Maritza Méndez-Reina

Soledad Estrella

Peirce's and Toulmin's perspectives to examine the informal inferential reasonings of preschool children

Keywords: Statistical inferential reasoning, informal statistical inference, Peirce, Toulmin, inductive reasoning

This paper reports on the preliminary results of a research that studies the informal inferential reasoning of preschool children that arises when they explore samples and the variability of sampling. For this purpose, some arguments that the children provide when they are required to advise other people in a hypothetical situation are analyzed. In the analysis of the argumentative exercise, the differentiating elements of the types of reasoning proposed by Peirce and the argumentative structure proposed by Toulmin are revealed. The results show how said structural analysis allows describe how the children detect regularities in the samples and declare an informal statistical inference as a conclusion in the decision-making process; in addition, it shows how preschool children are using modal qualifiers to justify generalizations that go beyond the evidence in the data when resolving problems in contexts of uncertainty.

Introduction

Statistical Inference (SI) is the heart of the statistic and reflects the power of its use since it is possible to formulate conclusions under uncertain conditions when only partial data are available [START_REF] Makar | Learning About Statistical Inference[END_REF]. In everyday life, citizens and researchers from different fields must face predictions, create estimates, and make decisions in the future without having all the data at hand, that is, considering samples of it.

Given this scenario, it is desirable to support the development of early statistical thinking in young children, in particular by promoting the generation of informal statistical inferences from data samples without resorting to the use of formal inference techniques, focusing their reasoning on the behavior of sample distributions and their contrast [START_REF] Pfannkuch | Informal inferential reasoning[END_REF], in authentic experiences of decision-making that allow coordinating their informal knowledge about the context and the problem data, and making reasonable statements beyond the data they possess [START_REF] Estrella | Early statistics in kindergarten: analysis of an educator's pedagogical content knowledge in lessons promoting informal inferential reasoning[END_REF].

With the interest of studying informal inferential reasoning of preschool children, an analysis of the structural aspects of such reasoning is carried out (Jeanotte & Kieran, 2017), using the ideas of Peirce and Toulmin to characterize elements of the structure of informal statistical inferences (Méndez-Reina & Estrella, 2023). It then uses the combined perspective of both the components of an argument proposed by [START_REF] Toulmin | The uses of argument[END_REF][START_REF] Toulmin | The uses of argument[END_REF] and the various elements of the reasoning proposed by [START_REF] Peirce | Deduction, induction, and hypothesis[END_REF] to describe how informal inferential reasoning develops through student conversations when they issue conclusions that go beyond data and when they provide persuasive arguments based on data analysis, the behavior of samples, and the variability of sampling.

Informal statistical inference

Statistical inference, according to de [START_REF] De Vetten | The development of informal statistical inference content knowledge of pre-service primary school teachers during a teacher college intervention[END_REF], can be considered as the outcome and process of creating or testing probabilistic generalizations from data, and it is statistical inferential reasoning that allows conclusions to be drawn from the analysis of the behavior of random samples. Several investigations have pointed out that this reasoning can be developed at an early age, and the term informal inferential reasoning (IIR) has been coined to refer to "the way in which students use their informal statistical knowledge to make arguments to support inferences about unknown populations based on observed samples" (Zieffler et al., 2008, p. 44). However, although IIR has been characterized as a cognitive activity required for formulating generalizations from random samples of data while considering evidence and articulating uncertainty [START_REF] Aridor | The co-emergence of aggregate and modeling reasoning[END_REF], further research is still required regarding how students establish arguments based on data analysis in situations of uncertainty and methodological tools to describe students' IIR and to classify the different aspects that structure the inferences generated in dealing with situations of uncertainty at the school level. For this report, the combined perspective of Peirce and Toulmin has been chosen as an alternative that could contribute to the characterization of the structural elements of IIR.

Peirce's and Toulmin's perspectives to examine informal inferential reasonings

The types of reasoning according to Peirce and the components it uses to distinguish them (case, rule, and result) and the argument structure proposed by Toulmin from its elements (data, warranty, support, exception, and conclusion) can be used to analyze different structural aspects of reasoning.

Types of reasoning according to Peirce

Peirce points out the distinctive characters ("rule", "case", and "result") that distinguish the structure of different classes of inference, understood as three elementary modes of reasoning: deduction, induction, and abduction. Peirce (1878) uses an example in which all beans are white and clarifies the elements rule, case, and result, explaining the distinctions between types of inference (CP 2.623): All beans in that bag are white. These beans are from that bag. Therefore, these beans are white.

Induction

Case

Result ⁂ Rule

These beans are from that bag.

These beans are white. Therefore, all beans in that bag are white.

Abduction

Rule

Result ⁂ Case

These beans are white. All beans in that bag are white. Therefore, these beans are from that bag.

In addition, Peirce considers that the reasoning of induction and making a hypothesis are essentially inferences from a sample (later, in 1901, he referred to making a hypothesis through the terms "retroduction" and "abduction").

Toulmin's argumentative model

According to Toulmin the conclusions are stated by the need to address a problem, in which a question has been posed to someone who must state something about it, who then establishes the possible solutions, usually attending to the possibilities that he considers most serious or plausible.

For [START_REF] Toulmin | The uses of argument[END_REF][START_REF] Toulmin | The uses of argument[END_REF], each argument has three base elements: data (D), warrants (W) and claims (C). However, it may not be enough simply to specify these elements; it may also be necessary to add some explicit reference to the degree of force that the data available to us confer on the assertion made under the warrants. Thus, three secondary elements can appear: modal qualifiers (Q), backing (B) and rebuttal (R), see Figure 1: The data (D) can be facts, information, figures, or examples that support the conclusion. The warrants (W) can be rules, principles, or statements that allow inferences to be drawn. The claims (C) are the statements that derive from the initial assertion. The backings (B) offer additional evidence or elements that justify the warrants. The rebuttal (R) outlines the circumstances in which the conclusion is false. Finally, the modal qualifiers (Q) qualify the conclusion by expressing degrees of confidence.

Combined perspective of Peirce's types of reasoning and Toulmin's argumentative structure

Various studies have used the combined perspective to analyze mathematical reasoning [START_REF] Conner | Identifying kinds of reasoning in a collective argumentation[END_REF][START_REF] Jeannotte | Mathematical reasoning: a conceptual model for learning and teaching at the primary and secondary levels of schooling[END_REF][START_REF] Pedemonte | The role of abduction in proving processes[END_REF]; however, Méndez-Reina & Estrella ( 2023) indicate that they have not found research to analyze structural aspects of informal inferential reasoning from the combined perspective of Peirce's types of reasoning and Toulmin's argumentative structure. Based on the position of the elements Case, Rule, and Result in the basic structure of "Data", "Warrant", and "Claim" as shown in Figure 2, the different types of reasoning can be distinguished [START_REF] Conner | Identifying kinds of reasoning in a collective argumentation[END_REF]Méndez-Reina & Estrella, 2023).

Methodology

To answer this question, -and based on consensus among the authors-, a qualitative, descriptive interpretative approach is adopted. Transcriptions of video recordings and the children's written productions were used, and one of the episodes of verbal argumentation was selected based on the manifestation of IIR in different arguments for resolving the task and interactions between the children and the teacher. For the analysis of argumentation episodes, in the first step, the assertions produced by children were identified, and the structure of the argumentative step was reconstructed, specifying data (D), warrants (W), and claims (C). On some occasions, it may also be necessary to use the secondary elements of modal qualifiers (Q), backing (B), and rebuttal (R). In the second step, an analytical, progressive, and consensual review among the authors is conducted of how the structure behaves vis-à-vis the elements proposed by Peirce (case, rule, and outcome) to determine the type of reasoning involved (deductive, inductive, or abductive); and finally, diagrams are designed with the key elements of the ideas of Toulmin and Peirce described, using the combined perspective. In addition, the diagrams show through the colors of the boxes whether the contribution of the elements corresponds to the contribution of the student, the teacher, or whether the teacher and the children worked together. The transcripts and the children's written productions have been translated from Spanish into English.

Participants

The case reported corresponds to one of the work teams: Alicia, Juan, and Francisco; they are identified as Group 2. They belonged to a group of 15 preschool children from a school in the Valparaíso region. The lesson was implemented by the teacher in her usual classroom. It was supported by an assistant and two participating professors-researchers (authors of the work) who were present to conduct the recorded video and stimulate preschool children to work together and talk out loud during the lesson.

Tasks

The implementation of this task is part of a sequence of lessons applied and improved through a Lesson Study process with nine preschool and elementary teachers and three researchers. Children experimented during lesson 1, a task in which they promoted the use of probabilistic language and a random game with concrete material. A detailed summary of this lesson can be found in Estrella and collaborators (2022). The task of lesson 2, called "the treasure of pearls and the pirates", was associated with the black box experiment [START_REF] Estrella | The Mystery of the Black Box: An Experience of Informal Inferential Reasoning[END_REF][START_REF] Van Dijke-Droogers | Repeated sampling with a black box to make informal statistical inference accessible[END_REF], this was employed in this study because these authors argue that the ideas of repeated sampling with a black box seem to be useful for the introduction of ideas of statistical inference. In this lesson, the black box experiment with three bottles, each of which contained 100 pearls of two different colors (gold and purple), was considered. In the first bottle, the pearls were in a proportion of 25 gold and 75 purple; in the second bottle, the pearls were in a proportion of 15 gold and 85 purple, and finally, in the third bottle, the pearls were in a proportion of 35 gold and 65 purple. In this experiment, the children shaken each bottle, and viewed it through a small window to investigate which of the 3 bottles can have the most golden pearls inside by collecting data and comparing samples, this means that the children must generate an informal inference. The key concepts were sample and sampling variability. The resources and some examples of these can be seen in Figure 3.

Results

The episodes chosen to illustrate, from the schematic diagrams, the structural elements manifested in the informal inferential reasoning of some preschool children. These episodes clearly show the collective argumentative expression (in the sense of [START_REF] Conner | Identifying kinds of reasoning in a collective argumentation[END_REF] of the children with the teacher, and according to IRI highlights and concepts typical of inferential statistics: (1) expression of probabilistic language and understanding of variation;

(2) evidence based on data and comparison between samples;

(3) regular sample behavior by assigning a confidence level to each event; and (4) assertions beyond data using expressions of uncertainty.

After the children shacking each bottle, they observe and record in three tables the results in their worksheets. They then visualize the behavior of the samples through dot plot (one for each Bottle), with this, each of the work teams should give recommendation on the bottle that could have more golden pearls inside. Thus, the teacher asks the full class a question: "What bottle would you recommend to the pirates to choose? Why?" Encouraging children to respond in groups and decide to give advice to pirates. Part of the statements of the children answering this question are presented below.

