# A short review on Improvements and stability for some interpolation inequalities

Jean Dolbeault, Maria J. Esteban, Alessio Figalli, Rupert Frank, Michael Loss

► To cite this version:

Jean Dolbeault, Maria J. Esteban, Alessio Figalli, Rupert Frank, Michael Loss. A short review on Improvements and stability for some interpolation inequalities. ICIAM Congress, Professor Shin'ichi Oishi, Aug 2023, Tokyo, Japan. hal-04456461

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# A short review on Improvements and stability for some interpolation inequalities

Jean Dolbeault, Maria J. Esteban, Alessio Figalli, Rupert Frank, Michael Loss

► To cite this version:

Jean Dolbeault, Maria J. Esteban, Alessio Figalli, Rupert Frank, Michael Loss. A short review on Improvements and stability for some interpolation inequalities. ICIAM Congress, Aug 2023, Tokyo, Japan. ICIAM2023 Springer Series 5 on "Various Methods for the Analysis of PDEs" (Vladimir Simeonov Georgiev and Tohru Ozawa eds.), Springer Nature Singapore, pp.37-60, 2025, 10.1007/978-981-96-9164-7_3. hal-04456461

HAL Id: hal-04456461

https://hal.science/hal-04456461v1

Submitted on 27 Apr 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>
# A SHORT REVIEW ON IMPROVEMENTS AND STABILITY FOR SOME INTERPOLATION INEQUALITIES

JEAN DOLBEAULT, MARIA J. ESTEBAN, ALESSIO FIGALLI, RUPERT L. FRANK, AND MICHAEL LOSS

**ABSTRACT.** In this paper, we present recent stability results with explicit and dimensionally sharp constants and optimal norms for the Sobolev inequality and for the Gaussian logarithmic Sobolev inequality obtained by the authors in [24]. The stability for the Gaussian logarithmic Sobolev inequality was obtained as a byproduct of the stability for the Sobolev inequality. Here we give a new, direct, alternative proof. We also discuss improved versions of interpolation inequalities based on the carré du champ method.

## 1. INTRODUCTION AND MAIN RESULTS

Let us assume that $d \ge 3$ without further notice. The classical Sobolev inequality on $\mathbb{R}^d$, can be written as follows:

$$ \|\nabla f\|_{L^2(\mathbb{R}^d)}^2 \ge S_d \|f\|_{L^{2^*}(\mathbb{R}^d)}^2 \quad \forall f \in \dot{H}^1(\mathbb{R}^d), \qquad (S) $$

where $2^* = \frac{2d}{d-2}$ is the Sobolev exponent, $S_d = \frac{1}{4}d(d-2)|\mathbb{S}^d|^{2/d}$ is the sharp Sobolev constant, and $|\mathbb{S}^d|$ denotes the volume of the unit sphere $\mathbb{S}^d \subset \mathbb{R}^{d+1}$. Here $\dot{H}^1(\mathbb{R}^d)$ denotes the closure of $C_c^\infty(\mathbb{R}^d)$ with respect to the seminorm $|f|_{\dot{H}^1(\mathbb{R}^d)} := \|\nabla f\|_{L^2(\mathbb{R}^d)}^2$. As proved in [1, 42] (see also [40, 41]), equality in (S) holds if $f$ is one of the Aubin-Talenti functions, that is, one of the functions belonging to the $(d+2)$-dimensional Aubin-Talenti manifold

$$ \mathcal{M} := \{g_{a,b,c} : (a,b,c) \in (0, +\infty) \times \mathbb{R}^d \times \mathbb{R}\} \quad \text{with} \quad g_{a,b,c}(x) := c(a + |x - b|^2)^{-\frac{d-2}{2}}. $$

In fact, there is equality in (S) if and only if $f$ is in $\mathcal{M}$ according to [37, 32, 15].

Applying the inverse of the stereographic projection and integrating on the sphere $\mathbb{S}^d$ with respect to the uniform probability measure $d\mu_d$, one can rewrite the above inequality as

$$ \|\nabla u\|_{L^2(\mathbb{S}^d)}^2 \ge \frac{1}{4} d(d-2) \left( \|u\|_{L^{2^*}(\mathbb{S}^d)}^2 - \|u\|_{L^2(\mathbb{S}^d)}^2 \right) \quad \forall u \in H^1(\mathbb{S}^d) \qquad (1) $$

and state, equivalently, that the only functions in $H^1(\mathbb{S}^d)$ for which there is equality in (1) are the functions $\omega \mapsto G_{b,c}(\omega) := c(1 + b \cdot \omega)^{-(d-2)/2}$ where $b \in B_1 := \{b \in \mathbb{R}^{d+1} : |b| < 1\}$ and $c \in \mathbb{R}$ are constants. One can embed (1) in the following family of Gagliardo-Nirenberg-Sobolev inequalities:

$$ \|\nabla u\|_{L^2(\mathbb{S}^d)}^2 \ge \frac{d}{p-2} \left( \|u\|_{L^p(\mathbb{S}^d)}^2 - \|u\|_{L^2(\mathbb{S}^d)}^2 \right) \quad \forall u \in H^1(\mathbb{S}^d), \forall p \in (1, 2) \cup (2, 2^*]. \qquad (\text{GNS}) $$

As proved by Bidaut-Véron and Véron [7] and Beckner [5] for $p > 2$ (see also [2, 3, 4] if $p \le 2^*$), the constant $d/(p-2)$ is the best possible constant in (GNS). Here $2^* := (2d^2 + 1)/(d-1)^2 < 2^*$ denotes the Bakry-Emery exponent. If $p < 2^*$, the only optimizers in $H^1(\mathbb{S}^d)$ for (GNS) are the constant functions. The carré du champ method used in [2, 3, 4] relies on the linear heat equation, which induces the limitation $p \le 2^*$. This limitation is not technical as shown in [27]. The whole range $p \in [1, 2) \cup (2, 2^*]$ was covered using a carré du champ method based on nonlinear diffusion

<footnote>Date: April 20, 2024.</footnote>

<footnote n="2020">2020 Mathematics Subject Classification. Primary: 49J40; Secondary: 26D10, 35A23, 58E35.</footnote>

<footnote>Key words and phrases. Sobolev inequality; logarithmic Sobolev inequality; interpolation; Gagliardo-Nirenberg inequalities; stability; rearrangement; Steiner symmetrization; heat equation; fast diffusion equations.</footnote>

<footnote>© 2024 by the authors. Reproduction of this article by any means permitted for noncommercial purposes. CC-BY 4.0.</footnote>
# A SHORT REVIEW ON IMPROVEMENTS AND STABILITY FOR SOME INTERPOLATION INEQUALITIES

JEAN DOLBEAULT, MARIA J. ESTEBAN, ALESSIO FIGALLI, RUPERT L. FRANK, AND MICHAEL LOSS

**ABSTRACT.** In this paper, we present recent stability results with explicit and dimensionally sharp constants and optimal norms for the Sobolev inequality and for the Gaussian logarithmic Sobolev inequality obtained by the authors in [24]. The stability for the Gaussian logarithmic Sobolev inequality was obtained as a byproduct of the stability for the Sobolev inequality. Here we give a new, direct, alternative proof. We also discuss improved versions of interpolation inequalities based on the carré du champ method.

## 1. INTRODUCTION AND MAIN RESULTS

Let us assume that $d \ge 3$ without further notice. The classical Sobolev inequality on $\mathbb{R}^d$, can be written as follows:

$$ \|\nabla f\|_{L^2(\mathbb{R}^d)}^2 \ge S_d \|f\|_{L^{2^*}(\mathbb{R}^d)}^2 \quad \forall f \in \dot{H}^1(\mathbb{R}^d), \qquad (S) $$

where $2^* = \frac{2d}{d-2}$ is the Sobolev exponent, $S_d = \frac{1}{4}d(d-2)|\mathbb{S}^d|^{2/d}$ is the sharp Sobolev constant, and $|\mathbb{S}^d|$ denotes the volume of the unit sphere $\mathbb{S}^d \subset \mathbb{R}^{d+1}$. Here $\dot{H}^1(\mathbb{R}^d)$ denotes the closure of $C_c^\infty(\mathbb{R}^d)$ with respect to the seminorm $|f|_{\dot{H}^1(\mathbb{R}^d)} := \|\nabla f\|_{L^2(\mathbb{R}^d)}^2$. As proved in [1, 42] (see also [40, 41]), equality in (S) holds if $f$ is one of the Aubin-Talenti functions, that is, one of the functions belonging to the $(d+2)$-dimensional Aubin-Talenti manifold

$$ \mathcal{M} := \{g_{a,b,c} : (a,b,c) \in (0, +\infty) \times \mathbb{R}^d \times \mathbb{R}\} \quad \text{with} \quad g_{a,b,c}(x) := c(a + |x - b|^2)^{-\frac{d-2}{2}}. $$

In fact, there is equality in (S) if and only if $f$ is in $\mathcal{M}$ according to [37, 32, 15].

Applying the inverse of the stereographic projection and integrating on the sphere $\mathbb{S}^d$ with respect to the uniform probability measure $d\mu_d$, one can rewrite the above inequality as

$$ \|\nabla u\|_{L^2(\mathbb{S}^d)}^2 \ge \frac{1}{4} d(d-2) \left( \|u\|_{L^{2^*}(\mathbb{S}^d)}^2 - \|u\|_{L^2(\mathbb{S}^d)}^2 \right) \quad \forall u \in H^1(\mathbb{S}^d) \qquad (1) $$

and state, equivalently, that the only functions in $H^1(\mathbb{S}^d)$ for which there is equality in (1) are the functions $\omega \mapsto G_{b,c}(\omega) := c(1 + b \cdot \omega)^{-(d-2)/2}$ where $b \in B_1 := \{b \in \mathbb{R}^{d+1} : |b| < 1\}$ and $c \in \mathbb{R}$ are constants. One can embed (1) in the following family of Gagliardo-Nirenberg-Sobolev inequalities:

$$ \|\nabla u\|_{L^2(\mathbb{S}^d)}^2 \ge \frac{d}{p-2} \left( \|u\|_{L^p(\mathbb{S}^d)}^2 - \|u\|_{L^2(\mathbb{S}^d)}^2 \right) \quad \forall u \in H^1(\mathbb{S}^d), \forall p \in (1, 2) \cup (2, 2^*]. \qquad (\text{GNS}) $$

As proved by Bidaut-Véron and Véron [7] and Beckner [5] for $p > 2$ (see also [2, 3, 4] if $p \le 2^*$), the constant $d/(p-2)$ is the best possible constant in (GNS). Here $2^* := (2d^2 + 1)/(d-1)^2 < 2^*$ denotes the Bakry-Emery exponent. If $p < 2^*$, the only optimizers in $H^1(\mathbb{S}^d)$ for (GNS) are the constant functions. The carré du champ method used in [2, 3, 4] relies on the linear heat equation, which induces the limitation $p \le 2^*$. This limitation is not technical as shown in [27]. The whole range $p \in [1, 2) \cup (2, 2^*]$ was covered using a carré du champ method based on nonlinear diffusion

<footnote>Date: April 20, 2024.</footnote>

<footnote n="2020">2020 Mathematics Subject Classification. Primary: 49J40; Secondary: 26D10, 35A23, 58E35.</footnote>

<footnote>Key words and phrases. Sobolev inequality; logarithmic Sobolev inequality; interpolation; Gagliardo-Nirenberg inequalities; stability; rearrangement; Steiner symmetrization; heat equation; fast diffusion equations.</footnote>

<footnote>© 2024 by the authors. Reproduction of this article by any means permitted for noncommercial purposes. CC-BY 4.0.</footnote>
equations by Demange and Dolbeault-Esteban-Loss in [22, 26]. The case $p = 2$ (that can be obtained from (GNS) by taking the limit $p \to 2$) is the logarithmic Sobolev inequality

$$ \|\nabla u\|_{L^2(\mathbb{S}^d)}^2 \geq \frac{d}{2} \int_{\mathbb{S}^d} |u|^2 \ln \left( \frac{|u|^2}{\|u\|_{L^2(\mathbb{S}^d)}^2} \right) d\mu_d \quad \forall u \in H^1(\mathbb{S}^d) \setminus \{0\}, \qquad (\text{LSI}) $$

where the only optimizers are the constant functions. Inequality (LSI) can also be proved by the carré du champ method based on the heat equation.

A remarkable feature of the carré du champ method is that additional terms appear in the computations. This has been used in [25] to establish, for all $p \in (2, 2^*)$, the improved interpolation inequalities

$$ \|\nabla u\|_{L^2(\mathbb{S}^d)}^2 - \frac{d}{p-2} \left( \|u\|_{L^p(\mathbb{S}^d)}^2 - \|u\|_{L^2(\mathbb{S}^d)}^2 \right) \geq \|u\|_{L^2(\mathbb{S}^d)}^2 \Psi_{d,p} \left( \frac{\|\nabla u\|_{L^2(\mathbb{S}^d)}^2}{\|u\|_{L^2(\mathbb{S}^d)}^2} \right) \quad \forall u \in H^1(\mathbb{S}^d) \setminus \{0\} \quad (2) $$

for some convex function $\Psi_{d,p}$ such that $\Psi_{d,p}(0) = \Psi'_{d,p}(0) = 0$ and $\Psi_{d,p}(s) > 0$ if $s > 0$. Inequality (2) is obviously stronger than (GNS) and can be used to prove that the equality case in (GNS) is realized only by constant functions. If we test this inequality with $u_{\varepsilon}(\omega) = 1 + \varepsilon \mathbf{b} \cdot \omega$ for a given $\mathbf{b} \in B_1$, an elementary computation shows that there is a cancellation of the $O(\varepsilon^2)$ terms as $\varepsilon \to 0$ and the first non-zero term is of the order of $O(\varepsilon^4)$.

If we denote by $\Pi_1$ the $L^2(\mathbb{S}^d)$ projection on the space generated by the coordinate functions $\omega_i$ with $i = 1, 2, \dots, d$, we learn from [10, Theorem 6] that there is an explicit constant $\kappa = \kappa(p, d) \in (0, 1)$, depending on $d$ and $p \in [1, 2) \cup (2, 2^*)$ such that, for all $u \in H^1(\mathbb{S}^d)$,

$$ \|\nabla u\|_{L^2(\mathbb{S}^d)}^2 - \frac{d}{p-2} \left( \|u\|_{L^p(\mathbb{S}^d)}^2 - \|u\|_{L^2(\mathbb{S}^d)}^2 \right) \geq \kappa \left( \frac{\|\nabla \Pi_1 u\|_{L^2(\mathbb{S}^d)}^4}{\|\nabla u\|_{L^2(\mathbb{S}^d)}^2 + \|u\|_{L^2(\mathbb{S}^d)}^2} + \|\nabla(\mathrm{Id} - \Pi_1) u\|_{L^2(\mathbb{S}^d)}^2 \right). \quad (3) $$

The proof relies on (2) and on a decomposition in spherical harmonics directly inspired by [31]. Inequality (3) improves over (GNS), but this improvement degenerates as $p \to 2^*$. Indeed, any constant $\kappa(p, d)$ for which (3) holds necessarily satisfies $\lim_{p \to 2^*} \kappa(p, d) = 0$, for otherwise we had an inequality at $p = 2^*$ whose right side vanishes only for constants while, according to our discussion of (1), the left side vanishes for all function $G_{b,c}$.

Let us notice that Inequality (2) can take various forms. For instance, if $p \in (2, 2^*)$ and $1/\theta = 1 + (d-1/(d+2))^2 (p-1) (2^* - p)/(p-2)$, we read from [23, Ineq. (2.4)] that (2) takes the form

$$ \|\nabla u\|_{L^2(\mathbb{S}^d)}^2 \geq \frac{d\theta}{p-2} \left( \|u\|_{L^p(\mathbb{S}^d)}^{2/\theta} \|u\|_{L^2(\mathbb{S}^d)}^{2-2/\theta} - \|u\|_{L^2(\mathbb{S}^d)}^2 \right) \quad \forall u \in H^1(\mathbb{S}^d), $$

which is a strict improvement compared to (GNS), as can be seen using Hölder inequalities. The case $p = 2$ is also covered, except that in the left-hand side of (2), the deficit of (GNS) has to be replaced by the deficit of (LSI).

Without entering into the details, let us quote some related results. By a direct variational approach, an improved inequality like (2) is proved in [31, Theorem 2], in the subcritical range, for some $\Psi_{d,p}(s) \sim s^2$. Using nonlinear flows and appropriate orthogonality constraints, improved inequalities with $\Psi_{d,p}(s) \sim s$ are known from [27]. Both results are unified in [10]. Let us finally mention that improved inequalities are proved in [8] with explicit constants, not on $\mathbb{S}^d$ but on $\mathbb{R}^d$, for the Gagliardo-Nirenberg-Sobolev inequalities using entropy methods and regularization effects for fast diffusion flows, but with some restrictions on the decay of the functions at infinity.

