Coercive Inequalities and U-Bounds on Step-Two Carnot Groups

We prove Poincaré and Logβ-Sobolev inequalities for a class of probability measures on step-two Carnot groups.


Introduction
Although the question of obtaining coercive inequalities such as the Poincaré or the Logarithmic Sobolev inequalities for a probability measure on a metric measure space has been a subject of numerous works, the literature on this topic in the setup of Carnot groups is scarce.
In [14], L. Gross obtained the following Logarithmic Sobolev inequality: where ∇ is the standard gradient on R n and dμ = e − |x| 2 2 Z dλ is the Gaussian measure.
In a setup of a more general metric space, a natural question would be to try to find similar inequalities with different measures of the form dμ = e −U (d) Z dλ, where U is a function of a metric d, and where the Euclidean gradient is replaced by a more general sub-gradient in R n .
Aside from their theoretical importance, such inequalities are needed because of their applications, some of which will be discussed briefly. L.Gross also pointed out [14] the importance of the inequality (1.1) in the sense that it can be extended to infinite dimensions with additional useful results. (See also works: [5,6,15,28,30,31,35].) He proved that if L is the non-positive self-adjoint operator on L 2 (μ) such that then (1.1) is equivalent to the fact that the semigroup P t = e tL generated by L is hypercontractive: i.e. for q (t) ≤ 1 + (q − 1) e 2t with q > 1, we have P t f q (t) f q for all f ∈ L q (μ) . [14] In [1], D. Bakry and M. Emery extended the Logarithmic Sobolev inequality for a larger class of probability measures defined on Riemaniann manifolds under an important Curvature-Dimension condition. More generally, if ( ) is a probability space, and L is a non-positive self-adjoint operator acting on L 2 (μ) , we say that the measure μ satisfies a Logarithmic Sobolev inequality if there is a constant c such that, for f ∈ D (L) , In this general setting, the connection between this inequality and the property of hypercontractivity was shown in [14].
Another generalisation, the so-called q-Logarithmic Sobolev inequality, in the setting of a metric measure space, was obtained by S. Bobkov and M. Ledoux in [4], in the form: where q ∈ (1,2] . Here, on a metric space, the magnitude of the gradient is defined by |f (x) − f (y) | d (x, y) .
In [5], S. Bobkov and B. Zegarliński showed that the q-Logarithmic Sobolev inequality is better than the classical q = 2 inequality in the sense that one gets a stronger decay of tail estimates. In addition, when the space is finite, and under weak conditions, they proved that the corresponding semigroup P t is ultracontractive i.e. P t f ∞ f p for all t ≥ 0 and p ∈ [1, ∞) . We point out, that in [21], M. Ledoux made a connection between the Logarithmic Sobolev inequality and the isoperimetric problem. (See also: [1,3,18]) The important q-Poincaré inequality can be obtained from the q-Logarithmic Sobolev inequality by simply replacing f by 1 + εf in that inequality, and letting ε → 0.
In this paper, our primary interest is to prove the existence of coercive inequalities for different measures in the setting of step-two nilpotent Lie groups, whose tangent space at every point is spanned by a family of degenerate and non-commuting vector fields where |R| is the cardinality of the index set R. These inequalities, when satisfied, give us information about the spectra of the associated generators of the form where |R| is strictly less than the dimension of the space. (See also [13] and references therein) Thus, by Hörmander's result in [7], the sub-Laplacian (1.2) is hypoelliptic; in other words, every distributional solution to Lu = f is of class C ∞ whenever f is of class C ∞ .
We point out that, according to [33], if we have a uniqueness of solution in the space of square integrable functions for the Cauchy problem ∂u ∂t = Lu u| t=0 = f, then the solution of the heat equation will be given by u = P t f .
In the setting of step-two nilpotent Lie groups, since the Laplacian is of Hörmander type and has some degeneracy, D. Bakry and M. Emery's Curvature-Dimension condition in [1] will no longer hold true. In [16], a method of studying coercive inequalities on general metric spaces that does not require a bound on the curvature of space was developed. Working on a general metric space equipped with non-commuting vector fields {X 1 , . . . , X n }, their method is based on U-bounds, which are inequalities of the form: dλ is a probability measure, U(d) and U (d) are functions having a suitable growth at infinity, λ is a natural measure like the Lebesgue measure for instance (which is the Haar measure for nilpotent Lie groups), d is a metric related to the gradient ∇ = (X 1 , . . . , X n ) , and q ∈ (1, ∞) .
It is worth mentioning that in the setting of nilpotent Lie groups, heat kernel estimates were studied to get a variety of coercive inequalities ( [2, 11, 22-27, 29, 34]). In our setting, we study coercive inequalities involving sub-gradients and probability measures on the group which is a difficult and much less explored subject. An approach, pioneered in [16], was used by J. Inglis to get Poincaré inequality in the setting of the Heisenberg-type group with measure as a function of Kaplan distance [17] and by M. Chatzakou et al. to get Poincaré inequality in the setting of the Engel-type group with a measure as a function of some homegenous norm [10].
In Section 2, we define the step-two Carnot group, and we introduce a family of homogeneous norms N, which generalise the Kaplan norm on Heisenberg-type groups. Section 3 contains the main theorem, which is a proof of a U-Bound of the form where g(N) satisfies some growth conditions. In Section 4, we apply this U-bound together with some results of [16] to get the q-Poincaré inequality with q ≥ 2 for the measures dμ = e −g(N) Z dλ. This generalises the result by J. Inglis [17] who, in the setting of the Heisenberg-type group, proved the q-Poincaré inequality for the measure dμ = e −αÑ p Z dλ, where p ≥ 2, α > 0, q is the finite index conjugate to p, and withÑ the Kaplan norm.
(Note that the examples studied in [16] do not include measures weighted by more general functions of the Kaplan norm.) In Section 5, we extend J. Inglis et al.'s Theorem 2.1 [18] who proved a Log β −Sobolev, β ∈ (0, 1) inequality in the context of the Heisenberg group. Recall that for density defined with a smooth homogenous norm, β = 1 is not allowed ( [16]). We extend the corresponding results to a φ−Logarithmic Sobolev inequality, where φ is concave, on step-two Carnot groups. Finally, we utilise the U-Bound to get to a Log β −Sobolev inequality for dμ = e −αN p Z dλ, where p ≥ 4, q ≥ 2, and 0 < β ≤ indicating also no-go zone of parameters where the corresponding inequality fails. We were able to use the methods of this paper to obtain similar results in the setting of the Higher-Dimensional Anisotropic Heisenberg group [8].

