# A conformal geometric point of view on the Caffarelli-Kohn-Nirenberg inequality

Louis Dupaigne, Ivan Gentil, Simon Zugmeyer

<figure><img src="image_1.png" /></figure>

## To cite this version:

Louis Dupaigne, Ivan Gentil, Simon Zugmeyer. A conformal geometric point of view on the Caffarelli-Kohn-Nirenberg inequality. 2024. hal-03456704v3

HAL Id: hal-03456704

https://hal.science/hal-03456704v3

Preprint submitted on 8 Jan 2024 (v3), last revised 3 Apr 2024 (v4)

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_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# A conformal geometric point of view on the Caffarelli-Kohn-Nirenberg inequality

Louis Dupaigne, Ivan Gentil, Simon Zugmeyer

<figure><img src="image_1.png" /></figure>

## To cite this version:

Louis Dupaigne, Ivan Gentil, Simon Zugmeyer. A conformal geometric point of view on the Caffarelli-Kohn-Nirenberg inequality. 2024. hal-03456704v3

HAL Id: hal-03456704

https://hal.science/hal-03456704v3

Preprint submitted on 8 Jan 2024 (v3), last revised 3 Apr 2024 (v4)

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_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# A conformal geometric point of view on the Caffarelli-Kohn-Nirenberg inequality

L. Dupaigne, I. Gentil and S. Zugmeyer

January 8, 2024

## Abstract

We are interested in the Caffarelli-Kohn-Nirenberg inequality (CKN in short), introduced by these authors in 1984. We explain why the CKN inequality can be viewed as a Sobolev inequality on a weighted Riemannian manifold. More precisely, we prove that the CKN inequality can be interpreted in this way on three different and equivalent models, obtained as weighted versions of the standard Euclidean space, round sphere and hyperbolic space. This result can be viewed as an extension of conformal invariance to the weighted setting. Since the spherical CKN model we introduce has finite measure, the $\Gamma$-calculus introduced by Bakry and Émery provides a way to prove the Sobolev inequalities. This method allows us to recover the optimality of the region of parameters describing symmetry-breaking of minimizers of the CKN inequality, introduced by Felli and Schneider and proved by Dolbeault, Esteban and Loss in 2016. Finally, we develop the notion of $n$-conformal invariants, exhibiting a way to extend the notion of scalar curvature to weighted manifolds such as the CKN models.

# 1 Introduction and main results

## 1.1 The CKN Euclidean space

In their seminal paper [CKN84], Caffarelli, Kohn and Nirenberg found the optimal range of real parameters $a, b, p$ for which the following inequality holds true:

$$ \left( \int_{\mathbf{R}^d} \frac{|v|^p}{|x|^{bp}} dx \right)^{2/p} \leq C_{a,b} \int_{\mathbf{R}^d} \frac{|\nabla v|^2}{|x|^{2a}} dx, \quad v \in \mathcal{C}_c^\infty(\mathbf{R}^d \setminus \{0\}). \quad (1) $$

Here, $|\cdot|$ is the Euclidean norm in $\mathbf{R}^d$, $d \in \mathbf{N}^*$ and $C_{a,b}$ denotes the optimal constant, depending on $a, b$ and $d$ only. Note that the case $a = b = 0$ (and $p = 2d/(d-2)$) corresponds to Sobolev's inequality, while the case $b = a + 1$ (and $p = 2$) is Hardy's inequality, so that (1) is sometimes called the Hardy-Sobolev inequality. Note also that the inequality is achieved in the former case, while it is not in the latter. Let us consider the measure

$$ d\mu_{\mathbf{E}}(x) = |x|^{-bp} dx. \quad (2) $$

Then, the left-hand side of (1) is simply the $L^p$-norm of $v$ with respect to the measure $\mu_{\mathbf{E}}$ (squared). In addition, if we consider the metric<fnref n="1" /> on the manifold $M = \mathbf{R}^d \setminus \{0\}$ given by

$$ \mathfrak{g}_{\mathbf{E}}^{ij} = |x|^{bp-2a} \delta^{ij}, \quad (3) $$

then (1) takes the simpler form

$$ \left( \int |v|^p d\mu_{\mathbf{E}} \right)^{2/p} \leq C_{a,b} \int |\nabla_{\mathfrak{g}_{\mathbf{E}}} v|^2_{\mathfrak{g}_{\mathbf{E}}} d\mu_{\mathbf{E}}. $$

<footnote n="1">¹If $(M, g)$ is $d$-dimensional Riemannian manifold whose metric $g$ is represented in a local system of coordinates at a point $x \in M$ by the matrix $G(x) = (g_{ij}(x))_{1 \leq i,j \leq d}$, we use the letter $\mathfrak{g}$ to denote the bilinear form on the cotangent space of $M$ represented by the inverse matrix $G(x)^{-1} = (g^{ij}(x))_{1 \leq i,j \leq d}$</footnote>
# A conformal geometric point of view on the Caffarelli-Kohn-Nirenberg inequality

L. Dupaigne, I. Gentil and S. Zugmeyer

January 8, 2024

## Abstract

We are interested in the Caffarelli-Kohn-Nirenberg inequality (CKN in short), introduced by these authors in 1984. We explain why the CKN inequality can be viewed as a Sobolev inequality on a weighted Riemannian manifold. More precisely, we prove that the CKN inequality can be interpreted in this way on three different and equivalent models, obtained as weighted versions of the standard Euclidean space, round sphere and hyperbolic space. This result can be viewed as an extension of conformal invariance to the weighted setting. Since the spherical CKN model we introduce has finite measure, the $\Gamma$-calculus introduced by Bakry and Émery provides a way to prove the Sobolev inequalities. This method allows us to recover the optimality of the region of parameters describing symmetry-breaking of minimizers of the CKN inequality, introduced by Felli and Schneider and proved by Dolbeault, Esteban and Loss in 2016. Finally, we develop the notion of $n$-conformal invariants, exhibiting a way to extend the notion of scalar curvature to weighted manifolds such as the CKN models.

# 1 Introduction and main results

## 1.1 The CKN Euclidean space

In their seminal paper [CKN84], Caffarelli, Kohn and Nirenberg found the optimal range of real parameters $a, b, p$ for which the following inequality holds true:

$$ \left( \int_{\mathbf{R}^d} \frac{|v|^p}{|x|^{bp}} dx \right)^{2/p} \leq C_{a,b} \int_{\mathbf{R}^d} \frac{|\nabla v|^2}{|x|^{2a}} dx, \quad v \in \mathcal{C}_c^\infty(\mathbf{R}^d \setminus \{0\}). \quad (1) $$

Here, $|\cdot|$ is the Euclidean norm in $\mathbf{R}^d$, $d \in \mathbb{N}^*$ and $C_{a,b}$ denotes the optimal constant, depending on $a, b$ and $d$ only. Note that the case $a = b = 0$ (and $p = 2d/(d-2)$) corresponds to Sobolev's inequality, while the case $b = a + 1$ (and $p = 2$) is Hardy's inequality, so that (1) is sometimes called the Hardy-Sobolev inequality. Note also that the inequality is achieved in the former case, while it is not in the latter. Let us consider the measure

$$ d\mu_{\mathbf{E}}(x) = |x|^{-bp} dx. \quad (2) $$

Then, the left-hand side of (1) is simply the $L^p$-norm of $v$ with respect to the measure $\mu_{\mathbf{E}}$ (squared). In addition, if we consider the metric<fnref n="1" /> on the manifold $M = \mathbf{R}^d \setminus \{0\}$ given by

$$ \mathfrak{g}_{\mathbf{E}}^{ij} = |x|^{bp-2a} \delta^{ij}, \quad (3) $$

then (1) takes the simpler form

$$ \left( \int |v|^p d\mu_{\mathbf{E}} \right)^{2/p} \leq C_{a,b} \int |\nabla_{\mathfrak{g}_{\mathbf{E}}} v|^2_{\mathfrak{g}_{\mathbf{E}}} d\mu_{\mathbf{E}}. $$