The Gaussian measure on $\mathbb{R}^N$ can be seen as an infinite dimensional limit $d \to +\infty$ of the uniform probability measure on the sphere of dimension $d$ and radius $\sqrt{d}$ tested against functions depending only on a finite number $N$ of coordinates: see for instance [39, 11, 24]. Since $\lim_{d \to +\infty} 2^* = 2$, it also turns out that (S) has to be replaced by a Gaussian interpolation inequality as follows. On $\mathbb{R}^N$, with $N \geq 1$, let us consider the Gaussian measure $d\gamma(x) := e^{-\pi |x|^2} dx$. With $L^2(\gamma) := L^2(\mathbb{R}^N, d\gamma)$,
equations by Demange and Dolbeault-Esteban-Loss in [22, 26]. The case $p = 2$ (that can be obtained from (GNS) by taking the limit $p \to 2$) is the logarithmic Sobolev inequality

$$ \|\nabla u\|_{L^2(\mathbb{S}^d)}^2 \geq \frac{d}{2} \int_{\mathbb{S}^d} |u|^2 \ln \left( \frac{|u|^2}{\|u\|_{L^2(\mathbb{S}^d)}^2} \right) d\mu_d \quad \forall u \in H^1(\mathbb{S}^d) \setminus \{0\}, \quad (\text{LSI}) $$

where the only optimizers are the constant functions. Inequality (LSI) can also be proved by the carré du champ method based on the heat equation.

A remarkable feature of the carré du champ method is that additional terms appear in the computations. This has been used in [25] to establish, for all $p \in (2, 2^*)$, the improved interpolation inequalities

$$ \|\nabla u\|_{L^2(\mathbb{S}^d)}^2 - \frac{d}{p-2} \left( \|u\|_{L^p(\mathbb{S}^d)}^2 - \|u\|_{L^2(\mathbb{S}^d)}^2 \right) \geq \|u\|_{L^2(\mathbb{S}^d)}^2 \Psi_{d,p} \left( \frac{\|\nabla u\|_{L^2(\mathbb{S}^d)}^2}{\|u\|_{L^2(\mathbb{S}^d)}^2} \right) \quad \forall u \in H^1(\mathbb{S}^d) \setminus \{0\} \quad (2) $$

for some convex function $\Psi_{d,p}$ such that $\Psi_{d,p}(0) = \Psi'_{d,p}(0) = 0$ and $\Psi_{d,p}(s) > 0$ if $s > 0$. Inequality (2) is obviously stronger than (GNS) and can be used to prove that the equality case in (GNS) is realized only by constant functions. If we test this inequality with $u_{\varepsilon}(\omega) = 1 + \varepsilon \mathbf{b} \cdot \omega$ for a given $\mathbf{b} \in B_1$, an elementary computation shows that there is a cancellation of the $O(\varepsilon^2)$ terms as $\varepsilon \to 0$ and the first non-zero term is of the order of $O(\varepsilon^4)$.

If we denote by $\Pi_1$ the $L^2(\mathbb{S}^d)$ projection on the space generated by the coordinate functions $\omega_i$ with $i = 1, 2, \dots, d$, we learn from [10, Theorem 6] that there is an explicit constant $\kappa = \kappa(p, d) \in (0, 1)$, depending on $d$ and $p \in [1, 2) \cup (2, 2^*)$ such that, for all $u \in H^1(\mathbb{S}^d)$,

$$ \|\nabla u\|_{L^2(\mathbb{S}^d)}^2 - \frac{d}{p-2} \left( \|u\|_{L^p(\mathbb{S}^d)}^2 - \|u\|_{L^2(\mathbb{S}^d)}^2 \right) \geq \kappa \left( \frac{\|\nabla \Pi_1 u\|_{L^2(\mathbb{S}^d)}^4}{\|\nabla u\|_{L^2(\mathbb{S}^d)}^2 + \|u\|_{L^2(\mathbb{S}^d)}^2} + \|\nabla(\mathrm{Id} - \Pi_1) u\|_{L^2(\mathbb{S}^d)}^2 \right). \quad (3) $$

The proof relies on (2) and on a decomposition in spherical harmonics directly inspired by [31]. Inequality (3) improves over (GNS), but this improvement degenerates as $p \to 2^*$. Indeed, any constant $\kappa(p, d)$ for which (3) holds necessarily satisfies $\lim_{p \to 2^*} \kappa(p, d) = 0$, for otherwise we had an inequality at $p = 2^*$ whose right side vanishes only for constants while, according to our discussion of (1), the left side vanishes for all function $G_{b,c}$.

Let us notice that Inequality (2) can take various forms. For instance, if $p \in (2, 2^\#)$ and $1/\theta = 1 + (\frac{d-1}{d+2})^2 (p-1) (2^\# - p)/(p-2)$, we read from [23, Ineq. (2.4)] that (2) takes the form

$$ \|\nabla u\|_{L^2(\mathbb{S}^d)}^2 \geq \frac{d\theta}{p-2} \left( \|u\|_{L^p(\mathbb{S}^d)}^{2/\theta} \|u\|_{L^2(\mathbb{S}^d)}^{2-2/\theta} - \|u\|_{L^2(\mathbb{S}^d)}^2 \right) \quad \forall u \in H^1(\mathbb{S}^d), $$

which is a strict improvement compared to (GNS), as can be seen using Hölder inequalities. The case $p = 2$ is also covered, except that in the left-hand side of (2), the deficit of (GNS) has to be replaced by the deficit of (LSI).

Without entering into the details, let us quote some related results. By a direct variational approach, an improved inequality like (2) is proved in [31, Theorem 2], in the subcritical range, for some $\Psi_{d,p}(s) \sim s^2$. Using nonlinear flows and appropriate orthogonality constraints, improved inequalities with $\Psi_{d,p}(s) \sim s$ are known from [27]. Both results are unified in [10]. Let us finally mention that improved inequalities are proved in [8] with explicit constants, not on $\mathbb{S}^d$ but on $\mathbb{R}^d$, for the Gagliardo-Nirenberg-Sobolev inequalities using entropy methods and regularization effects for fast diffusion flows, but with some restrictions on the decay of the functions at infinity.

The Gaussian measure on $\mathbb{R}^N$ can be seen as an infinite dimensional limit $d \to +\infty$ of the uniform probability measure on the sphere of dimension $d$ and radius $\sqrt{d}$ tested against functions depending only on a finite number $N$ of coordinates: see for instance [39, 11, 24]. Since $\lim_{d \to +\infty} 2^* = 2$, it also turns out that (S) has to be replaced by a Gaussian interpolation inequality as follows. On $\mathbb{R}^N$, with $N \geq 1$, let us consider the Gaussian measure $d\gamma(x) := e^{-\pi|x|^2} dx$. With $L^2(\gamma) := L^2(\mathbb{R}^N, d\gamma)$,
if $H^1(\gamma)$ denotes the space of all $u \in L^2(\gamma)$ with distributional gradient in $L^2(\gamma)$, the logarithmic Sobolev inequality is:

$$ \int_{\mathbb{R}^N} |\nabla u|^2 d\gamma \geq \pi \int_{\mathbb{R}^N} |u|^2 \ln \left( \frac{|u|^2}{\|u\|_{L^2(\gamma)}^2} \right) d\gamma \quad \forall u \in H^1(\gamma) \setminus \{0\}. \qquad (\text{LSI}) $$

According to the result of Carlen [16, Theorem 5], equality in (LSI) holds if and only if for some $a \in \mathbb{R}^N$ and $c \in \mathbb{R}$,

$$ u(x) = c e^{a \cdot x}. \qquad (4) $$

Improved forms of the inequality are also known for instance from [30, 28, 29, 10], under some restrictions. The article [33] makes a very interesting connection between deficit estimates for (LSI) (for certain restricted classes of functions) and the Mahler conjecture in convex geometry. A detailed list of earlier results and references can be found in [34].

Now let us turn our attention to stability issues for (S), (GNS) and (LSI).

* **Stability for the Sobolev inequality.** In [9] Brezis and Lieb asked the following question:

(Q) Do there exist constants $\kappa$, $\alpha > 0$ such that the Sobolev deficit $\delta$ controls some distance $d$ from the Aubin-Talenti manifold $\mathcal{M}$ according to

$$ \delta(f) := \|\nabla f\|_{L^2(\mathbb{R}^d)}^2 - S_d \|f\|_{L^{2^*}(\mathbb{R}^d)}^2 \geq \kappa \mathbf{d}(f, \mathcal{M})^\alpha \quad ? $$

The ‘best possible answer’ to this question would involve finding the strongest possible topology to define the distance $d$ and the best possible constant $\kappa$ and exponent $\alpha$. The first answer to Brezis and Lieb’s question was given by Bianchi and Egnell in [6]: there is a constant $C_{d,\text{BE}} > 0$ such that

$$ \delta(f) \geq C_{d,\text{BE}} \inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_{L^2(\mathbb{R}^d)}^2 \quad \forall f \in \dot{H}^1(\mathbb{R}^d). \qquad (5) $$

Similar results for other inequalities have been proved using the strategy of Bianchi and Egnell: see, for example, [20]. The main drawback of this strategy is that no explicit estimate of $C_{d,\text{BE}}$ is known nor its dependence on the dimension $d$. Recently, in [24], we proved the following result.

**Theorem 1 ([24, Theorem 1.1]).** There is an explicit constant $\beta > 0$ such that for any $d \geq 3$,

$$ \|\nabla f\|_{L^2(\mathbb{R}^d)}^2 - S_d \|f\|_{L^{2^*}(\mathbb{R}^d)}^2 \geq \frac{\beta}{d} \inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_{L^2(\mathbb{R}^d)}^2 \quad \forall f \in \dot{H}^1(\mathbb{R}^d). \qquad (6) $$

This result is dimensionally sharp. Indeed, Theorem 1 can be rewritten as $C_{d,\text{BE}} \geq \beta/d$. On the other hand, it was proved implicitly in [6] (see also [20]) that $C_{d,\text{BE}} \leq 4/(4+d)$. This inequality is in fact strict: $C_{d,\text{BE}} < 4/(4+d)$, according to [35], and we learn from [36] that equality in (5), written with the optimal value of $C_{d,\text{BE}}$, is achieved. Hence, Theorem 1 captures the dimensional behavior of $C_{d,\text{BE}}$. For completeness, let us mention the extension of Theorem 1 in [19] to fractional Sobolev inequalities.

Using the inverse stereographic projection, the stability result of Theorem 1 can be rewritten on $\mathbb{S}^d$ for any $u \in H^1(\mathbb{S}^d)$ as

$$ \|\nabla u\|_{L^2(\mathbb{S}^d)}^2 - \frac{1}{4} d(d-2) \left( \|u\|_{L^{2^*}(\mathbb{S}^d)}^2 - \|u\|_{L^2(\mathbb{S}^d)}^2 \right) \\ \geq \frac{\beta}{d} \inf_{(b,c) \in B_1 \times \mathbb{R}} \left( \|\nabla u - \nabla G_{b,c}\|_{L^2(\mathbb{S}^d)}^2 + \frac{1}{4} d(d-2) \|u - G_{b,c}\|_{L^2(\mathbb{S}^d)}^2 \right). $$

* **Stability for the Gaussian logarithmic Sobolev inequality.** The interpretation of the Gaussian measure as the limit of uniform probability measures on $d$-dimensional spheres as $d \to +\infty$ and the explicit dimensional dependence of the stability constant of Theorem 1 provides us with a stability result for (LSI).
if $H^1(\gamma)$ denotes the space of all $u \in L^2(\gamma)$ with distributional gradient in $L^2(\gamma)$, the logarithmic Sobolev inequality is:

$$ \int_{\mathbb{R}^N} |\nabla u|^2 d\gamma \geq \pi \int_{\mathbb{R}^N} |u|^2 \ln \left( \frac{|u|^2}{\|u\|_{L^2(\gamma)}^2} \right) d\gamma \quad \forall u \in H^1(\gamma) \setminus \{0\}. \quad (\text{LSI}) $$

According to the result of Carlen [16, Theorem 5], equality in (LSI) holds if and only if for some $a \in \mathbb{R}^N$ and $c \in \mathbb{R}$,

$$ u(x) = c e^{a \cdot x}. \quad (4) $$

Improved forms of the inequality are also known for instance from [30, 28, 29, 10], under some restrictions. The article [33] makes a very interesting connection between deficit estimates for (LSI) (for certain restricted classes of functions) and the Mahler conjecture in convex geometry. A detailed list of earlier results and references can be found in [34].

Now let us turn our attention to stability issues for (S), (GNS) and (LSI).

* **Stability for the Sobolev inequality.** In [9] Brezis and Lieb asked the following question:

(Q) Do there exist constants $\kappa$, $\alpha > 0$ such that the Sobolev deficit $\delta$ controls some distance $d$ from the Aubin-Talenti manifold $\mathcal{M}$ according to

$$ \delta(f) := \|\nabla f\|_{L^2(\mathbb{R}^d)}^2 - S_d \|f\|_{L^{2^*}(\mathbb{R}^d)}^2 \geq \kappa \mathbf{d}(f, \mathcal{M})^\alpha \quad ? $$

The ‘best possible answer’ to this question would involve finding the strongest possible topology to define the distance $d$ and the best possible constant $\kappa$ and exponent $\alpha$. The first answer to Brezis and Lieb’s question was given by Bianchi and Egnell in [6]: there is a constant $C_{d,\text{BE}} > 0$ such that

$$ \delta(f) \geq C_{d,\text{BE}} \inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_{L^2(\mathbb{R}^d)}^2 \quad \forall f \in \dot{\text{H}}^1(\mathbb{R}^d). \quad (5) $$

Similar results for other inequalities have been proved using the strategy of Bianchi and Egnell: see, for example, [20]. The main drawback of this strategy is that no explicit estimate of $C_{d,\text{BE}}$ is known nor its dependence on the dimension $d$. Recently, in [24], we proved the following result.

**Theorem 1 ([24, Theorem 1.1]).** There is an explicit constant $\beta > 0$ such that for any $d \geq 3$,

$$ \|\nabla f\|_{L^2(\mathbb{R}^d)}^2 - S_d \|f\|_{L^{2^*}(\mathbb{R}^d)}^2 \geq \frac{\beta}{d} \inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_{L^2(\mathbb{R}^d)}^2 \quad \forall f \in \dot{\text{H}}^1(\mathbb{R}^d). \quad (6) $$

This result is dimensionally sharp. Indeed, Theorem 1 can be rewritten as $C_{d,\text{BE}} \geq \beta/d$. On the other hand, it was proved implicitly in [6] (see also [20]) that $C_{d,\text{BE}} \leq 4/(4+d)$. This inequality is in fact strict: $C_{d,\text{BE}} < 4/(4+d)$, according to [35], and we learn from [36] that equality in (5), written with the optimal value of $C_{d,\text{BE}}$, is achieved. Hence, Theorem 1 captures the dimensional behavior of $C_{d,\text{BE}}$. For completeness, let us mention the extension of Theorem 1 in [19] to fractional Sobolev inequalities.

Using the inverse stereographic projection, the stability result of Theorem 1 can be rewritten on $\mathbb{S}^d$ for any $u \in H^1(\mathbb{S}^d)$ as

$$ \|\nabla u\|_{L^2(\mathbb{S}^d)}^2 - \frac{1}{4} d(d-2) \left( \|u\|_{L^{2^*}(\mathbb{S}^d)}^2 - \|u\|_{L^2(\mathbb{S}^d)}^2 \right) \\ \geq \frac{\beta}{d} \inf_{(b,c) \in B_1 \times \mathbb{R}} \left( \|\nabla u - \nabla G_{b,c}\|_{L^2(\mathbb{S}^d)}^2 + \frac{1}{4} d(d-2) \|u - G_{b,c}\|_{L^2(\mathbb{S}^d)}^2 \right). $$

* **Stability for the Gaussian logarithmic Sobolev inequality.** The interpretation of the Gaussian measure as the limit of uniform probability measures on $d$-dimensional spheres as $d \to +\infty$ and the explicit dimensional dependence of the stability constant of Theorem 1 provides us with a stability result for (LSI).
**Theorem 2** ([24, Corollary 1.2]). With $\beta > 0$ as in Theorem 1, for all $N \in \mathbb{N}$, we have

$$ \int_{\mathbb{R}^N} |\nabla u|^2 d\gamma - \pi \int_{\mathbb{R}^N} |u|^2 \ln \left( \frac{|u|^2}{\|u\|_{\text{L}^2(\gamma)}^2} \right) d\gamma \geq \frac{\beta \pi}{2} \inf_{b \in \mathbb{R}^N, c \in \mathbb{R}} \int_{\mathbb{R}^N} |u - c e^{b \cdot x}|^2 d\gamma \quad \forall u \in \text{H}^1(\gamma). \quad (7) $$

In [24] this result was obtained as a corollary of Theorem 1. In this paper we give a new, direct proof which highlights the strategy of [24] in a slightly simpler setting.

In the next sections, we briefly describe the strategies of [25] and [24] to prove the improvements in the case of subcritical inequalities (Section 2) and the stability results of Theorem 1 (Section 3) and Theorem 2 (Section 4). The new proof of Theorem 2 is given in Section 5.