Setup
Carnot groups are geodesic metric spaces that appear in many mathematical contexts like harmonic analysis in the study of hypoelliptic differential operators ( [1,33]) and in geometric measure theory (see extensive reference list in the survey paper [4]).
We will be working in the setting of G, a step-two Carnot group, i.e. a group isomorphic to R n+m with the group law ,..,n;j =1,..,m for x, x ∈ R n , z, z ∈ R m , where the matrices (j ) are n × n skew-symmetric and linearly independent and ., . R n stands for the inner product on R n . One can verify that (R n+m , •) is a Lie group whose identity is the origin and where the inverse is given by ( is an automorphism of (R n+m , •) for any λ > 0. Then, G = (R n+m , •, δ λ ) is a homogeneous Lie group.
The Jacobian matrix at (0, 0) of the left translation τ (x,z) i.e the map for fixed (x, z) ∈ G) takes the following form Then, the Jacobian basis of g, the Lie algebra of G, is given by with a ∈ (0, ∞). The motivation behind choosing such a norm is that in the setting of the Heisenbergtype groups (where we assume in addition that (j ) are orthogonal matrices and that is the Kaplan norm which arises from the fundamental solution of the sub-Laplacian. In other words, N 2−n−2m = 0 in G\{0}. Recall that J. Inglis, in [17], proved the q-Poincaré inequality in the setting of the Heisenberg-type group for the measure dμ = where λ is the Lebesgue measure, α > 0, p ≥ 2, q is the finite index conjugate to p, and N ≡ |x| 4 + 16|z| 2 1 4 . We extend, giving a simpler proof, the result of J. Inglis by obtaining a q-Poincaré inequality in the setting of step-two Carnot groups for Our first key result this paper is obtaining the following U-Bound (Section 3): under certain growth conditions for g(N). This U-bound is a useful tool to get a q-Poincaré inequality (Section 4) and a Log β -Sobolev inequality (Section 5) for q ≥ 2. We expect that this U-bound can be used to extend those coercive inequalities to (non-product) measures in an infinite dimensional setting. [36] 3 U-Bound holds for all locally Lipschitz functions f, supported outside the unit ball {N < 1} , with C and D positive constants depending on q and independent of f .
The main tools we use are Hardy's inequality (see [32] and references therein), which we will prove, and the Euclidean Coarea formula (see [12]). We also use the following properties of a smooth norm N proven in the Appendix.