<footnote n="1">¹If $(M, g)$ is $d$-dimensional Riemannian manifold whose metric $g$ is represented in a local system of coordinates at a point $x \in M$ by the matrix $G(x) = (g_{ij}(x))_{1 \leq i,j \leq d}$, we use the letter $\mathfrak{g}$ to denote the bilinear form on the cotangent space of $M$ represented by the inverse matrix $G(x)^{-1} = (g^{ij}(x))_{1 \leq i,j \leq d}$</footnote>
By a standard scaling argument<fnref n="2" />, the following relation is necessary for the inequality to hold true:

$$p = \frac{2d}{d-2+2(b-a)} = \frac{d}{a_c - a + b}, \quad (4)$$

where $a_c = \frac{d-2}{2}$. Through the property of modified inversion symmetry (see Theorem 1.4(ii) in [CW01]), we may always assume that $a < a_c = \frac{d-2}{2}$ since the case $a > a_c$ is dual to it and the inequality fails to be true if $a = a_c$ (see [CKN84]). For simplicity, we also focus on the case $d \ge 4$ and refer to [DEL14] for the remaining cases $d \in \{1, 2, 3\}$. Then, (1) holds true if and only if

$$a \le b \le a+1$$

For simplicity, we do not consider the limiting case $b = a+1$ (Hardy's inequality) and we define accordingly the set

$$\Theta = \{(a, b) \in \mathbb{R}^2, a \le b < a+1, a < a_c\} \quad (5)$$

so that the CKN inequality (1) is valid whenever $(a, b) \in \Theta$ (see Section A.3).

Observe that for $(a, b) \in \Theta$, $p \le \frac{2d}{d-2}$ and so $p$ can be rewritten as the critical Sobolev exponent associated to an intrinsic dimension $n \in [d, +\infty)$ through the relations

$$p = \frac{2n}{n-2}, \quad n = \frac{d}{1+a-b}. \quad (6)$$

The fact that $n$ is a meaningful number, entering in the classical Bakry-Emery curvature-dimension condition, will become transparent in a moment. To summarize, one can view inequality (1) exactly as Sobolev's inequality stated on the weighted Riemannian manifold<fnref n="3" /> that we introduce now.

**Definition 1.1** (The Euclidean CKN space). The Euclidean CKN space is the triple $(M, g_E, \mu_E)$, where the manifold is $M = \mathbb{R}^d \setminus \{0\}$, the metric<fnref n="4" /> is $g_E^{ij} = |x|^{2(1-\alpha)}\delta^{ij}$ and where the measure $\mu_E$ is given by (2). The corresponding Riemannian volume is given by $dV_{g_E} = |x|^{d(\alpha-1)}dx$, the weight $W_E$, verifying $d\mu_E = e^{-W_E}dV_{g_E}$, is given by $W_E = -\frac{\alpha(n-d)}{2} \log|x|^2$ and the generator<fnref n="5" /> is given by $L_E = \Delta_{g_E} - \nabla_{g_E} W_E \cdot \nabla = |x|^{2(1-\alpha)}(\Delta - a\nabla \log|x|^2 \cdot \nabla)$.

For notational convenience, we introduced above the parameter<fnref n="6" />:

$$\alpha = 1 + a - \frac{ph}{2}, \quad (7)$$

where $(a, b) \in \Theta$ (defined in (5)) and $p$ is the critical exponent given by (4). In other words, returning to the parameters $a, b, d$ (and $a_c = (d-2)/2$),

$$\alpha = \frac{(a_c - a)(a + 1 - b)}{a_c - a + b}.$$

Note that for any $(a, b) \in \Theta$, we have $\alpha \ge 0$, see Section A.3 and Figure 1 for more information about parameters. Equivalently, and this is the notation adopted in this paper<fnref n="7" />, one can see the Euclidean CKN space as a Markov triple $(M, \mu_E, \Gamma_E)$, where $\mu_E$ verifies (2) and the carré du champ operator is given by

$$\Gamma_E(v) = |\nabla_{g_E} v|_{g_E}^2 = |x|^{bp-2a} |\nabla v|^2 = |x|^{2(1-\alpha)} |\nabla v|^2$$

Its associated bilinear form is denoted by $\Gamma_E(u, v) = |x|^{2(1-\alpha)} \nabla u \cdot \nabla v$ for $u, v \in C_c^\infty(\mathbb{R}^d \setminus \{0\})$. Inequality (1) now reads

$$\left( \int |v|^p d\mu_E \right)^{2/p} \le C_{a,b} \int \Gamma_E(v) d\mu_E, \quad v \in C_c^\infty(\mathbb{R}^d \setminus \{0\}).$$

<footnote n="2">²To see this, apply (1) to the function $x \mapsto v(\lambda x)$ where $\lambda > 0$ and let $\lambda \to 0^+$ and $\lambda \to +\infty$</footnote>

<footnote n="3">³The words "smooth metric measure space" and "manifold with density" are also employed in the literature to designate the same object.</footnote>

<footnote n="4">⁴The given expression of $g_E$ is just a rewriting of (3)</footnote>

<footnote n="5">⁵i.e. the operator such that $-\int u L_E v d\mu_E = \int (\nabla_{g_E} u \cdot g \nabla_{g_E} v) d\mu_E$ for $u, v \in C_c^\infty(\mathbb{R}^d \setminus \{0\})$</footnote>

<footnote n="6">⁶The reader may check that $\alpha$ turns out to be the same parameter as the one introduced in [DEL14] (for different reasons).</footnote>

<footnote n="7">⁷See [BGL14] for an introduction to $\Gamma$-calculus.</footnote>
By a standard scaling argument<fnref n="2" />, the following relation is necessary for the inequality to hold true:

$$p = \frac{2d}{d-2+2(b-a)} = \frac{d}{a_c - a + b}, \qquad (4)$$

where $a_c = \frac{d-2}{2}$. Through the property of modified inversion symmetry (see Theorem 1.4(ii) in [CW01]), we may always assume that $a < a_c = \frac{d-2}{2}$ since the case $a > a_c$ is dual to it and the inequality fails to be true if $a = a_c$ (see [CKN84]). For simplicity, we also focus on the case $d \ge 4$ and refer to [DEL14] for the remaining cases $d \in \{1, 2, 3\}$. Then, (1) holds true if and only if

$$a \le b \le a+1$$

For simplicity, we do not consider the limiting case $b = a+1$ (Hardy's inequality) and we define accordingly the set

$$\Theta = \{(a, b) \in \mathbb{R}^2, a \le b < a+1, a < a_c\} \qquad (5)$$

so that the CKN inequality (1) is valid whenever $(a, b) \in \Theta$ (see Section A.3).