Teacher: Now, as a group you must decide what you recommend to the pirates Group 1:

The [bottle] 3 Teacher:

Why? Group 1:

[Bottle] 3, because it is [the bottle] that had more golden pearls than the 1 and 2. Teacher:

Group 2, did you choose bottle 3? Why? Group 2:

Because you may find many pearls. Teacher:

... is possible, or is highly likely, or is unlikely, to see put the little finger where you believe. The informal inferential reasoning put into play by Group 2, in a collective argumentative exercise is schematized in Figure 4, this illustrates how preschool children, from the Results (R1 to R6) detect the regularity in the variation of the samples, informally expressing the generalization "In the [bottle] 3 is highly likely that find many [golden] pearls ". This generalization is according to the warrant "The [Bottle] 3, because it has more golden pearls, it has up to 6 [golden pearls], the [Bottle] 3... instead, the other [Bottle 2] reached up to 2 and the other [Bottle 1] reached up to 5, instead this [Bottle 3] reached up to 6". The Rule expressed by Group 2 is stated as informal statistical inference and accounts for statistical inference in the context of the experiment proposed in the story. The reasoning analyzed, from the Peirce´s perspective is inductive, because it has reached to express the generalization of the results and the rule, from the detected regularity.

Discussion

The combination of Peirce´s and Toulmin's perspectives allowed us to recognize how informal statistical inferences, were given as a conclusion of the proposed tasks in order to detect regularities in the samples and in the variability of the sampling, either by taking into account the behavior of the samples collected within each group or by including in their warrants descriptions of the behavior of the samples throughout the course, in order to declare widespread ownership over which bottle would have the greatest chance of having golden pearls inside it in the form of informal statistical inference.

The study reported could provide a theoretical framework for understanding ISI and IIR, highlight the usefulness of combining Peirce's and Toulmin's perspectives to analyze the structural aspects of statistical reasoning, and provide insights into how informal inferential reasoning can be examined in young children. The research is still ongoing, and we are currently analyzing in more detail the progressive uses of modal qualifiers in the early years and how they are related to IIR. Some issues need further reflection. One crucial issue is: What are the structures of abductive IIR? Another aspect that needs careful consideration is the procedural aspect of this reasoning, in particular processes like conjecturing, identifying a pattern, generalizing, comparing, and classifying, among others.

Figure 1 .

 1 Figure 1. Expanded structure of Toulmin's argumentative model (1958/2003)

Figure 2 .

 2 Figure 2. Toulmin-style diagrams of arguments revealing different kinds of reasoningThis report focuses on the action of reasoning in formulating informal statistical inferences and is framed by the arguments generated by preschool children. Considering this, the following research

Figure 3 .

 3 Figure 3. Resources in task "The treasure of pearls and the pirates"

  more golden pearls, it has up to 6 [golden pearls], the [bottle] 3...instead, the other [Bottle 2] reached up to 2 and the other [Bottle 1] reached up to 5, instead this [Bottle 3] reached up to 6".

Figure 4 .

 4 Figure 4. Example of IIR schematized according to combined Peirce-Toulmin perspective

Table 1 : Types of elementary inference according to Peirce (1878) Deduction

 1

	Rule
	Case
	⁂Result

Acknowledgment

Support from National Doctorate Scholarship ANID 21210862 and ANID Fondecyt 1200346.


Giovanni Brigati

AND Jean Dolbeault

Nikita Simonov

LOGARITHMIC SOBOLEV AND INTERPOLATION INEQUALITIES ON THE SPHERE: CONSTRUCTIVE STABILITY RESULTS

Keywords: 2020 Mathematics Subject Classification. Primary: 26D10; Secondary: 58J99, 39B62, 43A90, 49J40, 46E35 logarithmic Sobolev inequality, Gagliardo-Nirenberg inequalities, stability, sphere, spectral decomposition

We consider Gagliardo-Nirenberg inequalities on the sphere which interpolate between the Poincaré inequality and the Sobolev inequality, and include the logarithmic Sobolev inequality as a special case. We establish explicit stability results in the subcritical regime using spectral decomposition techniques, and entropy and carré du champ methods applied to nonlinear diffusion flows.

INTRODUCTION AND MAIN RESULTS

Functional inequalities are essential in many areas of mathematics. The knowledge of optimal constants, or at least good estimates of them, is crucial for various applications. Whether optimality cases are achieved is a standard issue in analysis. The next natural question is to understand how the deficit, say the difference of the two sides of the functional inequality, measures the distance to the set of optimal functions. Such a question has been actively studied in critical Sobolev inequalities, but much less in subcritical interpolation inequalities. In the case of the sphere, a global stability result based on Bianchi-Egnell-type methods was recently obtained for a family of Gagliardo-Nirenberg inequalities by Frank in [START_REF] Frank | Degenerate stability of some Sobolev inequalities[END_REF], with the striking observation that only the power 4 of a natural distance is controlled by the deficit. Here we give a more detailed picture, which includes the logarithmic Sobolev inequality, and provide explicit estimates.

On the sphere S d with d ≥ 1, the logarithmic Sobolev inequality can be written as (LS)

S d ∇F 2 d µ ≥ d 2 S d F 2 log ⎛ ⎝ F 2 F 2 L 2 (S d ) ⎞ ⎠ d µ ∀F ∈ H 1 (S d , d µ) ,
where d µ denotes the uniform probability measure. The equality case is achieved by constant functions and d 2 is the optimal constant as shown by taking the test functions F ε (x) = 1 + ε x ⋅ ν, for some arbitrary ν ∈ S d , in the limit as ε → 0. Our first result is an improved inequality under an orthogonality constraint, which improves upon [START_REF]Interpolation inequalities on the sphere: linear vs. nonlinear flows[END_REF]Proposition 5.4].

Theorem 1. Let d ≥ 1. For any F ∈ H 1 (S d , d µ) such that (1.1) S d
x F d µ = 0 , we have and it is in fact Sobolev's inequality with optimal constant on S d , but this is out of the scope of our paper which focuses on the subcritical regime p < 2 * . The logarithmic Sobolev inequality (LS) is obtained from (GN) by taking the limit as p → 2, and the counterpart of the above results for p ≠ 2, in the subcritical range p < 2 * , goes as follows.

(1.2) S d ∇F 2 d µ - d 2 S d F 2 log ⎛ ⎝ F 2
Theorem 4. Assume that d ≥ 1 and p ∈ (1,2) ∪ (2,2 * ). For any function F ∈ H 1 (S d , d µ)

such that the orthogonality condition (1.1) holds, we have

(1.3) S d ∇F 2 d µ - d p -2 F 2 L p (S d ) -F 2 L 2 (S d ) ≥ C d ,p S d ∇F 2 d µ with C d ,p = 2 d -p (d-2)
2 (d+p) .

Taking F ε (x) = 1 + ε x ⋅ ν as above shows that (1.1) is needed in Theorem 4. We also have a higher-order estimate of the deficit as a consequence of the carré du champ method.

Proposition 5. Let d ≥ 1 and p ∈ (1,2) ∪ (2,2 * ). There is a convex function ψ on R + with ψ(0) = ψ ′ (0) = 0 such that, for any F ∈ H 1 (S d , d µ), we have

S d ∇F 2 d µ - d p -2 F 2 L p (S d ) -F 2 L 2 (S d ) ≥ F 2 L p (S d ) ψ ⎛ ⎝ ∇F 2 L 2 (S d ) F 2 L p (S d ) ⎞ ⎠ .
An explicit expression for ψ will be given in Section 3. The two results of Theorem 4 and Proposition 5 can be combined to prove the analogue of Theorem 3 for p ≠ 2, with an explicit constant: see Section 4.

Theorem 6. Let d ≥ 1 and p ∈ (1,2) ∪ (2,2 * ). For any F ∈ H 1 (S d , d µ), we have

S d ∇F 2 d µ - d p -2 F 2 L p (S d ) -F 2 L 2 (S d ) ≥ S d ,p ⎛ ⎝ ∇Π 1 F 4 L 2 (S d ) ∇F 2 L 2 (S d ) + F 2 L 2 (S d ) + ∇(Id -Π 1 )F 2 L 2 (S d ) ⎞ ⎠
for some explicit stability constant S d ,p > 0.

Let us give a brief account of the literature. In this paper, we address the distinction between improved inequalities (inequalities with improved constants under orthogonality constraints) and quantitative stability (as a measure of a distance to the set of optimal functions). There are many adjacent directions of research like, for instance, stability in weaker norms (see for instance [START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities[END_REF][START_REF] Nguyen | Fractional Sobolev and Hardy-Littlewood-Sobolev inequalities[END_REF] for Sobolev's inequality) or notions of stability with no explicit notion of distance. To our knowledge, not so much has been done in subcritical interpolation inequalities (see [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: Flows, regularity and the entropy method[END_REF][START_REF] Frank | Degenerate stability of some Sobolev inequalities[END_REF] and some references therein), except for the logarithmic Sobolev inequality, for which we refer to [START_REF]Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities[END_REF][START_REF] Fathi | Quantitative logarithmic Sobolev inequalities and stability estimates[END_REF] and [START_REF] Indrei | A quantitative log-Sobolev inequality for a two parameter family of functions[END_REF][START_REF] Feo | Some remarks on the stability of the log-Sobolev inequality for the Gaussian measure[END_REF][START_REF] Indrei | Deficit estimates for the logarithmic Sobolev inequality[END_REF][START_REF] Kim | Instability results for the logarithmic Sobolev inequality and its application to related inequalities[END_REF].

The Gagliardo-Nirenberg inequalities (GN) on the sphere have been established with optimal constant for any p ∈ (2,2 * ) in [10, Corollary 6.1] and in [START_REF]Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality[END_REF]. In dimension d = 2, Onofri's inequality is obtained from (GN) in the limit as p → 2 * = +∞: see [START_REF]Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality[END_REF][START_REF] Carlen | Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on S n[END_REF]. With p ∈ [1,2) or p > 2 but not too large (if d ≥ 2), inequality (GN) was known earlier from [START_REF] Émery | Diffusions hypercontractives[END_REF].

A Markovian point of view is presented in [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF], with many more references therein on related questions. On Euclidean space, similar inequalities go back to [START_REF] Soboleff | Sur un théorème d'analyse fonctionnelle[END_REF][START_REF] Gagliardo | Proprietà di alcune classi di funzioni in più variabili[END_REF][START_REF] Nirenberg | On elliptic partial differential equations[END_REF]. The logarithmic Sobolev inequality (LS) is a well known-limit case as p → 2 and can be considered in a common framework with (GN). Whenever possible, we shall adopt this point of view. For an overview of early results on the sphere, we refer to [START_REF]Logarithmic Sobolev inequalities and contractivity properties of semigroups[END_REF]Section 6,(iv)]. The literature on (LS) on the circle and on the sphere can be traced back at least to [START_REF] Weissler | Logarithmic Sobolev inequalities and hypercontractive estimates on the circle[END_REF], [46, Theorem 1, page 268] with computations based on the ultraspherical operator, and [START_REF] Rothaus | Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators[END_REF] for a more variational approach. The inequality with optimal constant is stated in [3, inequality (13) page 195] as a consequence of the carré du champ method. Also see [START_REF] Émery | Inégalités de Sobolev pour un semi-groupe symétrique[END_REF] and [14, page 342] for related results and [START_REF]Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature[END_REF][START_REF] Dolbeault | Nonlinear flows and rigidity results on compact manifolds[END_REF][START_REF]Interpolation inequalities on the sphere: linear vs. nonlinear flows[END_REF] for a PDE approach based on entropy estimates and the carré du champ method. After Schwarz foliated symmetrization, the problem is reduced to a simpler family of interpolation inequalities involving only the ultraspherical operator.

