the setting of general step-two Carnot groups.
Recently, T. Bieske [4] revisited this counterexample and proved that under a change of the inner product imposed on the vectors in the Lie Algebra, which now requires the generators to be orthogonal instead of orthonormal, the anisotropic Heisenberg group is turned into a group of Heisenberg-type i.e. it is now polarizable! The goal of this paper is to study coercive inequalities such as the q−Poincaré inequality and the β−Logarithmic Sobolev inequality in the setting of the anisotropic Heisenberg group with respect to measures as a function of the explicit fundamental solution. We will first show the computations, that were partially omitted in [2] in the R 5 setting, and use that to get an explicit fundamental solution for higher dimensions.
In the setting of nilpotent Lie groups, heat kernel estimates have been used to get coercive inequalities [35,24,1,31,25,26,27,28,29,12]. In our setting, we do not use heat kernel estimates; instead, we study coercive inequalities involving sub-gradients and probability measures depending on the group. An approach to study such problems was pioneered in [19]. It was later used by with functions U possessing suitable growth properties at infinity. Our key result in this paper is obtaining (in section 3) for a probability measure dµ = e −g(N ) Z dλ, defined with g(N ) satisfying suitable growth conditions, the following U-Bound in the setting of the higher-dimensional anisotropic Heisenberg group: where ▽ :=(X i ) 1≤i≤n , with constants C, D ∈ (0, ∞) independent of the function f for which the right-hand side is well defined. This U-bound is used (in section 4) to get a q-Poincaré inequality and a β−Logarithmic Sobolev inequality for q ≥ 2 . We expect that our results can be used to extend those coercive inequalities to an infinite dimensional setting, which is of interest. (See also works: [38,6,30,32,5,18].) In the second section, we will start by extending Z. Balogh and J. Tyson's anisotropic Heisenberg group in R 5 [2] to a higher-dimensional anisotropic Heisenberg group in R 2n+1 , and use R. Beals, B. Gaveau, and P. Greiner's [3] explicit intergal representation to compute the fundamental solution. We also compute bounds for |▽N | and x · ▽N (section 2), which are essential to get the U-Bound (1.3) (section 3). We remark that in our setting, unlike in the case studied in [8], x · ▽N can be negative, and we will need the dimension n > 5 to take care of the negative term. For n ≤ 5, some other method is yet to be explored to get U-Bounds.
Finally, in the fourth section, we apply the U-Bound to get coercive inequalities like the q-Poincaré inequality and the β−Logarithmic Sobolev inequality for q ≥ 2 .
For what follows, we denote |x| = to be the Euclidean norm. The following lemma is crucial in obtaining a U-Bound in section 3.
In the rest of this section we provide a proof of the above lemmata which involves lengthy calculations based on general formula for Green functions for type-2 Carnot groups provided in [3].
The reader interested in applications to coercive inequalities is invited to jump directly to section 3.

Derivation of the Formula for Homogeneous Norm N .
To compute explicitly the fundamental solution, we now use R. Beals, B. Gaveau, and P.
Greiner's Theorem 2 of [3]. The fundamental solution for the sub-laplacian △ = 2n j=1 X 2 j on G with singularity at 0 is: Here, w j (τ ) are the eigenvalues and e j (τ ) are the corresponding normalised eigenvectors of the The eigenvalues are: , w j (τ ) = −τ, for j = 2, 3, .., n, and w j (τ ) = τ, for j = n + 2, n + 3, ..., 2n. The corresponding normalised eigenvectors are: for j = 1, 2, ..., n, and for j = n + 1, n + 2, ..., 2n. Thus, j=1,j =n+1 x 2 j , we obtain: Replacing (2.6) and (2.7) in (2.5), we get: using the trigonometric identities csch(τ ) = (csch τ 2 ) 2 2coth τ 2 and coth τ 2 The next step is to compute Write I(A, B, t) = I 1 (A, B, t) − I 2 (A, B, t), where Remark that We now compute dx using complex theory. Write denominator in the integrand as a polynomial p(z) has four complex roots. If z is a root, then −z is a root. Since p(z) has real coefficients, then z is a root, and −z is the fourth root. Let α be a root in the upper half plane, so −ᾱ is also in the upper half plane. We thus have By identification with (2.10), we get that . In fact, Now, we go back to computing the integral: , by the residue theorem: . and Replacing in (2.12), we get: We can now compute I(A, B, t) using (2.9): The fundamental solution is given up to a constant multiple: using the last equation, Note that for n = 2, we get back the fundamental solution as calculated in [2].