Observe that for $(a, b) \in \Theta$, $p \le \frac{2d}{d-2}$ and so $p$ can be rewritten as the critical Sobolev exponent associated to an intrinsic dimension $n \in [d, +\infty)$ through the relations

$$p = \frac{2n}{n-2}, \quad n = \frac{d}{1+a-b}. \qquad (6)$$

The fact that $n$ is a meaningful number, entering in the classical Bakry-Emery curvature-dimension condition, will become transparent in a moment. To summarize, one can view inequality (1) exactly as Sobolev's inequality stated on the weighted Riemannian manifold<fnref n="3" /> that we introduce now.

**Definition 1.1** (The Euclidean CKN space). The Euclidean CKN space is the triple $(M, g_E, \mu_E)$, where the manifold is $M = \mathbb{R}^d \setminus \{0\}$, the metric<fnref n="4" /> is $g_E^{ij} = |x|^{2(1-\alpha)}\delta^{ij}$ and where the measure $\mu_E$ is given by (2). The corresponding Riemannian volume is given by $dV_{g_E} = |x|^{d(\alpha-1)}dx$, the weight $W_E$, verifying $d\mu_E = e^{-W_E}dV_{g_E}$, is given by $W_E = -\frac{\alpha(n-d)}{2} \log|x|^2$ and the generator<fnref n="5" /> is given by $L_E = \Delta_{g_E} - \nabla_{g_E}W_E \cdot \nabla = |x|^{2(1-\alpha)}(\Delta - a\nabla\log|x|^2 \cdot \nabla)$.

For notational convenience, we introduced above the parameter<fnref n="6" />:

$$\alpha = 1 + a - \frac{ph}{2}, \qquad (7)$$

where $(a, b) \in \Theta$ (defined in (5)) and $p$ is the critical exponent given by (4). In other words, returning to the parameters $a, b, d$ (and $a_c = (d-2)/2$),

$$\alpha = \frac{(a_c - a)(a + 1 - b)}{a_c - a + b}.$$

Note that for any $(a, b) \in \Theta$, we have $\alpha \ge 0$, see Section A.3 and Figure 1 for more information about parameters. Equivalently, and this is the notation adopted in this paper<fnref n="7" />, one can see the Euclidean CKN space as a Markov triple $(M, \mu_E, \Gamma_E)$, where $\mu_E$ verifies (2) and the carré du champ operator is given by

$$\Gamma_E(v) = |\nabla_{g_E} v|_{g_E}^2 = |x|^{bp-2a} |\nabla v|^2 = |x|^{2(1-\alpha)} |\nabla v|^2$$

Its associated bilinear form is denoted by $\Gamma_E(u, v) = |x|^{2(1-\alpha)} \nabla u \cdot \nabla v$ for $u, v \in C_c^\infty(\mathbb{R}^d \setminus \{0\})$. Inequality (1) now reads

$$\left( \int |v|^p d\mu_E \right)^{2/p} \le C_{a,b} \int \Gamma_E(v) d\mu_E, \quad v \in C_c^\infty(\mathbb{R}^d \setminus \{0\}).$$

<footnote n="2">²To see this, apply (1) to the function $x \mapsto v(\lambda x)$ where $\lambda > 0$ and let $\lambda \to 0^+$ and $\lambda \to +\infty$</footnote>

<footnote n="3">³The words "smooth metric measure space" and "manifold with density" are also employed in the literature to designate the same object.</footnote>

<footnote n="4">⁴The given expression of $g_E$ is just a rewriting of (3)</footnote>

<footnote n="5">⁵i.e. the operator such that $-\int u L_E v d\mu_E = \int (\nabla_{g_E} u \cdot g \nabla_{g_E} v) d\mu_E$ for $u, v \in C_c^\infty(\mathbb{R}^d \setminus \{0\})$</footnote>

<footnote n="6">⁶The reader may check that $\alpha$ turns out to be the same parameter as the one introduced in [DEL14] (for different reasons).</footnote>

<footnote n="7">⁷See [BGL14] for an introduction to $\Gamma$-calculus.</footnote>
## 1.2 Conformal invariance

As noticed earlier, when $a = b = 0$ ($\alpha = 1$), we recover the standard Sobolev inequality on the standard Euclidean space. In that case, since the metric of the $d$-dimensional sphere $S^d$ and the metric of the $d$-dimensional hyperbolic space $H^d$ are both conformally equivalent to the Euclidean metric, Sobolev's inequality takes equivalent forms on these three model spaces. More precisely, the Euclidean Sobolev inequality applied to the function $\varphi^{\frac{2-d}{2}}v$, where $\varphi(x) = \frac{1+|x|^2}{2}$ (respectively $\varphi(x) = \frac{1-|x|^2}{2}$) and $v \in C_c^\infty(\mathbf{R}^d)$ (resp. $v \in C_c^\infty(\mathbf{B})$) yields

$$ \left( \int |v|^p dV_g \right)^{2/p} \le C \left[ \int |\nabla_g v|^2 dV_g + \int S_g v^2 dV_g \right], \quad (8) $$

where $g$ is the round metric on the sphere $S^d$ expressed in stereographic coordinates (resp. the metric of the hyperbolic space in the Poincaré ball model), $dV_g$ the associated Riemannian volume, $S_g = \frac{d(d-2)}{4}$ (resp. $S_g = -\frac{d(d-2)}{4}$) and $C = \frac{4}{d(d-2)}|S^d|^{-\frac{2}{d}}$ the best constant in the standard Euclidean Sobolev inequality. By analogy, we can extend the conformal invariance property to the setting of weighted manifolds as described next.

### The spherical CKN and the hyperbolic CKN spaces

Recall that the metric and reference measure of the Euclidean CKN space read

$$ g_E^{ij} = |x|^{2(1-\alpha)} \delta^{ij} \quad \text{and} \quad d\mu_E = |x|^{-bp} dx. $$

Keeping in mind the expression of the standard stereographic projection, we define next the spherical and hyperbolic CKN spaces as follows.

**Definition 1.2** (The spherical and the hyperbolic CKN spaces).

* The spherical CKN space is the triple $(M, g_S, \mu_S)$, where $M = \mathbf{R}^d \setminus \{0\}$,

$$ g_S^{ij} = |x|^{2(1-\alpha)} \left( \frac{1+|x|^{2\alpha}}{2} \right)^2 \delta^{ij} \quad \text{and} \quad d\mu_S = |x|^{-bp} \left( \frac{2}{1+|x|^{2\alpha}} \right)^n dx. $$

Associated objects are given by the following formulae:

- Riemannian volume: $dV_{g_S} = 2^d \frac{|x|^{d(\alpha-1)}}{(1+|x|^{2\alpha})^2} dx$,

- weight: $W_S = (n-d) \log(1+|x|^{2\alpha}) - \frac{\alpha(n-d)}{2} \log|x|^2$,

- Carré du champ operator: $\Gamma_S(v) = |\nabla_{g_S} v|_{g_S}^2 = |x|^{2(1-\alpha)} \frac{(1+|x|^{2\alpha})^2}{4} |\nabla v|^2$,

- generator: $L_S(f) = |x|^{2(1-\alpha)} \frac{(1+|x|^{2\alpha})^2}{4} [\Delta f - a \nabla f \cdot \nabla \log|x|^2 - (n-2)\nabla f \cdot \nabla \log(1+|x|^{2\alpha})]$.