Lemma 2 There exist constants
and there exists a constant B ∈ (0, ∞) such that We note that the horizontal tangent space G is identified with R n , and the function x in (3.3) corresponds to horizontal vector fields Proof of Theorem 1 First, we prove the result for q = 2 : We note that using integration by parts, one gets Next, using (3.1) and (3.2), Replacing f by f 2 |x| 2 : As for the left-hand side of (3.4), Using the calculation of ∇N · x, from (3.3), we get: Combining with (3.4), }.
Applying Cauchy's inequality with where is a fixed constant to be chosen later, to obtain }.
The aim now is to estimate the first term on the right-hand side of (3.5). We use the following integration by parts formula: for F r := {g ∈ G :φ(g) < r} and I (y) = (y 1 , y 2 , ..., y n+m ) T the identity map on R n+m . Now consider F r = |x| g (N ) < r , where 1 < r < 2. Integrating by parts using (3.6), where in the last step we used Cauchy's inequality. Subtracting on both sides of the last inequality by 2 F r |f e −g(N) 2 | 2 N 2 |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 Euclidean Coarea formula ( [12], page 247): It remains to compute the right hand side of (3.7). The first term, Using (3.1) and taking into consideration that N > 1 and on We do not worry about the third term in this inequality since it is dominated by f 2 g (N) N 2 e −g(N) dλ for N > 1. For the second term of (3.7), x j X j I, ∇ euc |x| g (N ) R n+m dλ.
For e i the standard Euclidean basis on R n+m , Taking the dot product and summing, jl is skew symmetric.
Therefore, replacing, x j X j I, ∇ euc |x| g (N ) R n+m dλ Using the fact that we are integrating over {1 < |x| g (N ) < 2}, Using the condition of the theorem that g (N ) ≤ g (N ) 3 N 3 , we get Inserting bounds onÂ andB in (3.7), we get: Using this last bound to estimate (3.5), we get: So, choosing 10 n−2 + < A we get Secondly, for q > 2, replacing |f | by |f | q 2 ,we get: Remark We note that at this point we get the inequality which implies the necessary and sufficient condition for exponential decay in L p as described in [31].

Poincaré Inequality
We now have the U-Bound (2.1) at our disposal and are ready to prove the q-Poincaré inequality using the method [16]: 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 Note that we have this Poincaré inequality on balls in the setting of Nilpotent lie groups thanks to J. Jerison's celebrated paper [19]. With this we can use the following result: Note that the constants in the above Theorem depend on q. The role of η in Theorem 3 is played by g (N ) N 2 from the U-Bound of Theorem 1. Hence, we get the following corollaries: First, on {N ≥ 1}, g (N ) ≤ k 3 N 3k sinh 3 (N k ) = g (N ) 3 N 3 , so the condition of Theorem 1 is satisfied. Second, Thus, the conditions of Theorem 3 are satisfied for η = kN k−3 sinh N k , and k ≥ 3 2 . So, the Poincaré inequality holds for q ≥ 2.
The following corollary was proven in the setting of the Heisenberg-type group for the Kaplan norm N = |x| 4 + 16|z| 2 1 4 by J. Inglis, Theorem 4.5.5 of [17]. In the setting of the step-two Carnot group, we obtain a generalised version for a similar homogeneous norm N = |x| 4 + a|z| 2 Thus, the conditions of Theorem 3 are satisfied for η = kN k−3 , and k ≥ 4. So, the Poincaré inequality holds for q ≥ 2.
The following corollary improves Corollary 5 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.
so the condition of Theorem 1 is satisfied. Secondly, Thus, the conditions of Theorem 3 are satisfied for η = kN k−3 log(N + 1) + N k−2 N + 1 , and k ≥ 3. Hence, the Poincaré inequality holds for q ≥ 2.

