Improved interpolation inequalities and stability

For exponents in the subcritical range, we revisit some optimal interpolation inequalities on the sphere with carr\'e du champ methods and use the remainder terms to produce improved inequalities. The method provides us with lower estimates of the optimal constants in the symmetry breaking range and stability estimates for the optimal functions. Some of these results can be reformulated in the Euclidean space using the stereographic projection.

With λ = 1, inequalities (1.1) and (1.2) can be rewritten as ) for all u ∈ H 1 ( d , dμ), (1.3) for any p ∈ [1, 2) ∪ (2, 2 * ) if d = 1, 2, and for any p ∈ [1, 2) ∪ (2, 2 * ] if d ≥ 3. Since dμ is a probability measure, we know from Hölder's inequality that the right-hand side of (1.3) is nonnegative independently of the sign of (p − 2). We will call (1.3) the Gagliardo-Nirenberg-Sobolev interpolation inequality. In the case p > 2, it is usually attributed to Beckner [5] but can also be found in [7,Corollary 6.1]. However, an earlier version corresponding to the range p ∈ [1, 2) ∩ (2, 2 # ) was established in the context of continuous Markov processes and linear diffusion operators by Bakry and Emery in [2,3], using the carré du champ method, where 2 # is the Bakry-Emery exponent defined as for any d ≥ 2, and where we shall adopt the convention that 2 # = +∞ if d = 1. Notice that the case p = 2 # is also covered in [2,3] if d ≥ 2. By taking the limit in (1.3) as p → 2, we obtain the logarithmic Sobolev inequality on d : ) dμ for all u ∈ H 1 ( d , dμ) \ {0}. (1.4) For brevity, we shall consider it as the "p = 2 case" of the Gagliardo-Nirenberg-Sobolev interpolation inequality. Inequality (1.4) was known from earlier works, see, for instance, [20]. Various proofs of (1.3) have been published. By Schwarz foliated symmetrization, it is possible to reduce (1.3) to inequalities based on the ultraspherical operator, which simplifies a lot the computations, see [10,11,16] and references therein for earlier results on the ultraspherical operator. In this paper, we rely on the carré du champ method of Bakry and Emery and refer to [4] for a general overview of this technique. We also revisit some improved Gagliardo-Nirenberg-Sobolev inequalities that can be written as Here φ is a nonnegative convex function such that φ(0) = 0 and φ (0) = 1. As a consequence, φ(s) ≥ s and we recover (1.3) if φ(s) ≡ s, but in improved inequalities we will have φ(s) > s for all s ̸ = 0. Such improvements have been obtained in [9,11,14,16]. Here we write down more precise estimates and draw some interesting consequences of (1.5), such as lower estimates for the best constants in (1.1) and (1.2) or improved weighted Gagliardo-Nirenberg inequalities in the Euclidean space ℝ d .
The improved inequality (1.5), with φ(s) > s for s ̸ = 0, can also be considered as a stability result for (1.3) in the sense that it can also be rewritten as Here the right-hand side of the inequality is a measure of the distance to the optimal functions, which are the constant functions, see Appendix A for details.

