On Gaussian interpolation inequalities

This paper is devoted to Gaussian interpolation inequalities with endpoint cases corresponding to the Gaussian Poincar\'e and the logarithmic Sobolev inequalities, seen as limits in large dimensions of Gagliardo-Nirenberg-Sobolev inequalities on spheres. Entropy methods are investigated using not only heat flow techniques but also nonlinear diffusion equations as on spheres. A new stability result is established for the Gaussian measure, which is directly inspired by recent results for spheres.


Introduction and main results
Let us consider the Gagliardo-Nirenberg-Sobolev inequalities on the unit d -dimensional sphere By convention, we take 2 * = +∞ if d = 1 or 2. The purpose of this paper is to clarify the links of these interpolation inequalities with the family of Gaussian interpolation inequalities where the exponent p is taken in the range 1 ≤ p < 2. Inequality (2) is intermediate between the Poincaré inequality corresponding to p = 1 and the Gaussian logarithmic Sobolev inequality obtained as a limit case of (2) as p → 2 − .Here d γ(y) := (2 π) −n/2 e − 1 2 |y| 2 d y denotes the centred normalized Gaussian probability measure and the dimension n is any positive integer.
It is somewhat classical that, if we consider the Sobolev inequality on the sphere, i.e., Inequality (1) with p = 2 * and d ≥ 3, rescale it to the sphere of radius d and fix a function depending on n variables only on this sphere, since the curvature tends to 0 at the right order as one takes the limit as d → +∞, then the sphere tends to the flat space with Gaussian measure.For instance, we read (with adapted notation) from [9, p. 4818] that if we rescale this inequality so as to be on a sphere of radius d and take the limit d → ∞ or p → 2 we obtain in the Poincaré limit the Gross logarithmic inequality for the Gaussian measure since −(1/d ) ∆ on S d goes to −∆ + x • ∇ on the infinite-dimensional limit.The last part of the sentence refers to a result known as the Maxwell-Poincaré lemma : see [39] and [43,Remark 4,p. 254] for some historical comments.The statement of [9] has been made more precise later in [11,12] using a slightly different limit.However, to our knowledge, the infinite-dimensional limit has not been considered in the subcritical range p < 2 * .
Theorem 1.Let n be a positive integer, p ∈ [1, 2) and consider a function v ∈ H 1 (R n , d x) with compact support.For any d ≥ n, large enough, if u d (ω) = v ω 1 / d , ω 2 / d , . . ., ω n / d where ω = (ω 1 , ω 2 , . . ., The carré du champ method has frequently been applied to prove Gaussian interpolation inequalities ranging between the logarithmic Sobolev inequality and the Poincaré inequality like (2) using the linear flow associated with the Ornstein-Uhlenbeck operator; see [4], and [7] for an overview.Still in the case of the Gaussian measure, we adopt here a new point of view by using nonlinear diffusion equations in order to prove the same inequalities, but with different remainder terms.This is a very natural point of view when dealing with inequalities like (1) on the sphere, as shown for instance in [26] (see earlier references therein).In that case, linear flows are indeed limited to exponents p ≤ 2 # if d ≥ 2. To overcome this difficulty if either d = 2 and p > 2 # or d ≥ 3 and p ∈ (2 # , 2 * ], one has to consider fast diffusion flows.Before explaining the results for the Gaussian measure, let us summarize the main known results on the sphere.On S d , let us consider a positive solution u of where ∆ denotes the Laplace-Beltrami operator on S d .In the case m = 1, u p solves the heat equation and for this reason, we shall call it the linear case.Otherwise u p solves a nonlinear diffusion equation corresponding either to a fast diffusion flow with m < 1 or to a porous media equation with m > 1.In any case, we claim that Let us define The following result can be found in [22,26] with additional details in [18,24,27]. The limit, as t → +∞, of any solution of ( 3) is a constant.This means that the deficit, that is, the difference of the two sides in Inequality (1), converges to 0. Then, by Proposition 2, it follows