## 2. IMPROVED INTERPOLATION INEQUALITIES

In this section we describe the method used in [25] to prove (2) and similar improvements of the (GNS) inequalities in the subcritical case $p \in (2, 2^*)$. Inequalities (GNS) and their limiting case corresponding to $p = 2$ can be written as:

$$ i \geq d e \quad \text{where} \quad i := \|\nabla u\|_{\text{L}^2(\mathbb{S}^d)}^2 \quad \text{and} \quad e := \begin{cases} \frac{1}{p-2} \left( \|u\|_{\text{L}^p(\mathbb{S}^d)}^2 - \|u\|_{\text{L}^2(\mathbb{S}^d)}^2 \right) & \text{if } p \neq 2, \\ \frac{1}{2} \int_{\mathbb{S}^d} |u|^2 \ln \left( \frac{|u|^2}{\|u\|_{\text{L}^2(\mathbb{S}^d)}^2} \right) d\mu_d & \text{if } p = 2, \end{cases} $$

for any $u \in \text{H}^1(\mathbb{S}^d)$. By homogeneity we can assume that $\|u\|_{\text{L}^2(\mathbb{S}^d)} = 1$. Let $I_p := [0, +\infty)$ if $p \ge 2$ and $I_p := [0, 1/(2-p))$ if $p \in (1, 2)$.

**Theorem 3** ([25, Theorem 1.1]). Let $d \ge 3$ and $p \in (2, 2^*)$. With the above notation and conventions, there is an explicit function $\varphi$, such that $\varphi(0) = 0$, $\varphi'(0) = 1$, and $\varphi'' > 0$ on $I_p$, for which

$$ i \geq d \varphi(e). \quad (8) $$

Similar results can be proved in dimension $d = 1$ and $d = 2$: see [22, 25, 23, 10]. Since

$$ i - d e \geq d (\varphi(e) - e) \geq 0, $$

it is clear that the equality case in (2) holds if and only if $e = 0$, that is, if $u$ is a constant. Moreover $\varphi(e) - e$ measures a distance to the constants, for instance in $\text{L}^1(\mathbb{S}^d)$ norm using a generalized Ciszár-Kullback-Pinsker inequality, and $\varphi(s) - s \sim \varphi''(0) s^2/2$ as $s \to 0$. To prove (2) in the stronger homogeneous Sobolev norm $\dot{\text{H}}^1(\mathbb{R}^d)$, it is enough to define the convex function

$$ \Psi_{d,p}(s) := s - d \varphi^{-1} \left( \frac{s}{d} \right). $$

It is elementary to verify that $\Psi_{d,p}(0) = \Psi'_{d,p}(0) = 0$ and that $\Psi_{d,p}(s) > 0$ if $s \neq 0$.

Let us give a sketch of the proof of Theorem 3 based on the method of [25]. As a first step, using Schwarz foliated symmetrization (see for instance [25, Section 2] for references) and cylindrical coordinates on $\mathbb{S}^d \subset \mathbb{R}^{d+1}$, we can reduce the problem to functions depending only on one coordinate $z \in [-1, 1]$ corresponding to the South Pole – North Pole axis. In other words, we consider a function $u(\omega) = f(z)$, where $\omega = (\omega_1, \omega_2, \dots, \omega_{d+1})$ and $z = \omega_{d+1}$, and find that

$$ \|\nabla u\|_{\text{L}^2(\mathbb{S}^d)}^2 = \int_{-1}^{1} |f'(z)|^2 (1 - z^2) d\sigma_d(z) \quad \text{and} \quad \|u\|_{\text{L}^q(\mathbb{S}^d)}^q = \int_{-1}^{1} |f(z)|^q d\sigma_d(z), $$

where $'$ denotes the $z$-derivative and where

$$ d\sigma_d(z) := \frac{\Gamma(\frac{d+1}{2})}{\sqrt{\pi} \Gamma(\frac{d}{2})} (1 - z^2)^{\frac{d}{2}-1} dz. $$

In particular, the Laplace-Beltrami operator on $\mathbb{S}^d$ is reduced to the *ultraspherical* operator: $\Delta u = \mathcal{L} f := (1 - z^2) f'' - d z f'$ on $(-1, 1)$.
**Theorem 2** ([24, Corollary 1.2]). With $\beta > 0$ as in Theorem 1, for all $N \in \mathbb{N}$, we have

$$ \int_{\mathbb{R}^N} |\nabla u|^2 d\gamma - \pi \int_{\mathbb{R}^N} |u|^2 \ln \left( \frac{|u|^2}{\|u\|_{\text{L}^2(\gamma)}^2} \right) d\gamma \geq \frac{\beta \pi}{2} \inf_{b \in \mathbb{R}^N, c \in \mathbb{R}} \int_{\mathbb{R}^N} |u - c e^{b \cdot x}|^2 d\gamma \quad \forall u \in \text{H}^1(\gamma). \quad (7) $$

In [24] this result was obtained as a corollary of Theorem 1. In this paper we give a new, direct proof which highlights the strategy of [24] in a slightly simpler setting.

In the next sections, we briefly describe the strategies of [25] and [24] to prove the improvements in the case of subcritical inequalities (Section 2) and the stability results of Theorem 1 (Section 3) and Theorem 2 (Section 4). The new proof of Theorem 2 is given in Section 5.

## 2. IMPROVED INTERPOLATION INEQUALITIES

In this section we describe the method used in [25] to prove (2) and similar improvements of the (GNS) inequalities in the subcritical case $p \in (2, 2^*)$. Inequalities (GNS) and their limiting case corresponding to $p = 2$ can be written as:

$$ i \geq d e \quad \text{where} \quad i := \|\nabla u\|_{\text{L}^2(\mathbb{S}^d)}^2 \quad \text{and} \quad e := \begin{cases} \frac{1}{p-2} \left( \|u\|_{\text{L}^p(\mathbb{S}^d)}^2 - \|u\|_{\text{L}^2(\mathbb{S}^d)}^2 \right) & \text{if } p \neq 2, \\ \frac{1}{2} \int_{\mathbb{S}^d} |u|^2 \ln \left( \frac{|u|^2}{\|u\|_{\text{L}^2(\mathbb{S}^d)}^2} \right) d\mu_d & \text{if } p = 2, \end{cases} $$

for any $u \in \text{H}^1(\mathbb{S}^d)$. By homogeneity we can assume that $\|u\|_{\text{L}^2(\mathbb{S}^d)} = 1$. Let $I_p := [0, +\infty)$ if $p \ge 2$ and $I_p := [0, 1/(2-p))$ if $p \in (1, 2)$.

**Theorem 3** ([25, Theorem 1.1]). Let $d \ge 3$ and $p \in (2, 2^*)$. With the above notation and conventions, there is an explicit function $\varphi$, such that $\varphi(0) = 0$, $\varphi'(0) = 1$, and $\varphi'' > 0$ on $I_p$, for which

$$ i \geq d \varphi(e). \quad (8) $$

Similar results can be proved in dimension $d = 1$ and $d = 2$: see [22, 25, 23, 10]. Since

$$ i - d e \geq d (\varphi(e) - e) \geq 0, $$

it is clear that the equality case in (2) holds if and only if $e = 0$, that is, if $u$ is a constant. Moreover $\varphi(e) - e$ measures a distance to the constants, for instance in $\text{L}^1(\mathbb{S}^d)$ norm using a generalized Ciszár-Kullback-Pinsker inequality, and $\varphi(s) - s \sim \varphi''(0) s^2/2$ as $s \to 0$. To prove (2) in the stronger homogeneous Sobolev norm $\dot{\text{H}}^1(\mathbb{R}^d)$, it is enough to define the convex function

$$ \Psi_{d,p}(s) := s - d \varphi^{-1} \left( \frac{s}{d} \right). $$

It is elementary to verify that $\Psi_{d,p}(0) = \Psi'_{d,p}(0) = 0$ and that $\Psi_{d,p}(s) > 0$ if $s \neq 0$.

Let us give a sketch of the proof of Theorem 3 based on the method of [25]. As a first step, using Schwarz foliated symmetrization (see for instance [25, Section 2] for references) and cylindrical coordinates on $\mathbb{S}^d \subset \mathbb{R}^{d+1}$, we can reduce the problem to functions depending only on one coordinate $z \in [-1, 1]$ corresponding to the South Pole - North Pole axis. In other words, we consider a function $u(\omega) = f(z)$, where $\omega = (\omega_1, \omega_2, \dots, \omega_{d+1})$ and $z = \omega_{d+1}$, and find that

$$ \|\nabla u\|_{\text{L}^2(\mathbb{S}^d)}^2 = \int_{-1}^{1} |f'(z)|^2 (1 - z^2) d\sigma_d(z) \quad \text{and} \quad \|u\|_{\text{L}^q(\mathbb{S}^d)}^q = \int_{-1}^{1} |f(z)|^q d\sigma_d(z), $$

where $'$ denotes the $z$-derivative and where

$$ d\sigma_d(z) := \frac{\Gamma(\frac{d+1}{2})}{\sqrt{\pi} \Gamma(\frac{d}{2})} (1 - z^2)^{\frac{d}{2}-1} dz. $$

In particular, the Laplace-Beltrami operator on $\mathbb{S}^d$ is reduced to the *ultraspherical* operator: $\Delta u = \mathcal{L} f := (1 - z^2) f'' - d z f'$ on $(-1, 1)$.
The key idea is to prove that $i - d\varphi(\mathbf{e})$ is monotone nonincreasing under the action of

$$ \frac{\partial \rho}{\partial t} = \mathcal{L} \rho^m, \qquad (9) $$

where $m = 1$ corresponds to the heat flow, $m > 1$ to the porous medium flow and $m < 1$ to the fast diffusion flow. Here we choose $\rho = |f|^p$ so that $\|u\|_{L^p(\mathbb{R}^d)}$ is conserved under the action of (9). It is convenient to introduce the exponent $\beta$ such that

$$ m = 1 + \frac{2}{p} \left( \frac{1}{\beta} - 1 \right) $$

and consider the function $f = w^\beta$, such that $w^{\beta p} = \rho$, which solves

$$ \frac{\partial w}{\partial t} = w^{2-2\beta} \left( \mathcal{L}w + \kappa \frac{|w'|^2}{w} \right) \quad \text{with} \quad \kappa := \beta(p-2) + 1. \qquad (10) $$

The carré du champ method shows that $\frac{d}{dt}(i - d\mathbf{e}) \le 0$ if the function

$$ \gamma(\beta) := - \left( \frac{d-1}{d+2} (\kappa + \beta - 1) \right)^2 + \kappa (\beta - 1) + \frac{d}{d+2} (\kappa + \beta - 1) $$

takes nonnegative values, which amounts to $m_-(d, p) \le m \le m_+(d, p)$ where

$$ m_{\pm}(d, p) := \frac{1}{(d+2)p} \left( dp + 2 \pm \sqrt{d(d-2)(p-1)(2^* - p)} \right). $$

Using $\lim_{t \to +\infty} (i(t) - d\mathbf{e}(t)) = 0$, we conclude that $i(t) - d\mathbf{e}(t) \ge 0$ for any $t \ge 0$ and, as a special case, for $t = 0$: this proves (GNS) for an arbitrary initial datum. If $m_-(d, p) < m < m_+(d, p)$, we find that $\gamma(\beta) > 0$, which leaves some space for an improvement. A more detailed and quite lengthy computation along the flow (10) yields

$$ \frac{d}{dt}(i - d\varphi(\mathbf{e})) \le \frac{\gamma}{\beta^2} \frac{i - d\varphi(\mathbf{e})}{(1 - (p-2)\mathbf{e})^\delta} \frac{d\mathbf{e}}{dt} $$

where $\delta = 1$ if $1 \le p \le 2$ and $\delta := \frac{2-(4-p)\beta}{2\beta(p-2)}$ if $p > 2$, if $\varphi$ solves

$$ \frac{d\varphi}{ds} = 1 + \frac{\gamma}{\beta^2} \frac{\varphi(s)}{(1 - (p-2)s)^\delta}, \quad \varphi(0) = 0. $$

This proves (8) for any $t \ge 0$ as a consequence of the monotonicity of $i - d\varphi(\mathbf{e})$ and the fact that $\lim_{t \to +\infty} (i(t) - d\varphi(\mathbf{e}(t))) = 0$. See [10, Appendix B.4] for detailed justifications. $\square$

As a consequence of the carré du champ method, optimal constants in Gagliardo-Nirenberg-Sobolev inequalities determine optimal rates of decay in some associated fast diffusion equations. A stability result for a functional inequality can usually be rephrased as an improved inequality, under appropriate constraints. Measured by the dissipation of some entropy functional, this means a faster convergence rate towards an asymptotic profile for solutions of the associated evolution equation: see [17] for an application of stability for Gagliardo-Nirenberg-Sobolev inequalities to the proof of a quantitative rate of convergence for solutions of the Keller-Segel system, and [10] for explicit, improved, convergence rates based on (GNS) in the subcritical regime, in the case of the sphere. On the Euclidean space, improved convergence rates based on stability for Gagliardo-Nirenberg-Sobolev inequalities (subcritical and critical cases) and entropy methods are studied in [8]. Also see [30, 12] for the limiting case of the logarithmic Sobolev inequality and corresponding improved decay rates for linear heat flows. Notice that stability estimates can also be applied in the perspective of estimating a *posteriori* errors both from theoretical (test functions) and numerical (approximate numerical solutions) points of view.
The key idea is to prove that $i - d\varphi(\mathbf{e})$ is monotone nonincreasing under the action of

$$ \frac{\partial \rho}{\partial t} = \mathcal{L} \rho^m, \qquad (9) $$

where $m = 1$ corresponds to the heat flow, $m > 1$ to the porous medium flow and $m < 1$ to the fast diffusion flow. Here we choose $\rho = |f|^p$ so that $\|u\|_{L^p(\mathbb{R}^d)}$ is conserved under the action of (9). It is convenient to introduce the exponent $\beta$ such that

$$ m = 1 + \frac{2}{p} \left( \frac{1}{\beta} - 1 \right) $$

and consider the function $f = w^\beta$, such that $w^{\beta p} = \rho$, which solves

$$ \frac{\partial w}{\partial t} = w^{2-2\beta} \left( \mathcal{L}w + \kappa \frac{|w'|^2}{w} \right) \quad \text{with} \quad \kappa := \beta(p-2) + 1. \qquad (10) $$

The carré du champ method shows that $\frac{d}{dt}(i - d\mathbf{e}) \le 0$ if the function

$$ \gamma(\beta) := - \left( \frac{d-1}{d+2} (\kappa + \beta - 1) \right)^2 + \kappa (\beta - 1) + \frac{d}{d+2} (\kappa + \beta - 1) $$

takes nonnegative values, which amounts to $m_-(d, p) \le m \le m_+(d, p)$ where

$$ m_{\pm}(d, p) := \frac{1}{(d+2)p} \left( dp + 2 \pm \sqrt{d(d-2)(p-1)(2^* - p)} \right). $$

Using $\lim_{t \to +\infty} (i(t) - d\mathbf{e}(t)) = 0$, we conclude that $i(t) - d\mathbf{e}(t) \ge 0$ for any $t \ge 0$ and, as a special case, for $t = 0$: this proves (GNS) for an arbitrary initial datum. If $m_-(d, p) < m < m_+(d, p)$, we find that $\gamma(\beta) > 0$, which leaves some space for an improvement. A more detailed and quite lengthy computation along the flow (10) yields

$$ \frac{d}{dt}(i - d\varphi(\mathbf{e})) \le \frac{\gamma}{\beta^2} \frac{i - d\varphi(\mathbf{e})}{(1 - (p-2)\mathbf{e})^\delta} \frac{d\mathbf{e}}{dt} $$

where $\delta = 1$ if $1 \le p \le 2$ and $\delta := \frac{2-(4-p)\beta}{2\beta(p-2)}$ if $p > 2$, if $\varphi$ solves

$$ \frac{d\varphi}{ds} = 1 + \frac{\gamma}{\beta^2} \frac{\varphi(s)}{(1 - (p-2)s)^{\delta}}, \quad \varphi(0) = 0. $$

This proves (8) for any $t \ge 0$ as a consequence of the monotonicity of $i - d\varphi(\mathbf{e})$ and the fact that $\lim_{t \to +\infty} (i(t) - d\varphi(\mathbf{e}(t))) = 0$. See [10, Appendix B.4] for detailed justifications. $\square$

As a consequence of the carré du champ method, optimal constants in Gagliardo-Nirenberg-Sobolev inequalities determine optimal rates of decay in some associated fast diffusion equations. A stability result for a functional inequality can usually be rephrased as an improved inequality, under appropriate constraints. Measured by the dissipation of some entropy functional, this means a faster convergence rate towards an asymptotic profile for solutions of the associated evolution equation: see [17] for an application of stability for Gagliardo-Nirenberg-Sobolev inequalities to the proof of a quantitative rate of convergence for solutions of the Keller-Segel system, and [10] for explicit, improved, convergence rates based on (GNS) in the subcritical regime, in the case of the sphere. On the Euclidean space, improved convergence rates based on stability for Gagliardo-Nirenberg-Sobolev inequalities (subcritical and critical cases) and entropy methods are studied in [8]. Also see [30, 12] for the limiting case of the logarithmic Sobolev inequality and corresponding improved decay rates for linear heat flows. Notice that stability estimates can also be applied in the perspective of estimating a *posteriori* errors both from theoretical (test functions) and numerical (approximate numerical solutions) points of view.
### 3. STABILITY OF THE SOBOLEV INEQUALITY: PROOF OF THEOREM 1

In this section we explain the general ideas of the proof of Theorem 1 in [24].

#### 3.1. On the stability proof by Bianchi and Egnell.
The strategy of Bianchi-Egnell to prove (5) is based on two main steps:

(1) A local stability estimate in a neighborhood of $M$, obtained by a local spectral analysis.