The interest for stability issues was raised by [START_REF] Lieb | Sobolev inequalities with remainder terms[END_REF] and the stability result of Bianchi and Egnell in [START_REF] Bianchi | A note on the Sobolev inequality[END_REF], on Euclidean space. Over the years, various approaches have been developed, based on compactness methods and contradiction arguments as in [START_REF] Bianchi | A note on the Sobolev inequality[END_REF][START_REF] Chen | Remainder terms in the fractional Sobolev inequality[END_REF], spectral analysis and orthogonality conditions as in [START_REF]Interpolation inequalities on the sphere: linear vs. nonlinear flows[END_REF]Proposition 5.4] and [START_REF] Gui | The sphere covering inequality and its applications[END_REF], or entropy methods and improved inequalities as in [START_REF] Arnold | Refined convex Sobolev inequalities[END_REF][START_REF] Dolbeault | Improved interpolation inequalities, relative entropy and fast diffusion equations[END_REF][START_REF] Dolbeault | Improved interpolation inequalities on the sphere[END_REF][START_REF]Interpolation inequalities on the sphere: linear vs. nonlinear flows[END_REF][START_REF] Esteban | Improved interpolation inequalities and stability[END_REF]. For spectral methods, a fruitful strategy relies on the Funk-Hecke formula, which is behind (2.2), and the approach of [START_REF] Lieb | Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities[END_REF][START_REF]Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality[END_REF], which applies to the stability result for fractional interpolation inequalities of [START_REF] Chen | Remainder terms in the fractional Sobolev inequality[END_REF] and [START_REF] Dolbeault | Optimal functional inequalities for fractional operators on the sphere and applications[END_REF]Corollary 2.3]. This is the method we use in Section 2. Stability issues for (GN) have recently been discussed in [START_REF] Frank | Degenerate stability of some Sobolev inequalities[END_REF] with methods of Bianchi-Egnelltype, with the drawback that no estimate of the stability constant is known. This drawback can be cured by a carré du champ method as we shall see in Section 3. Without entering into details, let us mention some recent progress on stability in [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: Flows, regularity and the entropy method[END_REF][START_REF] Dolbeault | Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence[END_REF][START_REF] König | On the sharp constant in the Bianchi-Egnell stability inequality[END_REF][START_REF] Chen | Sharp stability of log-Sobolev and Moser-Onofri inequalities on the sphere[END_REF] for related critical inequalities. This paper is organized as follows. Section 2 is devoted to the proof by spectral methods of Theorem 7 (see below), which is an extension of Theorems 1 and 4: under orthogonality constraints, these results are reduced to estimates of improved constants in inequalities (LS) and (GN), with various refinements based on a decomposition in spherical harmonics. An explicit stability result without constraints corresponding to Propositions 2 and 5 is proved in Section 3. The proofs of Theorems 3 and 6, in Section 4, is based on the spectral decomposition method developed by Frank in [START_REF] Frank | Degenerate stability of some Sobolev inequalities[END_REF]. We collect the previous estimates (with and without orthogonality constraints) in global results, with explicit constants. Various additional results are stated in two appendices: the extension of the method to interpolation inequalities for the Gaussian measure on Euclidean space and a discussion of its limitations in Appendix A, the details of the computations of the carré du champ method on the sphere and its application in order to establish improved functional inequalities in Appendix B.

IMPROVEMENTS UNDER ORTHOGONALITY CONSTRAINTS

In this section, we prove Theorems 1 and 4 in the slightly more general framework of Theorem 7 below. Let us consider the generalized entropy functionals

E 2 [F ] ∶= 1 2 S d F 2 log ⎛ ⎝ F 2 F 2 L 2 (S d ) ⎞ ⎠ d µ and E p [F ] ∶= F 2 L p (S d ) -F 2 L 2 (S d ) p -2 if p ≠ 2 .
With this notation, we can rephrase (LS) and (GN) as

S d ∇F 2 d µ ≥ d E p [F ] ∀F ∈ H 1 (S d , d µ) ,
for any p ∈ [1,2 * ). The optimality case is achieved by considering the test function F ε = 1 + ε ϕ 1 in the limit as ε → 0, where ϕ 1 is an eigenfunction of the Laplace-Beltrami operator such that -∆ϕ 1 = d ϕ 1 , for instance ϕ 1 (x) = x ⋅ν for some ν ∈ S d as in Section 1.

Let us consider the decomposition into spherical harmonics of L 2 (S d , d µ),

L 2 (S d , d µ) = ∞ ⊕ =0 H ,
where H is the subspace of spherical harmonics of degree ≥ 0. See for instance [START_REF] Müller | Spherical harmonics[END_REF][START_REF] Stein | Introduction to Fourier analysis on Euclidean spaces[END_REF][START_REF] Berger | Le spectre d'une variété Riemannienne[END_REF][START_REF] Morimoto | Analytic functionals on the sphere[END_REF]

S d ∇F 2 d µ -d E p [F ] ≥ C d ,p,k S d ∇(Id -Π k )F 2 d µ for some explicit constant C d ,p,k ∈ (0,1) such that C d ,p,k ≤ C d ,p,1 = 2 d -p (d-2) 2 (d+p) .
The expression for C d ,p,k is given below in the proof. inequality (2.1) can be seen as an improvement of (LS) and (GN), namely

(1 -C d ,p,k ) S d ∇F 2 d µ ≥ d E p [F ]
for any F ∈ H 1 (S d , d µ) such that Π k F = 0. With k = 1, this establishes (1.2) and (1.3), thus proving Theorem 1 if p = 2, and Theorem 4 if p ≠ 2.

Proof of Theorem 7. Let (F j ) j ∈N be the decomposition of F along H j for any j ∈ N. We learn from [START_REF]Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality[END_REF]Ineq. (19)] or [START_REF] Dolbeault | Optimal functional inequalities for fractional operators on the sphere and applications[END_REF]Ineq. (1.6)] that the subcritical interpolation inequalities

(2.2) E p [F ] ≤ ∞ j =1 ζ j (p) S d F j 2 d µ ∀F ∈ H 1 (S d , d µ) hold for any p ∈ (1,2) ∪ (2,2 * ) with ζ j (p) ∶= γ j d p -1 p -2 and γ j (x) ∶= Γ(x) Γ( j + d -x) Γ(d -x) Γ(x + j ) .
This result is based on the Funk-Hecke theorem (see for instance [START_REF] Frank | A new, rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality[END_REF]Section 4]) and

Lieb's ideas in [START_REF] Lieb | Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities[END_REF]. Notice that ζ j (p) ≥ 0 for any p ∈ (1,2) ∪ (2,2 * ). According to [29, Lemma 2.2], the function ζ j is strictly monotone increasing on (1,∞) for any j ≥ 2 and the limits

λ j = d lim p→2 * ζ j (p)
are the eigenvalues of the Laplace-Beltrami on the sphere, with λ j = j ( j +d -1). Hence

d E p [F ] ≤ ∞ j =1 λ j S d F j 2 d µ = S d ∇F 2 d µ ,
which is the essence of the proof of (GN) in [START_REF]Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality[END_REF] and also the main idea for the proof of the stability result for fractional interpolation inequalities of [START_REF] Dolbeault | Optimal functional inequalities for fractional operators on the sphere and applications[END_REF]Corollary 2.3]. Here we draw some consequences in standard norms for nonfractional operators and identify estimates of the stability constant in the corresponding stability result.

⊳ The case p ≠ 2. Let x = d p ∈ ((d -2) 2, d ] if d ≥ 2 and x ∈ (0,d] if d = 1. We consider ξ j (x) ∶= γ j (x) -1 j ( j + d -1) and h j (x) = j ( j + d -1) (j + d -x) ( j + 1) (j + d )(j + x)
; notice that γ j (x) > 1 for x < d 2, while γ j (x) < 1 for x > d 2. An elementary computation shows that 0 < h j (x) < 1. Since γ j +1 (x)λ κ = h j (x)λ j +1 γ j (x), we obtain (2.3)

ξ j +1 (x) = h j (x)ξ j (x) + (1 -h j (x))ξ ⋆ j (x),
where

ξ ⋆ j (x) ∶= 1 1 -h j (x) h j (x) λ j - 1 λ j +1 = d -2 x j ( j + d )(2 x -d + 2) + d x .
Notice that (ξ ⋆ j (x)) j ≥2 is a monotone decreasing sequence for any fixed, admissible value of x. We start at j = 2 with the observation that ξ

⋆ 2 (x) < ξ 2 (x) if x is admissible.
This gives, by using (2.3), the following estimate

ξ 3 (x) = h 2 (x)ξ 2 (x) + 1 -h 2 (x) ξ ⋆ 2 (x) < ξ 2 (x). Using ξ ⋆ 3 (x) < ξ ⋆ 2 (x)
, we can iterate and conclude by induction that ξ j (x) < ξ 2 (x) for all j ≥ 3. As a consequence, we obtain

sup j ≥3 ζ j (p) j ( j + d -1) < ζ 2 (p) 2 (d + 1) = p 2 (d + p) < 1 d ∀ p ∈ (1,2) ∪ (2,2 * ).
We deduce from (2.2) that

E p [F ] ≤ S d F 1 2 d µ + p 2 (d + p) ∞ j =2 j ( j + d -1) S d F j 2 d µ = 1 d S d ∇F 2 d µ + 2 d -p (d -2) 2 d (d + p) S d ∇(Id -Π 1 )F 2 d µ ,
which proves the result with k = 1 and gives the expression for C d ,p,1 .

Let us consider the case k > 1. We already know that ξ 2 (x) > ξ ⋆ 2 (x). For any j ≥ 2, we deduce from (2.3) that

ξ j +1 (x) -ξ ⋆ j +1 (x) = h j (x) ξ j -ξ ⋆ j (x) + ξ ⋆ j (x) -ξ ⋆ j +1 (x) ≥ h j (x) ξ j -ξ ⋆ j (x) because j ↦ ξ ⋆ j (x) is monotone decreasing. By induction, this proves that ξ j (x) > ξ ⋆ j (x) for any j ≥ 2. As a consequence of (2.3), j ↦ ξ j (x) is also monotone decreasing and sup j ≥k+2 ζ j (p) j ( j + d -1) < ζ k+1 (p) (k + 1) (k + d ) < 1 d ∀ p ∈ (1,2) ∪ (2,2 * ).
Altogether, for any k ≥ 1, we have

d E p [F ] ≤ S d ∇Π k F 2 d µ + d ζ k+1 (p) (k + 1) (k + d ) S d ∇F 2 d µ ,
and the stability constant in (2.1) is estimated by

C d ,p,k = 1 - d ζ k+1 (p) (k + 1) (k + d ) .
In our method, this constant cannot be improved as shown by a test function such that F j = 0 for any j ∈ N such that j ≠ 0 and j ≠ k + 1, but this does not prove the optimality of C d ,p,k .