that the deficit is non-negative, which directly proves (1).The same monotonicity property applies to the deficit of the logarithmic Sobolev inequality on the sphere  In view of the results of Theorem 1, it is natural to ask whether there is also a monotonicity property of the deficit associated to the Gaussian interpolation inequalities (2) when we rely on a nonlinear diffusion flow on R n .Let m ± (p) := lim d →+∞ m ± (d , p) where m ± is given by ( 4) and notice that m The diffusion operator associated to the Gaussian measure is the Ornstein-Uhlenbeck operator L = ∆ − x • ∇ and we consider now the nonlinear parabolic equation In the definition of L , ∆ denotes the standard Laplacian on R n .The following result is new for m = 1 while the case m = 1 follows from the method of the carré du champ developed by Bakry and Emery in [4].
The limiting case p = 2 corresponding to the Gaussian logarithmic Sobolev inequality is also covered but it is obtained as a standard application of the linear carré du champ method known from [4] because m ± (2) = 1.See Fig. 2. Said in simple words, the result of Theorem 3 is that the admissible range of exponents of the nonlinear flow, for which the deficit associated to ( 2) is monotone non-increasing, is obtained as the limit of the range of the corresponding exponents on the sphere, in the large dimensions limit.Moreover, p = 2 appears as a critical exponent for the Gaussian measure.
Let us now focus on stability results.The main result of [18] for the sphere is a constructive stability estimate for Inequality (1), limited to the subcritical range p ∈ (1, 2 * ), which measures the distance to optimal functions, and distinguishes the subspaces generated by constant functions, spherical harmonic functions associated to the first positive eigenvalue of the Laplace-Beltrami operator, and the orthogonal directions.Optimal exponents in the stability estimate measuring the distance to the set of optimal functions differ, depending on the directions.Here we have the exact counterpart in the Gaussian case.Let Π 1 denote the orthogonal projection of L 2 (R n , d γ) onto the (n + 1)-dimensional function space generated by 1 and x i with i = 1, 2, . . .,n. Theorem 4. For all n ≥ 1, and all p ∈ (1, 2), there is an explicit constant c n,p > 0 such that, for all v ∈ H 1 (d γ), it holds Exponents 2 and 4 which appear in the right-hand side of the inequality are sharp and the constant c n,p has an explicit although complicated expression given in the proof.If p = 1, the last term drops and the distance to optimal functions is measured only by , and a decomposition on Hermite polynomials shows that the estimate cannot be improved.If p = 2 * and n ≥ 3, the recent stability result on Sobolev's inequality on R n of [25], which quantifies the estimate of Bianchi and Egnell in [14] can be translated into a stability result for (1) on S d , that can be recast in the form of [18,Theorem 6].By a large dimension argument, a stability on the Gaussian logarithmic Sobolev inequality is also shown in [25], although the distance is measured only by an L 2 (R n , d γ) norm.Whether a stronger estimate can be obtained in the limiting case p = 2, eventually under some restriction, is therefore so far an open question.This paper is organized as follows.In Section 2, we give a new proof of Inequality (2) as a consequence of Inequality (1) by taking a large dimensions limit, applied to the inequality written for a function depending only on a fixed number n of real variables.To our knowledge, this is new except for the limit case p = 2 of the logarithmic Sobolev inequality.Section 3 is devoted to the proof of Theorem 3: we characterize the nonlinear diffusion flows of porous medium or fast diffusion type such that the deficit is monotone non-increasing and recover the picture known on the sphere in the large dimensions limit.Moreover, by the carré du champ method, we establish improved inequalities that provide us with first stability results.The stability result of Theorem 4 for the Gaussian measure is proved in Section 4 using a detailed Taylor expansion and the improved inequalities of Section 3.