* The CKN hyperbolic space is the triple $(\mathbf{B} \setminus \{0\}, g_H, \mu_H)$, where $\mathbf{B}$ is the open unit ball in $\mathbf{R}^d$,

$$ g_H^{ij} = |x|^{2(1-\alpha)} \left( \frac{1-|x|^{2\alpha}}{2} \right)^2 \delta^{ij} \quad \text{and} \quad d\mu_H = |x|^{-bp} \left( \frac{2}{1-|x|^{2\alpha}} \right)^n dx. $$

Associated objects to this triple are given by the following formulae:

- Riemannian volume: $dV_{g_H} = 2^d \frac{|x|^{d(\alpha-1)}}{(1-|x|^{2\alpha})^d} dx$,

- weight: $W_H = (n-d) \log(1-|x|^{2\alpha}) - \frac{\alpha(n-d)}{2} \log|x|^2$,

- Carré du champ operator: $\Gamma_H(v) = |\nabla_{g_H} v|_{g_S}^2 = |x|^{2(1-\alpha)} \frac{(1-|x|^{2\alpha})^2}{4} |\nabla v|^2$,

- generator: $L_H(f) = |x|^{2(1-\alpha)} \frac{(1-|x|^{2\alpha})^2}{4} [\Delta f - a \nabla f \cdot \nabla \log|x|^2 - (n-2)\nabla f \cdot \nabla \log(1-|x|^{2\alpha})]$.
## 1.2 Conformal invariance

As noticed earlier, when $a = b = 0$ ($\alpha = 1$), we recover the standard Sobolev inequality on the standard Euclidean space. In that case, since the metric of the $d$-dimensional sphere $S^d$ and the metric of the $d$-dimensional hyperbolic space $\mathbf{H}^d$ are both conformally equivalent to the Euclidean metric, Sobolev's inequality takes equivalent forms on these three model spaces. More precisely, the Euclidean Sobolev inequality applied to the function $\varphi^{\frac{2-d}{2}}v$, where $\varphi(x) = \frac{1+|x|^2}{2}$ (respectively $\varphi(x) = \frac{1-|x|^2}{2}$) and $v \in C_c^\infty(\mathbf{R}^d)$ (resp. $v \in C_c^\infty(\mathbf{B})$) yields

$$ \left( \int |v|^p dV_{\mathfrak{g}} \right)^{2/p} \leq C \left[ \int |\nabla_{\mathfrak{g}} v|^2 dV_{\mathfrak{g}} + \int S_{\mathfrak{g}} v^2 dV_{\mathfrak{g}} \right], \quad (8) $$

where $\mathfrak{g}$ is the round metric on the sphere $S^d$ expressed in stereographic coordinates (resp. the metric of the hyperbolic space in the Poincaré ball model), $dV_{\mathfrak{g}}$ the associated Riemannian volume, $S_{\mathfrak{g}} = \frac{d(d-2)}{4}$ (resp. $S_{\mathfrak{g}} = -\frac{d(d-2)}{4}$) and $C = \frac{4}{d(d-2)}|S^d|^{-\frac{2}{d}}$ the best constant in the standard Euclidean Sobolev inequality. By analogy, we can extend the conformal invariance property to the setting of weighted manifolds as described next.

### The spherical CKN and the hyperbolic CKN spaces

Recall that the metric and reference measure of the Euclidean CKN space read

$$ \mathfrak{g}_{\mathbf{E}}^{ij} = |x|^{2(1-\alpha)} \delta^{ij} \quad \text{and} \quad d\mu_{\mathbf{E}} = |x|^{-bp} dx. $$

Keeping in mind the expression of the standard stereographic projection, we define next the spherical and hyperbolic CKN spaces as follows.

**Definition 1.2** (The spherical and the hyperbolic CKN spaces).

* The spherical CKN space is the triple $(M, \mathfrak{g}_{\mathbf{S}}, \mu_{\mathbf{S}})$, where $M = \mathbf{R}^d \setminus \{0\}$,

$$ \mathfrak{g}_{\mathbf{S}}^{ij} = |x|^{2(1-\alpha)} \left( \frac{1+|x|^{2\alpha}}{2} \right)^2 \delta^{ij} \quad \text{and} \quad d\mu_{\mathbf{S}} = |x|^{-bp} \left( \frac{2}{1+|x|^{2\alpha}} \right)^n dx. $$

Associated objects are given by the following formulae:

- Riemannian volume: $dV_{\mathfrak{g}_{\mathbf{S}}} = 2^d \frac{|x|^{d(\alpha-1)}}{(1+|x|^{2\alpha})^2} dx$,

- weight: $W_{\mathbf{S}} = (n-d) \log(1+|x|^{2\alpha}) - \frac{\alpha(n-d)}{2} \log|x|^2$,

- Carré du champ operator: $\Gamma_{\mathbf{S}}(v) = |\nabla_{\mathfrak{g}_{\mathbf{S}}} v|_{\mathfrak{g}_{\mathbf{S}}}^2 = |x|^{2(1-\alpha)} \frac{(1+|x|^{2\alpha})^2}{4} |\nabla v|^2$,

- generator: $L_{\mathbf{S}}(f) = |x|^{2(1-\alpha)} \frac{(1+|x|^{2\alpha})^2}{4} [\Delta f - a \nabla f \cdot \nabla \log|x|^2 - (n-2)\nabla f \cdot \nabla \log(1+|x|^{2\alpha})]$.

* The CKN hyperbolic space is the triple $(\mathbf{B} \setminus \{0\}, \mathfrak{g}_{\mathbf{H}}, \mu_{\mathbf{H}})$, where $\mathbf{B}$ is the open unit ball in $\mathbf{R}^d$,

$$ \mathfrak{g}_{\mathbf{H}}^{ij} = |x|^{2(1-\alpha)} \left( \frac{1-|x|^{2\alpha}}{2} \right)^2 \delta^{ij} \quad \text{and} \quad d\mu_{\mathbf{H}} = |x|^{-bp} \left( \frac{2}{1-|x|^{2\alpha}} \right)^n dx. $$

Associated objects to this triple are given by the following formulae:

- Riemannian volume: $dV_{\mathfrak{g}_{\mathbf{H}}} = 2^d \frac{|x|^{d(\alpha-1)}}{(1-|x|^{2\alpha})^d} dx$,

- weight: $W_{\mathbf{H}} = (n-d) \log(1-|x|^{2\alpha}) - \frac{\alpha(n-d)}{2} \log|x|^2$,

- Carré du champ operator: $\Gamma_{\mathbf{H}}(v) = |\nabla_{\mathfrak{g}_{\mathbf{H}}} v|_{\mathfrak{g}_{\mathbf{H}}}^2 = |x|^{2(1-\alpha)} \frac{(1-|x|^{2\alpha})^2}{4} |\nabla v|^2$,

- generator: $L_{\mathbf{H}}(f) = |x|^{2(1-\alpha)} \frac{(1-|x|^{2\alpha})^2}{4} [\Delta f - a \nabla f \cdot \nabla \log|x|^2 - (n-2)\nabla f \cdot \nabla \log(1-|x|^{2\alpha})]$.
**Remark 1.3.** • Note that in the case $\alpha = 1$ (which is achieved in $\Theta$ only when $a = b = 0$, see Lemma 1.9), the CKN sphere is the standard round sphere (punctured at both of its poles) viewed in the stereographic projection chart. Similarly, for $\alpha = 1$, the CKN hyperbolic space is the (punctured) hyperbolic space.