⊳ The case p = 2. By taking the limit as p → 2 + in (2.2), we obtain that

η j ∶= 2 d lim p→2 + ζ j (p) = ψ( j + d 2) -ψ(d 2) ,
where

ψ(z) = Γ ′ (z) Γ(z) is the digamma function, and 1
2 S d F 2 log ⎛ ⎝ F 2 F 2 L 2 (S d ) ⎞ ⎠ d µ ≤ d 2 ∞ j =1 η j S d F j 2 d µ ∀F ∈ H 1 (S d , d µ) .
From ψ(z +1) = ψ(z)+1 z obtained by differentiating the identity Γ(z +1) = z Γ(z) with respect to z, we learn that

η j +1 = η j + 2 d + 2 j .
We claim that

η 2 ≤ η j ≤ 2 λ j d (d + 2) ∀ j ≥ 2
because there is equality for j = 2 as η 2 = 4 (d+1) d (d+2) and λ 2 = 2 (d + 1) on the one hand, and

η j +1 -η j = 2 d + 2 j ≤ 2 (d + 2 j ) d (d + 2) = 2 (λ j +1 -λ j ) d (d + 2)
on the other hand, so that the result follows by induction.

Using λ j +1 = λ j + (d + 2 j ) = λ j + 2 z j where z j ∶= j + d 2, we also have

η j +1 λ j +1 = η j + 1 z j λ j + 2 z j < η j λ j ,
where the inequality follows from

z 2 j > λ j 2 η j ∀ j ≥ 1 .
This inequality is indeed true for j = 1 because η 1 = 2 d and we obtain the result for any j ≥ 1 by induction using

η j +1 -η j = 2 d + 2 j ≥ λ j +1 2 z 2 j +1 - λ j 2 z 2 j = 2 4 j 2 + 2 (d 2 + 2) j + d 3 (d + 2 j ) 2 (d + 2 + 2 j ) 2 .
Altogether, for any k ≥ 1, we have

d E 2 [F ] ≤ S d ∇Π k F 2 d µ + d η k+1 (k + 1) (k + d ) S d ∇F 2 d µ
and the constant in (2.1) is given by

C d ,2,k = 1 - d η k+1 (k + 1) (k + d ) .
In the framework of our method, this estimate of the constant cannot be improved as shown by a test function such that F j = 0 for any j ∈ N such that j ≠ 0 and j ≠ k + 1, but again this does not prove the optimality of C d ,p,k .

IMPROVEMENTS BY THE carré du champ METHOD

We improve upon Frank's stability result in [START_REF] Frank | Degenerate stability of some Sobolev inequalities[END_REF] by giving a constructive estimate based on the carré du champ method, without assuming any additional constraint. Various computations that are needed for a complete proof, most of them already known in the literature, are collected in Appendix B.

A simple estimate based on the heat flow, below the Bakry-Emery exponent.

Let us consider the constant γ given by (3.1)

γ ∶= d -1 d + 2 2 (p -1) (2 # -p) if d ≥ 2 , γ ∶= p -1 3 if d = 1 , where 2 # ∶= 2 d 2 +1 (d-1) 2 is the Bakry-Emery exponent. Notice that γ = 2 -p with 1 ≤ p ≤ 2 # means that d = 1 and p = 7 4 = p * (1), d > 1 and p = p * (d) ∶= 3 + d + 2 d 2 -2 √ 4 d + 4 d 2 + d 3 (d -1) 2 . Let us define (3.2) s ⋆ ∶= 1 p -2 if p > 2 and s ⋆ ∶= +∞ if p ≤ 2 .
For any s ∈ [0, s ⋆ ), let

(3.3) ϕ(s) = 1-(p-2) s-(1-(p-2) s) - γ p-2 2-p-γ if γ ≠ 2 -p and p ≠ 2 , ϕ(s) = 1 2-p (1 + (2 -p) s) log (1 + (2 -p) s) if γ = 2 -p ≠ 0 , ϕ(s) = 1 γ (e γ s -1) if p = 2 .
In [19, Theorem 2.1] (also see [START_REF] Dolbeault | Improved interpolation inequalities on the sphere[END_REF] and earlier related references therein) the improved Gagliardo-Nirenberg inequalities

(3.4) ∇F 2 L 2 (S d ) ≥ d ϕ ⎛ ⎝ E p [F ] F 2 L p (S d ) ⎞ ⎠ F 2 L p (S d ) ∀F ∈ H 1 (S d )
are stated with γ given by (3.1) under the conditions

d ≥ 1 and 1 ≤ p ≤ 2 # if d ≥ 2 , p ≥ 1 if d = 1 .
Why this estimate is based on the heat flow is explained in Appendix B. Additional justifications and discussion of the case p = 2 are also given in Appendix B. Since ϕ(0) = 0, ϕ ′ (0) = 1, and ϕ is convex increasing, with an asymptote at

s = s ⋆ if p ∈ (2,2 # ), we know that ϕ ∶ [0, s ⋆ ) → R + is invertible and ψ ∶ R + → [0, s ⋆ ), s ↦ ψ(s) ∶= s -ϕ -1 (s), is convex increasing with ψ(0) = ψ ′ (0) = 0, lim t →+∞ t -ψ(t ) = s ⋆ , and
ψ ′′ (0) = ϕ ′′ (0) = (d -1) 2 (d + 2) 2 2 # -p (p -1) > 0 ∀ p ∈ (1,2 # ).
Proposition 8. With the above notation, d ≥ 1 and p ∈ (1,2 # ), we have

∇F 2 L 2 (S d ) -d E p [F ] ≥ d F 2 L p (S d ) ψ ⎛ ⎝ 1 d ∇F 2 L 2 (S d ) F 2 L p (S d ) ⎞ ⎠ ∀F ∈ H 1 (S d ).
If p = 2, notice that ψ is explicit and given by ψ(t ) ∶= t -

1 γ log(1 + γ t ) ∀ t ≥ 0 .
The proof of Proposition 2 follows from the observation that ψ(t ) ≥ γ 2 t 2 1+γ t for any t ≥ 0.

ϕ m,p (s) ∶= s 0 exp -ζ (1 -(p -2) z) 1-δ -(1 -(p -2) s) 1-δ d z , provided m ∈ A p ∶= A p ∶= m ∈ [m -(d, p), m + (d, p)] ∶ 2 p ≤ m < 1 if p < 4 , where (3.6
)

m ± (d, p) ∶= 1 (d + 2) p d p + 2 ± d (p -1) 2 d -(d -2) p ,
while the parameters δ and ζ are defined by

δ ∶= 1 + (m -1) p 2 4 (p -2) , ζ ∶= (d + 2) 2 p 2 m 2 -2 p (d + 2) (d p + 2) m + d 2 5 p 2 -12 p + 8 + 4 d (3 -2 p) p + 4 (1 -m) (d + 2) 2 p 2 .
Let s ⋆ ∶= 1 (p -2) as in (3.2) and consider the inverse function ϕ -1 m,p ∶ R + → [0, s ⋆ ) and ψ m,p (s) ∶= s -ϕ -1 m,p (s). Exactly as in the case m = 1, we have the improved entropyentropy production inequality

∇F 2 L 2 (S d ) ≥ d F 2 L p (S d ) ϕ m,p ⎛ ⎝ E p [F ] F 2 L p (S d ) ⎞ ⎠ ∀F ∈ H 1 (S d ),
which provides us with the following stability estimate.

Proposition 9. With above notation, d ≥ 1, p ∈ (2,2 * ) and m ∈ A p , we have ∇F 2 L 2 (S d ) -d E p [F ] ≥ d F 2 L p (S d ) ψ m,p ⎛ ⎝ ∇F 2 L 2 (S d ) d F 2 L p (S d ) ⎞ ⎠ ∀F ∈ H 1 (S d ).
The function ϕ m,p can be expressed in terms of the incomplete Γ function, while ψ m,p is known only implicitly.

Comparison with other estimates.

Let us assume that p ∈ (2,2 * ). In [START_REF] Frank | Degenerate stability of some Sobolev inequalities[END_REF], Frank proves the existence of a positive constant c ⋆ (d, p) such that

∇F 2 L 2 (S d ) -d E p [F ] ≥ c ⋆ (d, p) ∇F 2 L 2 (S d ) + F -F 2 L 2 (S d ) 2 ∇F 2 L 2 (S d ) + d p-2 F 2 L 2 (S d ) ∀F ∈ H 1 (S d , d µ) ,
where F ∶= ∫ S d F d µ, which in particular implies the existence of a positive constant c(d, p) such that

(3.7) ∇F 2 L 2 (S d ) -d E p [F ] ≥ c(d, p) ∇F 4 L 2 (S d ) ∇F 2 L 2 (S d ) + d p-2 F 2 L 2 (S d ) ∀F ∈ H 1 (S d , d µ) ,
for all p ∈ (2,2 * ). The value of the constant c ⋆ (d, p) found in [START_REF] Frank | Degenerate stability of some Sobolev inequalities[END_REF] is unknown as it follows from a compactness argument, in the spirit of [START_REF] Bianchi | A note on the Sobolev inequality[END_REF], but the exponent 4 in the right-hand side of (3.7) is optimal. With the test functions

F ε (x) = 1 + ε x ⋅ ν for some arbitrary ν ∈ S d , we can indeed check that lim ε→0 1 ε 4 ∇F ε 2 L 2 (S d ) -d E p [F ε ] = (d + p) (p -1) 2 d (d + 3)
, which gives the upper bounds

c(d, p) ≤ (p -1) (d + p) 2 (p -2) (d + 3) and c ⋆ (d, p) ≤ d 2 (d + 1) 2 (p -1) (d + p) 2 (p -2) (d + 3) . Let us notice that ∇F 2 L 2 (S d ) ≥ d F -F 2 L 2 (S d
) by the Poincaré inequality, so that we have We claim that the carré du champ method provides us with a constructive estimate of c(d, p). Let

∇F 2 L 2 (S d ) + F -F 2 L 2 (S d ) 2 ≥ ∇F 4 L 2 (S d ) ≥ d 2 (d + 1) 2 ∇F 2 L 2 (S d ) + F -F 2 L 2 (S d )
φ c (s) ∶= d 2 (1 -c) 2 c s -s ⋆ + s 2 ⋆ + 4 c s (s -s ⋆ ) .
c = sup c > 0 ∶ ∃m ∈ A p such that φ c (s) ≤ ϕ m,p (s) ∀ s ∈ [0, s ⋆ ) .
Proof. With no loss of generality, let us assume that

F L p (S d ) = 1 and define i = ∇F 2 L 2 (S d ) and e ∶= 1 -F 2 L 2 (S d ) p -2 , so that F 2 L 2 (S d ) = 1 -(p -2) e. With c = c(d, p), we can rewrite (3.7) as i -d e ≥ c i 2 i + d p-2 -d e , which amounts to i -d e ≥ φ c (e).
Since we know that id e ≥ ϕ m,p (e), the conclusion follows for the largest possible c > 0 such that ϕ m,p ≥ φ c .