From subcritical interpolation inequalities on the sphere to Gaussian interpolation
In this section we explain how Inequality (2) can be seen as the limit of Inequality (1) in the large dimensions limit, that is, as d → +∞, and prove Theorem 1. Comments on the limit case p = 2 can be found at the end of this section.
The unit sphere S d is parametrized in terms of the stereographic coordinates by where ω = (ω 1 , ω 2 , . . ., ω d +1 ) denote the coordinates in R d +1 ⊃ S d and x = (x 1 , x 2 , . . ., x d ) are Cartesian coordinates in R d .To a function u on S d , we associate a function w on R d using the stereographic projection such that It is a standard result that and that where 〈x〉 := 1 + |x| 2 and δ(p) := 2 d −p (d −2).Using the stereographic projection, Inequality (1) can be written on the Euclidean space R d as the weighted interpolation inequality See [24] for details.Equality is achieved by the Aubin-Talenti function w (x) := 〈x〉 2−d .Assume that d ≥ 4. Let us consider f = w/w and notice that the inequality rewritten in terms of f is .
Since we are interested in the limit as d → +∞, we shall consider the case 1 ≤ p < 2, which is admissible for any d ≥ 2.
With g (x) = f x/ d , we obtain after changing variables that Let us assume that n ≥ 1 is a given integer and take we also assume that the function g depends only on y.In other words, we write g = g d where g d (y, z) = v(y) for some function v defined on R n , that is, Here we use a subscript d in order to emphasize that g d has to be considered as a function on R d .Let us define Lemma 5. Let n be a positive integer, p ∈ [1, 2), consider a function v ∈ H 1 (R n , d x) with compact support and define g d according to (9).Then we have In other words, we prove that the infinite dimensional limit of ( 8), for functions depending only on a finite number n of real variables, is (2).The assumption of compact support can be removed if v and ∇v have sufficient decay properties at infinity.
we can integrate with respect to z and obtain d y be the centred normalized Gaussian probability measure.We recall that .
Applying these formulas with a = b = k = d ≥ 2, we find that and, as a consequence we obtain lim Similar computations show that and This completes the proof of Lemma 5.
Proof of Theorem 1. Applied to the function u d , Inequality ( 1) is transformed into Inequality (8) applied to Notice that the factor 2 δ(p)/2 , which arises from the stereographic projection and appears in (7), plays a role in the computation of the constant C d ,p and is taken into account in the limit as d → +∞.Since the right-hand side uniformly converges to v(y) for any smooth and compactly supported function v, the same conclusion holds for Theorem 1 as for Lemma 5.
It is a natural question to ask what happens in (1) to the marginals depending only on a finite number n of variables if p = 2 or in the case 2 < p ≤ 2 * = 2 d /(d − 2).We may notice that lim d →+∞ 2 d /(d − 2) = 2 and it is known, for instance from [9], that one recovers the Gaussian logarithmic Sobolev inequality as a limit case of Sobolev's inequality on S d corresponding to p = 2 * when d → +∞.This is also true if we consider a sequence (p d ) d ∈N with 1 < p d < 2 * , depending on d , if its limit is also 2, as shown next.By convention, when p d = 2, we consider the Gaussian logarithmic Sobolev inequality instead of (2).

Proposition 6. Let n be a positive integer and consider a function
Proof.The proof is an adaptation of the proof of Theorem 1 and, in the case p d = 2, relies on the standard observation that As this computation raises no special difficulty, details are omitted.

Entropy methods and nonlinear flows for Gaussian measures
In this section, we prove the result of Theorem 3 for the Gaussian measure d γ and extend it to the slightly more general framework of a uniformly strictly log-concave measure d µ, before drawing some consequences.Most of the results are similar to computations usually done on the sphere, but we are not aware of the use of nonlinear flows (m = 1) in the context of Gaussian measures.This approach is very natural in the perspective of spheres in the large-dimensional limit.

