The One Dimensional Free Poincar\'e Inequality

In this paper we discuss the natural candidate for the one dimensional free Poincar\'e inequality. Two main strong points sustain this candidacy. One is the random matrix heuristic and the other the relations with the other free functional inequalities, namely, the free transportation and Log-Sobolev inequalities. As in the classical case the Poincar\'e is implied by the others. This investigation is driven by a nice lemma of Haagerup which relates logarithmic potentials and Chebyshev polynomials. The Poincar\'e inequality revolves around the counting number operator for the Chebyshev polynomials of first kind with respect to the arcsine law on $[-2,2]$. This counting number operator appears naturally in a representation of the minimum of the logarithmic potential with external fields as well as in the perturbation of logarithmic energy with external fields, which is the essential connection between all these inequalities.

for any smooth function f on the interval [−2, 2], where ω(dx dy) = 1 [−2,2] (x)1 [−2,2] (y) (4 − xy)dxdy Notice here that this statement has a different flavor as its classical counterpart. In the case of the standard Gaussian measure for example, inequality (0.1) is the expression of the spectral gap of the Ornstein-Uhlenbeck operator. In the free case, it was shown in [16] that (0.2) is equivalent to where (Lf )(x) = −(4 − x 2 )f (x) + xf (x) and N are respectively the Jacobi operator and the counting number operator for the orthonormal basis of Chebyshev polynomials T n (x/2) of L 2 (β), where β is the arcsine law 4−x 2 . At least at a first look, we are not comparing a second order operator with a projection, as in the classical case, but with an integro-dfferential operator. However, for this particular case, it is true that L = N 2 , and thus the above comparison is essentially the spectral gap for N .
Another natural interpretation of (0.2) is that the L 2 norm of the classical derivative f with respect to α is greater than the L 2 norm of the non-commutative derivative Df = f (x)−f (y) x−y with respect to a suitable measure ω. This non-commutative derivative is very natural in free probability theory and it is not the first time it appears in some form of Poincaré's inequality. In fact, Biane in [2] sets up a Poincaré inequality in several non-commuting variables which in the one dimensional case amounts to (0. 3) Var for any C 1 function f on [ −2, 2]. This is more in the classical spirit with the role of the derivative played by the non-commutative derivative f (x)−f (y) x−y . At any rate this inequality can be translated into the spectral gap for the counting number operator M associated to the scaled Chebyshev polynomials of the second kind for the semicircular law α. This fact makes (0.2) and (0.3) formally the same. However, this argument does not show a more structural tie between the two versions of Poincaré's. A more organic appearance of the counting number operator M in the life of (0.2) is revealed in Section 3. However, the only spectral properties of M which contributes is the mere non-negativity.
The main investigation of this work is actually to demonstrate that the operator point of view emphasized towards the description of the Poincaré inequalities (0.2) and (0.3) and their relationships may be pushed forward to similarly study Poincaré inequalities for large classes of equilibrium measures and not only the semicircular law. In particular, this analysis reveals the suitable Poincaré inequality in the free context, and allows for the connections with the other functional inequalities.
It is well known that (0.1) is valid for measures µ(dx) = e −V (x) dx, where V is strong convex. In fact, if V is strong convex, say V (x) ≥ ρ for some ρ > 0 and all x ∈ R, applying Poincaré's inequality (0.1) to the measure e −nTrV (X) dX on Hermitian n × n matrices and functions of the form Φ(X) = Tr(φ(X)) (for details see [16]) leads to: for any C 1 function on the support [−2c + b, 2c + b] of µ V . Here µ V is the equilibrium measure (i.e. the minimizer) of (0.5) E V (µ) = V dµ − log |x − y|µ(dx)µ(dy) over the sef of all probability measures on R. It is well known (see for example [19]) that the support of µ V is one interval in the case V is convex. The measure ω b,c on the left hand side of (0.4) is just a linear rescaling of the measure ω defined above, precisely (0.6) The point is now, (0.4) is a well-defined notion on its own for any given probability measure µ on the interval [−2c + b, 2c + b]. It defines the canonical free Poincaré inequality which will be investigated in this work. As it is in the case of (0.2) for the semicircular law, this inequality gravitates around the counting number operator N . The investigation is driven by a lemma by Haagerup which was extensively used in [10] to deal with the minimization of the logarithmic energy with external fields) providing an analytic description of the number operator N as which connects with free derivatives. In particular, this description produces concise and efficient interpretations of the equilibrium measure µ V and logarithmic energy E V associated to an external field V of independent interest. With these tools, the free Poincaré inequality for a measure µ may then be described at the operator level as the comparison of N with the operator with Dirichlet form (f ) 2 dµ.
In [16], another version of Poincaré's inequality inspired by Biane's version of (0.3) was presented, which states that for µ with compact support, there is a constant ρ > 0 such that (0. 7) ρVar as long as f is C 1 (R). Besides the example of the semicircular law, we do not know however if there is any (interesting) connection between the two Poincaré inequalities (0.4) and (0.7). As we will show, the Poincaré inequality (0.4) will be justified by its connection with the transportation and Log-Sobolev inequalities (which does not seem of the same nature for (0.7)). Indeed, once the proper free Poincaré inequality (0.4) has been identified, the next purpose is to investigate its relationships with the traditional free functional inequalities such as transportation and Log-Sobolev inequalities. The free transportation inequality associated to a potential V claims that there is a ρ > 0 with the property that (0.8) ρW 2 2 (ν, µ V ) ≤ E V (ν) − E V (µ V ) for any other probability measure ν on the real line. Free Log-Sobolev states that there is a ρ > 0 so that for any other (sufficiently nice) probability measure ν, where Hν = p.v. 2 x−y ν(dy) is the Hilbert transform of the measure ν. In this paper we show that under some mild assumptions, the transportation and Log-Sobolev inequalities imply the free Poincaré inequality (0.4). It should be pointed out that these implications are easy or standard in the classical case. That the Poincaré inequality follows from the Log-Sobolev is obtained by a simple Taylor expansion on (classical) entropy (see e.g. [1,20]). The implication from the transportation inequality is a bit more involved, the simplest argument going through Hamilton-Jacobi equations ( [4,20]). Actually, what the classical case puts forward is the necessity of suitable perturbation properties of both the logarithmic energy and equilibrium measure in the free context. This will be achieved in the second part of this paper. At the heart of the argument is a perturbation argument for the logarithmic energy E V , which is given by the counting number operator N , the same one which plays the key role in understanding the free Poincaré inequality. Again, this perturbation property might be of independent interest.
Here is how the paper is organized. In Section 1 we introduce the preliminary material, namely the logarithmic potentials, Chebyshev polynomials and we briefly discuss Haagerup's Lemma. We also introduce and study several related operators, the most important one being the counting number operator N and its analytic description.
Section 2 is the one introducing the Poincaré inequality (0.4) and several associated properties, while Section 3 investigates various equivalent characterizations of this. The main ones are equivalent via some sort of duality, which is somewhat reminiscent of the duality associated to the Monge-Kantorovich distance in the theory of mass transportation.
In Section 4 we give the perturbation results which is the backbone for the connection between the other functional inequalities and Poincaré. This last connection is discussed in Section 5 together with a discussion about why the perturbation used in the classical case to go from the Log-Sobolev and transportation is not enough.