• Note that, letting $\varphi(x) = \frac{1+|x|^{2\alpha}}{2}$, we have $\Gamma_S = \varphi^2\Gamma_E$ and $\mu_S = \varphi^{-n}\mu_E$. We shall say that the CKN Euclidean and spherical spaces belong to the same $n$-conformal class ($n$ not necessarily being equal to the topological dimension). Similarly, with $\psi(x) = \frac{1-|x|^{2\alpha}}{2}$, we have $\Gamma_H = \psi^2\Gamma_E$ and $\mu_H = \psi^{-n}\mu_E$, so that the hyperbolic CKN space also belongs to the same $n$-conformal class.

• When $(a, b) \in \Theta$, $\mu_S$ has finite mass (see Remark A.10 in Section A.3). In this Section, we prefer not to normalize the measure $\mu_S$, a choice which makes the conformal invariance of Sobolev's inequality more transparent.

With these definitions at hand, we prove

**Theorem 1.4** (Conformal invariance of the three model spaces). Let $C > 0$ be an arbitrary constant. The three following Sobolev inequalities associated to each CKN model are equivalent:

$$ (i) \quad \forall v \in C_c^\infty(\mathbf{R}^d \setminus \{0\}), \quad \left( \int |v|^p d\mu_{\mathbf{E}} \right)^{2/p} \le C \int \Gamma_{\mathbf{E}}(v) d\mu_{\mathbf{E}}, \quad (9) $$

$$ (ii) \quad \forall v \in C_c^\infty(\mathbf{R}^d \setminus \{0\}), \quad \left( \int |v|^p d\mu_S \right)^{2/p} \le C \left( \int \Gamma_S(v) d\mu_S + \frac{n(n-2)}{4} \alpha^2 \int v^2 d\mu_S \right), \quad (10) $$

$$ (iii) \quad \forall v \in C_c^\infty(\mathbf{B} \setminus \{0\}), \quad \left( \int |v|^p d\mu_H \right)^{2/p} \le C \left( \int \Gamma_H(v) d\mu_H - \frac{n(n-2)}{4} \alpha^2 \int v^2 d\mu_H \right). \quad (11) $$

**Remark 1.5.** • Inequality (9) is valid for some constant $C = C_{a,b}$ if and only if $(a, b) \in \Theta$ as proved in [CKN84]. Hence, so are (10) and (11).

• As we shall see, the value of the optimal constant $C$ is known only in a restricted range of parameters, see Theorem 1.7 below.

• Since its set of test functions is smaller, inequality (11) need not be optimal even though (9) and (10) are. For example, in the absence of weights, $C = \frac{4}{d(d-2)|\mathbf{S}^d|^{2/d}}$ is the optimal constant in (9) and (10) and extremals exist (and are classified), see Theorem 1.7 below. In contrast, inequality (11) holds with the same constant $C = \frac{4}{d(d-2)|\mathbf{S}^d|^{2/d}}$ but when $n = d = 3$ (and again $\alpha = 1$), the constant $-\frac{n(n-2)}{4}$ can be improved to $-\frac{(n-1)^2}{4}$, see [BFL08]. Using this fact and the proof of Theorem 1.4, it follows that the standard Sobolev inequality in $\mathbf{R}^3$ improves to

$$ \left( \int_{\mathbf{R}^3} |v|^6 dx \right)^{1/3} \le \frac{1}{3} \left( \frac{2}{\pi} \right)^{4/3} \left( \int_{\mathbf{R}^3} |\nabla v|^2 dx - \int_{\mathbf{R}^3} \frac{v^2}{(1 - |x|^2)^2} dx \right), $$

when restricted to functions $v \in H_0^1(\mathbf{B})$. When $d \ge 4$ (and $\alpha = 1$), inequality (11) is again optimal, but contrary to (9) and (10), the inequality is never attained$^8$. We have not yet investigated the optimality of (11) in the general case.

## 1.3 Curvature-dimension conditions on the spherical CKN space

We just saw that the CKN inequality takes the forms (9), (10), (11) on the three CKN spaces. But what is the value of the best constant $C$? To answer this question, let us first recall the following classical definition and result: a smooth weighted manifold $(M, \mathfrak{g}, \mu)$ is said to satisfy the CD$(\rho, n)$ condition if for every $f \in C^\infty(M)$,

$$ \Gamma_2(f) \ge \rho \Gamma(f) + \frac{1}{n} (Lf)^2, $$

<footnote>$^8$If it were, then (9) would also be attained by a compactly supported function.</footnote>
**Remark 1.3.** • Note that in the case $\alpha = 1$ (which is achieved in $\Theta$ only when $a = b = 0$, see Lemma 1.9), the CKN sphere is the standard round sphere (punctured at both of its poles) viewed in the stereographic projection chart. Similarly, for $\alpha = 1$, the CKN hyperbolic space is the (punctured) hyperbolic space.

• Note that, letting $\varphi(x) = \frac{1+|x|^{2\alpha}}{2}$, we have $\Gamma_S = \varphi^2\Gamma_E$ and $\mu_S = \varphi^{-n}\mu_E$. We shall say that the CKN Euclidean and spherical spaces belong to the same $n$-conformal class ($n$ not necessarily being equal to the topological dimension). Similarly, with $\psi(x) = \frac{1-|x|^{2\alpha}}{2}$, we have $\Gamma_H = \psi^2\Gamma_E$ and $\mu_H = \psi^{-n}\mu_E$, so that the hyperbolic CKN space also belongs to the same $n$-conformal class.

• When $(a, b) \in \Theta$, $\mu_S$ has finite mass (see Remark A.10 in Section A.3). In this Section, we prefer not to normalize the measure $\mu_S$, a choice which makes the conformal invariance of Sobolev's inequality more transparent.

With these definitions at hand, we prove

**Theorem 1.4** (Conformal invariance of the three model spaces). Let $C > 0$ be an arbitrary constant. The three following Sobolev inequalities associated to each CKN model are equivalent:

$$ (i) \quad \forall v \in C_c^\infty(\mathbf{R}^d \setminus \{0\}), \quad \left( \int |v|^p d\mu_{\mathbf{E}} \right)^{2/p} \le C \int \Gamma_{\mathbf{E}}(v) d\mu_{\mathbf{E}}, \quad (9) $$

$$ (ii) \quad \forall v \in C_c^\infty(\mathbf{R}^d \setminus \{0\}), \quad \left( \int |v|^p d\mu_S \right)^{2/p} \le C \left( \int \Gamma_S(v) d\mu_S + \frac{n(n-2)}{4} \alpha^2 \int v^2 d\mu_S \right), \quad (10) $$

$$ (iii) \quad \forall v \in C_c^\infty(\mathbf{B} \setminus \{0\}), \quad \left( \int |v|^p d\mu_H \right)^{2/p} \le C \left( \int \Gamma_H(v) d\mu_H - \frac{n(n-2)}{4} \alpha^2 \int v^2 d\mu_H \right). \quad (11) $$

**Remark 1.5.** • Inequality (9) is valid for some constant $C = C_{a,b}$ if and only if $(a, b) \in \Theta$ as proved in [CKN84]. Hence, so are (10) and (11).

• As we shall see, the value of the optimal constant $C$ is known only in a restricted range of parameters, see Theorem 1.7 below.