Gaussian interpolation inequalities: a proof by the carré du champ method
On R n , let us consider the probability measure and redefine the Ornstein-Uhlenbeck operator by This generalizes the case of the harmonic potential φ(y) = 1 2 |y| 2 considered in the introduction.We assume that φ satisfies the Bakry-Emery condition for some λ > 0. The harmonic potential corresponds to the equality case with λ = 1.Under Assumption (12), it is well known (see for instance [7, Section 7.6.2])that, with λ = λ and for any p ∈ [1, 2), and also, by taking the limit as p → 2 − , that The classical proof by the carré du champ, as in [3][4][5], relies on the Ornstein-Uhlenbeck flow Here we consider a more general strategy and compute as in the carré du champ method using the nonlinear diffusion There is no difficulty in proving global existence of a positive bounded solution.The case of a non-negative solution is also covered by a standard approximation of the initial datum.In the case of the harmonic potential, the key of the existence proof is to combine an L 1 -contraction property and the Maximum Principle as in the classical theory of nonlinear diffusions: see for instance [42,Chapter 9] for a general reference on this topic.In the case of a general potential φ, for existence and uniqueness we refer to [19, Section 3] and to [33] for a general overview of the topic.Concerning the t → +∞ limit, it is enough to consider initial data bounded from above and from below by two positive constants: this property is preserved along the flow and the nonlinear diffusion term can then be estimated using linear operators, so that the solution converges to a unique constant determined by mass conservation.The proofs can be adapted from [17,19] and from the results on the sphere on the other hand: see [18] and references therein.As they present no essential difficulty, they will be omitted.Our goal here is to understand the range of m for which we have a monotonicity property of the deficit as in the case m = 1.Let m ± (p) be defined as in (5).
We consider the measure d µ as in (10) such that (12) holds for some λ > 0. If ρ > 0 solves (14) with L defined by (11) for an initial datum ρ(t = 0, The result of Theorem 7 is new for m = 1.Theorem 3 corresponds to the special case of the harmonic potential φ(y) = 1 2 |y| 2 with v = ρ 1/p in Theorem 7. The range of the admissible parameters (m, p) is the same in Theorems 3 and 7, and shown in Fig. 2. With the additional observation that ρ(t , •) converges to a constant as t → +∞ so that the limit of the deficit is 0, the monotonicity of the deficit of Theorem 7 provides us with a proof of (13).Indeed, as a monotone non-increasing function with limit 0 as t → +∞, the deficit is non-negative for any t ≥ 0 and, as a special case, for the deficit written for the initial datum.This, written for ρ(t = 0, •) = | f | p , is precisely Inequality (13).
Proof of Theorem 7. In order to do computations, a very convenient reformulation is obtained with the substitution for any t ≥ 0, with initial datum w(t = 0, A second useful computation goes as follows: The purely technical purpose of introducing the exponent β is to make the last line of the above computation 2-homogeneous in w, which makes the discussion easier to read.Inserting the two following estimates, if, for any function w, we have Altogether, this gives the condition See Fig. 3. Equivalently, written in terms of m, the condition is m − (p) ≤ m ≤ m + (p) with m ± (p) defined by (5).Summarizing our computations, we learn that To make these computations rigorous, one has to justify all integrations by parts.This is by now rather standard and can be done using the following scheme.
(1) Let h > 0 be large enough and consider Ω h := x ∈ R n : φ(x) < h .We can consider the evolution equation restricted to Ω h , with no flux boundary conditions.Then apply the carré du champ method and keep track of the boundary terms.Since Ω h is a bounded convex domain, these terms have a sign due to Grisvard's lemma: see for instance [29, Lemma 5.2], [38, Proposition 4.2], [23,31] and [33,Lemma A.3]. (2) Extend Inequality (13) written on Ω h to R d by taking the limit as h → +∞ and then argue by density.
This completes the proof of Theorem 7.  16) is such that See [27] for a similar statement on the sphere and its proof.