PRELIMINARIES
In this section we introduce some basic notions we are going to use in this paper. A potential on a closed subset S of the real line is simply a function V : S → R. For our investigation of the logarithmic potentials with external fields, we will assume that V is of class C 3 on the interior of S and that if S is unbounded, We will call such a potential admissible.
For a probability measure µ the logarithmic energy with external field V is given by It is known that given a closed subset S and an admissible potential V (see [19] or [6]) there is a unique minimizer µ V in the set of probability measures on S. In addition this measure also has compact support. We will denote for simplicity E V = E V (µ V ). The support of the measure µ is denoted by suppµ.
The equilibrium measure µ V of (1.
For the definition of the notion of quasi-everywhere, we refer the reader to [19]. What we will need from this is in particular that the equality on suppµ is almost surely with respect to any probability measure of finite logarithmic energy.
If (X, X ), (Y, Y) are two measurable spaces, µ is a measure on X and φ : X → Y , is a measurable map, we set φ # µ to stand for the push forward measure It is easy to verify that changing the variable of integration to x → cx + b and y → cy which in turn results with Alternatively, they are given by the recursion relation with the generating function is the sequence of orthogonal polynomials for the arcsine 1−x 2 . The Chebyshev polynomials of second kind U n are defined by (1.6) U n (cos θ) = sin(n + 1)θ sin θ .
These satisfy the recurrence and the generating function is These are the orthogonal polynomials for the semicircular distribution The main connection between the Chebyshev polynomials of the first and second kind is given by (1.8) T n (x) = nU n−1 (x).
In the sequel we will use the following notation φ n (x) = T n x 2 and ψ n (x) = U n x 2 for n ≥ 0.
We mention that these are the orthogonal polynomials for the arcsine and semicircular on [−2, 2]. It is easy to check the following relations between φ n and ψ n : where here and throughout this paper, sign(x) = 1 for x > 0, sign(x) = −1 for x < 0 and sign(x) = 0 for x = 0.
The following Lemma which will play an important role in the subsequent analysis appears in some seminar notes of Haagerup [12]. Lemma 1 (Haagerup).
(1) For any real x, y ∈ [−2, 2], x = y, we have where the series here are convergent on x = y.
(2) For x > 2 and y ∈ [−2, 2], a similar expansion takes place, where the series is absolutely convergent. (3) The logarithmic potential of a probability measure µ on [−2, 2] is given by where this series makes sense pointwise. Therefore, the logarithmic energy of the measure µ is given by In particular Proof. A full scale proof is given in [10], here we only outline the main calculation leading to (1.10). Write x = 2 cos u and y = 2 cos v, and observe Hence Notice that in the middle of this we used the fact that for a complex number z, with |z| = 1, z = 1, the usual logarithmic formula which computes the logarithm is still valid: It is this simple lemma which gives the theme of dealing with logarithmic energies of measures by reducing them via rescaling to measures on the interval [−2, 2]. The next statement is a simple consequence.
If µ is a signed measure on [−2, 2] with finite total variation and finite logarithmic energy, then if and only if µ(dx) = β(dx). Here, "almost everywhere" is understood with respect to the Lebesgue measure. Additionally, the constant c must be 0.
We define the following probability measures related to the interval [−2c + b, 2c + b] which are used throughout this note.
. Throughout this paper we use ·, · γ to denote the scalar product in L 2 (γ) and reserve ·, · for the inner product in L 2 (β).