Main Results
Our first result goes as follows. Let Notice that for all d ≥ 1, 1 < p * (d) < 2 and lim d→+∞ p * (d) = 2. For any admissible s ≥ 0, i.e., for any s Written in terms of ‖u‖ 2 L 2 ( d ) and ‖u‖ 2 L p ( d ) , we shall prove in Section 3 that (1.5) holds with φ given by (2.2) and gives rise to a following, new interpolation inequality.
and let γ be given by (2.1). Then we have In inequalities (2.4) and (2.5), the equality case is achieved by constant functions only, and the constants d 2−p−γ in (2.4) and 2d p−2 in (2.5) are sharp as can be shown by testing the inequality with u = 1 + εv, with v such that −∆v = dv in the limit as ε → 0. Now, let us come back to (1.1) and (1.2). We deduce from Theorem 2.1 the following estimates of the best constants in (1.1) and (1.2), see Figure 1 for an illustration.
Our third result has to do with stability for inequalities in the Euclidean space ℝ d with d ≥ 2. For all x ∈ ℝ d , let us define ⟨x⟩ := √ 1 + |x| 2 and recall that | d | = 2π d+1 2 /Γ( d+1 2 ). Using the stereographic projection of d onto ℝ d (see Appendix B), inequality (1.3) can be written as a weighted interpolation inequality in ℝ d : Notice that δ(2 * ) = 0 for any d ≥ 3, so that the inequality is the Sobolev inequality with sharp constant if p = 2 * . However, for any p ∈ [1, 2) ∪ (2, 2 * ] and d ≥ 3, equality is obtained with v ⋆ (x) = ⟨x⟩ 2−d and this function is, up to an arbitrary multiplicative constant, the only one to realize the equality case if p < 2 * . Equality is achieved by v ⋆ = 1 in dimension d = 2 for any p ∈ [1, 2) ∪ (2, +∞). Let us notice that ∇v ⋆ is not in Again, the right-hand side of the inequality is a measure of the distance to v ⋆ . The proof is elementary. With φ given by (2.2) and ψ(s) = φ(s) − s, we notice that As a consequence, we have The result of Theorem 2.3 follows by applying the stereographic projection. A sharper result, valid also if p ∈ [1, 2), will be given in Proposition 3.4.
As noticed in [ where the infimum is taken on the set of the functions u ∈ H 1 + ( d , dμ), with ∫ d u dμ = 1 and ∫ d x|u| p dμ = 0. Then for any p ∈ (2, 2 # ), we have for any function u ∈ H 1 ( d , dμ) such that ∫ d x i |u| p dμ = 0, with i = 1, 2, . . . , d. We know from [16] that Λ ⋆ (p) > d but the value is not explicit except for the limit case p = 2. In this case, the inequality becomes a logarithmic Sobolev inequality, which has been stated in [16,Proposition 5.4]. Using the stereographic projection, we obtain new inequalities on ℝ d which are as follows.
Theorem 2.4. Let d ≥ 2 and assume that p ∈ (2, 2 # ). Then Under the same conditions on v, we also have Notice that the right-hand side of each of the two inequalities is proportional to the corresponding entropy and not to the square of the entropy as in Theorem 2.3. This result is a counterpart for p ∈ (2, 2 # ), with a quantitative constant, of the result of Bianchi and Egnell in [6] for the critical exponent p = 2 * . See Remark 3.5. The constant Λ ⋆ (p) can be estimated explicitly in the limit case as p = 2, see [16,Proposition 5.4] for further details. So far, all results have been limited to the Bakry-Emery range and rely on heat flow estimates on the sphere. However, using nonlinear flows as in [16], improvements and stability results can also be achieved when p ∈ [2 # , 2 * ). This will be the topic of Section 4 while all results of Section 2 are proved in Section 3 using the heat flow and the carré du champ method on the sphere.

Heat Flow and Carré du Champ Method
In this section, our goal is to prove that (1.5) holds with φ given by (2.2).
In its simplest version, the carré du champ method goes as follows. We define the entropy and the Fisher information, respectively, by Then we shall assume that these quantities are driven by the flow such that u p is evolved by the heat equation, that is, we shall assume that u > 0 solves where ∆ denotes the Laplace-Beltrami operator on d . In the next result, the prime denotes a derivative with respect to t.
Proof. Since (3.1) amounts to ∂u p ∂t = ∆u p , it is straightforward to check that Let us summarize results that can be found in [9,11,14,16]. We adopt the presentation of the proof of [17,Lemma 4.3]. With d considered as a d-dimensional compact manifold with metric g and measure dμ, let us introduce some notation. If A ij and B ij are two tensors, then A : B := g im g jn A ij B mn and ‖A‖ 2 := A : A.
Here g ij is the inverse of the metric tensor, i.e., g ij g jk = δ i k . We use the Einstein summation convention and δ i k denotes the Kronecker symbol. Let us denote the Hessian by Hu and define the trace-free Hessian by We also define the trace-free tensor An elementary but lengthy computation that can be found in [17] shows that where γ is given by (2.1). In the framework of the carré du champ method of Bakry and Emery applied to a solution u of (3.1), the admissible range for p is therefore (2.3) as shown in [3,16]; this is the range in which we know that γ ≥ 0. Since lim t→+∞ e(t) = lim t→+∞ i(t) = 0 and d dt (i − d e) = i + 2d i ≤ 0, it is straightforward to deduce that i − d e ≥ 0 for any t ≥ 0 and, as a special case, at t = 0 for an arbitrary initial datum. This completes the proof of (1.3), after replacing u by |u| and removing the assumption u > 0 by a density argument.
Following an idea of [1], it has been observed in [11] that an improvement is achieved for any p ∈ [1, 2) ∪ (2, 2 # ) using where the last equality holds if we impose that ‖u‖ L p ( d ) = 1 at t = 0. The proof of Lemma 3.1 is complete.
Proof. The solution of (3.3) is unique and it is a straightforward computation that φ, given by (2.2), solves (3.3). Proof. With the notation of Lemma 3.1, we compute using (3.3) in the equality and then (3.2) in the inequality. Since lim t→+∞ e(t) = lim t→+∞ i(t) = 0 and i − d φ(e) ∼ i − d e ≥ 0 in the asymptotic regime as t → +∞, this proves that for functions u satisfying By homogeneity, this proves (1.5) for an arbitrary function u.  The fact that can be recovered using Hölder's inequality. For instance, if p > 2, we know that ‖u‖ L 2 ( d ) ≤ ‖u‖ L p ( d ) . By homogeneity, we can assume without loss of generality that ‖u‖ L 2 ( d ) = 1 and t = ‖u‖ 2 which is obviously satisfied for any t ≥ 1 because θ is nonnegative. Similar arguments apply if p < 2, p ̸ = p * (d) and the case p = p * (d) is obtained as a limit case. The difference of the two sides in the inequality is the measure of the distance to the constants.
As in [6], the stability can also be obtained in the stronger semi-norm u → ∫ d |∇u| 2 dμ. We can indeed rewrite the improved inequality as for any u satisfying ‖u‖ 2 L p ( d ) = 1, and obtain that An explicit lower bound for μ(λ) has been obtained in [12,Proposition 8]. Let us recall it with a sketch of the proof for completeness.