Improved inequalities based on the carré du champ method
In the proof of Theorem 7, using only Q β [w] ≥ 0 is a crude estimate.Let us explain how one can obtain improved estimates by making a better use of Q β [w] ≥ 0. Under Condition (17), we can indeed rewrite Q β [w] as an integral of a sum of squares, As in [2, Theorem 2], let us consider the special case m = β = 1 of the linear flow.Let us define the entropy and the Fisher information by and Inequality (13) amounts simply to We can now state a first improved entropy-entropy production inequality.

Proposition 9.
Let n be any positive integer.We consider the measure d µ as in (10) such that (12) holds for some λ > 0. For any p ∈ (1, 2), let ϕ(s) := 1 + s − (1 + s) p−1 .With the above notation, we have Since p ∈ (1, 2), the function ϕ is convex with ϕ(0) = 0 and ϕ (0) = 2−p, which implies in particular that ϕ(s) ≥ (2 − p) s for any s ≥ 0, so that Inequality ( 21) is stronger than (13).Inequality (21) amounts to and can also be rewritten as We claim no originality in either Proposition 9 or Inequality (22)    By integrating from t = 0 to +∞ using the facts that e ≤ 0, e + 2 λ 2−p e ≤ 0 and lim t →+∞ e (t ) = lim t →+∞ e(t) = 0, we conclude that Inspired once more by the results on the sphere (see [24] and earlier references therein), it is a very natural question to wonder if improved entropy-entropy production inequalities can be achieved with β = 1.The answer goes as follows.Let us consider the function ϕ β given by 2 − p λ and δ(β) is defined by (20).
Proposition 10.Let n be any positive integer.We consider the measure d µ as in (10) such that (12) holds for some λ > 0. For any p ∈ (1, 2), take β ∈ 1, β + (p) .With the same notation as in Proposition 9 and χ β as above, we have Proof.With the notation e := and i := , we have now to consider the differential inequality w 2 d µ , which directly comes from the carré du champ computation (18) with Q β and δ defined respectively in (19) and (20).The key ingredient is to replace an estimate due to Demange in the case of the sphere for p > 2 (see [21,22] and also [Lemma 15, (iii)]) by its counterpart for log-concave measures and p ∈ (1, 2).Compared to the linear case, the Cauchy-Schwarz inequality in the proof of Proposition 9 has to be replaced by the Hölder inequalities where the last equality is a consequence of the definition of i and (23).Hence, on the condition that χ solves The solution χ(s) = (1−b) ϕ(s) is such that ϕ solves ( 24) with b = δ β −2 (2−p)/λ , which shows that χ = χ β .The proof then follows from the same considerations as for Proposition 9 (also see [18,Appendix B.4] for details).
If φ(y) = |y| 2 /2 is the harmonic potential so that d µ = d γ, by testing (2) with f ε (y) = 1 + ε y 1 where y 1 denotes the first coordinate of y = (y 1 , y 2 , . . ., y n ) ∈ R n , we find that This is the standard computation for checking that λ = λ = 1 is the optimal constant in (2).Since we also learn that ( 21) involves the optimal exponent at least in the limit as ε → 0. After observing that ∇ f ε 2 L 2 (R n ,d γ) = ε 2 , we may wonder whether the deficit . The detailed answer is not limited to the case µ = γ and goes as follows.For simplicity, we take β = 1 and consider ϕ(s) := 1 + s − (1 + s) p−1 as in Proposition 9. We recall that ϕ is monotone increasing and convex on R + , such that ϕ (0) = 2 − p, hence invertible of inverse ϕ −1 such that ψ(t ) := t − (2 − p) ϕ −1 (t ) is also a convex, non-negative, monotone increasing function.
Corollary 11.Let p ∈ (1, 2) and n be a positive integer.We consider the measure d µ as in (10) such that (12) holds for some λ > 0. With ψ as above, for any f ∈ H 1 (d γ) we have Moreover, there is some κ > 0 such that The constant κ depends only on p and its value is estimated in the proof below.