Using Lemma 1 we prove the first result of this note which appears partially in [10]. It will naturally lead to the operator formulation of the Poincaré inequality next. Theorem 1. Assume that V is a C 3 function on [−2, 2] and A ∈ R a constant. Then, there is a unique signed measure µ on [−2, 2] of finite total variation which solves where almost everywhere is with respect to the Lebesgue measure on [−2, 2]. The solution µ is given by In addition, the constant C must be given by C = − 2 −2 V (x) β(dx). Moreover, for any C 1 function φ on [−2, 2] we have that Proof. In the first place, the uniqueness is clear. To prove the rest we first write the function V in terms of Chebysev polynomials of the first kind Assuming (µ, V ) solve (1.16) and invoking Haagerup's representation, results now with Thus, equating the coefficients, we must have C = − V (y)β(dy) and To prove equality (1.17), our task is therefore to show that Notice that both sides of this equation are linear functions of V and thus by a simple approximation argument it suffices to check it for the case of V (x) = φ n (x) for some n ≥ 1, which boils down to There are several ways of doing this. The straightforward way is to look at the generating functions of both sides and use (1.5). We pause now and give a more general statement which will be used later on.
As usual, for simplicity we denote U = U 0,1 . Then Proof. The idea is to use the generating functions (1.5) and (1.7) and compute the operator U of these generating functions. To carry this out, let which are the generating functions of φ n , respectively ψ n . Then it is easy to compute which immediately implies the first half of (1.21). On the other hand r .
Coming back to the proof of Theorem 1. Armed with (1.21) and (1.8) and the simple fact that 2 −2 yφ n (y)β(dy) = n (n = even) 0 (n = odd) and it is now an easy task to verify (1.20), and in turn (1.19). To prove equality (1.18), we need to check that To this end, notice that for −1 < r < 1, (1.26) Now to compute the kernel inside the integration, notice that (here we inspire from [5]) with x = 2 cos u and y = 2 cos v, Taking the derivative with respect to r, gives Using Lebesgue's dominated convergence combined with (1.26), after letting r ↑ 1, the rest follows from Theorem 1 motivates the introduction of the following operators. (1.27) Using the above theorem it is clear that N φ is the unique solution ψ which satisfies where almost everywhere is with respect to the Lebesgue measure on [−2, 2]. We collect the main properties of the operators E and N in the following. (1.28) (2) One has Eφ 0 = 0, while for n ≥ 1, Eφ n = 1 n φ n and N φ n = nφ n for any n ≥ 0. In other words, N is the counting number operator for the Chebyshev basis in L 2 (β).
In particular, N φ, ψ = φ, N ψ . (4) If we take Lφ = N 2 φ for C 2 functions, then The operator L is actually the Jacobi operator with invariant measure the arcsine law β. Moreover, L has a unique selfadjoint extension, still denoted by L and defined on (6) If the minimizer of E V on [−2, 2] has full support, then Proof.
(1) We need to settle the fact that if φ is C 2 on [−2, 2], then Eφ is again C 2 on [−2, 2]. This is needed to give consistency to the second line of (1.28). To this end, we try to remove the singularity in E, by invoking (1.13) and (1.11) for the measure µ(dy) = yβ(dy) to justify the following It is obvious from this writing and Lebesgue's dominated convergence that Eφ is actually a continuous function on [−2, 2]. Taking the derivative with respect to x it is straightforward to deduce (again using (1.13) and (1.11) and dominated convergence) that Taking again the derivative with respect to x reveals that The rest now follows from Definition 1 and Theorem 1.
In turn, it is sufficient to do this for φ = φ n , ψ = φ m . Thus we need only show that using (1.8) which is just the orthogonality of the polynomials U n x 2 with respect to α. The formula for L is just an integration by parts.
The selfadjoint extension can be easily demonstrated by the fact that L has the eingenvalues {n 2 } n≥0 with eigenfunctions φ n . Indeed, it is easy to see that there is an isometry A : L 2 (β) → 2 (N) = {(a n ) n≥0 : n≥0 |a n | 2 < ∞}, which sends φ = n≥0 a n φ n into (a n ) n≥0 . This isometry sends the operator L defined on the linear span of φ n into the multiplication operator R(a n ) n≥0 = (n 2 a n ) n≥0 on the space of sequences with finitely many nonzero entries. Since the operator R has a unique selfadjoint extension, the same is true for L. The domain of R is pushed back by the inverse of A into H. (5) Just a rewriting of (1.17).
We collect here some integration by parts properties of the operator N which will be used later on.