GLOBAL STABILITY RESULTS

We collect the statements of Theorems 3 and 6 into a single result. The whole section is devoted to its proof.

S d ∇F 2 d µ -d E p [F ] ≥ S d ,p ⎛ ⎝ ∇Π 1 F 4 L 2 (S d ) ∇F 2 L 2 (S d ) + F 2 L 2 (S d ) + ∇(Id -Π 1 )F 2 L 2 (S d ) ⎞ ⎠
for some explicit stability constant S d ,p > 0.

The value of S d ,p is elementary and explicit but its expression is lengthy. We explain in the proof how to compute it with all necessary details to obtain a numerical expression for S d ,p for given p and d , if needed.

Proof of Theorem 11. By homogeneity of (4.1), we can assume that F L 2 (S d ) = 1 without loss of generality. For clarity, we subdivide the proof into various steps. Let us start with the case p > 2.

• An estimate based on the carré du champ method.

If ∇F 2 L 2 (S d ) F 2 L p (S d ) ≥ ϑ 0 > 0,
we know by the convexity of ψ m,p that (4.2)

∇F 2 L 2 (S d ) -d E p [F ] ≥ d F 2 L p (S d ) ψ m,p 1 d ∇F 2 L 2 (S d ) F 2 L p (S d ) ≥ d ϑ 0 ψ m,p ϑ 0 d ∇F 2 L 2 (S d ) .
In that case, we conclude from ∇F

2 L 2 (S d ) = ∇Π 1 F 2 L 2 (S d ) + ∇(Id -Π 1 )F 2 L 2 (S d )
and

∇Π 1 F 2 L 2 (S d ) ≥ ∇Π 1 F 4 L 2 (S d ) ∇F 2 L 2 (S d ) + F 2 L p (S d )
.

Let us assume now that ∇F

2 L 2 (S d ) < ϑ 0 F 2 L p (S d )
. By taking into account (GN), we obtain

∇F 2 L 2 (S d ) < ϑ 0 F 2 L p (S d ) ≤ ϑ 0 F 2 L 2 (S d ) + p -2 d ∇F 2 L 2 (S d ) .
Using F L 2 (S d ) = 1, under the assumption that ϑ 0 < d (p -2), we know that

(4.3) ϑ ∶= ∇F 2 L 2 (S d ) < d ϑ 0 d -(p -2) ϑ 0 .
Notice that the parameter ϑ 0 still has to be chosen.

• An estimate of the average. Let us estimate Π 0 F ∶= ∫ S d F d µ. By the Poincaré inequality, we have

1 = F 2 L 2 (S d ) = S d F d µ 2 + (Id -Π 0 )F 2 L 2 (S d ) ≤ S d F d µ 2 + ϑ d ,
and on the other hand we know that

∫ S d F d µ 2 ≤ F 2 L 2 (S d ) = 1 by the Cauchy-Schwarz inequality, so that (4.4) d -ϑ d < S d F d µ 2 ≤ 1 .
We assume in the sequel that (4.5)

ϑ < d .
• Partial decomposition on spherical harmonics. With no loss of generality, let us write (4.6)

F = M (1 + ε Y + ηG) such that M = Π 0 F and Π 1 F = ε M Y where Y (x) = d +1 d x ⋅ ν for some given ν ∈ S d .
Here the functions Y and G are normalized so that ∇Y L 2 (S d ) = ∇G L 2 (S d ) = 1 and

M -2 ∇F 2 L 2 (S d ) = ε 2 + η 2 = ϑ and M -2 F 2 L 2 (S d ) = 1 + 1 d ε 2 + η 2 G 2 L 2 (S d ) .
We observe that Π 0 (F -M ) = 0. Using (GN) and the Poincaré inequality, we have

F -M 2 L p (S d ) ≤ F -M 2 L 2 (S d ) + p -2 d ∇F 2 L 2 (S d ) ≤ p -1 d ∇F 2 L 2 (S d ) .
Similarly, by (2.1), i.e., d p

2 (d + p) ∇G 2 L 2 (S d ) = 1 - 2 d -p (d -2) 2 (d + p) ∇G 2 L 2 (S d ) ≥ d E p [G],
and the improved Poincaré inequality (2.1) written with p = 1 and k = 1

G 2 L 2 (S d ) ≤ 1 2 (d + 1) ∇G 2 L 2 (S d ) = 1 2 (d + 1)
, we have

G 2 L p (S d ) ≤ G 2 L 2 (S d ) + p (p -2) 2 (d + p) ∇G 2 L 2 (S d ) ≤ C p,d using ∇G L 2 (S d ) = 1, with C p,d ∶= 1 2 (d+1) + p (p-2)
2 (d+p) . By the Cauchy-Schwarz inequality, we also have

G L 1 (R d ) ≤ 1 2 (d + 1)

,

We recall that the eigenvalues of -∆ on S d are λ k = k (k + d -1) with k ∈ N. In preparation for a detailed Taylor expansion as in [START_REF] Frank | Degenerate stability of some Sobolev inequalities[END_REF], let us consider the function

Y (x) ∶= d + 1 d x ⋅ ν , which is such that -∆Y = λ 1 Y with λ 1 = d and ∇Y 2 L 2 (S d ) = 1 , Y 2 L 2 (S d ) = 1 d , Y 4 L 4 (S d ) = 3 (d + 1) (d + 3) d 2 , Y 6 L 6 (S d ) = 15 (d + 1) 2 (d + 3) (d + 5) d 2 .
The function

Y 2 ∶= Y 2 -1 d is such that -∆Y 2 = λ 2 Y 2 with λ 2 = 2 (d + 1)
and

Y 2 2 L 2 (S d ) = 2 d (d + 3) , ∇Y 2 2 L 2 (S d ) = 4 (d + 1) d (d + 3) . The function Y 3 ∶= Y 3 - 3 (d+1) d (d+3) Y is such that -∆Y 3 = λ 3 Y 3 with λ 3 = 3 (d + 2) and Y 3 2 L 2 (S d ) = 6 (d + 1) 2 (d + 5) (d + 3) 2 d 2 , ∇Y 3 2 L 2 (S d ) = 18 (d + 2) (d + 1) 2 (d + 5) (d + 3) 2 d 2 .
As a consequence of (4.6), we know that

Π 0 G = Π 1 G = 0 and ∇G L 2 (S d ) = 1. Let g 2 ∶= ∫ S d ∇Y 2 ⋅ ∇G d µ ∇Y 2 L 2 (S d ) and g 3 ∶= ∫ S d ∇Y 3 ⋅ ∇G d µ ∇Y 3 L 2 (S d ) . With k = 1, 2, using -∆Y k = λ k Y k with λ k = ∇Y k 2 L 2 (S d ) Y k 2 L 2 (S d ) , we compute S d Y k G d µ = S d Y k G d µ = Y k 2 L 2 (S d ) ∇Y k 2 L 2 (S d ) S d ∇Y k ⋅ ∇G d µ = g k Y k 2 L 2 (S d ) ∇Y k L 2 (S d )
and obtain

S d Y 2 G d µ = S d Y 2 G d µ = g 2 d (d + 1) (d + 3) , S d Y 3 G d µ = S d Y 3 G d µ = c 3 g 3 with c 3 ∶= d + 1 d (d + 3) 2 (d + 2) (d + 5)
.

• Taylor expansions [START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF]. Let us start with elementary estimates of 1 + ε Y L p (S d ) . If it holds that 2 ≤ p < 3 and s < 1, we have 1 2

(1 + s) p + (1 -s) p ≤ 1 + p 2 (p -1) s 2 1 + 1 12 (p -2) (p -3) s 2
because all other terms in the series expansion of the left-hand side around s = 0 correspond to even powers of s and appear with nonpositive coefficients. If either 1 ≤ p < 2 or p > 3 and s < 1 2, let We adapt the convention that c

f p (s) ∶= 1 2 (1 + s) p + (1 -s) p -1 + p 2 (p -1) s 2 and notice that f ′′ p (s) = p 2 (p -1) (1 + s) p-2 + (1 -s) p-2 -2 ≥
(+) p = 0 if p ∈ [2,3). Using the fact that Y (-x) = -Y (x), 1 + ε Y p L p (S d ) = 1 2 1 + ε Y p L p (S d ) + 1 -ε Y p L p (S d )
.

For any ε ∈ (0,1 2) we use (4.7) to write

1 + ε Y p L p (S d ) -1 + p 2 (p -1) Y 2 L 2 (S d ) + 1 12 (p -2) (p -3) Y 4 L 4 (S d ) ε 2 ε 2 ≤ c (+) p Y 6 L 6 (S d ) ε 6 .
For similar reasons, one can prove that there is another constant c

p,d ε 6 ≤ 1 + ε Y p L p (S d ) -1 + a p,d ε 2 + b p,d ε 4 ≤ c (+) p,d ε 6 , with a p,d ∶= p (p -1) 2 d , b p,d ∶= 1 4 (p -2) (p -3) d + 1 d (d + 3) a p,d , c (±) p,d ∶= 15 (d + 1) 2 (d + 3) (d + 5) d 2 c (±) p .
Estimate (4.8) is valid under the condition that ε < 1 2. We shall therefore request that (4.9)

ϑ < 1 4 ,
which is an obvious sufficient condition according to (4.3). Now we draw two consequences of (4.8). First, let us give an upper estimate of 1

+ ε Y 2 L p (S d ) .
Using

(1 + s) 2 p ≤ 1 + 2 s p -(p -2) s 2 p 2 + 2 3 (p -1) (p -2)
s 3 p 3 ,
we obtain (4.10)

1 + ε Y 2 L p (S d ) ≤ 1 + 2 p a p,d ε 2 + 1 p 2 2 p b p,d -(p -2) a 2 p,d ε 4 + r (+) ε 6 ,
where the remainder term r (+) is explicitly estimated by

96 p 3 r (+) = 64 a 3 p,d p 2 -3 p + 2 + 48 a 2 p,d p 2 -3 p + 2 (2b p,d + c p,d ) + 12 a p,d (p -2) (2b p,d + c p,d )(2b p,d (p -1) + c p,d (p -1) -8 p) + 8 b 3 p,d p 2 -3 p + 2 + 12 b 2 p,d (p -2) (c p,d (p -1) -4 p) + 6 b p,d c p,d (p -2) (c p,d (p -1) -8 p) + c p,d c 2 p,d p 2 -3 p + 2 -12 c p,d (p -2) p + 192 p 2 .
To do this estimate, we simply write that ε α ≤ 2 6-α ε 6 for any α > 6 using the (nonoptimal) bound ε 2 < 1 2. Similarly, using

(1 + s) 2 p -1 ≤ 1 -(p -2) s p + (p -1) (p -2) s 2 p 2 - 1 3 (p -1) (p -2) (3 p -2)
s 3 p 3 + 1 6 (p -1) (p -2) (3 p -2) (2 p -1)
s 4 p 4 ,
we obtain (4.11)

1 + ε Y 2-p L p (S d ) ≤ 1 + p -2 p a p,d ε 2 - p -2 p 2 p b p,d -(p -1) a 2 p,d ε 4 + r (-) ε 6 ,
where the remainder term r (-) also has an explicit expression in terms of a p,d , b p,d and c (-) p,d , which is not given here.