(2) A reduction of the global estimate to the local estimate by the concentration-compactness method based on Lions' analysis (see [38]).

Theorem 1 is a significant improvement of Bianchi-Egnell's result, as it contains a dimensionally sharp lower estimate for the best stability constant. Our strategy is to make the second step constructive and the first one explicit, with much more detailed estimates.

#### 3.2. Strategy of the proof of Theorem 1.
The proof is divided into several steps:

(1) Local analysis: prove the inequality for nonnegative functions close to $M$ with an explicit remainder term. The analysis is quite involved: it relies on “cuttings” at various heights, the use of uniform bounds on spherical harmonics and some delicate concavity properties.

(2) Local to global extension: prove the inequality for nonnegative functions far from $M$ using the competing symmetries of [18] and a continuous Steiner symmetrization.

(3) Deduce the inequality for sign-changing functions from the inequality for nonnegative functions by a concavity argument.

* In the first step, in order to obtain uniform estimates as $d \to +\infty$, we need to expand

$$ (1+r)^{2*} - 1 - 2^* r $$

with an accurate remainder term, for all $r \ge -1$. To do that, we “cut $r$ into pieces” by defining

$$ r_1 := \min\{r, \gamma\}, \quad r_2 := \min\{(r - \gamma)_+, M - \gamma\} \quad \text{and} \quad r_3 := (r - M)_+ $$

where $\gamma$ and $M$ are suitable parameters satisfying $0 < \gamma < M$. Furthermore, define

$$ \theta := 2^* - 2 = \frac{4}{d-2}. $$

Notice that $\theta \in (0, 1]$ if $d \ge 6$ and $\lim_{d \to +\infty} \theta(d) = 0$.

**Lemma 4** ([24, Proposition 2.9]). Given $d \ge 6$, $r \in [-1, \infty)$, and $\bar{M} \in [\sqrt{e}, +\infty)$, we have

$$ (1+r)^{2*} - 1 - 2^* r \le \frac{1}{2} 2^* (2^* - 1) (r_1 + r_2)^2 + 2 (r_1 + r_2) r_3 + \left(1 + C_M \theta \bar{M}^{-1} \ln \bar{M}\right) r_3^{2*} \\ + \left(\frac{3}{2} \gamma \theta r_1^2 + C_{M,\bar{M}} \theta r_2^2\right) \mathbb{1}_{\{r \le M\}} + C_{M,\bar{M}} \theta M^2 \mathbb{1}_{\{r > M\}}, $$

where all the constants in the above inequality are explicit.

One can then prove that there exist computable constants $\epsilon_1, \epsilon_2, k_0$, and $\epsilon_0 \in (0, 1/\theta)$, such that

$$ \|\nabla r\|_{L^2(\mathbb{S}^d)}^2 + A \|r\|_{L^2(\mathbb{S}^d)}^2 - A \|1+r\|_{L^{2^*}(\mathbb{S}^d)}^2 \ge \frac{4\epsilon_0}{d-2} \left( \|\nabla r\|_{L^2(\mathbb{S}^d)}^2 + A \|r\|_{L^2(\mathbb{S}^d)}^2 \right) + \sum_{k=1}^3 I_k, $$

with $A := \frac{1}{4} d (d - 2)$ and

$$ I_1 := (1 - \theta \epsilon_0) \int_{\mathbb{S}^d} (|\nabla r_1|^2 + A r_1^2) d\mu_d - A (2^* - 1 + \epsilon_1 \theta) \int_{\mathbb{S}^d} r_1^2 d\mu_d + A k_0 \theta \int_{\mathbb{S}^d} (r_2^2 + r_3^2) d\mu_d, $$

$$ I_2 := (1 - \theta \epsilon_0) \int_{\mathbb{S}^d} (|\nabla r_2|^2 + A r_2^2) d\mu_d - A (2^* - 1 + (k_0 + C_{\epsilon_1, \epsilon_2}) \theta) \int_{\mathbb{S}^d} r_2^2 d\mu_d, $$

$$ I_3 := (1 - \theta \epsilon_0) \int_{\mathbb{S}^d} (|\nabla r_3|^2 + A r_3^2) d\mu_d - \frac{2}{2^*} A (1 + \epsilon_2 \theta) \int_{\mathbb{S}^d} r_3^{2*} d\mu_d - A k_0 \theta \int_{\mathbb{S}^d} r_3^2 d\mu_d. $$

Next, one can use spectral gap estimates to prove $I_1 \ge 0$ and the Sobolev inequality to prove $I_3 \ge 0$, noting that the extra coefficient $2/2^* < 1$ gives enough room to accommodate all error terms. Finally, using that $\mu(\{r_2 > 0\})$ is small, an improved spectral gap inequality allows us to show $I_2 \ge 0$.
### 3. STABILITY OF THE SOBOLEV INEQUALITY: PROOF OF THEOREM 1

In this section we explain the general ideas of the proof of Theorem 1 in [24].

#### 3.1. On the stability proof by Bianchi and Egnell.
The strategy of Bianchi-Egnell to prove (5) is based on two main steps:

(1) A local stability estimate in a neighborhood of $M$, obtained by a local spectral analysis.

(2) A reduction of the global estimate to the local estimate by the concentration-compactness method based on Lions' analysis (see [38]).

Theorem 1 is a significant improvement of Bianchi-Egnell's result, as it contains a dimensionally sharp lower estimate for the best stability constant. Our strategy is to make the second step constructive and the first one explicit, with much more detailed estimates.

#### 3.2. Strategy of the proof of Theorem 1.
The proof is divided into several steps:

(1) Local analysis: prove the inequality for nonnegative functions close to $M$ with an explicit remainder term. The analysis is quite involved: it relies on “cuttings” at various heights, the use of uniform bounds on spherical harmonics and some delicate concavity properties.

(2) Local to global extension: prove the inequality for nonnegative functions far from $M$ using the competing symmetries of [18] and a continuous Steiner symmetrization.

(3) Deduce the inequality for sign-changing functions from the inequality for nonnegative functions by a concavity argument.

* In the first step, in order to obtain uniform estimates as $d \to +\infty$, we need to expand

$$ (1+r)^{2*} - 1 - 2^* r $$

with an accurate remainder term, for all $r \ge -1$. To do that, we “cut $r$ into pieces” by defining

$$ r_1 := \min\{r, \gamma\}, \quad r_2 := \min\{(r-\gamma)_+, M-\gamma\} \quad \text{and} \quad r_3 := (r-M)_+ $$

where $\gamma$ and $M$ are suitable parameters satisfying $0 < \gamma < M$. Furthermore, define

$$ \theta := 2^* - 2 = \frac{4}{d-2}. $$

Notice that $\theta \in (0, 1]$ if $d \ge 6$ and $\lim_{d \to +\infty} \theta(d) = 0$.

**Lemma 4** ([24, Proposition 2.9]). Given $d \ge 6$, $r \in [-1, \infty)$, and $\bar{M} \in [\sqrt{e}, +\infty)$, we have

$$ (1+r)^{2*} - 1 - 2^* r \le \frac{1}{2} 2^* (2^* - 1) (r_1+r_2)^2 + 2(r_1+r_2)r_3 + \left(1+C_M \theta \bar{M}^{-1} \ln \bar{M}\right) r_3^{2*} \\ + \left(\frac{3}{2} \gamma \theta r_1^2 + C_{M,\bar{M}} \theta r_2^2\right) \mathbb{1}_{\{r \le M\}} + C_{M,\bar{M}} \theta M^2 \mathbb{1}_{\{r > M\}}, $$

where all the constants in the above inequality are explicit.

One can then prove that there exist computable constants $\epsilon_1, \epsilon_2, k_0$, and $\epsilon_0 \in (0, 1/\theta)$, such that

$$ \|\nabla r\|_{L^2(\mathbb{S}^d)}^2 + A \|r\|_{L^2(\mathbb{S}^d)}^2 - A \|1+r\|_{L^{2^*}(\mathbb{S}^d)}^2 \ge \frac{4\epsilon_0}{d-2} \left( \|\nabla r\|_{L^2(\mathbb{S}^d)}^2 + A \|r\|_{L^2(\mathbb{S}^d)}^2 \right) + \sum_{k=1}^3 I_k, $$

with $A := \frac{1}{4} d (d-2)$ and

$$ I_1 := (1 - \theta \epsilon_0) \int_{\mathbb{S}^d} (|\nabla r_1|^2 + A r_1^2) d\mu_d - A (2^* - 1 + \epsilon_1 \theta) \int_{\mathbb{S}^d} r_1^2 d\mu_d + A k_0 \theta \int_{\mathbb{S}^d} (r_2^2 + r_3^2) d\mu_d, $$

$$ I_2 := (1 - \theta \epsilon_0) \int_{\mathbb{S}^d} (|\nabla r_2|^2 + A r_2^2) d\mu_d - A (2^* - 1 + (k_0 + C_{\epsilon_1, \epsilon_2}) \theta) \int_{\mathbb{S}^d} r_2^2 d\mu_d, $$

$$ I_3 := (1 - \theta \epsilon_0) \int_{\mathbb{S}^d} (|\nabla r_3|^2 + A r_3^2) d\mu_d - \frac{2}{2^*} A (1 + \epsilon_2 \theta) \int_{\mathbb{S}^d} r_3^{2*} d\mu_d - A k_0 \theta \int_{\mathbb{S}^d} r_3^2 d\mu_d. $$

Next, one can use spectral gap estimates to prove $I_1 \ge 0$ and the Sobolev inequality to prove $I_3 \ge 0$, noting that the extra coefficient $2/2^* < 1$ gives enough room to accommodate all error terms. Finally, using that $\mu(\{r_2 > 0\})$ is small, an improved spectral gap inequality allows us to show $I_2 \ge 0$.
If $d = 3, 4$, or $5$, we can rely on a simpler Taylor expansion that can be found in [24, Proposition 2.7]. As a consequence, the following result has been proved.

**Theorem 5 ([24, Theorem 2.1]).** There are explicit constants $\epsilon_0 \in (0, 1/3)$ and $\delta \in (0, 1/2)$ such that for all $d \ge 3$ and for all nonnegative $u = 1 + r \in H^1(\mathbb{S}^d)$ with

$$ \|r\|_{L^{2^*}(\mathbb{S}^d)}^2 \le \frac{\delta}{1-\delta}, \quad \int_{\mathbb{S}^d} r \, d\mu_d = 0 \quad \text{and} \quad \int_{\mathbb{S}^d} \omega r \, d\mu_d = 0, $$

one has

$$ \|\nabla u\|_{L^2(\mathbb{S}^d)}^2 + A \|u\|_{L^2(\mathbb{S}^d)}^2 - A \|u\|_{L^{2^*}(\mathbb{S}^d)}^2 \ge \frac{4\epsilon_0}{d-2} \left( \|\nabla r\|_{L^2(\mathbb{S}^d)}^2 + A \|r\|_{L^2(\mathbb{S}^d)}^2 \right). $$

Let us define the stability quotient

$$ \mathcal{E}(f) := \frac{\|\nabla f\|_{L^2(\mathbb{R}^d)}^2 - S_d \|f\|_{L^{2^*}(\mathbb{R}^d)}^2}{\inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_{L^2(\mathbb{R}^d)}^2} $$

and consider the infimum

$$ \mathcal{I}(\delta) := \inf \left\{ \mathcal{E}(f) : 0 \le f \in \dot{H}^1(\mathbb{R}^d) \setminus \mathcal{M}, \quad \inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_{L^2(\mathbb{R}^d)}^2 \le \delta \|\nabla f\|_{L^2(\mathbb{R}^d)}^2 \right\}. $$

For a given $f$, up to a conformal transformation, we can assume that $\inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_{L^2(\mathbb{R}^d)}^2$ is realized by the Aubin-Talenti function $g = g_\star$ with

$$ g_{\star}(x) := |\mathbb{S}^d|^{-\frac{d-2}{2d}} \left( \frac{2}{1+|x|^2} \right)^{\frac{d-2}{2}} \quad \forall x \in \mathbb{R}^d, \qquad (11) $$

use the inverse stereographic projection to transform $f$ and $g$ respectively into $u = 1 + r$ and $1$, and notice that

$$ \begin{aligned} A \|r\|_{L^{2^*}(\mathbb{S}^d)}^2 &\le \|\nabla r\|_{L^2(\mathbb{S}^d)}^2 + A \|r\|_{L^2(\mathbb{S}^d)}^2 = \inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_{L^2(\mathbb{R}^d)}^2, \\ \|\nabla f\|_{L^2(\mathbb{R}^d)}^2 &= \|\nabla u\|_{L^2(\mathbb{S}^d)}^2 + A \|u\|_{L^2(\mathbb{S}^d)}^2 = \|\nabla r\|_{L^2(\mathbb{S}^d)}^2 + A \|r\|_{L^2(\mathbb{S}^d)}^2 + A, \end{aligned} $$

where the first line follows from (1). We deduce from the condition $\inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_{L^2(\mathbb{R}^d)}^2 \le \delta \|\nabla f\|_{L^2(\mathbb{R}^d)}^2$ that

$$ A \|r\|_{L^{2^*}(\mathbb{S}^d)}^2 \le \|\nabla r\|_{L^2(\mathbb{S}^d)}^2 + A \|r\|_{L^2(\mathbb{S}^d)}^2 \le \frac{\delta A}{1-\delta} $$

and apply Theorem 5 to obtain

$$ \mathcal{E}(f) = \frac{\|\nabla u\|_{L^2(\mathbb{S}^d)}^2 + A \|u\|_{L^2(\mathbb{S}^d)}^2 - A \|u\|_{L^{2^*}(\mathbb{S}^d)}^2}{\|\nabla r\|_{L^2(\mathbb{S}^d)}^2 + A \|r\|_{L^2(\mathbb{S}^d)}^2} \ge \frac{4\epsilon_0}{d-2}. $$

This completes the *local* analysis where, by homogeneity, the scale is fixed in terms of $\|\nabla f\|_{L^2(\mathbb{R}^d)}^2$. With the notation of Theorem 5,

$$ \mathcal{I}(\delta) \ge \frac{4\epsilon_0}{d-2}. $$

• In Step 2, we deal with nonnegative functions $f$ that are not close to the manifold $\mathcal{M}$, i.e., such that

$$ \inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_{L^2(\mathbb{R}^d)}^2 > \delta \|\nabla f\|_{L^2(\mathbb{R}^d)}^2. \qquad (12) $$

The first ingredient is the method of competing symmetries [18] of Carlen and Loss. Consider any nonnegative function $f \in \dot{H}^1(\mathbb{R}^d)$ and let

$$ (Uf)(x) := \left( \frac{2}{|x - e_d|^2} \right)^{\frac{d-2}{2}} f \left( \frac{x_1}{|x - e_d|^2}, \dots, \frac{x_{d-1}}{|x - e_d|^2}, \frac{|x|^2 - 1}{|x - e_d|^2} \right) \quad \text{where} \quad e_d = (0, \dots, 0, 1) \in \mathbb{R}^d, $$
If $d = 3, 4$, or $5$, we can rely on a simpler Taylor expansion that can be found in [24, Proposition 2.7]. As a consequence, the following result has been proved.