φ−Logarithmic Sobolev Inequality
After proving the q-Poincaré inequality for measures as a function of the homogeneous norm N = |x| 4 + a|z| 2 We will first extend their theorem, and then we will use Theorem 1 to get more general coercive inequalities.
Proof First of all, we remark that for a concave function φ as in our assumptions, we have Suppose first that |f | q dμ = 1, and let E = {x ∈ R N : log |f | q > U}.
where the last inequality uses (5.1) on E, and uses the fact that φ is non-decreasing on where the last inequality is true since U is bounded below and d = 1dμ. Using the classical Sobolev inequality, Using the U-bound in the Theorem's condition Finally, replace |f | q by |f | q μ|f | q to get the desired inequality.

Corollary 9
Let h (1) Then, for all n ≥ 1, h (n) (x) satisfies the hypotheses on φ(x) from Theorem 7. Therefore, we obtain where log * (n) is the positive part of log (n) .
Proof The proof proceeds by induction. For n = 1, h (1) (1) (0) = log(α) > 0, and h (1) is non-negative, non-decreasing, and concave; hence the conditions of Theorem 7 are satisfied. Assume it is true for n = k, prove it is true for n = k + 1 : The result follows directly.
Returning to the measure as a function of the homogeneous norm N = |x| 4 + a|z| 2 1 4 , dμ = e −N p Z dλ, we will prove using Theorems 1 and 7, that the Log β -Sobolev inequality (0 < β ≤ 1) (Corollary 8) holds for q ≥ 2, yet fails for 1 < q < 2pβ p − 1 . To start with, we will show why the Log β -Sobolev inequality fails for 1 < q < 2pβ p − 1 . The proof uses the idea of Theorem 6.3 of [16].

Theorem 10
Let G be a stratified group, and φ be a smooth homogeneous norm on G\{0}. For Proof The proof is by contradiction. Denote by d(g, h) the control distance between two points g, h ∈ G, Q the homogeneous dimension of G, and δ t the dilation by t > 0. Let g 0 ∈ G be such that (∇φ)(g 0 ) = 0 (by Theorem 6.2 of [16], where it is required that G is at least of step-two). For t > 0 put r = t −p+1 2 , and f = max min 2 − d(g,δ t (g 0 )) r , 1 , 0 . Eventually, we will be taking t → ∞.

Theorem 11
Let G be an step-two Carnot group. Consider the probability measure given by where Z is the normalization constant and N = |x| 4 + a|z| 2 1 4 with a ∈ (0, ∞). Let g : [0, ∞) → [0, ∞) be a differentiable increasing function such that g is increasing, for some β > 0, and g (N ) < dg (N ) 2 By the condition g(N) ≤ c g (N ) N 2 1 β , we obtain φ(g(N)) = (1 + g(N)) β ≤ g (N ) N 2 on {N ≥ 1}. Hence, since g(N) is increasing and using the U-bound, we have In order to use Theorem 7, it remains to prove: On {N < 1}, since g (N ) is increasing and using (3.1), It remains to compute ∇ · V . Using | | ≤ B |x| 2 N 3 , (3.2), and All terms can be absorbed by the first term in (5.7). Using (5.5) and (5.6), the condition of Theorem 7 is satisfied, and we obtain Log β -Sobolev inequality: for C and D positive constants.
Let the probability measure be dμ = e −βN p Z dλ, where Z is the normalization constant.
Then, for p ≥ 4 and 0 < β ≤ p − 3 p , for C and D positive constants and for q ≥ 2.

Appendix: Proof of Lemma 2
Proof We first compute ∇N = (X i N) i=1,...,n . (1) where we used that for each skew-symmetric matrix (k) , all k ∈ {1, ..., m}, we have that From (1), that with some constants A, C ∈ (0, ∞), we have By choosing a ∈ (0, ∞) sufficiently small, we can ensure that C ≤ 1. We note that using antisymmetry of matrices (k) il we get (k ) il x l δ kk .
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