• Taylor expansions [START_REF] Arnold | Refined convex Sobolev inequalities[END_REF]. With u ≥ 0, u + r ≥ 0 and p > 2, we claim that

(u + r ) p ≤ u p + p u p-1 r + p 2 (p -1) u p-2 r 2 + 2<k<p C p k u p-k r k + K p r p
for some constant K p > 0, where the coefficients

C p k ∶= Γ(p + 1) Γ(k + 1) Γ(p -k + 1)
are the binomial coefficients if p is an integer. It is proved in [START_REF] Dolbeault | Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence[END_REF] that K p = 1 if p ∈ (2,4] ∪ {6}. The proof is similar to the above analysis and is left to the reader. Let us integrate this inequality and raise both sides to the power 2 p to get

u + r 2 L p (S d ) ≤ u 2 L p (S d ) (1 + s) 2 p , with s = 1 u p L p (S d ) p S d u p-1 r d µ + p 2 (p -1) S d u p-2 r 2 d µ + 2<k<p C p k S d u p-k r k d µ + K p S d r p d µ ⎞ ⎠ .
By assumption 2 p < 1 so that we may use the identity (1 + s) 2 p ≤ 1 + 2 s p for any s ≥ -1. Notice that we can assume that u + r ≥ 0 and deduce from (1) that s ≥ -1. As a consequence, we have

u + r 2 L p (S d ) ≤ u 2 L p (S d ) + 2 p u 2-p L p (S d ) p S d u p-1 r d µ + p 2 (p -1) S d u p-2 r 2 d µ + 2<k<p C p k S d u p-k r k d µ + K p S d r p d µ .
We apply these computations to u = 1 + ε Y and r = ηG to obtain

M -2 F 2 L p (S d ) -1 + ε Y 2 L p (S d ) ≤ 2 p 1 + ε Y 2-p L p (S d ) η p S d (1 + ε Y ) p-1 G d µ + p 2 (p -1) η S d (1 + ε Y ) p-2 G 2 d µ + 2<k<p C p k η k S d (1 + ε Y ) p-k G k d µ + K p η p S d G p d µ .
Let us detail the expansion of each of the terms involving G in the right-hand side of this estimates. For any s ∈ (-1 2, 1 2), using the expansion

(1 + s) p-1 ≤ 1 + (p -1) s + 1 2 (p -1) (p -2) s 2 + 1 6 (p -1) (p -2) (p -3) s 3 + R p s 4
for some constant R p > 0 applied with s = 1 + ε Y , we obtain

η S d (1 + ε Y ) p-1 G d µ ≤ 1 2 (p -1) (p -2) g 2 d (d + 1) (d + 3) η ε 2 + 1 6 (p -1) (p -2) (p -3) c 3 g 3 η ε 3 + R p η ε 4 2 (d + 1)
.

The other terms admit simpler expansions:

η 2 S d (1 + ε Y ) p-2 G 2 d µ ≤ η 2 (1 + ε) p-2 G 2 L 2 (S d ) ≤ G 2 L 2 (S d ) η 2 + 1 2 (d + 1) η 2 (1 + ε) p-2 -1 and 2<k<p C p k η k S d (1 + ε Y ) p-k G k d µ + K p η p S d G p d µ ≤ 2<k<p C p k η k (1 + ε) p-k G k L p (S d ) + K p η p G p L p (S d ) ≤ 2<k<p C p k η k (1 + ε) p-k C k p p,d + K p η p C p,d .
Collecting (4.10) and (4.11) with the above estimates, we arrive at

M -2 F 2 L p (S d ) ≤ 1 + 2 p a p,d ε 2 + 1 p 2 2 p b p,d -(p -2) a 2 p,d ε 4 + r (+) ε 6 + 1 + p -2 p a p,d ε 2 - p -2 p 2 p b p,d -(p -1) a 2 p,d ε 4 + r (-) ε 6 ⋅ (p -1) (p -2)
g 2 d (d + 1) (d + 3) η ε 2 + 1 3 (p -1) (p -2) (p -3) c 3 g 3 η ε 3 + 2 R p η ε 4 2 (d + 1) + (p -1) G 2 L 2 (S d ) η 2 + 1 2 (d + 1) η 2 (1 + ε) p-2 -1 + 2 p ⎛ ⎝ 2<k<p C p k η k (1 + ε) p-k C k p p,d + K p η p C p,d ⎞ ⎠
.

Using g 2 < 1, g 3 < 1, and 2 (d + 1) G 2 L 2 (S d ) < 1, this gives rise to an explicit although lengthy expression for a positive constant R p,d such that

M -2 S d ∇F 2 d µ -d E p [F ] ≥ A ε 4 -B ε 2 η +C η 2 -R p,d ϑ p + ϑ 5 2 , with A ∶= (p-1) (d+p) 2 d (d+3) , B ∶= d (p-1) d (d+1) (d+3) and C ∶= d +2 2 (d+1) . The discriminant B 2 -4 A C = - 1 d (d + 3) (p -1) 2 d -p (d -2)
is negative if (and only if) p ∈ (1,2 * ), so that we can write

A s 2 -B s +C = (A -λ) s 2 -B s + (C -λ) + λ s 2 + 1 ≥ λ s 2 + 1 ,
where

λ ∶= 1 2 A +C + (A -C ) 2 + B 2
is given by the condition that B 2 -4 (A -λ) (C -λ) = 0. Altogether, we obtain

M -2 S d ∇F 2 d µ -d E p [F ] ≥ λ ε 4 + η 2 -R p,d ϑ p + ϑ 5 2 .
• Conclusion if p > 2. We choose ϑ > 0 such that (4.5) and (4.9) are fulfilled. With the additional assumption that

ϑ ≤ ϑ p,d ∶= θ > 0 ∶ R p,d θ p + θ 5 2 = λ 4 θ 2 , using η 4 ≤ η 2 and 2 ε 2 η 2 ≤ ε 4 + η 2 if η < 1, we have R p,d ϑ p + ϑ 5 2 ≤ λ 4 ϑ 2 = λ 4 ε 2 + η 2 2 ≤ λ 2 ε 4 + η 2 ≤ λ 2 ε 4 ε 2 + η 2 + 1 + η 2 .
For any

F such that ∇F 2 L 2 (S d ) = ϑ, we obtain S d ∇F 2 d µ -d E p [F ] ≥ λ 2 ⎛ ⎝ ∇Π 1 F 4 L 2 (S d ) ∇F 2 L 2 (S d ) + F 2 L p (S d ) + ∇(Id -Π 1 )F 2 L 2 (S d ) ⎞ ⎠ .
Using (4.2) and (4.4), this completes the proof of Theorem 11 if p > 2 with

ϑ ≤ min d 2 , 1 4 , ϑ p,d = d ϑ 0 d -(p -2) ϑ 0 and S d ,p = min d ϑ 0 ψ m,p ϑ 0 d , λ
2 .

• The case p ≤ 2. The strategy is the same, with some simplifications, so we only sketch the proof and emphasize the changes compared to the case p > 2. Let us notice that

(1 + s) p ≤ 1 + p s + p 2 (p -1) s 2 if 1 ≤ p < 2 and (1+ s) 2 log (1+ s) 2 ≤ 2 s +2 s 2 + 2 3 s 3 in the limit case p = 2.
The estimates involving 1+ε Y are therefore essentially the same if we assume ε < 1 2, while the computation of u + r 2 L p (S d ) is in fact simpler, when applied to u = 1 + ε Y and r = ηG. The estimate on the average is simplified because

F L p (S d ) ≤ F L 2 (S d
) by Hölder's inequality, since d µ is a probability measure on S d . Spectral estimates are exactly the same and the Taylor expansions present no additional difficulty, as we can use (GN) for some exponent q ∈ (2,2 * ) to control the remainder terms if p = 2, so that (1 + s) 2 log (1 + s) 2 ≤ 2 s + 2 s 2 + κ q s q for some κ q > 0. The conclusion is the same as for p > 2 except that we have to replace ψ m,p by ψ defined as in Proposition 8.

APPENDIX A. IMPROVED GAUSSIAN INEQUALITIES, HYPERCONTRACTIVITY AND STABILITY

Whether the results of Theorems 1, 4 and 7 can be extended to Euclidean case with the Gaussian measure is a very natural question. Spherical harmonics can indeed be replaced by Hermite polynomials and there is a clear correspondence for spectral estimates. The answer is yes for a whole family of interpolation inequalities, but it is no for the logarithmic Sobolev inequality, which is an endpoint of the family.

Let us consider the normalized Gaussian measure on R d defined by

d σ(x) = (2π) -d 2 e -1 2 x 2 d x .
For any p ∈ [1,2), Beckner in [START_REF] Beckner | A generalized Poincaré inequality for Gaussian measures[END_REF] established the family of interpolation inequalities (A.1)

f 2 L 2 (R d ,d σ) -f 2 L p (R d ,d σ) 2 -p ≤ ∇f 2 L 2 (R d ,d σ) ∀ f ∈ H 1 (R d , d σ) .
With p = 1, inequality (A.1) is the Gaussian Poincaré inequality while one recovers the Gaussian logarithmic Sobolev inequality of [START_REF] Gross | Logarithmic Sobolev inequalities[END_REF] in the limit as p → 2. For any p ∈ [1,2), the inequality is optimal: using f ε ∶= 1 + ε ϕ as a test function, where ϕ is such that ∫ R d ϕ d σ = 0, we recover the Gaussian Poincaré inequality with optimal constant in the limit as ε → 0, so that the constant in (A.1) cannot be improved. Based on [START_REF] Latała | Between Sobolev and Poincaré[END_REF][START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF], the improved version of the inequality

(A.2) f 2 L 2 (R d ,d σ) -f 2 L p (R d ,d σ) 2 -p ≤ p 2 ∇ f 2 L 2 (R d ,d σ) ∀ f ∈ H 1 (R d , d σ)
holds under the additional condition

(A.3) R d x f (x)dσ = 0 .
Let us give a short proof of (A.2). Assume that f = ∑ k∈N f k is a decomposition on Hermite functions such that Lf k = -k f k where L = ∆x ⋅ ∇ is the Ornstein-Uhlenbeck operator, and let

a k ∶= f k 2 L 2 (R d ,d σ) for any k ∈ N, so that f 2 L 2 (R d ,d σ) = k∈N a k and ∇ f 2 L 2 (R d ,d σ) = k∈N k a k .
Let us consider the solution of (A. [START_REF] Émery | Inégalités de Sobolev pour un semi-groupe symétrique[END_REF])

∂u ∂t = Lu with initial datum u(t = 0, ⋅) = f and notice that u(t , ⋅) 2 L 2 (R d ,d σ) = k∈N a k e -2 k t .
Hence, if (A.3) holds, a 1 = 0 and