**Theorem 5** ([24, Theorem 2.1]). There are explicit constants $\epsilon_0 \in (0, 1/3)$ and $\delta \in (0, 1/2)$ such that for all $d \ge 3$ and for all nonnegative $u = 1 + r \in H^1(\mathbb{S}^d)$ with

$$ \|r\|_{L^{2^*}(\mathbb{S}^d)}^2 \le \frac{\delta}{1-\delta}, \quad \int_{\mathbb{S}^d} r \, d\mu_d = 0 \quad \text{and} \quad \int_{\mathbb{S}^d} \omega r \, d\mu_d = 0, $$

one has

$$ \|\nabla u\|_{L^2(\mathbb{S}^d)}^2 + A \|u\|_{L^2(\mathbb{S}^d)}^2 - A \|u\|_{L^{2^*}(\mathbb{S}^d)}^2 \ge \frac{4\epsilon_0}{d-2} \left( \|\nabla r\|_{L^2(\mathbb{S}^d)}^2 + A \|r\|_{L^2(\mathbb{S}^d)}^2 \right). $$

Let us define the stability quotient

$$ \mathcal{E}(f) := \frac{\|\nabla f\|_{L^2(\mathbb{R}^d)}^2 - S_d \|f\|_{L^{2^*}(\mathbb{R}^d)}^2}{\inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_{L^2(\mathbb{R}^d)}^2} $$

and consider the infimum

$$ \mathcal{I}(\delta) := \inf \left\{ \mathcal{E}(f) : 0 \le f \in \dot{H}^1(\mathbb{R}^d) \setminus \mathcal{M}, \quad \inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_{L^2(\mathbb{R}^d)}^2 \le \delta \|\nabla f\|_{L^2(\mathbb{R}^d)}^2 \right\}. $$

For a given $f$, up to a conformal transformation, we can assume that $\inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_{L^2(\mathbb{R}^d)}^2$ is realized by the Aubin-Talenti function $g = g_\star$ with

$$ g_{\star}(x) := |\mathbb{S}^d|^{-\frac{d-2}{2d}} \left( \frac{2}{1 + |x|^2} \right)^{\frac{d-2}{2}} \quad \forall x \in \mathbb{R}^d, \qquad (11) $$

use the inverse stereographic projection to transform $f$ and $g$ respectively into $u = 1 + r$ and $1$, and notice that

$$ \begin{aligned} A \|r\|_{L^{2^*}(\mathbb{S}^d)}^2 &\le \|\nabla r\|_{L^2(\mathbb{S}^d)}^2 + A \|r\|_{L^2(\mathbb{S}^d)}^2 = \inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_{L^2(\mathbb{R}^d)}^2, \\ \|\nabla f\|_{L^2(\mathbb{R}^d)}^2 &= \|\nabla u\|_{L^2(\mathbb{S}^d)}^2 + A \|u\|_{L^2(\mathbb{S}^d)}^2 = \|\nabla r\|_{L^2(\mathbb{S}^d)}^2 + A \|r\|_{L^2(\mathbb{S}^d)}^2 + A, \end{aligned} $$

where the first line follows from (1). We deduce from the condition $\inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_{L^2(\mathbb{R}^d)}^2 \le \delta \|\nabla f\|_{L^2(\mathbb{R}^d)}^2$ that

$$ A \|r\|_{L^{2^*}(\mathbb{S}^d)}^2 \le \|\nabla r\|_{L^2(\mathbb{S}^d)}^2 + A \|r\|_{L^2(\mathbb{S}^d)}^2 \le \frac{\delta A}{1-\delta} $$

and apply Theorem 5 to obtain

$$ \mathcal{E}(f) = \frac{\|\nabla u\|_{L^2(\mathbb{S}^d)}^2 + A \|u\|_{L^2(\mathbb{S}^d)}^2 - A \|u\|_{L^{2^*}(\mathbb{S}^d)}^2}{\|\nabla r\|_{L^2(\mathbb{S}^d)}^2 + A \|r\|_{L^2(\mathbb{S}^d)}^2} \ge \frac{4\epsilon_0}{d-2}. $$

This completes the local analysis where, by homogeneity, the scale is fixed in terms of $\|\nabla f\|_{L^2(\mathbb{R}^d)}^2$. With the notation of Theorem 5,

$$ \mathcal{I}(\delta) \ge \frac{4\epsilon_0}{d-2}. $$

• In Step 2, we deal with nonnegative functions $f$ that are not close to the manifold $\mathcal{M}$, i.e., such that

$$ \inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_{L^2(\mathbb{R}^d)}^2 > \delta \|\nabla f\|_{L^2(\mathbb{R}^d)}^2. \qquad (12) $$

The first ingredient is the method of competing symmetries [18] of Carlen and Loss. Consider any nonnegative function $f \in \dot{H}^1(\mathbb{R}^d)$ and let

$$ (Uf)(x) := \left( \frac{2}{|x - e_d|^2} \right)^{\frac{d-2}{2}} f \left( \frac{x_1}{|x - e_d|^2}, \dots, \frac{x_{d-1}}{|x - e_d|^2}, \frac{|x|^2 - 1}{|x - e_d|^2} \right) \quad \text{where} \quad e_d = (0, \dots, 0, 1) \in \mathbb{R}^d, $$
and notice that $\mathcal{E}(Uf) = \mathcal{E}(f)$. We also consider the symmetric decreasing rearrangement $\mathcal{R}f = f^*$, with the properties that $f$ and $f^*$ are equimeasurable, and that $\|\nabla f^*\|_{L^2(\mathbb{R}^d)} \le \|\nabla f\|_{L^2(\mathbb{R}^d)}$. The following result is taken from the proof of [18, Theorem 3.3].

**Theorem 6** ([18]). Let $f \in L^{2^*}(\mathbb{R}^d)$ be a nonnegative function with $\|f\|_{L^{2^*}(\mathbb{R}^d)} = 1$. The sequence $f_n = (\mathcal{R}U)^n f$ is such that $\lim_{n \to +\infty} \|f_n - g_*\|_{L^{2^*}(\mathbb{R}^d)} = 0$. If $f \in \dot{H}^1(\mathbb{R}^d)$, then $(\|\nabla f_n\|_{L^2(\mathbb{R}^d)})_{n \in \mathbb{N}}$ is a nonincreasing sequence.

Whether $f_n$ satisfies (12) for all $n \in \mathbb{N}$ or not, we face an alternative.

**Lemma 7** ([24, Lemma 3.5]). Let $f$ be as in Theorem 6 and such that (12) holds and let $f_n = (\mathcal{R}U)^n f$. Then either $\inf_{g \in \mathcal{M}} \|\nabla f_n - \nabla g\|_{L^2(\mathbb{R}^d)}^2 \ge \delta \|\nabla f_n\|_{L^2(\mathbb{R}^d)}^2$ for all $n \in \mathbb{N}$, or there exists $n_0 \in \mathbb{N}$ such that

$$ \inf_{g \in \mathcal{M}} \|\nabla f_{n_0} - \nabla g\|_{L^2(\mathbb{R}^d)}^2 > \delta \|\nabla f_{n_0}\|_{L^2(\mathbb{R}^d)}^2 \quad \text{and} \quad \inf_{g \in \mathcal{M}} \|\nabla f_{n_0+1} - \nabla g\|_{L^2(\mathbb{R}^d)}^2 < \delta \|\nabla f_{n_0+1}\|_{L^2(\mathbb{R}^d)}^2. $$

In the first case, that is if $f_n$ satisfies (12) for all $n \in \mathbb{N}$, we have

$$ \lim_{n \to +\infty} \|\nabla f_n\|_2^2 \le \frac{1}{\delta} \lim_{n \to +\infty} \inf_{g \in \mathcal{M}} \|\nabla f_n - \nabla g\|_2^2 = \frac{1}{\delta} \left( \lim_{n \to +\infty} \|\nabla f_n\|_2^2 - S_d \|f\|_{2^*}^2 \right) $$

where the last equality arises as a consequence of the properties of $(f_n)_{n \in \mathbb{N}}$ (see [24, Lemma 3.4]). Combined with the simple estimate

$$ \mathcal{E}(f) = \frac{\|\nabla f\|_2^2 - S_d \|f\|_{2^*}^2}{\inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_2^2} \ge \frac{\|\nabla f\|_2^2 - S_d \|f\|_{2^*}^2}{\|\nabla f\|_2^2} \ge \frac{\|\nabla f_n\|_2^2 - S_d \|f\|_{2^*}^2}{\|\nabla f_n\|_2^2}, $$

we can take the limit as $n \to +\infty$ and obtain $\mathcal{E}(f) \ge \delta$. In the second case, we adapt a strategy that has some similarities to one used by Christ in [21] and is based on a continuous rearrangement flow $(f_\tau)_{n_0 \le \tau < n_0+1}$ with $f_{n_0} = Uf_{n_0}$ such that

$$ \|f_\tau\|_{L^{2^*}(\mathbb{R}^d)} = \|f_{n_0}\|_{L^{2^*}(\mathbb{R}^d)}, \quad \tau \mapsto \|\nabla f_\tau\|_{L^2(\mathbb{R}^d)} \text{ is nonincreasing, and } \lim_{\tau \to n_0+1} f_\tau = f_{n_0+1}. $$

Choosing the smallest $\tau_0 \in (n_0, n_0 + 1)$ such that $\inf_{g \in \mathcal{M}} \|\nabla(f_{\tau_0} - g)\|_{L^2(\mathbb{R}^d)}^2 = \delta \|\nabla f_{\tau_0}\|_{L^2(\mathbb{R}^d)}^2$ and using $\|f\|_{L^{2^*}(\mathbb{R}^d)} = \|f_{n_0}\|_{L^{2^*}(\mathbb{R}^d)} = \|f_{\tau_0}\|_{L^{2^*}(\mathbb{R}^d)}$, this gives:

$$ 1 - S_d \frac{\|f\|_{L^{2^*}(\mathbb{R}^d)}^2}{\|\nabla f\|_{L^2(\mathbb{R}^d)}^2} \ge 1 - S_d \frac{\|f_{\tau_0}\|_{L^{2^*}(\mathbb{R}^d)}^2}{\|\nabla f_{\tau_0}\|_{L^2(\mathbb{R}^d)}^2} = \delta \mathcal{E}(f_{\tau_0}) \ge \delta \mathcal{I}(\delta). $$

As a consequence, we obtain $C_{d,\text{BE}}^{\text{pos}} := \inf_{f \ge 0} \mathcal{E}(f) \ge \min\{\delta, \delta \mathcal{I}(\delta)\} = \delta \mathcal{I}(\delta)$ because $\mathcal{I}(\delta) < 1$. To build the flow, we refer to [13, 14] and to [24] for further references. The existence of $\tau_0$ requires a discussion that can be found in [24, Section 3.1.2].

• The third step is to remove the positivity assumption of Theorem 1 as in [24, Section 3.2]. Take $f = f_+ - f_-$ with $\|f\|_{L^{2^*}(\mathbb{R}^d)} = 1$ and define $m := \|f_-\|_{L^{2^*}(\mathbb{R}^d)}^{2^*}$. Without loss of generality one may assume that $1 - m = \|f_+\|_{L^{2^*}(\mathbb{R}^d)}^{2^*} > 1/2$. The positive concave function

$$ h_d(m) := m^{\frac{d-2}{d}} + (1-m)^{\frac{d-2}{d}} - 1 $$

satisfies

$$ 2 h_d(1/2) m \le h_d(m), \quad h_d(1/2) = 2^{2/d} - 1. $$

With $\delta(f) = \|\nabla f\|_{L^2(\mathbb{R}^d)}^2 - S_d \|f\|_{L^{2^*}(\mathbb{R}^d)}^2$, one finds $g_+ \in \mathcal{M}$ such that

$$ \delta(f) \ge C_{d,\text{BE}}^{\text{pos}} \|\nabla f_+ - \nabla g_+\|_{L^2(\mathbb{R}^d)}^2 + \frac{2 h_d(1/2)}{h_d(1/2) + 1} \|\nabla f_-\|_{L^2(\mathbb{R}^d)}^2, $$
and notice that $\mathcal{E}(Uf) = \mathcal{E}(f)$. We also consider the symmetric decreasing rearrangement $\mathcal{R}f = f^*$, with the properties that $f$ and $f^*$ are equimeasurable, and that $\|\nabla f^*\|_{L^2(\mathbb{R}^d)} \le \|\nabla f\|_{L^2(\mathbb{R}^d)}$. The following result is taken from the proof of [18, Theorem 3.3].

**Theorem 6** ([18]). Let $f \in L^{2^*}(\mathbb{R}^d)$ be a nonnegative function with $\|f\|_{L^{2^*}(\mathbb{R}^d)} = 1$. The sequence $f_n = (\mathcal{R}U)^n f$ is such that $\lim_{n \to +\infty} \|f_n - g_*\|_{L^{2^*}(\mathbb{R}^d)} = 0$. If $f \in \dot{H}^1(\mathbb{R}^d)$, then $(\|\nabla f_n\|_{L^2(\mathbb{R}^d)})_{n \in \mathbb{N}}$ is a nonincreasing sequence.

Whether $f_n$ satisfies (12) for all $n \in \mathbb{N}$ or not, we face an alternative.

**Lemma 7** ([24, Lemma 3.5]). Let $f$ be as in Theorem 6 and such that (12) holds and let $f_n = (\mathcal{R}U)^n f$. Then either $\inf_{g \in \mathcal{M}} \|\nabla f_n - \nabla g\|_{L^2(\mathbb{R}^d)}^2 \ge \delta \|\nabla f_n\|_{L^2(\mathbb{R}^d)}^2$ for all $n \in \mathbb{N}$, or there exists $n_0 \in \mathbb{N}$ such that

$$ \inf_{g \in \mathcal{M}} \|\nabla f_{n_0} - \nabla g\|_{L^2(\mathbb{R}^d)}^2 > \delta \|\nabla f_{n_0}\|_{L^2(\mathbb{R}^d)}^2 \quad \text{and} \quad \inf_{g \in \mathcal{M}} \|\nabla f_{n_0+1} - \nabla g\|_{L^2(\mathbb{R}^d)}^2 < \delta \|\nabla f_{n_0+1}\|_{L^2(\mathbb{R}^d)}^2. $$

In the first case, that is if $f_n$ satisfies (12) for all $n \in \mathbb{N}$, we have

$$ \lim_{n \to +\infty} \|\nabla f_n\|_2^2 \le \frac{1}{\delta} \lim_{n \to +\infty} \inf_{g \in \mathcal{M}} \|\nabla f_n - \nabla g\|_2^2 = \frac{1}{\delta} \left( \lim_{n \to +\infty} \|\nabla f_n\|_2^2 - S_d \|f\|_{2^*}^2 \right) $$

where the last equality arises as a consequence of the properties of $(f_n)_{n \in \mathbb{N}}$ (see [24, Lemma 3.4]). Combined with the simple estimate

$$ \mathcal{E}(f) = \frac{\|\nabla f\|_2^2 - S_d \|f\|_{2^*}^2}{\inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_2^2} \ge \frac{\|\nabla f\|_2^2 - S_d \|f\|_{2^*}^2}{\|\nabla f\|_2^2} \ge \frac{\|\nabla f_n\|_2^2 - S_d \|f\|_{2^*}^2}{\|\nabla f_n\|_2^2}, $$

we can take the limit as $n \to +\infty$ and obtain $\mathcal{E}(f) \ge \delta$. In the second case, we adapt a strategy that has some similarities to one used by Christ in [21] and is based on a continuous rearrangement flow $(f_\tau)_{n_0 \le \tau < n_0+1}$ with $f_{n_0} = Uf_{n_0}$ such that

$$ \|f_{\tau}\|_{L^{2^*}(\mathbb{R}^d)} = \|f_{n_0}\|_{L^{2^*}(\mathbb{R}^d)}, \quad \tau \mapsto \|\nabla f_{\tau}\|_{L^2(\mathbb{R}^d)} \text{ is nonincreasing, and } \lim_{\tau \to n_0+1} f_{\tau} = f_{n_0+1}. $$

Choosing the smallest $\tau_0 \in (n_0, n_0 + 1)$ such that $\inf_{g \in \mathcal{M}} \|\nabla(f_{\tau_0} - g)\|_{L^2(\mathbb{R}^d)}^2 = \delta \|\nabla f_{\tau_0}\|_{L^2(\mathbb{R}^d)}^2$ and using $\|f\|_{L^{2^*}(\mathbb{R}^d)} = \|f_{n_0}\|_{L^{2^*}(\mathbb{R}^d)} = \|f_{\tau_0}\|_{L^{2^*}(\mathbb{R}^d)}$, this gives:

$$ 1 - S_d \frac{\|f\|_{L^{2^*}(\mathbb{R}^d)}^2}{\|\nabla f\|_{L^2(\mathbb{R}^d)}^2} \ge 1 - S_d \frac{\|f_{\tau_0}\|_{L^{2^*}(\mathbb{R}^d)}^2}{\|\nabla f_{\tau_0}\|_{L^2(\mathbb{R}^d)}^2} = \delta \mathcal{E}(f_{\tau_0}) \ge \delta \mathcal{I}(\delta). $$

As a consequence, we obtain $C_{d,\text{BE}}^{\text{pos}} := \inf_{f \ge 0} \mathcal{E}(f) \ge \min\{\delta, \delta \mathcal{I}(\delta)\} = \delta \mathcal{I}(\delta)$ because $\mathcal{I}(\delta) < 1$. To build the flow, we refer to [13, 14] and to [24] for further references. The existence of $\tau_0$ requires a discussion that can be found in [24, Section 3.1.2].