Recall that by |x|
we denote the Euclidean norm. In the setting of the anisotropic Heisenberg group R 2n+1 , we have the fundamental solution In this subsection, we are going to show the following relations: (2.14) x · ▽N ≥ − |x| 2 4nN , Proof. We first calculate ∂ xj N and ∂ t N.
For j = 1 and j = n + 1 : To get a bound from above for Now we calculate ∂ xj N for j = 2, .., n, n + 2, .., 2n : We will now be able to get a lower bound for x · ▽N. In fact, x j ∂ xj N using (2.18) and (2.21), Using the above defined Euclidean norm |x| = It remains to bound |▽N |.

U-Bound
The following U-Bound will be used in section 4 to prove the q−Poincaré inequality for q ≥ 2 for the measure dµ = e −g(N ) Z dλ under the condition that g ′ (N ) N 2 is an increasing function on the anisotropic Heisenberg group where n > 5. In addition, in section 4, we will prove a β−Logarithmic Sobolev inequality for dµ = e −αN p Z dλ, for p ≥ 4, q ≥ 2, and 0 < β ≤ p−3 p .
Z dλ be a probability measure and Z the normalization constant. Then, for q ≥ 2, holds outside the unit ball {N < 1} with C and D positive constants independent of a function f for which the right hand side is well defined.
Proof. First, we prove the result for q = 2 : Using integration by parts, Since N −2n is the fundamental solution, we have Hence, Replacing f by f 2 |x| 2 and using (2.3), the left-hand side of (3.1) becomes: Where the last inequality is true since N > 1. As for the right-hand side of (3.1), Using the bound on ▽N · x, from (2.2), we get: using the bound on |▽N |, from (2.4), we get Combining the last inequality with (3.2), we get For n large enough, to be determined later, we need to bound 1 2n f 2 N |x| 2 e −g(N ) dλ from the right hand side of (3.4) by 1 N 2 e −g(N ) dλ by using Hardy's inequality (see [33] and references therein) and the Coarea formula (page 468 of [7]). Since β could be chosen to be arbitrarily small and since N > 1, we do not worry about the term β f 2 N 2 |x| 2 e −g(N ) dλ since it follows the same procedure as 1 2n α is to be chosen later. The aim now is to estimate the first term of (3.5). Consider F r = α |x| 2 g ′ (N ) N < r , where 1 < r < 2. Integrating by parts: Where the last step uses Cauchy's inequality. Subtracting on both sides of the last inequality by 1 4n Fr |f e −g(N ) 2 | 2 N |x| 2 dλ, and using the fact that 1 < r < 2, we get: Integrating both sides of the inequality from r = 1 to r = 2, we get: To recover the full measure in the boundary term, we use the Coarea formula: It remains to compute the right hand side of (3.6). The first term,