From the carré du champ method to Obata's theorem
As a side result, we consider an improvement of the carré du champ method as in [36, Theorem 2.1] or [26], which goes as follows.Let us consider the optimal constant λ 1 > 0 in the Poincaré inequality On the other hand, by the computation of Section 3.1 with β = p = 1, we know that which proves, for instance using (18), that This result goes back to the work of Obata and the carré du champ method is in fact very close to the spirit of the historical proof: see [41, p. 327].
Proof.The carré du champ method applied as in Section 3.1 with p = 1 shows that λ 1 ≥ λ .
With β = 1 and p = 1, the computation in the proof of Theorem 7 shows that, if the initial datum w(t = 0, •) is an optimal function for the Poincaré inequality (26), then Here we keep all terms and in particular do not use the fact that Hess φ − λ Id ≥ 0 a.e. in the sense of positive matrices.Since ∇w = 0 a.e. and Q 1 [w] ≥ 0, we find that Hess φ−λ Id = 0 a.e.This completes the proof in the equality case λ = λ 1 .
Remark 14.The proof of Lemma 13 is reminiscent of [26,36].The result when λ = λ 1 points in the direction of Obata's theorem (also known as the Obata-Lichnerowicz theorem) and in some sense, it is the analogue for Gaussian measures of the result of [35, p. 135] (also see for instance [13, p. 179]) on the sphere.The case λ = λ 1 is easy to understand in dimension d = 1: with β = 1 and p = 1, we apply the computation of the proof of Lemma 13 to a function u in the eigenspace associated with λ 1 and obtain that u = 0 almost everywhere.This means that u(y) = a y +b for some real constants a = 0 and b, and there is no loss of generality if we take a = 1.Using now the eigenvalue equation L u + λ 1 u = 0, we read that φ (y) = λ 1 (y − b), which means that φ is a harmonic potential.In higher dimensions, one has to remember that Inequality (13) can be tensorized on product spaces: see for instance [20,28,34].This is however responsible for some technicalities, which are dealt with in greatest generality, e.g., in [30].

Improved inequalities under orthogonality conditions
Let Π 1 be the L 2 (R n , d γ)orthogonal projection onto the space generated by the constants and the coordinate functions, corresponding to the Hermite polynomials of order less or equal than 1.The following result was recently proved in [18, Appendix A] on the basis of Nelson's hypercontractivity estimate in [40,Theorem 3] and its relation with Gross' logarithmic Sobolev inequality in [32] (also see [1,34] for earlier results).
Proposition 15.Let n ≥ 1 and p ∈ [1, 2).For any f ∈ H 1 (R n , d γ), we have Compared to (2), this result provides us with an improved entropy-entropy production inequalities under orthogonality conditions.As noted in [18], such an improvement is not optimal.There are other possible approaches.For instance, a finer analysis of entropy methods has been used in [27] on the sphere, that could probably be adapted to the case of the Gaussian measure.Alternatively, one could use the convex interpolation of [34], with the possible advantage that the result would not degenerate in the limit as p → 2 using the recent stability result of [25,Theorem 2].