• Since its set of test functions is smaller, inequality (11) need not be optimal even though (9) and (10) are. For example, in the absence of weights, $C = \frac{4}{d(d-2)|\mathbf{S}^d|^{2/d}}$ is the optimal constant in (9) and (10) and extremals exist (and are classified), see Theorem 1.7 below. In contrast, inequality (11) holds with the same constant $C = \frac{4}{d(d-2)|\mathbf{S}^d|^{2/d}}$ but when $n = d = 3$ (and again $\alpha = 1$), the constant $-\frac{n(n-2)}{4}$ can be improved to $-\frac{(n-1)^2}{4}$, see [BFL08]. Using this fact and the proof of Theorem 1.4, it follows that the standard Sobolev inequality in $\mathbf{R}^3$ improves to

$$ \left( \int_{\mathbf{R}^3} |v|^6 dx \right)^{1/3} \le \frac{1}{3} \left( \frac{2}{\pi} \right)^{4/3} \left( \int_{\mathbf{R}^3} |\nabla v|^2 dx - \int_{\mathbf{R}^3} \frac{v^2}{(1 - |x|^2)^2} dx \right), $$

when restricted to functions $v \in H_0^1(\mathbf{B})$. When $d \ge 4$ (and $\alpha = 1$), inequality (11) is again optimal, but contrary to (9) and (10), the inequality is never attained$^8$. We have not yet investigated the optimality of (11) in the general case.

## 1.3 Curvature-dimension conditions on the spherical CKN space

We just saw that the CKN inequality takes the forms (9), (10), (11) on the three CKN spaces. But what is the value of the best constant $C$? To answer this question, let us first recall the following classical definition and result: a smooth weighted manifold $(M, \mathfrak{g}, \mu)$ is said to satisfy the $CD(\rho, n)$ condition if for every $f \in C^\infty(M)$,

$$ \Gamma_2(f) \ge \rho \Gamma(f) + \frac{1}{n} (Lf)^2, $$

<footnote>$^8$If it were, then (9) would also be attained by a compactly supported function.</footnote>
where $(\rho, n) \in \mathbf{R} \times (\mathbf{R} \cup \{+\infty\})$, $d\mu = e^{-W}dV_g$ for some $W \in \mathcal{C}^{\infty}(M)$, $\Gamma(f, h) = \langle \nabla^g f, \nabla^g h \rangle_g$, $\Gamma(f) = \Gamma(f, f) = |\nabla^g f|^2$, $Lf = \Delta_g f - \Gamma(W, f)$ and $\Gamma_2(f) = \frac{1}{2}L(\Gamma(f)) - \Gamma(f, Lf)$, for smooth functions $f, g$ on $M$. The following theorem, generalizing the earlier work [BV91], holds true:

**Theorem A.** ([BL96] and [BGL14, Thm. 6.8.3]) Let $(M, \mathfrak{g}, \mu)$ be a smooth weighted manifold satisfying the CD$(\rho, n)$ condition with $\rho > 0$, $n > d = \dim(M)$ ($n > 2$). Assume in addition that the associated operator $L$ is essentially self-adjoint in $L^2(\mu)$. Let $p = \frac{2n}{n-2}$ and normalize the measure $\mu$ so that $\mu(M) = 1$. Then,

$$ \left( \int |v|^p d\mu \right)^{2/p} \le \frac{4}{n(n-2)} \frac{n-1}{\rho} \int \Gamma(v) d\mu + \int v^2 d\mu, \quad v \in \mathcal{C}_c^\infty(M). \qquad (12) $$

Let us remark that this theorem can also be stated in the more general context of full Markov triples as proposed in [BGL14] and also on metric measure spaces as proved in [Pro15]. Thanks to Theorem A, it suffices to determine whether the CKN sphere is a smooth weighted manifold satisfying the CD$(\rho, n)$ condition and that the associated operator is essentially self-adjoint, in order to obtain an explicit value (which turns out to be optimal in our case) for the constant $C$ in (10). This is what we do next.

**Proposition 1.6** (Curvature-dimension condition for the spherical CKN space). Let $(a, b) \in \Theta$ and

$$ \rho = \alpha^2(n-1). \qquad (13) $$

Then, the spherical CKN space satisfies the curvature-dimension condition CD$(\rho, n)$ if and only if

$$ \alpha^2 \le \frac{d-2}{n-2}. \qquad (14) $$

One implication of this proposition has been proved in [Ket15, Thm. 3.9]. The proof proposed here is different and based on tensors, which is a useful method to prove the equivalence between the two conditions. Note that $(\mathbf{R}^d \setminus \{0\}, \mathfrak{g}, \mu_{\mathbf{S}})$ is a smooth weighted manifold and that its operator is essentially self-adjoint if and only if $n \ge 3$, see [Ket15, Thm. 3.12], whence Sobolev's inequality (10) holds under the condition (14). In fact, more can be said. Revisiting the proofs of Theorem A given in [DGZ20, BGL14], we find that if the following weaker integrated form of the curvature-dimension condition

$$ \int \left( \Gamma_{\mathbf{S}}(f) - \rho \Gamma_{\mathbf{S}}(f) - \frac{1}{n} (L_{\mathbf{S}} f)^2 \right) f^{1-n} d\mu_{\mathbf{S}} \ge 0 \qquad (15) $$

holds for functions $f \in \mathcal{C}^{\infty}(\mathbf{R}^d \setminus \{0\})$ such that $\inf f > 0$ and $\sup f < +\infty$, then the sharp Sobolev inequality is valid on the CKN sphere. More precisely, letting $H_0^1(\mu_{\mathbf{S}})$ denote the closure of $\mathcal{C}_c^{\infty}(\mathbf{R}^d \setminus \{0\})$ with respect to the norm

$$ \|f\|_{H_0^1(\mu_{\mathbf{S}})}^2 = \int (\Gamma_{\mathbf{S}}(f) + f^2) d\mu_{\mathbf{S}}. $$

**Theorem 1.7** (Sobolev inequality for the spherical CKN space). Let $(a, b) \in \Theta$. Whenever

$$ 0 < \alpha < 1, \qquad (16) $$

the following optimal Sobolev inequality holds

$$ \left( \int v^p d\mu \right)^{2/p} \le \frac{4}{n(n-2)\alpha^2} \int \Gamma_{\mathbf{S}}(v) d\mu + \int v^2 d\mu, \qquad (17) $$

for any $v \in H_0^1(\mu_{\mathbf{S}})$, where $\mu = \frac{1}{2}\mu_{\mathbf{S}}$ and $Z$ is a normalization constant⁹ such that $\mu$ is a probability measure. That is, inequality (10) is valid with optimal constant

$$ C = \frac{4}{n(n-2)\alpha^2 Z^{\frac{2}{n}}}. $$

$$ {}^9Z = \mu_{\mathbf{S}}(\mathbf{R}^d \setminus \{0\}) = \frac{2}{\alpha} |\mathbf{S}^{d-1}| \int_0^{+\infty} (\cosh t)^{-n} dt $$
where $(\rho, n) \in \mathbf{R} \times (\mathbf{R} \cup \{+\infty\})$, $d\mu = e^{-W}dV_g$ for some $W \in \mathcal{C}^{\infty}(M)$, $\Gamma(f, h) = \langle \nabla^g f, \nabla^g h \rangle_g$, $\Gamma(f) = \Gamma(f, f) = |\nabla^g f|^2$, $Lf = \Delta_g f - \Gamma(W, f)$ and $\Gamma_2(f) = \frac{1}{2}L(\Gamma(f)) - \Gamma(f, Lf)$, for smooth functions $f, g$ on $M$. The following theorem, generalizing the earlier work [BV91], holds true:

**Theorem A.** ([BL96] and [BGL14, Thm. 6.8.3]) Let $(M, \mathfrak{g}, \mu)$ be a smooth weighted manifold satisfying the CD$(\rho, n)$ condition with $\rho > 0$, $n > d = \dim(M)$ ($n > 2$). Assume in addition that the associated operator $L$ is essentially self-adjoint in $L^2(\mu)$. Let $p = \frac{2n}{n-2}$ and normalize the measure $\mu$ so that $\mu(M) = 1$. Then,

$$ \left( \int |v|^p d\mu \right)^{2/p} \le \frac{4}{n(n-2)} \frac{n-1}{\rho} \int \Gamma(v) d\mu + \int v^2 d\mu, \quad v \in \mathcal{C}_c^\infty(M). \qquad (12) $$

Let us remark that this theorem can also be stated in the more general context of full Markov triples as proposed in [BGL14] and also on metric measure spaces as proved in [Pro15]. Thanks to Theorem A, it suffices to determine whether the CKN sphere is a smooth weighted manifold satisfying the CD$(\rho, n)$ condition and that the associated operator is essentially self-adjoint, in order to obtain an explicit value (which turns out to be optimal in our case) for the constant $C$ in (10). This is what we do next.

**Proposition 1.6** (Curvature-dimension condition for the spherical CKN space). Let $(a, b) \in \Theta$ and

$$ \rho = \alpha^2(n-1). \qquad (13) $$

Then, the spherical CKN space satisfies the curvature-dimension condition CD$(\rho, n)$ if and only if

$$ \alpha^2 \le \frac{d-2}{n-2}. \qquad (14) $$

One implication of this proposition has been proved in [Ket15, Thm. 3.9]. The proof proposed here is different and based on tensors, which is a useful method to prove the equivalence between the two conditions. Note that $(\mathbf{R}^d \setminus \{0\}, \mathfrak{g}_S, \mu_S)$ is a smooth weighted manifold and that its operator is essentially self-adjoint if and only if $n \ge 3$, see [Ket15, Thm. 3.12], whence Sobolev's inequality (10) holds under the condition (14). In fact, more can be said. Revisiting the proofs of Theorem A given in [DGZ20, BGL14], we find that if the following weaker integrated form of the curvature-dimension condition

$$ \int \left( \Gamma_{\mathbf{S}}(f) - \rho \Gamma_{\mathbf{S}}(f) - \frac{1}{n} (L_{\mathbf{S}} f)^2 \right) f^{1-n} d\mu_{\mathbf{S}} \ge 0 \qquad (15) $$

holds for functions $f \in \mathcal{C}^{\infty}(\mathbf{R}^d \setminus \{0\})$ such that $\inf f > 0$ and $\sup f < +\infty$, then the sharp Sobolev inequality is valid on the CKN sphere. More precisely, letting $H_0^1(\mu_S)$ denote the closure of $\mathcal{C}_c^\infty(\mathbf{R}^d \setminus \{0\})$ with respect to the norm

$$ \|f\|_{H_0^1(\mu_S)}^2 = \int (\Gamma_S(f) + f^2) d\mu_S. $$

**Theorem 1.7** (Sobolev inequality for the spherical CKN space). Let $(a, b) \in \Theta$. Whenever

$$ 0 < \alpha < 1, \qquad (16) $$

the following optimal Sobolev inequality holds

$$ \left( \int v^p d\mu \right)^{2/p} \le \frac{4}{n(n-2)\alpha^2} \int \Gamma_S(v) d\mu + \int v^2 d\mu, \qquad (17) $$

for any $v \in H_0^1(\mu_S)$, where $\mu = \frac{1}{2}\mu_S$ and $Z$ is a normalization constant⁹ such that $\mu$ is a probability measure. That is, inequality (10) is valid with optimal constant

$$ C = \frac{4}{n(n-2)\alpha^2 Z^{\frac{2}{n}}}. $$

$$ {}^9Z = \mu_S(\mathbf{R}^d \setminus \{0\}) = \frac{2}{\alpha} |\mathbf{S}^{d-1}| \int_0^{+\infty} (\cosh t)^{-n} dt $$
In addition, equality holds in (17) if and only if

$$v(x) = (\lambda + \gamma \tanh(\alpha s))^{-\frac{n-2}{2}}, \quad s = \log|x|,$$

where $\lambda, \gamma$ are arbitrary constants such that $\lambda > |\gamma|$. In particular optimal functions for both inequalities (1) and (17) are radial.

**Remark 1.8.** • Note that $\varphi_1(x) = \tanh(\alpha s)$ is a radial eigenfunction of $L_S$ associated to the eigenvalue $\lambda = \alpha^2 n$. So, except for the round sphere (corresponding to the case $\alpha = 1$), the extremals of Sobolev's inequality are obtained as a linear combination of radial extremals of Poincaré's inequality (20) (raised to the power $-\frac{n-2}{2}$), provided this combination is bounded below by a positive constant.

• As we shall prove in Lemma 1.9, condition (16) is equivalent to $\alpha^2 \le \frac{d-1}{n-1}$. In the limiting case $\alpha^2 = \frac{d-1}{n-1}$, extremals of Sobolev's inequality are radial, while extremals of Poincaré's inequality need not be, see Proposition 1.10 below.

• The extremals of Sobolev's inequality on the round sphere (i.e. the limiting case $\alpha = 1$) were discovered by T. Aubin, see e.g. Theorem 5.1 in [Heb00]. They are more often written as constant multiples of

$$v = (\beta - \cos(r))^{-\frac{d-2}{2}},$$

where $\beta > 1$ and $r$ is the geodesic distance to an arbitrary point $\omega_0 \in S^d$. With our notations, they take the form

$$v = (\lambda + \gamma\varphi_{1,d})^{-\frac{d-2}{2}},$$

where $\varphi_{1,d}$ is any eigenfunction of $-L_S = -\Delta_{S^d}$ associated to the first nonzero eigenvalue $\lambda_1 = d$ and $\lambda, \gamma$ are arbitrary constants such that $\lambda > |\gamma||\varphi_{1,d}|_\infty$.

Our notation puts forward the connection between the extremals of Sobolev's inequality and the extremals of Poincaré's inequality in Proposition 1.10: the former are obtained as a linear combination of the latter (raised to the power $-\frac{n-2}{2}$), provided this combination is bounded below by a positive constant.

• Hardy's inequality (i.e. the case $\alpha = 0$ in (9)) is optimal for the constant

$$C = \lim_{\alpha \to 0^+} \frac{4}{n(n-2)\alpha^2 Z_n^{\frac{2}{n}}} = \left( \frac{2}{d-2-2a} \right)^2,$$

but equality is never achieved.

• It follows from Theorem 1.4 and Theorem 1.7 that for $\alpha \in (0, 1)$, extremal functions for Sobolev's inequality on the Euclidean CKN space take the form

$$v(x) = \left( \frac{1 + |x|^{2\alpha}}{2} \right)^{-\frac{n-2}{2}},$$

up to normalization and dilation, providing thereby an alternative proof of the main result in [DEL14].