• The third step is to remove the positivity assumption of Theorem 1 as in [24, Section 3.2]. Take $f = f_+ - f_-$ with $\|f\|_{L^{2^*}(\mathbb{R}^d)} = 1$ and define $m := \|f_-\|_{L^{2^*}(\mathbb{R}^d)}^{2*}$. Without loss of generality one may assume that $1 - m = \|f_+\|_{L^{2^*}(\mathbb{R}^d)}^{2*} > 1/2$. The positive concave function

$$ h_d(m) := m^{\frac{d-2}{d}} + (1-m)^{\frac{d-2}{d}} - 1 $$

satisfies

$$ 2 h_d(1/2) m \le h_d(m), \quad h_d(1/2) = 2^{2/d} - 1. $$

With $\delta(f) = \|\nabla f\|_{L^2(\mathbb{R}^d)}^2 - S_d \|f\|_{L^{2^*}(\mathbb{R}^d)}^2$, one finds $g_+ \in \mathcal{M}$ such that

$$ \delta(f) \ge C_{d,\text{BE}}^{\text{pos}} \|\nabla f_+ - \nabla g_+\|_{L^2(\mathbb{R}^d)}^2 + \frac{2 h_d(1/2)}{h_d(1/2) + 1} \|\nabla f_-\|_{L^2(\mathbb{R}^d)}^2, $$
and therefore

$$C_{d,\text{BE}} \geq \frac{1}{2} \min \left\{ \delta \mathcal{I}(\delta), \frac{2 h_d(1/2)}{h_d(1/2) + 1} \right\}.$$

• Combining all estimates of the three previous steps completes the proof of Theorem 1. □

### 4. STABILITY OF THE GAUSSIAN LOGARITHMIC SOBOLEV INEQUALITY: PROOF OF THEOREM 2

In this section we describe the main steps in the proof of the stability estimate for the Gaussian logarithmic Sobolev inequality (LSI) by considering the large dimensional limit of (6), as it appears in [24, Section 4]. With $u = f/g_\star$, where $g_\star$ is given by (11), Inequality (6) can be rewritten as

$$\int_{\mathbb{R}^d} |\nabla u|^2 g_\star^2 dx + d(d-2) \int_{\mathbb{R}^d} |u|^2 g_\star^{2*} dx - d(d-2) \|g_\star\|_{L^{2^*(\mathbb{R}^d)}}^{2^*-2} \left( \int_{\mathbb{R}^d} |u|^{2^*} g_\star^{2*} dx \right)^{2/2^*} \\ \geq \frac{\beta}{d} \left( \int_{\mathbb{R}^d} |\nabla u|^2 g_\star^2 dx + d(d-2) \int_{\mathbb{R}^d} |u - g_d/g_\star|^2 g_\star^{2*} dx \right),$$

where $g_d \in \mathcal{M}$ realizes $\inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_{L^2(\mathbb{R}^d)}^2$. For some parameters $a_d, b_d$ and $c_d$, we can write that $g_d(x) = c_d (a_d + |x - b_d|^2)^{1-d/2}$. We rescale the function $u$ according to

$$u(x) = v(r_d x) \quad \forall x \in \mathbb{R}^d, \quad r_d = \sqrt{\frac{d}{2\pi}}$$

and consider the function $w_v^d$ such that $w_v^d(r_d x) = g_d(x)/g_\star(x)$. Hence

$$\int_{\mathbb{R}^d} |\nabla v|^2 \left(1 + \frac{1}{r_d^2} |x|^2\right)^2 d\boldsymbol{\mu}_d \geq 2\pi(d-2) \left( \left( \int_{\mathbb{R}^d} |v|^{2^*} d\boldsymbol{\mu}_d \right)^{2/2^*} - \int_{\mathbb{R}^d} |v|^2 d\boldsymbol{\mu}_d \right) \\ + \frac{\beta}{d} \left( \int_{\mathbb{R}^d} |\nabla v|^2 d\boldsymbol{\mu}_d + 2\pi(d-2) \int_{\mathbb{R}^d} |v - w_v^{d/2}|^2 d\boldsymbol{\mu}_d \right)$$

where $d\boldsymbol{\mu}_d = Z_d^{-1} g_\star^{2*} dx$ is the probability measure given by

$$d\boldsymbol{\mu}_d(x) := \frac{1}{Z_d} \left(1 + \frac{1}{r_d^2} |x|^2\right)^{-d} dx \quad \text{with} \quad Z_d = \frac{2^{1-d} \sqrt{\pi}}{\Gamma\left(\frac{d+1}{2}\right)} \left(\frac{d}{2}\right)^{\frac{d}{2}}.$$

Our goal is to take the limit $d \to +\infty$ when one considers functions $v(x)$ depending only on $y \in \mathbb{R}^N$, with $x = (y, z) \in \mathbb{R}^N \times \mathbb{R}^{d-N} \approx \mathbb{R}^d$, for some fixed integer $N$. With $|x|^2 = |y|^2 + |z|^2$, notice that

$$1 + \frac{1}{r_d^2} |x|^2 = 1 + \frac{1}{r_d^2} (|y|^2 + |z|^2) = \left(1 + \frac{1}{r_d^2} |y|^2\right) \left(1 + \frac{|z|^2}{r_d^2 + |y|^2}\right)$$

and, as a consequence,

$$\lim_{d \to +\infty} \left(1 + \frac{1}{r_d^2} |y|^2\right)^{-\frac{N+d}{2}} = e^{-\pi |y|^2},$$

$$\lim_{d \to +\infty} \int_{\mathbb{R}^d} |v(y)|^2 d\boldsymbol{\mu}_d = \int_{\mathbb{R}^N} |v|^2 d\gamma,$$

$$\lim_{d \to +\infty} \int_{\mathbb{R}^d} |\nabla v|^2 \left(1 + \frac{1}{r_d^2} |x|^2\right)^2 d\boldsymbol{\mu}_d = 4 \int_{\mathbb{R}^N} |\nabla v|^2 d\gamma,$$

where $d\gamma(y) = e^{-\pi |y|^2} dy$. However, the function $w_v^d$ depends on $d$ and the main difficulty is to obtain enough estimates on the parameters $a_d, b_d$ and $c_d$ to pass to the limit after integrating in all integrals with respect to $z$, which completes the proof of (7). See [24, Section 4] for further details. □
and therefore

$$C_{d,\text{BE}} \geq \frac{1}{2} \min \left\{ \delta \mathcal{I}(\delta), \frac{2 h_d(1/2)}{h_d(1/2) + 1} \right\}.$$

• Combining all estimates of the three previous steps completes the proof of Theorem 1. □

## 4. STABILITY OF THE GAUSSIAN LOGARITHMIC SOBOLEV INEQUALITY: PROOF OF THEOREM 2

In this section we describe the main steps in the proof of the stability estimate for the Gaussian logarithmic Sobolev inequality (LSI) by considering the large dimensional limit of (6), as it appears in [24, Section 4]. With $u = f/g_\star$, where $g_\star$ is given by (11), Inequality (6) can be rewritten as

$$\int_{\mathbb{R}^d} |\nabla u|^2 g_\star^2 dx + d(d-2) \int_{\mathbb{R}^d} |u|^2 g_\star^{2*} dx - d(d-2) \|g_\star\|_{L^{2^*(\mathbb{R}^d)}}^{2^*-2} \left( \int_{\mathbb{R}^d} |u|^{2^*} g_\star^{2*} dx \right)^{2/2^*} \\ \geq \frac{\beta}{d} \left( \int_{\mathbb{R}^d} |\nabla u|^2 g_\star^2 dx + d(d-2) \int_{\mathbb{R}^d} |u - g_d/g_\star|^2 g_\star^{2*} dx \right),$$

where $g_d \in \mathcal{M}$ realizes $\inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_{L^2(\mathbb{R}^d)}^2$. For some parameters $a_d, b_d$ and $c_d$, we can write that $g_d(x) = c_d (a_d + |x - b_d|^2)^{1-d/2}$. We rescale the function $u$ according to

$$u(x) = v(r_d x) \quad \forall x \in \mathbb{R}^d, \quad r_d = \sqrt{\frac{d}{2\pi}}$$

and consider the function $w_v^d$ such that $w_v^d(r_d x) = g_d(x)/g_\star(x)$. Hence

$$\int_{\mathbb{R}^d} |\nabla v|^2 \left(1 + \frac{1}{r_d^2} |x|^2\right)^2 d\boldsymbol{\mu}_d \geq 2\pi(d-2) \left( \left( \int_{\mathbb{R}^d} |v|^{2^*} d\boldsymbol{\mu}_d \right)^{2/2^*} - \int_{\mathbb{R}^d} |v|^2 d\boldsymbol{\mu}_d \right) \\ + \frac{\beta}{d} \left( \int_{\mathbb{R}^d} |\nabla v|^2 d\boldsymbol{\mu}_d + 2\pi(d-2) \int_{\mathbb{R}^d} |v - w_v^{d/2}|^2 d\boldsymbol{\mu}_d \right)$$

where $d\boldsymbol{\mu}_d = Z_d^{-1} g_\star^{2*} dx$ is the probability measure given by

$$d\boldsymbol{\mu}_d(x) := \frac{1}{Z_d} \left(1 + \frac{1}{r_d^2} |x|^2\right)^{-d} dx \quad \text{with} \quad Z_d = \frac{2^{1-d} \sqrt{\pi}}{\Gamma\left(\frac{d+1}{2}\right)} \left(\frac{d}{2}\right)^{\frac{d}{2}}.$$

Our goal is to take the limit $d \to +\infty$ when one considers functions $v(x)$ depending only on $y \in \mathbb{R}^N$, with $x = (y, z) \in \mathbb{R}^N \times \mathbb{R}^{d-N} \approx \mathbb{R}^d$, for some fixed integer $N$. With $|x|^2 = |y|^2 + |z|^2$, notice that

$$1 + \frac{1}{r_d^2} |x|^2 = 1 + \frac{1}{r_d^2} (|y|^2 + |z|^2) = \left(1 + \frac{1}{r_d^2} |y|^2\right) \left(1 + \frac{|z|^2}{r_d^2 + |y|^2}\right)$$

and, as a consequence,

$$\lim_{d \to +\infty} \left(1 + \frac{1}{r_d^2} |y|^2\right)^{-\frac{N+d}{2}} = e^{-\pi |y|^2},$$

$$\lim_{d \to +\infty} \int_{\mathbb{R}^d} |v(y)|^2 d\boldsymbol{\mu}_d = \int_{\mathbb{R}^N} |v|^2 d\gamma,$$

$$\lim_{d \to +\infty} \int_{\mathbb{R}^d} |\nabla v|^2 \left(1 + \frac{1}{r_d^2} |x|^2\right)^2 d\boldsymbol{\mu}_d = 4 \int_{\mathbb{R}^N} |\nabla v|^2 d\gamma,$$

where $d\gamma(y) = e^{-\pi |y|^2} dy$. However, the function $w_v^d$ depends on $d$ and the main difficulty is to obtain enough estimates on the parameters $a_d, b_d$ and $c_d$ to pass to the limit after integrating in all integrals with respect to $z$, which completes the proof of (7). See [24, Section 4] for further details. □
5. STABILITY OF THE LOGARITHMIC SOBOLEV INEQUALITY: A NEW PROOF OF THEOREM 2

As explained in the Introduction and in Section 4, in [24] we proved Theorem 2 as a corollary of Theorem 1. In this section we give a new and complete proof of Theorem 2 that is independent of Theorem 1.

This gives us the opportunity to present some of the details both of the infinite dimensional limit mentioned in Section 4 and of the combination of continuous symmetrization and discrete competing symmetries flows mentioned in Section 3. In both cases the arguments in the present section are slightly simpler, but illustrate well the main underlying ideas. We also emphasize that, while it may have seemed that the latter argument relies on the conformal invariance of the Sobolev inequality (which enters through the operator $U$ in the competing symmetries argument), this is not the case: in the setting of log-Sobolev (LSI), where there is no such conformal symmetry, there is another operator that serves the same purpose.

Just like in Section 3, we prove the quantitative version of the sharp logarithmic Sobolev inequality (Theorem 2) in two steps, one close and one far from the set of optimizers. Let us start with the following result that replaces Theorem 5 here. In fact, it is a consequence of Theorem 5.

**Theorem 8.** There are explicit constants $\eta > 0$ and $\delta \in (0, 1/2)$ such that for all $N \in \mathbb{N}$ and for all for all nonnegative $u = 1 + r \in H^1(\mathbb{S}^d)$ satisfying

$$ \int_{\mathbb{R}^N} r^2 d\gamma \le \frac{\delta}{1-\delta} \qquad (13) $$

and

$$ \int_{\mathbb{R}^N} r d\gamma = 0 = \int_{\mathbb{R}^N} x_j r d\gamma, \quad j = 1, 2, \dots, N, \qquad (14) $$

one has

$$ \int_{\mathbb{R}^N} |\nabla u|^2 d\gamma - \pi \int_{\mathbb{R}^N} |u|^2 \ln \left( \frac{|u|^2}{\|u\|_{L^2(\gamma)}^2} \right) d\gamma \ge \eta \int_{\mathbb{R}^N} r^2 d\gamma. $$

The constant $\delta$ coincides with the corresponding constant in Theorem 5 and $\eta = 2\pi\epsilon_0$.

*Proof.* Notice that $x \in L^2(\gamma)$, so the orthogonality constraints raise no integration issues. We denote $\Sigma_d := \{x \in \mathbb{R}^{d+1} : |x| = \rho_d\}$ with $\rho_d := \sqrt{d/(2\pi)}$. The factor of $1/(2\pi)$ in the definition of $\rho_d$ is necessary to get the $\pi$ in the exponent of the Gaussian density. We integrate on $\Sigma_d$ with respect to the uniform probability measure $d\mu_d$. Here, with a straightforward abuse, we use the same notation for the uniform probability measure on $\Sigma_d$ as on the unit sphere. By rescaling our result in Theorem 5 we find that

$$ \int_{\Sigma_d} |\nabla R|^2 d\mu_d - \pi \frac{d-2}{2} \left( \left( \int_{\Sigma_d} (1+R)^{\frac{2d}{d-2}} d\mu_d \right)^{\frac{d-2}{d}} - \int_{\Sigma_d} (1+R)^2 d\mu_d \right) \\ \ge 2\pi \epsilon_0 \int_{\Sigma_d} \left( \frac{1}{\pi} \frac{2}{d-2} |\nabla R|^2 + R^2 \right) d\mu_d. \quad (15) $$

This inequality is valid for all $R \in H^1(\Sigma_d)$ such that

$$ \left( \int_{\Sigma_d} R^{\frac{2d}{d-2}} d\mu_d \right)^{\frac{d-2}{d}} \le \frac{\delta}{1-\delta} \qquad (16) $$

and

$$ \int_{\Sigma_d} R d\mu_d = 0 = \int_{\Sigma_d} x_j R d\mu_d, \quad j = 1, \dots, d+1. \qquad (17) $$
5. STABILITY OF THE LOGARITHMIC SOBOLEV INEQUALITY: A NEW PROOF OF THEOREM 2

As explained in the Introduction and in Section 4, in [24] we proved Theorem 2 as a corollary of Theorem 1. In this section we give a new and complete proof of Theorem 2 that is independent of Theorem 1.

This gives us the opportunity to present some of the details both of the infinite dimensional limit mentioned in Section 4 and of the combination of continuous symmetrization and discrete competing symmetries flows mentioned in Section 3. In both cases the arguments in the present section are slightly simpler, but illustrate well the main underlying ideas. We also emphasize that, while it may have seemed that the latter argument relies on the conformal invariance of the Sobolev inequality (which enters through the operator $U$ in the competing symmetries argument), this is not the case: in the setting of log-Sobolev (LSI), where there is no such conformal symmetry, there is another operator that serves the same purpose.

Just like in Section 3, we prove the quantitative version of the sharp logarithmic Sobolev inequality (Theorem 2) in two steps, one close and one far from the set of optimizers. Let us start with the following result that replaces Theorem 5 here. In fact, it is a consequence of Theorem 5.

**Theorem 8.** There are explicit constants $\eta > 0$ and $\delta \in (0, 1/2)$ such that for all $N \in \mathbb{N}$ and for all for all nonnegative $u = 1 + r \in H^1(\mathbb{S}^d)$ satisfying

$$ \int_{\mathbb{R}^N} r^2 d\gamma \le \frac{\delta}{1-\delta} \qquad (13) $$

and

$$ \int_{\mathbb{R}^N} r d\gamma = 0 = \int_{\mathbb{R}^N} x_j r d\gamma, \quad j = 1, 2, \dots, N, \qquad (14) $$

one has

$$ \int_{\mathbb{R}^N} |\nabla u|^2 d\gamma - \pi \int_{\mathbb{R}^N} |u|^2 \ln \left( \frac{|u|^2}{\|u\|_{L^2(\gamma)}^2} \right) d\gamma \ge \eta \int_{\mathbb{R}^N} r^2 d\gamma. $$

The constant $\delta$ coincides with the corresponding constant in Theorem 5 and $\eta = 2\pi\epsilon_0$.

*Proof.* Notice that $x \in L^2(\gamma)$, so the orthogonality constraints raise no integration issues. We denote $\Sigma_d := \{x \in \mathbb{R}^{d+1} : |x| = \rho_d\}$ with $\rho_d := \sqrt{d/(2\pi)}$. The factor of $1/(2\pi)$ in the definition of $\rho_d$ is necessary to get the $\pi$ in the exponent of the Gaussian density. We integrate on $\Sigma_d$ with respect to the uniform probability measure $d\mu_d$. Here, with a straightforward abuse, we use the same notation for the uniform probability measure on $\Sigma_d$ as on the unit sphere. By rescaling our result in Theorem 5 we find that

$$ \int_{\Sigma_d} |\nabla R|^2 d\mu_d - \pi \frac{d-2}{2} \left( \left( \int_{\Sigma_d} (1+R)^{\frac{2d}{d-2}} d\mu_d \right)^{\frac{d-2}{d}} - \int_{\Sigma_d} (1+R)^2 d\mu_d \right) \\ \ge 2\pi \epsilon_0 \int_{\Sigma_d} \left( \frac{1}{\pi} \frac{2}{d-2} |\nabla R|^2 + R^2 \right) d\mu_d. \quad (15) $$

This inequality is valid for all $R \in H^1(\Sigma_d)$ such that

$$ \left( \int_{\Sigma_d} R^{\frac{2d}{d-2}} d\mu_d \right)^{\frac{d-2}{d}} \le \frac{\delta}{1-\delta} \qquad (16) $$

and

$$ \int_{\Sigma_d} R d\mu_d = 0 = \int_{\Sigma_d} x_j R d\mu_d, \quad j = 1, \dots, d+1. \qquad (17) $$
Given a function $r \in H^1(\gamma)$ and $d > N$, we apply this inequality to the function

$$R_d(x) := r(x_1, \dots, x_N) - \int_{\Sigma_d} r \, d\mu_d - 2\pi \frac{d+1}{d} \sum_{n=1}^{N} x_n \int_{\Sigma_d} y_n r(y_1, \dots, y_N) \, d\mu_d(y)$$

for $x \in \Sigma_d$. This function satisfies the orthogonality conditions (17). Note here that the functions $\sqrt{2\pi} \sqrt{(d+1)/d} x_j$ are $L^2$-normalized on $\Sigma_d$.