Theorem 2.
If N is the operator defined in (1.27), then for any two C 2 functions φ, ψ : Here we use the notation Π(φ) = φ dβ and the convention that x k φ is a shortcut for the function f (x) = x k φ(x). In addition, (1.33) The relation between N and U is that, for any This easily follows from To quickly see this, take the generating functions (here 0 < r, w < 1) g r and h w already introduced in the proof of Lemma 2 in (1.22) and observe that combined with the derivative of (1.13) and a little algebra gives ∞ n,m=0 which yields (1.35).
To get the rest of the proof, notice that if we set, then a simple scaling argument together with (1.29) and (1.32) imply To deal with (1.34), it suffices to do this for f = φ n and in fact in order to check the identity for each n, we take the generating functions instead of the left and right hand side, thus we need only check the following Now, since N is the counting number generator for φ n , the left hand side is actually equal to ∂ r g r (x), while the right hand side, from (1.23), gives Ug r = rh r (x)/2, in which case both sides give the same answer, namely − 4r 2 −rx(1+r 2 ) 2(1+rx+r 2 ) 2 . This completes the proof of Theorem 2.

POINCARÉ INEQUALITY, GENERAL PROPERTIES
This section introduces the natural candidate for the free Poincaré inequality which is investigated throughout this note.
It should be observed that the left hand side in (2.1) only depends on the measure µ through its support. Actually, the first assertion of Proposition 2 below shows that µ has support The next statement collects some of the properties of this free Poincaré inequality.
The following are true.
In fact, this inequality is equivalent to P (1/2) for the semicircular α b,c with equality in (2.2) or (2.1) only for linear functions f .
is actually a classical Poincaré inequality (spectral gap) for the operator N on L 2 (β) and it is equivalent (by item 3) to the free Poincaré for the semicircular. Proof.
(1) It is pretty obvious that if J is an interval with the property that µ(J) = 0, then choosing a function f such that f is constant outside the interval J and is equal to x on some smaller subinterval K ⊂ J leads to a contradiction.
One can not conclude that there is a density of µ with respect to the Lebesgue measure or for that matter with respect to the semicircular. Indeed, for instance if we take µ = 1 2 α b,c + 1 2 γ, with γ a singular measure with respect to α b,c , then µ still satisfies a free Poincare and it is not absolutely continuous with respect to α b,c .
Thus assume that µ = w α b,c with w a continuous function. We assume that b = 0, c = 1. In order to show that w(a) > 0, for any a ∈ (−2, 2), we assume on the contrary that w(a) = 0 for some a ∈ (−2, 2). Since w(x) ≥ 0 it follows that a is a minimum point and thus, from the smoothness of w, Now we choose an approximation of the identity constructed as follows. First consider Apply then the free Poincaré inequality to the function f (x) = φ((x − a)/δ) to obtain that Now, changing the variable x = a + δx and y = a + δy , for small enough δ, results with where C a > 0 is a constant depending on a and ρ. Hence we get a contradiction as we let δ → 0. Therefore on (−2, 2), the density w must be positive. Now we deal with the behavior at the edge. Assume w(−2) = 0. The vanishing of w near −2 is no longer of order 2, but of order 1. Thus w(x) = (x + 2)w (−2) + o((x + 2) 2 ). Take again f (x) = φ((x + 2)/δ) and apply the free Poincaré, to obtain Make the change of variables x = −2 + δx , y = −2 + δy and deduce that for a constant C > 0, where we used that 4 − (−2 + δx )(−2 + δy ) uniformly for x , y ∈ [0, 1] and small δ. Consequently, letting δ → 0, we arrive to a contradiction.
Then the Poincaré implies that for any r ∈ R, In particular we can rewrite this in terms of the operators, N and L and the notations from Proposition 1 On the other hand, since N φ 1 = φ 1 combined with (1.30) give which shows that µ = α. The free Poincaré's is actually equivalent to the statement N ≤ N 2 . As N is a non-negative operator, this is in fact equivalent to (2.2). (4) Follows from P (1/2) for α.

Remark 2.
Poincaré's inequality and the C 2 condition on the density w imply that w must be positive. Also, if w is positive everywhere and continuous then P (ρ) holds for some ρ. It is interesting to see what happens if the C 2 condition on w is dropped. Is it still true that there is a Poincaré inequality satisfied for some ρ > 0? And if so, under what are the regularity conditions on w?

Remark 3.
A natural question in this context is about the extension of the Poincaré to the case where the measure µ has more then one interval support. These arise naturally as equilibrium measures µ V for potentials V with several wells. Indeed, it was shown in [18] that if V is analytic near the support of µ V , then the support of µ V must be a finite union of intervals. If a probability measure µ is supported on a finite number of intervals, say I 1 ∪ I 2 · · · ∪ I k , and satisfies for all smooth functions on R, then it can be shown, that each restriction of µ m to each connected component I m satisfies an inequality of the form with γ m supported on I m × I m .

EQUIVALENT FORMS OF POINCARÉ'S INEQUALITY
In this section we discuss the various equivalent forms of the free Poincaré inequality (2.1). Before we do this, let us introduce some operators.
For a given measure µ = w dα, with w ∈ C 1 ([−2, 2]) let L w be the operator acting on L 2 (β) with the Dirichlet form given by 2 (f ) 2 µ. Then an integration by parts gives Notice that for the case w = 1, the operator L w becomes L given in part (4) of Proposition 1.
Here is a statement which will be used in the sequel. Proof. It is clear that L w sends the constant functions to 0 and thus we restrict our attention to the restriction of L w on the orthogonal to constants in L 2 (β) 0 , which the set of functions in L 2 (β) of mean 0.
There is another way of representing this operator as for any C 2 function f on [−2, 2]. It is not hard to check that the operator A w can be extended to a bounded operator on L 2 0 (β) due to the fact that w is C 1 . In addition, it maps H 0 = H ∩ L 2 0 (β) into itself and has the inverse on L 2 0 (β) given by A 1/w . In particular, we can use this to extend the operator L w to H 0 . The claim is now that this operator is actually selfadjoint. Indeed, if ψ ∈ L 2 0 (β), which in the domain of L * w , then by definition, φ → LA w φ, ψ extends to a bounded functional from L 2 0 (β) into R. Thus, there is a constant C > 0 such that LA w φ, ψ ≤ C φ , say for any C 2 function φ ∈ C 2 ([−2, 2]) ∩ L 2 0 (β) and then replacing φ by A 1/w φ and the fact that this is bounded we obtain that Lφ, ψ ≤ C A 1/w φ for any C 1 function φ on [−2, 2] in L 2 0 (β). Hence ψ is in the domain of L * , which is H 0 by the fourth item of Proposition 1. In particular this means that the domain of L * w is H 0 . On the other hand, since L w on H 0 is the closure of the same operator restricted to C 2 ([−2, 2]) ∩ H 0 , it follows that L w and L * w have the same domain of definition, namely H 0 and thus L w on H 0 is selfadjoint.
Recall the operator U, which is defined in Lemma 1 and for which Uφ n = 1 2 ψ n−1 . It is natural to look at this operator between L 2 (β) and L 2 (α). In this form, Var β (f ).
Now we define the inverse operator of U by (3.1) Vψ n = 2φ n+1 for n ≥ 0. where Π is as above the projection on constant functions in L 2 (β). On smooth functions φ, the operator V has an explicit form as