Stability results for the Gaussian measure in the subcritical range
The whole Section is devoted to the proof of Theorem 4. We split it into four lemmas.The key estimate is obtained in Lemma 19.
1 Let us start with the easy case, far away from the optimizers of (2) in the sense that for some θ > 0, we assume By homogenity of the inequalities, we can fix f 2 L 2 (R n ,d γ) = 1 without loss of generality.
Lemma 16.Let n ≥ 1 and θ ∈ (0, 1).For any function f In case (29), this already proves the result of Theorem 4 with c n,p ≤ c (1)  n,p := κ (θ).The result of Lemma 16 directly follows from ( 25), but we give a proof based on the convexity properties of the entropy, which directly explains where the constant comes from.
Proof.From Corollary 11 with λ = 1 and Hölder's inequality, we obtain 2 From now on we work in a neighbourhood of the constants which, by homogeneity of the inequalities, is defined as Lemma 17.Let n ≥ 1 and θ ∈ (0, 1).For any non-negative function we have the estimate Proof.With f := R n f d γ, the result follows from the Gaussian Poincaré inequality according to 3 Assume that f is as in (30) and let us decompose By homogeneity, if (30) holds and η > t ε 2 for some given t > 0, using Lemma 17, we obtain This case already covers the result of Theorem 4 with Proof.The result follows from Proposition 15 and from the chain of elementary inequalities 4 The next part of the proof relies on a Taylor expansion of u f 2 L p (R n ,d γ) .With no loss of generality, by rotational invariance, we can assume that ν = (1, 0, . . ., 0) so that with Cartesian coordinates x = (x 1 , x 2 , . . ., x n ) ∈ R n , we write u f (x) = 1 + ε x 1 + η r (x).The following result is at the core of our strategy.It heavily relies on the Gaussian logarithmic Sobolev inequality and new estimates for the remainder terms based on the boundedness of R n r 2 log r 2 d γ.
Lemma 19.Let n ≥ 1 and f ∈ H 1 (R n , d γ) be a non-negative function such that (30) holds.We keep the same notation as in Lemma 18 and further assume that η ≤ t ε 2 for some t > 0. Then there is a constant C > 0, depending only on n, p and t , such that Proof.This proof is elementary although a little bit lengthy.Several steps are similar to those of [18, Section 4] and will be only sketched.Let us split it into three steps.
Step 1.Let us start with a list of preliminary remarks.
• Let B ε be the centred ball of radius 1/(2 ε), that is, for any g ∈ L 2 (R n , d γ), by Hölder's inequality and, as a consequence, From now on, we assume without further notice that x ∈ B ε unless it is specified.

in the limit case corresponding to p = 2 .
The admissible values of the parameters are limited to m − (d , p) ≤ m ≤ m + (d , p) and 1 ≤ p ≤ 2 * if d ≥ 3.Moreover, at the endpoints, we have m ± (d , 1) = 1 and m ± (d , 2 * ) = (d − 1)/d if d ≥ 3, while m + (d , 2 # ) = 1 so that m = 1 is admissible if and only if 1 ≤ p ≤ 2 # when d ≥ 2. See Fig. 1.For appropriate initial data, it is shown in [27] that the monotonicity property of the deficit along the flow of (3) is violated for any p ∈ [2, 2 * ) or p = 2 * if d ≥ 3 as soon as either m < m − (d , p) or m > m + (d , p).The case p = 2 corresponding to the logarithmic Sobolev inequality is included.

Figure 1 .
Figure 1.Case d = 5.The admissible parameters p and m correspond to the grey area.The boundary of the admissible set is tangent to the vertical lines p = 1 at (m, p) = (1, 1) and p = 2 * = 10/3 at (m, p) = (4/5, 10/3).Qualitatively, this figure does not change as d increases but gets squeezed in the interval 1 ≤ p ≤ 2 as d → +∞.

Figure 2 .
Figure 2. The admissible parameters p and m correspond to the grey area and are independent of the dimension n.The boundary of the admissible set is tangent to the vertical lines p = 1 at (m, p) = (1, 1) and p = 2 at (m, p) = (1, 2).It is the limit set of the admissible parameters for Proposition 2 as d → +∞.

Figure 3 .Remark 8 .
Figure 3.The admissible parameters p and β correspond to the grey area and are independent of the dimension n.The boundary of the admissible set is tangent to the vertical lines p = 1 at (β, p) = (1, 1) and p = 2 at (β, p) = (1, 2).This figure corresponds to Fig. 2 up to the transformation of m → β = 2/ 2 − p (1 − m) according to (16).