Using (2.4),
A ≤ 1 2n Using the fact that F 2 = α |x| 2 g ′ (N ) N < 2 and that |x| ≤ C n N, Using Cauchy's inequality with γ: ab 2 n 2 (n − 1) 2 |∇f | and Cauchy's inequality with γ on (2n + 1)C n Hence, For the second term of (3.6), On the anisotropic Heisenberg group, for e i the standard Euclidean basis on R 2n+1 , Taking the dot product and summing, 2n j=1 x j < X j I, ∇ euc α Λ jl x l x j = 0 since Λ is skew symmetric. Hence, using (2.4), 2n j=1 x j < X j I, ∇ euc α Therefore, replacing, Using the fact that we are integrating over {1 < α |x| 2 g ′ (N ) Using the condition of the theorem that g ′′ (N ) ≤ g ′ (N ) 2 on {N ≥ 1}, we bound the second term on the right hand side of (3.8). Now we go back to (3.4): From (3.5) we have that 1 2n From (3.7) and (3.8) we have that We combine the last three inequalities and repeat the same procedure to β f 2 N 2 |x| 2 e −g(N ) dλ to get: To get the U-Bound, we need the left hand side of (3.9) to be positive i.e. and find suitable α, n, γ, and β. Since γ and β can be chosen to be arbitrarily small, we need to find solutions to the following inequality: First, we determine α : Let f (α) = 1 .
This holds true if we choose n > 5. Hence, we get: Second, for q > 2,replacing |f | by |f | q 2 ,we get: Calculating, Using Hölder's inequality, Where the last inequality uses ab≤ , and p ′ and q ′ are conjugates. Choosing p ′ = q q − 2 , we obtain Using the inequalities (3.11) and (3.12), we get,

q−Poincaré and β−Logarithmic Sobolev Inequalities
We now have the U-Bound (1.3) at our disposal and are ready to prove the q-Poincaré inequality using the U-Bound method of [19]: Let λ be a measure satisfying the q-Poincaré inequality for every ball B R = {x : N (x) < R}, i.e. there exists a constant C R ∈ (0, ∞) such that where 1 ≤ q < ∞. Note that we have this Poincaré inequality on balls in the setting of Nilpotent lie groups thanks to J. Jerison's celebrated paper [22]. Later on, we apply the following result of Theorem 4 (Hebisch, Zegarliński [19]). Let µ be a probability measure on R m which is absolutely continuous with respect to the measure λ and such that with some non-negative function η and some constants C, D ∈ (0, ∞) independent of a function f.

If for any
for some R ∈ (0, ∞) (depending on L), we have {η < L} ⊂ B R , then µ satisfies the q-Poincaré The role of η in Theorem 4 is played by g ′ (N ) N 2 from the U-Bound of Theorem 3. Hence, we get the following corollaries: where λ is the Lebesgue measure, and k≥ 1 in the setting of the anisotropic Heisenberg group R 2n+1 with n > 5.
Proof. g (N ) = cosh N k , so g ′ (N ) = kN k−1 sinh N k , and on the set {N > 3 2 },so the condition of Theorem 2 is satisfied. Secondly, Thus, the conditions of Theorem 4 are satisfied for η = kN k−3 sinh N k , and k ≥ 1. So, the Poincaré inequality holds for q ≥ 2.
Corollary 6. The Poincaré inequality for q ≥ 1 holds for the measure dµ = exp −N k Z dλ, where λ is the Lebesgue measure, and k ≥ 4 in the setting of the anisotropic Heisenberg group R 2n+1 with n > 5.
The following corollary improves Corollary 6 in an interesting way. Namely, at a cost of a logarithmic factor, we now get the Poincaré inequality for polynomial growth of order k ≥ 3.
Corollary 7. The Poincaré inequality for q ≥ 2 holds for the measure dµ = exp −N k log (N + 1) Z dλ, where λ is the Lebesgue measure, and k ≥ 3 in the setting of the anisotropic Heisenberg group R 2n+1 with n > 5.
To get the β−Logarithmic Sobolev inequality, we use the following theorem by the authors of this paper in [8] (which generalises J.Inglis et al.'s Theorem 2.1 [21]).
We will use the following theorem: for C and D positive constants and for q ≥ 2.
The proof follows closely that of Theorem 11 in [8], with some modification.
In order to use Theorem 8, it remains to prove: for C and D positive constants and for q ≥ 2.