We now use the well-known fact that, as $d \to +\infty$, the marginal of $d\mu_d$ corresponding to the first $N$ coordinates converges to $d\gamma$. Thus,

$$\lim_{d \to +\infty} \int_{\Sigma_d} |\nabla r|^2 d\mu_d = \int_{\mathbb{R}^N} |\nabla r|^2 d\gamma, \quad \lim_{d \to +\infty} \int_{\Sigma_d} r^2 d\mu_d = \int_{\mathbb{R}^N} r^2 d\gamma, \\ \lim_{d \to +\infty} \int_{\Sigma_d} r \, d\mu_d = \int_{\mathbb{R}^N} r \, d\gamma = 0, \quad \lim_{d \to +\infty} \int_{\Sigma_d} y_n r(y_1, \dots, y_N) \, d\mu_d(y) = \int_{\mathbb{R}^N} y_n r \, d\gamma = 0.$$

From this we conclude easily that

$$\lim_{d \to +\infty} \int_{\Sigma_d} |\nabla R_d|^2 d\mu_d = \int_{\mathbb{R}^N} |\nabla r|^2 d\gamma, \quad \lim_{d \to +\infty} \int_{\Sigma_d} R_d^2 d\mu_d = \int_{\mathbb{R}^N} r^2 d\gamma.$$

With some modest amount of effort one also finds that

$$\lim_{d \to +\infty} \int_{\Sigma_d} R_d^{\frac{2d}{d-2}} d\mu_d = \int_{\mathbb{R}^N} r^2 d\gamma.$$

Assuming that the inequality in (13) is strict, the same is true for the left side when $d$ is sufficiently large, and consequently the smallness condition (16) holds when $d$ is sufficiently large. Thus, inequality (15) is valid for all sufficiently large $d$. The equality case in (13) can be obtained at the very end by a simple approximation argument.

Now, we drop the gradient term in the right side and letting $d \to +\infty$ we infer that

$$\int_{\mathbb{R}^N} |\nabla r|^2 d\gamma - \pi \limsup_{d \to +\infty} \frac{d-2}{2} \left( \left( \int_{\Sigma_d} (1+R_d)^{\frac{2d}{d-2}} d\mu_d \right)^{\frac{d-2}{d}} - \int_{\Sigma_d} (1+R_d)^2 d\mu_d \right) \geq 2\pi \epsilon_0 \int_{\mathbb{R}^N} r^2 d\gamma.$$

Finally, we verify that

$$\limsup_{d \to +\infty} \frac{d-2}{2} \left( \left( \int_{\Sigma_d} (1+R_d)^{\frac{2d}{d-2}} d\mu_d \right)^{\frac{d-2}{d}} - \int_{\Sigma_d} (1+R_d)^2 d\mu_d \right) = \int_{\mathbb{R}^N} (1+r)^2 \ln \left( \frac{(1+r)^2}{\|1+r\|_{L^2(\gamma)}^2} \right) d\gamma.$$

In fact, if the orthogonality conditions were not present and the marginals would already be equal to their limit, this would follow from the fact that

$$\lim_{p \to 1^+} \frac{1}{p-1} \left( \left( \int_{\mathbb{R}^N} h^p d\gamma \right)^{1/p} - \int_{\mathbb{R}^N} h d\gamma \right) = \int_{\mathbb{R}^N} h \ln \left( \frac{h}{\int_{\mathbb{R}^N} h d\gamma} \right) d\gamma,$$

valid on any measure space for any nonnegative function $h$ that satisfies $h \in L^1 \cap L^{p_0}(\gamma)$ for some $p_0 > 1$. Proving the latter fact is simple, as well as including the effect of the orthogonality conditions and the convergence of the marginals, so we shall omit it. These remarks complete the proof Theorem 8. □

We emphasize that in the previous proof we did not use Theorem 1, but rather Theorem 5. In this way we avoid having to control the distance to the set of optimizers in the high-dimensional limit, which seems harder than verifying the orthogonality conditions.
Given a function $r \in H^1(\gamma)$ and $d > N$, we apply this inequality to the function

$$R_d(x) := r(x_1, \dots, x_N) - \int_{\Sigma_d} r \, d\mu_d - 2\pi \frac{d+1}{d} \sum_{n=1}^{N} x_n \int_{\Sigma_d} y_n r(y_1, \dots, y_N) \, d\mu_d(y)$$

for $x \in \Sigma_d$. This function satisfies the orthogonality conditions (17). Note here that the functions $\sqrt{2\pi} \sqrt{(d+1)/d} x_j$ are $L^2$-normalized on $\Sigma_d$.

We now use the well-known fact that, as $d \to +\infty$, the marginal of $d\mu_d$ corresponding to the first $N$ coordinates converges to $d\gamma$. Thus,

$$\lim_{d \to +\infty} \int_{\Sigma_d} |\nabla r|^2 d\mu_d = \int_{\mathbb{R}^N} |\nabla r|^2 d\gamma, \quad \lim_{d \to +\infty} \int_{\Sigma_d} r^2 d\mu_d = \int_{\mathbb{R}^N} r^2 d\gamma, \\ \lim_{d \to +\infty} \int_{\Sigma_d} r \, d\mu_d = \int_{\mathbb{R}^N} r \, d\gamma = 0, \quad \lim_{d \to +\infty} \int_{\Sigma_d} y_n r(y_1, \dots, y_N) \, d\mu_d(y) = \int_{\mathbb{R}^N} y_n r \, d\gamma = 0.$$

From this we conclude easily that

$$\lim_{d \to +\infty} \int_{\Sigma_d} |\nabla R_d|^2 d\mu_d = \int_{\mathbb{R}^N} |\nabla r|^2 d\gamma, \quad \lim_{d \to +\infty} \int_{\Sigma_d} R_d^2 d\mu_d = \int_{\mathbb{R}^N} r^2 d\gamma.$$

With some modest amount of effort one also finds that

$$\lim_{d \to +\infty} \int_{\Sigma_d} R_d^{\frac{2d}{d-2}} d\mu_d = \int_{\mathbb{R}^N} r^2 d\gamma.$$

Assuming that the inequality in (13) is strict, the same is true for the left side when $d$ is sufficiently large, and consequently the smallness condition (16) holds when $d$ is sufficiently large. Thus, inequality (15) is valid for all sufficiently large $d$. The equality case in (13) can be obtained at the very end by a simple approximation argument.

Now, we drop the gradient term in the right side and letting $d \to +\infty$ we infer that

$$\int_{\mathbb{R}^N} |\nabla r|^2 d\gamma - \pi \limsup_{d \to +\infty} \frac{d-2}{2} \left( \left( \int_{\Sigma_d} (1+R_d)^{\frac{2d}{d-2}} d\mu_d \right)^{\frac{d-2}{d}} - \int_{\Sigma_d} (1+R_d)^2 d\mu_d \right) \geq 2\pi \epsilon_0 \int_{\mathbb{R}^N} r^2 d\gamma.$$

Finally, we verify that

$$\limsup_{d \to +\infty} \frac{d-2}{2} \left( \left( \int_{\Sigma_d} (1+R_d)^{\frac{2d}{d-2}} d\mu_d \right)^{\frac{d-2}{d}} - \int_{\Sigma_d} (1+R_d)^2 d\mu_d \right) = \int_{\mathbb{R}^N} (1+r)^2 \ln \left( \frac{(1+r)^2}{\|1+r\|_{L^2(\gamma)}^2} \right) d\gamma.$$

In fact, if the orthogonality conditions were not present and the marginals would already be equal to their limit, this would follow from the fact that

$$\lim_{p \to 1^+} \frac{1}{p-1} \left( \left( \int_{\mathbb{R}^N} h^p d\gamma \right)^{1/p} - \int_{\mathbb{R}^N} h d\gamma \right) = \int_{\mathbb{R}^N} h \ln \left( \frac{h}{\int_{\mathbb{R}^N} h d\gamma} \right) d\gamma,$$

valid on any measure space for any nonnegative function $h$ that satisfies $h \in L^1 \cap L^{p_0}(\gamma)$ for some $p_0 > 1$. Proving the latter fact is simple, as well as including the effect of the orthogonality conditions and the convergence of the marginals, so we shall omit it. These remarks complete the proof Theorem 8. □

We emphasize that in the previous proof we did not use Theorem 1, but rather Theorem 5. In this way we avoid having to control the distance to the set of optimizers in the high-dimensional limit, which seems harder than verifying the orthogonality conditions.
*Proof of Theorem 2.* As in the proof of Theorem 1, we first prove the result for nonnegative functions and then extend it to sign changing solutions. Let us denote by $\kappa^{\text{Pos}}$ the stability constant in the stability inequality restricted to nonnegative functions.

*Step 1.* Let $\eta$ and $\delta$ be as in Theorem 8. For $0 \le u \in H^1(\gamma)$ we distinguish two cases.

• The first case is where

$$ \inf_{a \in \mathbb{R}^N, c \in \mathbb{R}} \int_{\mathbb{R}^N} |u - c e^{a \cdot x}|^2 d\gamma \le \delta \int_{\mathbb{R}^N} |u|^2 d\gamma. $$

The infimum on the left-hand side is attained at some $a \in \mathbb{R}^N$ and $c \in \mathbb{R}$, as can be checked by optimizing $\int_{\mathbb{R}^N} |v - c e^{a|v|^2/(2\pi) - \pi|x-a/\pi|^2/2}|^2 dx$ where $v(x) := u(x) e^{-\pi|v|^2/2}$. Let

$$ \tilde{u}(y) := e^{-y \cdot a - \frac{|a|^2}{2\pi}} u\left(y + \frac{a}{\pi}\right). $$

Then, by a simple computation involving an integration by parts and a change of variables,

$$ \int_{\mathbb{R}^N} |\nabla \tilde{u}|^2 d\gamma - \pi \int_{\mathbb{R}^N} |\tilde{u}|^2 \ln \left( \frac{|\tilde{u}|^2}{\|\tilde{u}\|_{L^2(\gamma)}^2} \right) d\gamma = \int_{\mathbb{R}^N} |\nabla u|^2 d\gamma - \pi \int_{\mathbb{R}^N} |u|^2 \ln \left( \frac{|u|^2}{\|u\|_{L^2(\gamma)}^2} \right) d\gamma. $$

Therefore, the deficit of $\tilde{u}$ coincides with that of $u$, while the infimum for $\tilde{u}$ among all functions of the form (4) is attained at the constant $c \exp(|a|^2/(2\pi))$. Finally, by multiplying $\tilde{u}$ with a constant, we may assume that this constant is equal to one. To summarize, we may assume without loss of generality that the infimum in the theorem is attained at $a = 0$ and $c = 1$.

Let us set $r := u - 1$. Then the minimality implies that $r$ satisfies the orthogonality conditions (14). Moreover, we have

$$ \int_{\mathbb{R}^N} r^2 d\gamma \le \delta \int_{\mathbb{R}^N} |u|^2 d\gamma = \delta \left( 1 + \int_{\mathbb{R}^N} r^2 d\gamma \right), $$

so the smallness condition (13) is satisfied and we can apply Theorem 8. This yields the inequality in the theorem with a stability constant $\eta$.

• Next, we consider the case where

$$ \inf_{a \in \mathbb{R}^N, c \in \mathbb{R}} \int_{\mathbb{R}^N} (u - c e^{x \cdot a})^2 d\gamma > \delta \int_{\mathbb{R}^N} |u|^2 d\gamma. $$

We argue similarly as we did in Section 3 concerning the Sobolev inequality, but there are some differences in this case.

For $f \in L^2(\gamma)$ we denote by $Uf$ its Gaussian rearrangement, that is, the function on $\mathbb{R}^N$ whose superlevel sets have the form $\{x \in \mathbb{R}^N : x_1 < \mu\}$ for some $\mu \in \mathbb{R}$ and have the same $\gamma$-measure as the corresponding superlevel sets of $f$. Moreover, we denote

$$ Vf := e^{\frac{\pi}{2}|x|^2} \mathcal{R} \left( e^{-\frac{\pi}{2}|x|^2} f \right), $$

where $\mathcal{R}$ is, as before, the Euclidean rearrangement. Then, as shown in [18, Theorem 4.1], for any $0 \le f \in L^2(\gamma)$ one has

$$ f_n := (VU)^n f \to \|f\|_{L^2(\gamma)} \quad \text{in } L^2(\gamma). $$

Moreover, $\|f_n\|_{L^2(\gamma)} = \|f\|_{L^2(\gamma)}$ and

$$ n \mapsto \int_{\mathbb{R}^N} |\nabla f_n|^2 d\gamma - \pi \int_{\mathbb{R}^N} f_n^2 \ln \left( \frac{f_n^2}{\|f_n\|_{L^2(\gamma)}^2} \right) d\gamma $$

is nonincreasing. This is the analogue of Theorem 6 in the present case.

We apply this procedure to our function $u$, which we assume here to be nonnegative, and obtain a sequence of functions $u_n$ with constant $L^2(\gamma)$-norm. Moreover, since

$$ \inf_{a,c} \|u_n - c e^{a \cdot x}\|_{L^2(\gamma)} \le \|u_n - \|u\|_{L^2(\gamma)}\|_{L^2(\gamma)} \to 0 \quad \text{as } n \to +\infty, $$
*Proof of Theorem 2.* As in the proof of Theorem 1, we first prove the result for nonnegative functions and then extend it to sign changing solutions. Let us denote by $\kappa^{\text{Pos}}$ the stability constant in the stability inequality restricted to nonnegative functions.

*Step 1.* Let $\eta$ and $\delta$ be as in Theorem 8. For $0 \le u \in H^1(\gamma)$ we distinguish two cases.

• The first case is where

$$ \inf_{a \in \mathbb{R}^N, c \in \mathbb{R}} \int_{\mathbb{R}^N} |u - c e^{a \cdot x}|^2 d\gamma \le \delta \int_{\mathbb{R}^N} |u|^2 d\gamma. $$

The infimum on the left-hand side is attained at some $a \in \mathbb{R}^N$ and $c \in \mathbb{R}$, as can be checked by optimizing $\int_{\mathbb{R}^N} |v - c e^{a|v|^2/(2\pi) - \pi|x-a/\pi|^2/2}|^2 dx$ where $v(x) := u(x) e^{-\pi|v|^2/2}$. Let

$$ \tilde{u}(y) := e^{-y \cdot a - \frac{|a|^2}{2\pi}} u\left(y + \frac{a}{\pi}\right). $$

Then, by a simple computation involving an integration by parts and a change of variables,

$$ \int_{\mathbb{R}^N} |\nabla \tilde{u}|^2 d\gamma - \pi \int_{\mathbb{R}^N} |\tilde{u}|^2 \ln \left( \frac{|\tilde{u}|^2}{\|\tilde{u}\|_{L^2(\gamma)}^2} \right) d\gamma = \int_{\mathbb{R}^N} |\nabla u|^2 d\gamma - \pi \int_{\mathbb{R}^N} |u|^2 \ln \left( \frac{|u|^2}{\|u\|_{L^2(\gamma)}^2} \right) d\gamma. $$

Therefore, the deficit of $\tilde{u}$ coincides with that of $u$, while the infimum for $\tilde{u}$ among all functions of the form (4) is attained at the constant $c \exp(|a|^2/(2\pi))$. Finally, by multiplying $\tilde{u}$ with a constant, we may assume that this constant is equal to one. To summarize, we may assume without loss of generality that the infimum in the theorem is attained at $a = 0$ and $c = 1$.

Let us set $r := u - 1$. Then the minimality implies that $r$ satisfies the orthogonality conditions (14). Moreover, we have

$$ \int_{\mathbb{R}^N} r^2 d\gamma \le \delta \int_{\mathbb{R}^N} |u|^2 d\gamma = \delta \left( 1 + \int_{\mathbb{R}^N} r^2 d\gamma \right), $$

so the smallness condition (13) is satisfied and we can apply Theorem 8. This yields the inequality in the theorem with a stability constant $\eta$.

• Next, we consider the case where

$$ \inf_{a \in \mathbb{R}^N, c \in \mathbb{R}} \int_{\mathbb{R}^N} (u - c e^{x \cdot a})^2 d\gamma > \delta \int_{\mathbb{R}^N} |u|^2 d\gamma. $$

We argue similarly as we did in Section 3 concerning the Sobolev inequality, but there are some differences in this case.