Proof. From Hölder's inequality ‖u‖
After dropping ‖∇u‖ 2 L 2 ( d ) in the second parenthesis of the right-hand side and observing that 1/(p − 2) ≥ (d − 2)/4, the conclusion holds using the Sobolev inequality in the first parenthesis. We indeed recall that μ(λ) = 1 4 d(d − 2) for any λ ≥ 1 if p = 2 * . We may notice that the estimate of Proposition 3.6 captures the order in λ of μ(λ) as λ → +∞ but is not accurate close to λ = 1 and limited to the case p ∈ (2, 2 * ) and d ≥ 3. It turns out that the whole range (2.3) for any d ≥ 1 can be covered as a consequence of Theorem 2.1 with a lower bound for μ(λ) which is increasing with respect to λ ≥ 1 and such that it takes the value 1 if λ = 1. This is essentially the contents of Theorem 2.2 for p ∈ (2, 2 # ), which also covers the range p ∈ [1, 2).

Inequalities Based on Nonlinear Flows
In this section, the range of p is This range includes, in particular, the case 2 # < p < 2 * , which was not covered in Section 3. As in [9,11,16], let us replace (3.1) by the nonlinear diffusion equation The parameter β has to be chosen appropriately as we shall see below. Notice that m > 0 can be larger or smaller than 1, depending on β, d and p. The entropy and the Fisher information are redefined, respectively, by The equation e = −2i holds true only if β = 1, in which case (4.2) coincides with (3.1). Here we have e = −2β 2 ‖∇u‖ 2 L 2 ( d ) ̸ = −2i if β ̸ = 1, but we can still compute d dt (i − d e) and obtain that 1 To guarantee that γ(β) ≥ 0 for some β ∈ ℝ, a discussion has to be made, see Lemma 4.3 below for a detailed statement and also [11]. Notice that the value of γ given by (2.1) in Sections 2 and 3 corresponds to (4.5) with β = 1. In the sequel let us denote by B(p, d) the set of β such that γ(β) ≥ 0 with p in the range (4.1).

Corollary 4.2.
Let d ≥ 1 and assume that p is in the range (4.1). For any β ∈ B(p, d), any solution of (4.2) is such that i − d e is monotone non-increasing with limit 0 as t → +∞.
As a consequence, we know that i ≥ d e, which proves (1.3) in the range (4.1). Let us define by The precise description of B(p, d) goes as follows.  As observed in [9,11,16], an improved inequality can also be obtained. Since the case p ∈ [1, 2) is covered in Section 3, we shall assume from now on that p > 2. With Proof. Using the identity 1 2 + β−1 β(p−2) + ζ = 1, Hölder's inequality shows that 1

Lemma 4.3. Let d ≥ 1 and assume that p is in the range (4.1). The set B(p, d) with p is defined by
On the other hand, by using the identity 1 2 + β−1 2β + 1 2β = 1, and Hölder's inequality again, we have also , since dμ is a probability measure on d . Therefore, from (4.4), we get the inequality For every β > 1, it is possible to find a function ψ β satisfying the ODE from which we conclude that i ≥ d φ β (e) with φ β := ψ β /ψ β . It is then elementary to check that φ β satisfies the ODE and that φ β (0) = 0. Solving this linear ODE, we find the expression of φ β . Notice that φ β is defined for any s ∈ [0, 1/(p − 2)) and that φ β (s) > 0 for any s ̸ = 0. From the equation satisfied by φ β , we get that φ β (s) > 1 and φ β (s) > 0, hence φ β (s) > s for any admissible β and any s ∈ (0, 1/(p − 2)).