It is clear in this case that
It is easy to see that one has to check this on the generating function of ψ n , which is h r (x) = 1 1−rx+r 2 , 0 < r < 1. For such a particular function, (cf. (1.24)) r which gives the formula. The point of the formula is that for a C 2 function f on [−2, 2], Vf is at least C 1 . Now take M = UN V − I, where I is the identity operator. It is very easy to check that M is the counting number operator for the {ψ n } n≥0 basis of L 2 (α) for the semicircle law. Indeed, on the basis ψ n , both sides give nψ n . With this definition, it is easy to check that We also have which stems from the fact that ψ n (x) − ψ n (y) x − y = n−1 k=0 ψ k (x)ψ n−k−1 (y) (a consequence of the generating function for ψ n 's) used in conjunction with the orthogonality of {ψ n } n≥0 with respect to the measure α. The next theorem describes equivalent description of the free Poincaré inequality P (ρ) which follow from the preceding operator-theoretic tools. Recall that U b,c appearing below is the one defined in Lemma 2.
Proof. We prove that (1) implies (2) implies (3) implies (1) and that (2) is equivalent to (4). In addition, even though it is not needed, we will also prove that (2) implies (1) with the duality argument which shows (1) implies (2). This last implication makes more transparent the duality behind P (ρ) and P 2 (ρ). By scaling it can be assumed that b = 0, c = 1.
(1) =⇒ (2) From Proposition 2 we learn that w > 0 on [−2, 2]. Write P (ρ) in the equivalent form 2ρN ≤ L w as (unbounded) selfadjoint operators on L 2 (β). Since w > 0, then we can find two positive constants c 1 , c 2 > 0, such that c 1 L ≤ L w ≤ c 2 L. Notice that the kernel of both N and L w is the space of constant functions and therefore the restrictions of N , L w to L 2 0 (β), the orthogonal to constant functions, are invertible. We will assume for the rest of this implication that the operators N , L w are taken on L 2 0 (β). As the inverse of N is E and this is bounded, it follows that L −1 w is also bounded.
After these preliminaries, we use some sort of duality. More precisely, the main idea is that for each fixed Indeed, the first equality is a consequence of This last equality may not be attained for g ∈ L 2 0 (β) ∩ C 2 ([−2, 2]), but L −1 w N f can be approximated by such functions. Poincaré's inequality P (ρ) implies in this case that 2ρ N L −1 w N f, f ≤ N f, f . A simpler argument of this inequality was suggested by the reviewer of this paper and is based on the fact that from 2ρN ≤ L w on L 2 0 (β), we get first 2ρL −1 w ≤ N −1 and then 2ρN L −1 w N ≤ N N −1 N = N . To get to (3.4), it suffices to observe that for It remains now to show that N L −1 w N = V 1 w U on C 2 ([−2, 2]) ∩ L 2 0 (β). Passing to the inverses, this follows from the following result which is remarkable enough to be called a Lemma.

Lemma 3. For any
Proof. It suffices to do this for w = φ n . Therefore we need to check that Vφ n Uφ m = EL φn Eφ m for all m ≥ 1 and n ≥ 0. It is clear that

Now we can continue with
Vφ n Uφ m = EL φn Eφ m , or From (1.9), this is equivalent to which becomes obvious based on (3.1) and part 2 of Proposition (1). Just as a clarification, sign(x) is −1 for x < 0, 0 for x = 0 and 1 for x > 0.
(2) =⇒ (1) We present two proofs for this implication. The first one is a duality argument like the one used in the previous implication and the second one is based on (1.34) and integration by parts.
The first proof is based on the following duality similar to (3.7). For any f ∈ L 2 The first two equalities can be justified as in the previous proof, the last line being just the consequence of the above Lemma. As the second form in Theorem 3 is written as 2ρ V 1 w Ug, g ≤ N g, g , P (ρ) is immediate. The second proof is based on the idea that from (1.29) and (1.34), a simple integration by parts yields Therefore, (3.4) implies P (ρ) from the following sequence where the second inequality is justified by 2ab ≤ a 2 + b 2 with a = f / √ ρ and b = 2 √ ρ Uf / √ w. Notice here that we need to know that w does not vanish on (−2, 2) and we have the suitable integrability of 1/w at ±2 to ensure the integrals are well defined.
(2) =⇒ (4) Take now g = Uf , with f = Vg. Therefore, if we replace f in (3.4) by Vg, then where we used that Π is the projection onto the constant functions which is also the kernel of N , thus N Π = ΠN = 0.

Remark 4.
It is interesting that the equivalence of the first and second part of Theorem 3 can be seen as some sort of duality. As we will see in Theorem 7, the second form of Poincaré P 2 (ρ) is naturally derived from the transportation inequality and this is the reason why we discuss this equivalent form. At first we arrived from the transportation inequality to (3.11) f which is a rewriting of P 2 (ρ) from which a straightforward application of the Cauchy's inequality implies P (ρ). This makes one believe that the second form is actually stronger than P (ρ) but the above theorem says that they are equivalent. The third form is (3.11) plus Cauchy-Schwarz. This actually appears naturally from the HWI inequality discussed in Section 5.
The fourth form is closer in spirit to the classical form of Poincaré as a spectral gap, though a little different. For example in the case of the semicircular on [−2, 2], w = 1 and this inequality becomes, which is nothing but non-negativity of M on L 2 (α). This is to be put in contrast with Biane's version (0.3) of Poincaré's which is actually a measure of the spectral gap of M.
Remark 5 (The optimality of the constant ρ in P (ρ)). P (ρ) becomes 2ρN ≤ L w . This inequality gives in particular that if 0 = λ 0 < λ 1 ≤ λ 2 . . . are the eigenvalues of L w ordered non-decreasingly, then 2ρn ≤ λ n . The optimal ρ is the infimum of λ n /n over n ≥ 1. On the other hand, if inf w > 0, then λ n grow at least quadratically and as such, there is a finite n, for which λ n = 2ρn, λ m > 2ρm for m = 1, 2, . . . , n − 1 and λ m ≥ 2ρm for all m ≥ n + 1. In some sense, the optimality constant is fitting the best linear growth for the spectrum of L w . From the point of view of P 2 (ρ), we are looking at the best constant of something which resembles a classical Poincaré inequality, as the left hand side of (3.4) is some sort of variance. However, unless w is constant, the isometric property of U between L 2 (β) and L 2 (α) is disturbed. P 4 (ρ) is comparing M + I with respect to the identity on a different L 2 .