(A.5) f 2 L 2 (R d ,d σ) -u(t , ⋅) 2 L 2 (R d ,d σ) = k≥2 a k 1 -e -2 k t ≤ 1 2 1 -e -4 t k∈N k a k = 1 2 1 -e -4 t ∇f 2 L 2 (R d ,d σ)
because k ↦ 1e -2 k t k is monotone nonincreasing for any given t ≥ 0. Next, we use Nelson's hypercontractivity estimate in [START_REF] Nelson | The free Markoff field[END_REF]Theorem 3] to find t * > 0 such that

u(t * , ⋅) 2 L 2 (R d ,d σ) ≤ f 2 L p (R d ,d σ) .
As noted in [START_REF] Gross | Logarithmic Sobolev inequalities[END_REF], this estimate can be seen as a consequence of the Gaussian logarithmic Sobolev inequality

(A.6) R d v 2 log ⎛ ⎝ v 2 v 2 L 2 (R d ,d σ) ⎞ ⎠ d σ ≤ 2 R d ∇v 2 d σ ∀ v ∈ H 1 (R d , d σ) ,
and the argument goes as follows. With h(t

) ∶= u(t , ⋅) L q(t ) (R d ,d σ)
for some exponent q depending on t and u solving (A.4), we have

h ′ h = q ′ q 2 R d u q h q log u q h q d σ - 4 h q q -1 q 2 R d ∇ u q 2 2 d σ ≤ 0 by (A.6) applied to v = u q 2 , if t ↦ q(t ) solves the ordinary differential equation q ′ = 2 (q -1) .
With q(0) = p < 2, we obtain q(t ) = 1 + (p -1) e 2t and find that Nelson's time t * is determined by the condition q(t * ) = 2 which means e -2t * = p -1. Replacing t = t * in (A.5) completes the proof of (A.2), which can be recast in the form of a stability result for (A.1).

Theorem 12. Let d ≥ 1 and p ∈ [1,2). For any f ∈ H 1 (R d , d σ) such that (A.3) holds, ∇f 2 L 2 (R d ,d σ) - 1 2 -p f 2 L 2 (R d ,d σ) -f 2 L p (R d ,d σ) ≥ 2 -p 2 ∇f 2 L 2 (R d ,d σ) .
As a byproduct of the proof, with t = t * in (A.5), we have the mode-by-mode interpolation inequality

f 2 L 2 (R d ,d σ) -f 2 L p (R d ,d σ) 2 -p ≤ k≥1 1 -(p -1) k k (2 -p) ∇ f k 2 L 2 (R d ,d σ) ∀ f ∈ H 1 (R d , d σ) ,
without imposing condition (A.3), for any p ∈ [START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF][START_REF] Arnold | Refined convex Sobolev inequalities[END_REF].

For any k ≥ 1, lim p→2 - 1 -(p -1) k k (2 -p) = lim p→2 - 1 -1 -(2 -p) k k (2 -p) = 1 ,
so that no improvement should be expected by this method. This is very similar to the case of the critical exponent on the sphere of dimension d ≥ 3. In this sense p = 2 is the critical case in the presence of a Gaussian weight, as all modes are equally involved in the estimate of the constant. This is a limitation of the method which does not forbid a stability result for (A.6), to be established by other methods.

Let us conclude this appendix with some bibliographic comments on the literature on inequality (A.1), for the Gaussian measure. The analogue of Proposition 5 in the Gaussian case is known from [START_REF] Arnold | Refined convex Sobolev inequalities[END_REF]; also see [START_REF] Dolbeault | ϕ-entropies: convexity, coercivity and hypocoercivity for Fokker-Planck and kinetic Fokker-Planck equations[END_REF]Section 2.5]). Assuming that not only condition (A.3) is satisfied, but also orthogonality conditions with all modes up to order k 0 ≥ 2, then an improvement of the order of

1 -(p -1) k 0 k 0 (2 -p)
can be achieved for inequality (A.1), which is the counterpart of Theorem 4 in the Gaussian case. This has been studied in [START_REF] Latała | Between Sobolev and Poincaré[END_REF] but we can refer to [START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF] for a more abstract setting and later papers, e.g., to [START_REF]From super Poincaré to weighted log-Sobolev and entropy-cost inequalities[END_REF][START_REF] Wang | A generalization of Poincaré and log-Sobolev inequalities[END_REF] for results on compact manifolds and generalizations involving weights. For an overview of interpolation between Poincaré and logarithmic Sobolev inequalities from the point of view of Markov processes, and for some spectral considerations, we refer to [START_REF]Functional inequalities, Markov semigroups and spectral theory[END_REF]Chapter 6]. Notice that hypercontractivity appears as one of the main motivations of the founding paper [START_REF] Émery | Diffusions hypercontractives[END_REF] of the carré du champ method.

APPENDIX B. Carré du champ METHOD AND IMPROVED INEQUALITIES

For sake of completeness, we collect various results of [START_REF] Dolbeault | Nonlinear flows and rigidity results on compact manifolds[END_REF][START_REF] Dolbeault | Improved interpolation inequalities on the sphere[END_REF][START_REF] Dolbeault | Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities[END_REF][START_REF] Esteban | Improved interpolation inequalities and stability[END_REF] and draw some new consequences. Computations similar to those of Section B.1 can be found in [START_REF] Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF] for the study of rigidity results in elliptic equations. For nonlinear parabolic flows, also see [START_REF] Demange | Des équations à diffusion rapide aux inégalités de Sobolev sur les modèles de la géométrie[END_REF][START_REF]Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature[END_REF]. Other sections of this appendix collects results which are scattered in the literature, but additional details needed in Section 3 are given, for instance a sketch of the proof Proposition 16 or the computations in the case p = 2. We also consider the trace-free tensor

Mv ∶= ∇v ⊗ ∇v v - 1 d ∇v 2 v g d ,
where (∇v ⊗ ∇v) i j ∶= ∂ i v ∂ j v and ∇v ⊗ ∇v 2 = ∇v 4 = (g

i j d ∂ i v ∂ j v) 2 using Einstein's convention. Using L ∶ g d = 0 , M ∶ g d = 0 ,
where a ∶ b denotes a i j b i j and a 2 ∶= a ∶ a, and

Lv 2 = Hv 2 - 1 d (∆v) 2 , Mv 2 = ∇v ⊗ ∇v v 2 - 1 d ∇v 4 v 2 = d -1 d ∇v 4 v 2 ,
we deduce from

S d ∆v ∇v 2 v d µ = S d ∇v 4 v 2 d µ -2 S d Hv ∶ ∇v ⊗ ∇v v d µ = d d -1 S d Mv 2 d µ -2 S d Lv ∶ ∇v ⊗ ∇v v d µ - 2 d S d ∆v ∇v 2 v d µ
a first identity that reads (B.1)

S d ∆v ∇v 2 v d µ = d d + 2 d d -1 S d Mv 2 d µ -2 S d Lv ∶ ∇v ⊗ ∇v v d µ .
The Bochner-Lichnerowicz-Weitzenböck formula on S d takes the simple form Hence

K [v] ∶= S d ∆v + κ ∇v 2 v ∆v + (β -1) ∇v 2 v d µ = S d (∆v) 2 d µ + (κ + β -1) S d ∆v ∇v 2 v d µ + κ (β -1) S d ∇v 4
v 2 d µ can be rewritten using (B.1) and (B.2) as

K [v] = d d -1 S d Lv 2 d µ + d S d ∇v 2 d µ + (κ + β -1) d d + 2 d d -1 S d Mv 2 d µ -2 S d Lv ∶ Mv d µ + κ (β -1) d d -1 S d Mv 2 d µ = d d -1 S d Lv 2 -2 b Lv ∶ Mv + c Mv 2 d µ + d S d ∇v 2 d µ = d d -1 S d Lv -b Mv 2 + c -b 2 Mv 2 d µ + d S d ∇v 2 d µ = d d -1 S d Lv -b Mv 2 d µ + c -b 2 S d ∇v 4 v 2 d µ + d S d ∇v 2 d µ where b = (κ + β -1) d -1 d + 2 and c = d d + 2 (κ + β -1) + κ (β -1) . Let κ = β (p -2) + 1. The condition γ ∶= c -b 2 ≥ 0 amounts to (B.3) γ = d d + 2 β (p -1) + 1 + β (p -2) (β -1) - d -1 d + 2 β (p -1) 2 , where γ = -A β 2 -2 B β +C with A = d -1 d + 2 (p -1) 2 + 2 -p , B = d + 3 -p d + 2 and C = 1 .
A necessary and sufficient condition for the existence of a β such that γ ≥ 0 is that the reduced discriminant is nonnegative, which amounts to

B 2 -A C = 4 d (d -2) (d + 2) 2 p -1 (2 * -p) ≥ 0 .
Summarizing, we have the following result, which can be found in [START_REF] Dolbeault | Nonlinear flows and rigidity results on compact manifolds[END_REF] for a general manifold with positive Ricci curvature.

Lemma 13. With the above notation, for any smooth function v on S d , we have

K [v] ≥ γ S d ∇v 4 v 2 d µ + d S d ∇v 2 d µ for some γ > 0 given in terms of β by (B.3) if p ∈ (1,2 * ).
Notice that we recover the expression for γ in (3.1) if we take β = 1. The case p = 2 does not add any difficulty compared to p ≠ 2. u .

In the linear case m = 1, u p solves the heat equation. Otherwise we deal with the nonlinear case either of a fast diffusion flow with m < 1 or of a solution of the porous media equation with m > 1. We claim that

d d t u 2 L p (S d ) = 0 and d d t u 2 L 2 (S d ) = 2 (p -2) S d u -p (1-m) ∇u 2 d µ .
Let us assume that the parameters β and m are related by

(B.5) m = 1 + 2 p 1 β -1 . If v is a function such that u = v β , then v solves ∂v ∂t = v 2-2 β ∆v + κ ∇v 2 v ,
with κ = β (p -2) + 1 and as a consequence we find that

d d t u 2 L 2 (S d ) = 2 (p -2) β 2 S d ∇v 2 d µ .
Similarly, we find that

(B.6) d d t ∇u 2 L 2 (S d ) = -2 S d β v β-1 ∂v ∂t ∆v β d µ = -2β 2 K [v].
By eliminating β in (B.3) using (B.5), we obtain (B.7)

γ = γ 0 + γ 1 d + γ 2 d 2 (d + 2) 2 2 -p (1 -m) 2 , with γ 0 = 4 (m p-1) 2 , γ 1 = -4 p m-3+p (2-m)(1+m) , γ 2 = (m 2 -2 m + 5) p 2 -
δ ∶= p -(4 -p) β 2 β (p -2) if p > 2 , δ ∶= 1 if p ∈ [1,2].
Lemma 15. If one of the conditions

(i) p ∈ (1,2 # ) and β = 1 (so that δ = 1), (ii) p ∈ (2,2 * ), β > 1, and β ≤ 2 (4 -p) if p < 4, is satisfied, then u = v β is such that (B.10) S d ∇v 4 v 2 d µ ≥ 1 β 2 ∫ S d ∇u 2 d µ ∫ S d ∇v 2 d µ ∫ S d u 2 d µ δ ∫ S d u p d µ β-1 β (p-2)
. Case (ii) was originally proved in [START_REF] Demange | Des équations à diffusion rapide aux inégalités de Sobolev sur les modèles de la géométrie[END_REF][START_REF]Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature[END_REF] and we refer to [START_REF] Dolbeault | Improved interpolation inequalities on the sphere[END_REF] for a proof in the case of the ultraspherical operator.