Let us define
(4.7) By arguing exactly as in the proof of Theorem 2.2, we obtain an estimate of the optimal constant in (1.1), which is valid, for instance, if 2 # < p < 2 * . Another consequence is that one can write an improved inequality on ℝ d in the spirit of Proposition 3.4, for any p ∈ (1, 2 * ), p ̸ = 2. Since the expression involves φ as defined in Theorem 4.4, we do not get any fully explicit expression, so we shall leave it to the interested reader. A major drawback of our method is that φ is defined through a primitive. With some additional work, φ can be written as an incomplete Γ function, which is however not of much practical interest. This is why it is interesting to consider a special case, for which we obtain an explicit control of the remainder term. For completeness, let us state the following result which applies to a particular class of functions u. Theorem 4.6 ([16]). Let d ≥ 3. If p ∈ (1, 2) ∪ (2, 2 * ), we have for any u ∈ H 1 ( d , dμ) with antipodal symmetry, i.e., (4.8) The limit case p = 2 corresponds to the improved logarithmic Sobolev inequality ) dμ for any u ∈ H 1 ( d , dμ) \ {0} such that (4.8) holds.
We refer to [16,Theorem 5.6] and its proof for details. Instead of (4.8), one can use any symmetry which guarantees that d dt ∫ d u(t, ⋅ ) βp dμ = 0 if we evolve u according to (4.2). Using the stereographic projection, one can obtain a weighted inequality with the same constant on ℝ d , for solutions which have the inversion symmetry corresponding to (4.8).
The interpolation inequalities (1.1) and (1.2) are equivalent to Keller-Lieb-Thirring estimates for the principal eigenvalue of Schrödinger operators, respectively, −∆ − V on d with V ≥ 0 in L q ( d ) for some q > 1, and −∆ + V on d with V > 0 such that V −1 ∈ L q ( d ), again for some q > 1. See, for instance, [12,13] and references therein.
Let us conclude with a summary and some considerations on open problems. This paper is devoted to improvements of (1.3) and (1.4) by taking into account additional terms in the carré du champ method. The stereographic projection then induces improved weighted inequalities on the Euclidean space ℝ d . Alternatively, various improvements have been obtained on ℝ d using the scaling invariance, see for instance [19] and references therein. It is to be expected that these two approaches are not unrelated as well as nonlinear diffusion flows on d and nonlinear diffusion flows on ℝ d can probably be related. The self-similar changes of variables based on the so-called Barenblatt solutions also points in this direction, see [15]. Concerning stability issues, we have been able to establish various estimates with explicit constants, which are all limited to the subcritical range p < 2 * when d ≥ 3. This is clearly not optimal (see [6,16]). A last point deserves to be mentioned: improved entropy production estimates like i ≥ d φ(e) mean increased convergence rates in evolution problems like (3.1) or (4.2); how to connect an initial time layer with large entropy e to an asymptotic time layer with an improved spectral gap obtained, for instance, by best matching (which amounts to impose additional orthogonality conditions for large time asymptotics), is a topic of active research.

A Estimating the Distance to the Constants
In Section 1, we claimed that the entropy is an estimate of the distance of the function u to the constant functions. Let us give some details. If p ∈ [1, 2), we know that with u = ‖u‖ L p ( d ) , for any u ∈ L p ∩ L 2 ( d ), by the generalized Csiszár-Kullback-Pinsker inequality, see [8,21] or [18, Proposition 2.1], and references therein. If p > 2, let us define the constant for any q > 1. Let q = p/2 and use the above constant to get, with t = u 2 /‖u‖ 2 L 2 ( d ) , the estimate and deduce that with u = ‖u‖ L 2 ( d ) , for any u ∈ L p ∩ L 2 ( d ). Although there is no good homogeneity property because of the definition of the function ν p/2 , the right-hand side is clearly a measure of the distance of u to the constant u.

B Stereographic Projection
Let x ∈ ℝ d , r = |x|, ω = x |x| and denote by (ρω, z) ∈ ℝ d × (−1, 1) the cartesian coordinates on the unit sphere d ⊂ ℝ d+1 given by Let u be a function defined on d and consider its counterpart v on ℝ d given by