PERTURBATION OF LOGARITHMIC POTENTIALS
In this section we provide some results related to logarithmic potentials which are the building blocks for the connection of transportation and Poincaré. The goal is to study the result of a perturbation of V on E V . First recall the following result from [10] which gives an expression for E V , rewritten here within the notations introduced so far.

Theorem 4.
Assume V is a C 3 potential. Then the equilibrium measure on R associated to V has support the interval and only if (c, b) is the unique absolute maximizer of and The equilibrium measure in this case is If this is the case, (b, c) is a solution of The first part of the theorem is well known and can be seen for example in [19, Theorems 1.10 and 1.11, Chapter IV], while (4.4) is a combination of (1 .3) and (1.31).
For the rest of this paper we will use the perturbation result for which the following assumptions on the potential V suffice.

Assumption 1.
( Remark 6. The first two conditions plus (4.2) are part of the existence of a single interval for the support of the equilibrium measure µ V as we presented here, while the third assumption is an improved version of (4.2). Moreover, in order to obtain a Poincaré inequality, we must have this third condition satisfied as it was shown in Proposition 2. Thus what is written here is just the minimal conditions in order to assure the well posedness of the Poincaré inequality.
Under the conditions of Assumption 1, if we perturb the potential V by V t = V + tf + t 2 g + o(t 2 ) (uniformly on R), where f, g are C 3 function with all bounded derivative, then V t itself, for small t, satisfies the conditions in Assumption 1, and thus its equilibrium measure has a one interval support The fact that the support of the equilibrium measure for the perturbed potential is still one interval follows roughly from the fact that the associated H t in Theorem 4 does not change much with t and thus it still has a unique maximum which is close to the one at time t = 0. Also the positivity condition (4.2) with V replaced by V t is satisfied for small t.
The fact that the endpoints of the support of the equilibrium measure, or otherwise stated, c t and b t are C 2 follows from the implicit function theorem applied to the system (4.3) with V replaced by V t . For a detailed argument on this perturbation, the reader is referred to the perturbation section in [10].
The main result in this section is the following description of how E V behaves under perturbations.
Theorem 5. Let V : R → R be a potential on R such that the equilibrium measure µ V has support [−2c + b, 2c + b]. In addition, assume V t , t ∈ (− , ) is a perturbation of V such that where f, g : R → R are C 3 on R with bounded derivatives, and o(t 2 ) is uniform on R. If E t = E Vt , then Proof. Assume for simplicity (without loss of generality) that c = 1, b = 0. The critical point system (4.3), reads as To simplify the writing in this proof, for any smooth functions h, k : [−2, 2] → R, set Π(h) = h dβ as in Theorem 2 and Recast the critical point system in this notation as (4.7) Π(xV ) = 2, Π(V ) = 0. Now, we notice that for small t, the equilibrium measure of V t has support [−2c t + b t , 2c t + b t ], where c t and b t depend C 2 on t. Thus we can write Continuing, from (4.4), Next, a simple Taylor expansion gives .
Expanding E t to second order yields and after regrouping the terms according to the power of t it becomes Equation (1.18) gives and thus the first line of (4.8) is precisely (4.5) modulo o(t 2 ). Our remaining task is to prove that the rest of (4.8) is zero (up to o(t 2 )).
while using the first equality in (1.32) for φ = V and ψ = V combined with (4.7), provides These show that the forth and fifth lines of (4.8) are 0. Finally, using (1.32) for φ = V and ψ = f together with (4.7), yields that which concludes that the last line in (4.8) is o(t 2 ). This completes the proof.

Remark 7.
Notice that Theorem 5 has a simpler proof in the case the equilibrium measure of V t has a support which is independent of t. Assuming b = 0, c = 1, conform to (4.3), this amounts to The simpler proof alluded to in this case follows directly from the formula (1.31) with V replaced by V t plus expansion in t.
The content of this theorem says that in fact the same formula holds true even without the constraints from (4.10) but one has to go through a careful examinations of the dependence on the coefficients c 1 , c 2 , b 1 and b 2 in (4.8) and notice that their contributions disappear due to some remarkable and non-trivial cancellations.
Next we study how the equilibrium measure changes under perturbation of the potential. Theorem 6. Let V satisfy the Assumptions 1 and let f ∈ C 3 b (R). Then, in the sense of distributions, where "almost every" is with respect to the Lebesgue measure. If b = 0, c = 1, this can be written in simpler terms as In addition, for x ∈ [−2c + b, 2c + b], Proof. As in the proof of Theorem 5, for small t, the equilibrium measure of V + tf has a one interval support In addition, assuming for simplicity that c 0 = 1 and b 0 = 0, then we know that for For a smooth function, φ, using equation (1.18), one gets Using Taylor's expansion in t, after a little calculation and with the notation from (4.6) we continue the above identity with After using (1.18), (1.36) and (4.9) the latter can be simplified further into Furthermore, from (1.17), we have Now, (4.12) follows from (1.34). The proof of the theorem is complete.

POINCARÉ'S INEQUALITY AND OTHER FUNCTIONAL INEQUALITIES
This section is devoted to the relationship of the free Poincaré inequality with the transportation and Log-Sobolev inequalities on the basis of the perturbation properties developed in the preceding section. As mentioned in the Introduction, the implications from the transportation and Log-Sobolev inequalities to the Poincaré inequality in the classical case are standard (cf. [1,17,4,20]). Their analogues in the free case are surprisingly more involved.
First recall the main functional inequalities to be compared with the free Poincaré inequality (see [16]).