Proof of Lemma 15. In case (i), v = u and inequality (B.10) is a consequence of the Cauchy-Schwarz inequality

S d ∇v 2 d µ = S d ∇v 2 v ⋅ v d µ ≤ S d ∇v 4 v 2 d µ 1 2 S d u 2 d µ 1 2
, Cases (i) and (ii) follow from two Hölder inequalities.

(1) With 1

2 + β-1 2 β + 1 2 β = 1, we deduce from S d ∇v 2 d µ = S d ∇v 2 v ⋅ 1 ⋅ v d µ ≤ S d ∇v 4 v 2 d µ 1 2 S d 1 d µ β-1 2 β S d u 2 d µ 1 2 β
, and the assumption that d µ is a probability measure, the first estimate .

S d ∇v 4 v 2 d µ 1 2 ≥ ∫ S d ∇v 2 d µ ∫ S d u 2 d µ 1 2 β . ( 2
The combination of our two estimates proves (B.10) with δ = δ 0 + 1 (2β).

Using (B.5), condition (ii) in Lemma 15 is changed into the condition that 2 p ≤ m < 1 and we may notice in [START_REF] Demange | Des équations à diffusion rapide aux inégalités de Sobolev sur les modèles de la géométrie[END_REF][START_REF]Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature[END_REF] that it is always satisfied if we choose β = 4 The reader is invited to check that the solution of (B.12) with initial datum ϕ(0) = 0 is given by (3. Proof. Let us give the scheme of a proof. Let γ ∶= γ β 2 , in order to simplify notation. We can argue as follows:

(1) i ′ + 2 d i = (id e) ′ ≤ 0 shows that 0 ≤ i(t) ≤ i(0)e -2 d t and in particular lim t →+∞ i(t) = 0.

(2) As t → +∞, e converges to a constant, hence lim t →+∞ e(t) = 0. ′ ≤ 0 that i ≥ d e using an integration from any t ≥ 0 to +∞. (5) Unless u is a constant, we read from (id e)

′ ≤ 1 2 γ d (e 2 ) ′ that id e > 1 2 γ d e 2 , using again an integration from any t ≥ 0 to +∞. (6) Take some ϑ ∈ (0,1) and consider the solution of (B.13)

ϕ ′ (s) = 1 + ϑ γ ϕ(s) (1 -(p -2) s) δ , ϕ(0) = 0 .
In the spirit of (B.11), we have a following chain of elementary estimates: for e = e(t) small enough, i.e, for t > 0 large enough. [START_REF]Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality[END_REF] It is simple to check from (B.14) that id ϑ ϕ(e) cannot change sign. (8) We conclude as above that id ϑ ϕ(e) ≥ 0 using an integration from any t ≥ 0 to +∞. (9) Finally, we consider the limit as ϑ → 1 -. Altogether, we conclude that i ≥ d ϕ(e), where ϕ solves (B.12). This completes the scheme of the proof of Proposition 16.

F 2 L 2 (

 22 S d ) for any p ∈ [1,2) ∪ (2,2 * ). Here, d µ again denotes the uniform probability measure on S d , the critical Sobolev exponent is 2 * ∶= 2 d (d -2) if d ≥ 3 and we adopt the convention that 2 * = +∞ if d = 1 or d = 2. Inequality (GN) with p = 1 is equivalent to the Poincaré inequality. If d ≥ 3, inequality (GN) also holds for the critical exponent p = 2 *

3. 2 .

 2 An estimate based on the fast diffusion flow, valid up to the critical exponent.The subcritical range p ∈ [2 # , 2 * ) corresponding to exponents between the Bakry-Emery exponent and the critical Sobolev exponent is not covered in Section 3.1. In that case, we rely on entropy methods based on a fast diffusion or porous medium equation of exponent m, which are detailed in Appendix B (with corresponding references), to establish that an improved inequality (3.4) holds for any ϕ = ϕ m,p , where(3.5)

2 and,

 2 at least if c ⋆ (d, p) and c(d, p) are the optimal constants, d 2 (d + 1) 2 c(d, p) ≤ c ⋆ (d, p) ≤ c(d, p) .

Corollary 10 .

 10 Let p ∈ (2,2 * ). With the notation of Proposition 9, inequality (3.7) holds with

Theorem 11 .

 11 Let d ≥ 1 and p ∈ (1,2 * ). For any F ∈ H 1 (S d , d µ), we have (4.1)

2 ( 1 + 2 (p - 1 ) s 2 1 + 1 12 (p - 2 )

 2121122 0 by convexity of the function y ↦ y p-2 so that c (+) p defined as the maximum of s ↦ f p (s) s 6 on [-1 2, 1 2] ∋ s is finite and we have (4.7) 1 s) p + (1s) p ≤ 1 + p (p -3) s 2 + c

6 L 6 (S d ) ε 6 .

 666

B. 1 . 1 d

 11 Algebraic preliminaries. Let us denote the Hessian by Hv and define the trace-free Hessian by Lv ∶= Hv -(∆v) g d .

1 2∆

 1 ( ∇v 2 ) = Hv 2 + ∇(∆v) ⋅ ∇v + (d -1) ∇v 2 , where the last term, i.e., Ric(∇v, ∇v) = (d -1) ∇v 2 , accounts for the Ricci curvature tensor contracted with ∇v ⊗ ∇v. An integration of this formula on S d shows a Lv 2 d µ + d S d ∇v 2 d µ .

B. 2 .

 2 Diffusion flow and monotonicity. Assume that u is a positive solution of (B.4) ∂u ∂t = u -p (1-m) ∆u + (m p -1) ∇u 2

2 ∫ S d ∇u 2 d µ ∫ S d u 2 d µ δ 0 ∫ S d u p d µ β- 1 β

 201 (p-2)

B. 4 . 1 p - 2 u 2 L 2 L 2 (

 412222 (6p) corresponding to an admissible fast diffusion exponent m = (p + 2) (2 p), for any p ∈ (2,2 * ). By "admissible", one should understand m -(d, p) ≤ m ≤ m + (d, p), so that γ is nonnegative. With the choice of m = (p + 2) (2 p), we find δ = 1p 8. Improved functional inequalities. Let us denote the entropy and the Fisher information respectively by e ∶= p (S d )u S d ) and i ∶= ∇u

2 L 2 2 e ′ 1 -

 2221 (S d ) , and let γ and δ be given respectively by (B.3) and (B.9). Up to the replacement of u by u u L p (S d ) , with no loss of generality, we shall assume that u L p (S d ) = 1 . We learn from (B.8) and (B.10) that (B.11) (id e) ′ ≤ γ ie ′ β 2 1 -(p -2) e δ . Solving the ordinary differential equation in the equality case of (B.11) is equivalent to solving d d t id ϕ(e) = γ β (p -2) e δ (id ϕ(e)) ,

3 ) 1 -Proposition 16 .

 3116 if m = 1 and by (3.5) with ζ = 2 γ β (1-β) in the nonlinear case. We learn from (B.11) that(id ϕ(e)) (p -2) e δ (id ϕ(e)) .This is enough to prove the following result. With the above notation, we claim that i ≥ d ϕ(e) .

( 3 ) 2 d

 32 From (B.11), we learn that(id e) ′ ≤ d γ ee ′ = 1 γ (e 2 ) ′ ,where the inequality follows from 1 -(p -2) e ≤ 1 and i ≥ d e. (4) It follows from (id e)

1 - 2 γ

 12 (id ϑ ϕ(e)) ′ ≤ (id ϕ(e)) ′ + d (1 -ϑ) (ϕ(e)) ′ ≤ (id ϕ(e)) (p -2) e δ (id ϑ ϕ(e)) .We know that ϕ(0) = 0 and read from (B.13) that ϕ ′ (0) = 1 andϕ ′′ (0) = ϑ γ ϕ ′ (0) = ϑ γ ,so that ϕ(e) -e ∼ 1 2 ϑ γ e 2 as e → 0. Using id e > 1 2 γ d e 2 , we learn that id ϕ(e) ≥ 1 d (1 -ϑ) e 2 1 + O(e)

(

  G. Brigati) CEREMADE (CNRS UMR N ○ 7534), PSL UNIVERSITY, UNIVERSITÉ PARIS-DAUPHINE, PLACE DE LATTRE DE TASSIGNY, 75775 PARIS 16, FRANCE Email address: brigati@ceremade.dauphine.fr (J. Dolbeault) CEREMADE (CNRS UMR N ○ 7534), PSL UNIVERSITY, UNIVERSITÉ PARIS-DAUPHINE, PLACE DE LATTRE DE TASSIGNY, 75775 PARIS 16, FRANCE Email address: dolbeaul@ceremade.dauphine.fr (N. Simonov) LJLL (CNRS UMR N ○ 7598) SORBONNE UNIVERSITÉ, 4 PLACE JUSSIEU, 75005 PARIS, FRANCE Email address: nikita.simonov@sorbonne-universite.fr

  . For any integer k ≥ 1, let us define Π k as the orthogonal projection with respect to L 2 (S d , d µ) onto ⊕ k =1 H . The following statement extends Theorems 1 and 4.

	Theorem 7. Assume that d ≥ 1, p ∈ (1,2 * ) and let k ≥ 1 be an integer. For any function
	F ∈ H 1 (S d , d µ), we have
	(2.1)

  12 p + 8. The condition γ ≥ 0 determines the range m -(d, p) ≤ m ≤ m + (d, p) of admissible parameters m, where m ± (d, p) is given by(3.6). Summarizing, we have the following result (also see[START_REF] Dolbeault | Nonlinear flows and rigidity results on compact manifolds[END_REF]).Notice that the case of the linear flow corresponds to the case m = β = 1 and v = u.Proof of Lemma 14. For a smooth solution, the result follows from (B.6) and Lemma 13. The result for a general solution is obtained by standard regularization procedures.

	Lemma 14. Assume that p ∈ (1,2 * ) and m ∈ [m -(d, p), m + (d, p)]. If u solves (B.4), then
	we have
	(B.8)	1 2 β 2	d d t	∇u	2 L 2 (S d ) -d E p [u] ≤ -γ	S d	∇v 4 v 2 d µ ,

where v = u 1 β with β and γ given in terms of m by (B.3) and (B.7) respectively. B.3. Interpolation. Depending on the value of p, we shall consider various interpolation inequalities. Let us define (B.9)

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ACKNOWLEDGEMENTS

This work has been partially supported by the Project EFI (ANR-17-CE40-0030) of the French National Research Agency (ANR). G.B. has been funded by the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 754362.