For $f \in L^2(\gamma)$ we denote by $Uf$ its Gaussian rearrangement, that is, the function on $\mathbb{R}^N$ whose superlevel sets have the form $\{x \in \mathbb{R}^N : x_1 < \mu\}$ for some $\mu \in \mathbb{R}$ and have the same $\gamma$-measure as the corresponding superlevel sets of $f$. Moreover, we denote

$$ Vf := e^{\frac{\pi}{2}|x|^2} \mathcal{R} \left( e^{-\frac{\pi}{2}|x|^2} f \right), $$

where $\mathcal{R}$ is, as before, the Euclidean rearrangement. Then, as shown in [18, Theorem 4.1], for any $0 \le f \in L^2(\gamma)$ one has

$$ f_n := (VU)^n f \to \|f\|_{L^2(\gamma)} \quad \text{in } L^2(\gamma). $$

Moreover, $\|f_n\|_{L^2(\gamma)} = \|f\|_{L^2(\gamma)}$ and

$$ n \mapsto \int_{\mathbb{R}^N} |\nabla f_n|^2 d\gamma - \pi \int_{\mathbb{R}^N} f_n^2 \ln \left( \frac{f_n^2}{\|f_n\|_{L^2(\gamma)}^2} \right) d\gamma $$

is nonincreasing. This is the analogue of Theorem 6 in the present case.

We apply this procedure to our function $u$, which we assume here to be nonnegative, and obtain a sequence of functions $u_n$ with constant $L^2(\gamma)$-norm. Moreover, since

$$ \inf_{a,c} \|u_n - c e^{a \cdot x}\|_{L^2(\gamma)} \le \|u_n - \|u\|_{L^2(\gamma)}\|_{L^2(\gamma)} \to 0 \quad \text{as } n \to +\infty, $$
there is an $n_0 \in \mathbb{N}$ such that

$$ \inf_{a,c} \|u_{n_0} - c e^{a \cdot x}\|_{L^2(\gamma)}^2 \geq \delta \|u\|_{L^2(\gamma)}^2 > \inf_{a,c} \|u_{n_0+1} - c e^{a \cdot x}\|_{L^2(\gamma)}^2. $$

This replaces Lemma 7. We have

$$ \begin{aligned} \frac{\int_{\mathbb{R}^N} |\nabla u|^2 d\gamma - \pi \int_{\mathbb{R}^N} |u|^2 \ln \left( \frac{|u|^2}{\|u\|_{L^2(\gamma)}^2} \right) d\gamma}{\inf_{a,c} \|u - c e^{a \cdot x}\|_{L^2(\gamma)}^2} &\geq \frac{\int_{\mathbb{R}^N} |\nabla u|^2 d\gamma - \pi \int_{\mathbb{R}^N} |u|^2 \ln \left( \frac{|u|^2}{\|u\|_{L^2(\gamma)}^2} \right) d\gamma}{\|u\|_{L^2(\gamma)}^2} \\ &\geq \frac{\int_{\mathbb{R}^N} |\nabla u_{n_0}|^2 d\gamma - \pi \int_{\mathbb{R}^N} u_{n_0}^2 \ln \left( \frac{u_{n_0}^2}{\|u_{n_0}\|_{L^2(\gamma)}^2} \right) d\gamma}{\|u\|_{L^2(\gamma)}^2}. \end{aligned} $$

We now use a continuous rearrangement flow to connect $u_{n_0}$ to $u_{n_0+1}$. More precisely, we consider a family of functions $(u_\tau)_{n_0 \le \tau \le n_0+1}$, where $u_{n_0} := U u_{n_0}$ and $u_{n_0+1} := u_{n_0+1}$. We define $u_\tau$ as $e^{\frac{\pi}{2}|x|^2}$ times the continuous (Euclidean) rearrangement of $e^{-\frac{\pi}{2}|x|^2} U u_{n_0}$ at parameter $\tau$. In the same way as in [24, Lemma 36], one sees that

$$ \tau \mapsto \inf_{a,c} \|u_{\tau} - c e^{a \cdot x}\|_{L^2(\gamma)}^2 $$

is continuous, and therefore there is a $\tau_0 \in [n_0, n_0 + 1)$ such that

$$ \inf_{a,c} \|u_{\tau_0} - c e^{a \cdot x}\|_{L^2(\gamma)}^2 = \delta \|u\|_{L^2(\gamma)}^2. $$

It follows that

$$ \begin{aligned} \frac{\int_{\mathbb{R}^N} |\nabla u_{n_0}|^2 d\gamma - \pi \int_{\mathbb{R}^N} u_{n_0}^2 \ln \left( \frac{u_{n_0}^2}{\|u_{n_0}\|_{L^2(\gamma)}^2} \right) d\gamma}{\|u\|_{L^2(\gamma)}^2} &\geq \frac{\int_{\mathbb{R}^N} |\nabla u_{\tau_0}|^2 d\gamma - \pi \int_{\mathbb{R}^N} u_{\tau_0}^2 \ln \left( \frac{u_{\tau_0}^2}{\|u_{\tau_0}\|_{L^2(\gamma)}^2} \right) d\gamma}{\|u\|_{L^2(\gamma)}^2} \\ &= \delta \frac{\int_{\mathbb{R}^N} |\nabla u_{\tau_0}|^2 d\gamma - \pi \int_{\mathbb{R}^N} u_{\tau_0}^2 \ln \left( \frac{u_{\tau_0}^2}{\|u_{\tau_0}\|_{L^2(\gamma)}^2} \right) d\gamma}{\inf_{a,c} \|u_{\tau_0} - c e^{a \cdot x}\|_{L^2(\gamma)}^2}. \end{aligned} $$

We can apply the result in the first case to the function $u_{\tau_0}$ and infer that the right side is larger or equal than $\kappa^{\text{pos}} := \delta \eta$. This concludes the proof in the case of nonnegative functions.

**Step 2.** We now prove the theorem in the general case, that is, for sign-changing functions.

We shall use the notation

$$ D(v) := \int_{\mathbb{R}^N} |\nabla v|^2 d\gamma - \pi \int_{\mathbb{R}^N} v^2 \ln \left( \frac{v^2}{\|v\|_{L^2(\gamma)}^2} \right) d\gamma \quad \text{for } v \in H^1(\gamma). $$

Let $u = u_+ - u_- \in H^1(\gamma)$. By homogeneity we can assume $\|u\|_{L^2(\gamma)} = 1$. Replacing $u$ by $-u$ if necessary, we can also assume that

$$ m := \|u_-\|_{L^2(\gamma)}^2 \in [0, \frac{1}{2}]. $$

Then

$$ D(u) = D(u_+) + D(u_-) + \pi h(m) $$

with

$$ h(p) := -(p \ln p + (1-p) \ln(1-p)). $$
there is an $n_0 \in \mathbb{N}$ such that

$$ \inf_{a,c} \|u_{n_0} - c e^{a \cdot x}\|_{L^2(\gamma)}^2 \geq \delta \|u\|_{L^2(\gamma)}^2 > \inf_{a,c} \|u_{n_0+1} - c e^{a \cdot x}\|_{L^2(\gamma)}^2. $$

This replaces Lemma 7. We have

$$ \begin{aligned} \frac{\int_{\mathbb{R}^N} |\nabla u|^2 d\gamma - \pi \int_{\mathbb{R}^N} |u|^2 \ln \left( \frac{|u|^2}{\|u\|_{L^2(\gamma)}^2} \right) d\gamma}{\inf_{a,c} \|u - c e^{a \cdot x}\|_{L^2(\gamma)}^2} &\geq \frac{\int_{\mathbb{R}^N} |\nabla u|^2 d\gamma - \pi \int_{\mathbb{R}^N} |u|^2 \ln \left( \frac{|u|^2}{\|u\|_{L^2(\gamma)}^2} \right) d\gamma}{\|u\|_{L^2(\gamma)}^2} \\ &\geq \frac{\int_{\mathbb{R}^N} |\nabla u_{n_0}|^2 d\gamma - \pi \int_{\mathbb{R}^N} u_{n_0}^2 \ln \left( \frac{u_{n_0}^2}{\|u_{n_0}\|_{L^2(\gamma)}^2} \right) d\gamma}{\|u\|_{L^2(\gamma)}^2}. \end{aligned} $$

We now use a continuous rearrangement flow to connect $u_{n_0}$ to $u_{n_0+1}$. More precisely, we consider a family of functions $(u_\tau)_{n_0 \le \tau \le n_0+1}$, where $u_{n_0} := U u_{n_0}$ and $u_{n_0+1} := u_{n_0+1}$. We define $u_\tau$ as $e^{\frac{\pi}{2}|x|^2}$ times the continuous (Euclidean) rearrangement of $e^{-\frac{\pi}{2}|x|^2} U u_{n_0}$ at parameter $\tau$. In the same way as in [24, Lemma 36], one sees that

$$ \tau \mapsto \inf_{a,c} \|u_\tau - c e^{a \cdot x}\|_{L^2(\gamma)}^2 $$

is continuous, and therefore there is a $\tau_0 \in [n_0, n_0 + 1)$ such that

$$ \inf_{a,c} \|u_{\tau_0} - c e^{a \cdot x}\|_{L^2(\gamma)}^2 = \delta \|u\|_{L^2(\gamma)}^2. $$

It follows that

$$ \begin{aligned} \frac{\int_{\mathbb{R}^N} |\nabla u_{n_0}|^2 d\gamma - \pi \int_{\mathbb{R}^N} u_{n_0}^2 \ln \left( \frac{u_{n_0}^2}{\|u_{n_0}\|_{L^2(\gamma)}^2} \right) d\gamma}{\|u\|_{L^2(\gamma)}^2} &\geq \frac{\int_{\mathbb{R}^N} |\nabla u_{\tau_0}|^2 d\gamma - \pi \int_{\mathbb{R}^N} u_{\tau_0}^2 \ln \left( \frac{u_{\tau_0}^2}{\|u_{\tau_0}\|_{L^2(\gamma)}^2} \right) d\gamma}{\|u\|_{L^2(\gamma)}^2} \\ &= \delta \frac{\int_{\mathbb{R}^N} |\nabla u_{\tau_0}|^2 d\gamma - \pi \int_{\mathbb{R}^N} u_{\tau_0}^2 \ln \left( \frac{u_{\tau_0}^2}{\|u_{\tau_0}\|_{L^2(\gamma)}^2} \right) d\gamma}{\inf_{a,c} \|u_{\tau_0} - c e^{a \cdot x}\|_{L^2(\gamma)}^2}. \end{aligned} $$

We can apply the result in the first case to the function $u_{\tau_0}$ and infer that the right side is larger or equal than $\kappa^{\text{pos}} := \delta \eta$. This concludes the proof in the case of nonnegative functions.

**Step 2.** We now prove the theorem in the general case, that is, for sign-changing functions.

We shall use the notation

$$ D(v) := \int_{\mathbb{R}^N} |\nabla v|^2 d\gamma - \pi \int_{\mathbb{R}^N} v^2 \ln \left( \frac{v^2}{\|v\|_{L^2(\gamma)}^2} \right) d\gamma \quad \text{for } v \in H^1(\gamma). $$

Let $u = u_+ - u_- \in H^1(\gamma)$. By homogeneity we can assume $\|u\|_{L^2(\gamma)} = 1$. Replacing $u$ by $-u$ if necessary, we can also assume that

$$ m := \|u_-\|_{L^2(\gamma)}^2 \in [0, \frac{1}{2}]. $$

Then

$$ D(u) = D(u_+) + D(u_-) + \pi h(m) $$

with

$$ h(p) := -(p \ln p + (1-p) \ln(1-p)). $$
Since the function $p \mapsto h(p)$ is monotone increasing and concave on $[0, 1/2]$, it holds that

$$h(p) \geq (2 \ln 2) p \quad \forall p \in [0, \frac{1}{2}].$$

Thus, with $\kappa^{\text{pos}}$ denoting the constant from Step 1,

$$ \begin{aligned} D(u) &\geq D(u_+) + (2 \pi \ln 2) m \geq \kappa^{\text{pos}} \inf_{a,c} \|u_+ - c e^{a \cdot x}\|_{L^2(\gamma)}^2 + (2 \pi \ln 2) \|u_-\|_{L^2(\gamma)}^2 \\ &\geq \frac{1}{2} \min \{\kappa^{\text{pos}}, 2 \pi \ln 2\} \inf_{a,c} \|u - c e^{a \cdot x}\|_{L^2(\gamma)}^2. \end{aligned} $$

This proves the inequality in the general case, with $\kappa = \frac{1}{2} \min \{\kappa^{\text{pos}}, 2 \pi \ln 2\}$. It is straightforward to verify that $\kappa = \beta \pi/2$, $\beta$ being the constant in the statement of Theorem 1. This ends our second proof of Theorem 2. □

**Acknowledgements.** Partial support through US National Science Foundation grants DMS-1954995 (R.L.F.) and DMS-2154340 (M.L.), as well as through the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Germany's Excellence Strategy EXC-2111-390814868 (R.L.F.) and through TRR 352 – Project-ID 470903074 (R.L.F.), the French National Research Agency (ANR) project *Conviviality* (ANR-23-CE40-0003, J.D.) and the European Research Council under the Grant Agreement No. 721675 (RSPDE) *Regularity and Stability in Partial Differential Equations* (A.F.).

The authors are grateful to an anonymous referee for a careful rereading which helped to improve the manuscript.

© 2024 by the authors. Reproduction of this article by any means permitted for noncommercial purposes. CC-BY 4.0

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Since the function $p \mapsto h(p)$ is monotone increasing and concave on $[0, 1/2]$, it holds that

$$h(p) \geq (2 \ln 2) p \quad \forall p \in [0, \frac{1}{2}].$$

Thus, with $\kappa^{\text{pos}}$ denoting the constant from Step 1,

$$D(u) \geq D(u_+) + (2 \pi \ln 2) m \geq \kappa^{\text{pos}} \inf_{a,c} \|u_+ - c e^{a \cdot x}\|_{L^2(\gamma)}^2 + (2 \pi \ln 2) \|u_-\|_{L^2(\gamma)}^2 \\ \geq \frac{1}{2} \min \{\kappa^{\text{pos}}, 2 \pi \ln 2\} \inf_{a,c} \|u - c e^{a \cdot x}\|_{L^2(\gamma)}^2.$$

This proves the inequality in the general case, with $\kappa = \frac{1}{2} \min \{\kappa^{\text{pos}}, 2 \pi \ln 2\}$. It is straightforward to verify that $\kappa = \beta \pi/2$, $\beta$ being the constant in the statement of Theorem 1. This ends our second proof of Theorem 2. $\square$

**Acknowledgements.** Partial support through US National Science Foundation grants DMS-1954995 (R.L.F.) and DMS-2154340 (M.L.), as well as through the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Germany's Excellence Strategy EXC-2111-390814868 (R.L.F.) and through TRR 352 – Project-ID 470903074 (R.L.F.), the French National Research Agency (ANR) project *Conviviality* (ANR-23-CE40-0003, J.D.) and the European Research Council under the Grant Agreement No. 721675 (RSPDE) *Regularity and Stability in Partial Differential Equations* (A.F.).

The authors are grateful to an anonymous referee for a careful rereading which helped to improve the manuscript.

© 2024 by the authors. Reproduction of this article by any means permitted for noncommercial purposes. CC-BY 4.0

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(J. Dolbeault, M. J. Esteban) CEREMADE (CNRS UMR No. 7534), PSL UNIVERSITY, UNIVERSITÉ PARIS-DAUPHINE, PLACE DE LATTRE DE TASSIGNY, 75775 PARIS 16, FRANCE.

E-mails: dolbeaul@ceremade.dauphine.fr, esteban@ceremade.dauphine.fr

(A. Figalli) MATHEMATICS DEPARTMENT, ETH ZÜRICH, RAMISTRASSE 101, 8092 ZÜRICH, SWITZERLAND.

E-mail: alessio figalli@math.ethz.ch

(R. L. Frank) DEPARTMENT OF MATHEMATICS, LMU MUNICH, THERESIENSTR. 39, 80333 MÜNCHEN, GERMANY, AND MUNICH CENTER FOR QUANTUM SCIENCE AND TECHNOLOGY, SCHELLINGSTR. 4, 80799 MÜNCHEN, GERMANY, AND MATHEMATICS 253-37, CALTECH, PASADENA, CA 91125, USA.

E-mail: r.frank@lmu.de

(M. Loss) SCHOOL OF MATHEMATICS, GEORGIA INSTITUTE OF TECHNOLOGY ATLANTA, GA 30332, UNITED STATES

OF AMERICA. E-mail: loss@math.gatech.edu
(J. Dolbeault, M. J. Esteban) CEREMADE (CNRS UMR No. 7534), PSL UNIVERSITY, UNIVERSITÉ PARIS-DAUPHINE, PLACE DE LATTRE DE TASSIGNY, 75775 PARIS 16, FRANCE.

E-mails: dolbeaul@ceremade.dauphine.fr, esteban@ceremade.dauphine.fr

(A. Figalli) MATHEMATICS DEPARTMENT, ETH ZÜRICH, RAMISTRASSE 101, 8092 ZÜRICH, SWITZERLAND.

E-mail: alessio figalli@math.ethz.ch

(R. L. Frank) DEPARTMENT OF MATHEMATICS, LMU MUNICH, THERESIENSTR. 39, 80333 MÜNCHEN, GERMANY, AND MUNICH CENTER FOR QUANTUM SCIENCE AND TECHNOLOGY, SCHELLINGSTR. 4, 80799 MÜNCHEN, GERMANY, AND MATHEMATICS 253-37, CALTECH, PASADENA, CA 91125, USA.

E-mail: r.frank@lmu.de

(M. Loss) SCHOOL OF MATHEMATICS, GEORGIA INSTITUTE OF TECHNOLOGY ATLANTA, GA 30332, UNITED STATES

OF AMERICA. E-mail: loss@math.gatech.edu