Definition 3.
(1) The probability measure µ V , or more appropriately, V satisfies a transportation inequality with parameter ρ > 0, if for any other measure µ, where the infimum is taken over all probability measures π, with marginals µ and ν (i.e. π(dx, R) = ν(dx) and π(R, dy) = µ(dy)). In short we refer to the inequality (5.1) as T (ρ) which was introduced by Biane and Voiculescu [3] for the semicircular and in this form by [13]. (2) Similarly we say that µ V satisfies a Log-Sobolev, in short LSI(ρ), ρ > 0, if for any other (sufficiently nice) probability measure ν, taken in the principal value sense. This inequality was introduced in this form by Biane [2]. (3) At last we say that µ V satisfies an HW I(ρ), ρ ∈ R, if for all sufficiently nice probability measure ν, . We should mention that Log-Sobolev implies transportation [15] and that HWI implies Log-Sobolev for ρ > 0. In particular, although the theorem below provides independent proofs, one main implication is the one from the transportation inequality to the Poincaré inequality.
A short description of the transportation map is in place here. For any probability measures, µ, ν on the real line, with µ absolutely continuous with respect to the Lebesgue, then W 2 2 (µ, ν) = (θ(x) − x) 2 µ(dx) with θ being the unique non-decreasing transportation map of µ into ν. In addition, if µ and ν have densities g µ and g ν , then (5.5) θ (x)g ν (θ(x)) = g µ (x) for all x ∈ supp(µ).
Before we proceed to the proof of the main theorem, we want to give a result about the behavior of the transport map of the equilibrium measure of a perturbed potential.

Proposition 4.
Assume that V is a potential satisfying the Assumption 1 and let V t = V + tf , where f is a C 3 function with all bounded derivatives and let µ V , µ t be the equilibrium measures of V , respectively V t . If θ t is the transport map from µ V into µ t , then there is a C 1 function ζ on the support of µ V such that uniformly in x on the support of µ V .
Proof. By rescaling, we may assume that the support of µ V is [−2, 2]. As we pointed out in the remark following Assumption 1, the support of the measure µ t is where c t and b t are of C 2 class in t.
From the above presentation of the transportation map, it is clear that θ t maps [−2, 2] into [−2c t +b t , 2c t +b t ] with θ t (−2) = −2c t + b t and θ t (2) = 2c t + b t . In order to remove the varying endpoints, we rescale θ t (x) = 2c t ψ t (x) + b t and with the help of (5.5), Assumption 1 and Theorem 4 we learn that where w(t, ·) is the density of the equilibrium measure of V (c t x + b t ) with respect to the semicircular law. The important fact to be spelled out here is that w : and we get the claimed expansion as soon as we prove that Ψ 0 can be extended to a continuous function on [−2, 2] and also that sup x∈(−2,2) |Ψ s (x) − Ψ 0 (x)| converges to 0 when s converges to 0.
If we take the behavior of the solution ψ t to (5.7) at points x ∈ (−2, 2), then standard results of perturbation of ordinary differential equations tell us that the perturbation with respect to t is of class C 2 . However, at the endpoints ±2 this becomes problematic and for this case, one needs a separate analysis. At least we know that ∂ t ψ t (x)| t=0 is well defined and uniformly continuous on compact sets of (−2, 2). In particular this justifies the writing of (5.6) uniformly on any compact interval in (−2, 2) for some continuous function ζ on (−2, 2).
Notice that F is a nice positive and C 2 function in all variables t, u, y. In particular, standard results in ordinary differential equations guarantee that (t, u) → ξ t (u) is a C 2 function in both (t, x) on [−t 0 , t 0 ] × (0, 1]. With this ξ t replacing ψ t it suffices to show that the writing (5.6) holds true uniformly for u in the interval [0, 1]. To do this, set η t (x) = ∂ t ξ t (x) and write as in (5.8) ξ t (u) = ξ 0 (u) + tη 0 (u) + t 0 (η s (u) − η 0 (u))ds.
We know that ξ t is a continuous function on [0, 1] for any small t with ξ t (0) = 0. From (5.9), whose first consequence is that ξ t (u)/u has a limit as u converges to 0, or otherwise stated that the derivative ξ t is well defined at 0 (for any t ∈ [−t 0 , t 0 ]) and in particular can be computed as ξ t (0) = (F (t, 0, 0)) 1/3 .
What this gives in terms of ξ t is that from (5.9) and the F (0, 0, 0) = 1, there is a positive constant C > 0 such that for any t ∈ [−t 0 , t 0 ] and u ∈ [0, 1], C −1 ≤ ξ t (u) < C and also C −1 ≤ ξt(u) u ≤ C. Now we look at our main interest, the derivative η t (u) = ∂ t ξ t (u). Observe that from (5.9), it is easy to deduce that and on the account of (5.9), we can rewrite this in the form with a t (u) = ∂ y F (t, u, ξ t (u)) and b t (u) = ∂ t F (t, u, ξ t (u)). This implies that there is a constant L t with the property that for t ∈ [−t 0 , t 0 ] and u ∈ (0, 1), Since for any fixed u > 0, the left hand side is continuous in t, it follows that L t is also a continuous function of t.
s (u)e As(u) ds, from which, multiplication by u 2 and passing to the limit u → 0 with the help of (5.10) yields t 0 L s (F (s, 0, 0)) 2/3 ds = 0 for all small t, which returns that L t = 0 for all t small enough. Hence, it is pretty clear now that can be extended to a continuous function at 0 for each t ∈ [−t 0 , t 0 ]. In particular η 0 is continuous on the interval [0, 1]. In fact a stronger statement holds true here, namely, that which follows from the fact that there is a constant C > 0 such that which in turn yields that for some K > 0, What is left to prove here is that sup u∈(0,1] |η s (u) − η 0 (u)| converges to 0 as s converges to 0. If this were not the case, then there would be > 0, s n −−−→ n→∞ 0 and u n ∈ [0, 1] such that |η sn (u n ) − η 0 (u n )| ≥ . Without loss of generality we may assume that u n is convergent to some v ∈ [0, 1] and then if v = 0 contradicts (5.11), while v = 0 contradicts the continuity of η at (0, v).
Proof. If ρ = 0, then (5.12) and (5.13) are trivial so that we assume below that ρ > 0. Assume furthermore that b = 0, c = 1. We will give here two proofs of (5.12). One is inspired by the classical case and uses the Hamilton-Jacobi semigroup and the dual formulation of the Wasserstein distance, while the other is based directly on the perturbation of the potential. For the first proof, we employ the tools from the infimum convolution semigroup used in [4] for the classical case. More precisely, take an arbitrary smooth function f : [−2, 2] → R and extend it to a smooth compactly supported function on the whole R. Now use the dual formulation of the Wasserstein distance, which now make the transportation inequality equivalent to for any pair of functions with g( For a given f , the optimal choice of g is given by g = Qf , where for any measure ν. In particular, minimizing over all measures ν, one obtains that Next, we point out that if we set Therefore, invoking (4.5) and (4.11), we obtain which combined with (1.29) gives (5.13). Now, from HW I(ρ), (5.19) and (5.21), (5.14) follows at once.

Remark 8.
It is interesting to point out that the dual formulation of the transportation implies P (ρ), while working with the transportation itself (basically the Wasserstein distance) yields P 2 (ρ) which as we noticed after the proof of Theorem 3, is in some sense the dual form of P (ρ). This is reminiscent of the discussion of Otto-Villani [17] about the Poincaré inequality in the classical case. We should also mention that HW I(ρ) for a real ρ, gives some sort of "defective" version of Poincaré.

Remark 9.
We know that the transportation and the Log-Sobolev are satisfied in the case of potentials V which are convex. The natural question is to see other cases where these functional inequalities are satisfied. As it was pointed out in [2], there are examples of double well potentials V for which the Log-Sobolev does not hold. These are cases where the equilibrium measure is supported on two intervals. It is not clear (at least we do not have any example) if the functional inequalities hold for cases where the measures are supported on several intervals.

Remark 10.
Note that in [17], the linearization of classical HW I(ρ) with ρ > 0 implies a seemingly stronger inequality than Poincaré's with constant ρ > 0. Even though Otto and Villani do not point this out, this is in fact equivalent to Poincaré's with constant ρ > 0.

Remark 11.
We pointed out in [16,Theorem 2] that if the potential V is such that V (x) − ρ|x| p for some p > 1, then the following transportation inequality holds Unfortunately, it turns out that for 1 < p < 2, c p = 0 and thus this inequality does not say anything. On the other hand, for p > 2 it implies a Poincaré's inequality with ρ = 0. Indeed, due to the fact that W p p (µ V +tf , µ V ) = o(t p ), for p > 2 this order is higher than 2, thus nothing interesting is seen from this inequality as t goes to 0.
The reader might wonder why the classical perturbation argument does not work. This is what we discuss in the remaining of this section.
The standard perturbation used in the classical case to linearize the Log-Sobolev or the transportation inequalities in order to reach the Poincaré inequality is ν t = (1 + tF )µ V for small t and a function F with F dµ V = 0. We show here that while this gives the free Poincaré's for a large class of functions it is not the whole story.
For simplicity we will assume that b = 0, c = 1. Take a continuous function F on [−2, 2] such that F dµ V = 0. This in particular means that for small t, ν t = (1 + tF )dµ V is again a probability measure. Thus applying the transportation, we get which, after the use of the fact that V (x) = 2 log |x − y|µ V (dy) + C on the support of µ V , leads to (5.23) ρ (ζ F (x)) 2 µ V (dx) ≤ − log |x − y|F (x)F (y)µ V (dx)µ V (dy).
Here in between we used that θ t , the transport map of µ V into µ t , is given by using essentially the same proof as in Proposition 4. Now we proceed as in the second proof of T CI(ρ) =⇒ P (ρ) from Theorem 7 to deduce that for any C 2 function on [−2, 2], φ(θ t (x))µ V (dx) = φ(x)(1 + tF (x))µ V (dx) and so expansion in t produces, with G(x) = x −2 F (y)µ V (dy). Consequently, ζ(x)g V (x) = C + G(x), from which at −2 and the continuity of ζ, we produce C = 0, thus, where here g V = (4 − x 2 ) U(V ) is the density of µ V with respect to the Lebesgue measure.
In order to make this look like (3.4) (P 2 (ρ)), we should take now F such that F dµ V = ν f = N f 2 β or equivalently, Hence, for those F which can be represented in this form, the right hand side of (5.23) becomes where we used the first equation of Proposition 1. Furthermore, now appealing to (1.29) and (1.34), it results with which is (3.4). However, in order to make sure that F with the choice (5.24) is continuous, we need to guarantee that N f (±2) = 0, which otherwise stated (cf. Definition 1) is the same as (5.26) f (x) β(dx) = 0 and xf (x) β(dx) = 0.
This means that we get Poincaré's inequality however on a set of functions f satisfying two constraints. It is not clear to us how to extend (5.25) from functions obeying (5.26) to any C 1 function. Perhaps a more interesting remark here is that the obstructions from (5.26) guarantee that the potential V t = V + tf satisfies, V t (x) β(dx) = 0 and xV t (x) β(dx) = 2.
These two equations ensure that (cf. (4.3)) the endpoints of the equilibrium measure of V t are −2 and 2, in other words we are just in the situation discussed in Remark 7. It seems that in order to overcome this obstruction, a nontrivial argument is needed and this is to some extent the content of Theorem 5 which is also reflected in the different perturbation we used in Section 4. A similar argument applies to the implication of free Poincaré by the free Log-Sobolev.

ACKNOWLEDGMENTS
The second author would like to thank University of Toulouse for its warm and inspiring hospitality where part of this work was carried out.
We also would like to express true appreciation for the scholar, careful, pertinent and sharp remarks of the anonymous reviewer which transformed the present paper into a better one.