Condition (16) is strictly weaker than condition (14). It turns out to be equivalent to

$$\alpha^2 \le \frac{d-1}{n-1}.$$

More precisely, consider the following Felli-Schneider region

$$\Theta_{FS} = \{(a, b) \in \Theta, b \ge b_{FS}(a) \text{ if } b \le 0\}, \quad \text{where} \quad b_{FS}(a) = \frac{d(a_c - a)}{2\sqrt{(a_c - a)^2 + d - 1}} - (a_c - a) \quad (18)$$

Let as well $\Theta_{DGZ} \subset \Theta$ be the domain where Proposition 1.6 is valid (condition (14)):

$$\Theta_{DGZ} = \{(a, b) \in \Theta, \mathfrak{B}_{DGZ} \ge 0\},$$

where

$$\mathfrak{B}_{DGZ} = (d-2) - (n-2)\alpha^2. \quad (19)$$

Then, we prove that the region $\Theta_{FS}$ corresponds exactly to the domain where Theorem 1.7 is valid:
In addition, equality holds in (17) if and only if

$$v(x) = (\lambda + \gamma \tanh(\alpha s))^{-\frac{n-2}{2}}, \quad s = \log|x|,$$

where $\lambda, \gamma$ are arbitrary constants such that $\lambda > |\gamma|$. In particular optimal functions for both inequalities (1) and (17) are radial.

**Remark 1.8.** • Note that $\varphi_1(x) = \tanh(\alpha s)$ is a radial eigenfunction of $L_S$ associated to the eigenvalue $\lambda = \alpha^2 n$. So, except for the round sphere (corresponding to the case $\alpha = 1$), the extremals of Sobolev's inequality are obtained as a linear combination of radial extremals of Poincaré's inequality (20) (raised to the power $-\frac{n-2}{2}$), provided this combination is bounded below by a positive constant.

• As we shall prove in Lemma 1.9, condition (16) is equivalent to $\alpha^2 \le \frac{d-1}{n-1}$. In the limiting case $\alpha^2 = \frac{d-1}{n-1}$, extremals of Sobolev's inequality are radial, while extremals of Poincaré's inequality need not be, see Proposition 1.10 below.

• The extremals of Sobolev's inequality on the round sphere (i.e. the limiting case $\alpha = 1$) were discovered by T. Aubin, see e.g. Theorem 5.1 in [Heb00]. They are more often written as constant multiples of

$$v = (\beta - \cos(r))^{-\frac{d-2}{2}},$$

where $\beta > 1$ and $r$ is the geodesic distance to an arbitrary point $\omega_0 \in S^d$. With our notations, they take the form

$$v = (\lambda + \gamma\varphi_{1,d})^{-\frac{d-2}{2}},$$

where $\varphi_{1,d}$ is any eigenfunction of $-L_S = -\Delta_{S^d}$ associated to the first nonzero eigenvalue $\lambda_1 = d$ and $\lambda, \gamma$ are arbitrary constants such that $\lambda > |\gamma||\varphi_{1,d}|_\infty$.

Our notation puts forward the connection between the extremals of Sobolev's inequality and the extremals of Poincaré's inequality in Proposition 1.10: the former are obtained as a linear combination of the latter (raised to the power $-\frac{n-2}{2}$), provided this combination is bounded below by a positive constant.

• Hardy's inequality (i.e. the case $\alpha = 0$ in (9)) is optimal for the constant

$$C = \lim_{\alpha \to 0^+} \frac{4}{n(n-2)\alpha^2 Z_n^{\frac{2}{n}}} = \left( \frac{2}{d-2-2a} \right)^2,$$

but equality is never achieved.

• It follows from Theorem 1.4 and Theorem 1.7 that for $\alpha \in (0, 1)$, extremal functions for Sobolev's inequality on the Euclidean CKN space take the form

$$v(x) = \left( \frac{1 + |x|^{2\alpha}}{2} \right)^{-\frac{n-2}{2}},$$

up to normalization and dilation, providing thereby an alternative proof of the main result in [DEL14].

Condition (16) is strictly weaker than condition (14). It turns out to be equivalent to

$$\alpha^2 \le \frac{d-1}{n-1}.$$

More precisely, consider the following Felli-Schneider region

$$\Theta_{FS} = \{(a, b) \in \Theta, b \ge b_{FS}(a) \text{ if } b \le 0\}, \quad \text{where} \quad b_{FS}(a) = \frac{d(a_c - a)}{2\sqrt{(a_c - a)^2 + d - 1}} - (a_c - a) \quad (18)$$

Let as well $\Theta_{DGZ} \subset \Theta$ be the domain where Proposition 1.6 is valid (condition (14)):

$$\Theta_{DGZ} = \{(a, b) \in \Theta, \mathfrak{B}_{DGZ} \ge 0\},$$

where

$$\mathfrak{B}_{DGZ} = (d-2) - (n-2)\alpha^2. \quad (19)$$

Then, we prove that the region $\Theta_{FS}$ corresponds exactly to the domain where Theorem 1.7 is valid:
**Lemma 1.9** (Comparison of the two regions).

$$ \Theta_{FS} = \left\{ (a,b) \in \Theta, \mathfrak{B}_{DGZ} + \frac{n-d}{n-1} \ge 0 \right\} = \left\{ (a,b) \in \Theta, \alpha^2 \le \frac{d-1}{n-1} \right\} = \{(a,b) \in \Theta, \alpha \in (0,1]\}. $$

Hence, $\Theta_{DGZ} \subsetneq \Theta_{FS}$. Moreover, for any $(a,b) \in \Theta$, $\alpha = 1$ if and only if $a = b = 0$ (that is the standard Euclidean case).

The two regions are represented in Figure 1 with $d = 4$. The condition $\theta \in \Theta_{FS}$ is known in the literature as

<figure><img src="image_3.png" /><figcaption>Figure 1: $b_{FS}(a)$ (dashed style in red) and the curve $\mathfrak{B}_{DGZ} = 0$ (dotted style in blue) with $d = 4$.</figcaption></figure>

the Felli-Schneider condition. Felli and Schneider [FS03], building on the work of Catrina and Wang [CW01] initially proved that extremal functions for the optimal CKN inequality (1) cannot be radial whenever (16) fails. Conversely, in their work [DEL14], Dolbeault, Esteban and Loss computed the optimal constant in (1) and proved that extremals for the optimal CKN inequality (1) are radial and explicit whenever (16) holds. Combining Theorem 1.7 with Theorem 1.4 gives an immediate alternative proof of these latter facts. Our point of view may further clarify why the Felli-Schneider condition is optimal. Indeed, it is well-known that a tight Sobolev inequality implies a Poincaré inequality: precisely, applying (17) with $v = 1 + \epsilon f$ and letting $\epsilon \to 0$ leads to

$$ \int f^2 d\mu - \left( \int f d\mu \right)^2 \le \frac{1}{n\alpha^2} \int \Gamma_S(f) d\mu. \qquad (20) $$

We prove the following.

**Proposition 1.10** (Poincaré inequality for the spherical CKN space). Let $(a,b) \in \Theta$. The Poincaré inequality (20) holds with optimal constant $C = \frac{1}{n\alpha^2}$ if and only if the Felli-Schneider condition (16) holds. In addition, equality holds in (20) if and only if $f$ is an eigenfunction associated to the first nonzero eigenvalue of $L_S$,

* if $0 < \alpha < \frac{d-1}{n-1}$, 
$f(x) = \lambda + \gamma \tanh(\alpha s)$, where $s = \log|x|$, 
for some constants $(\lambda, \gamma) \in \mathbb{R}^2$.

* otherwise, if $\alpha = \frac{d-1}{n-1}$, 
$f(x) = \lambda + \gamma \tanh(\alpha s) + \nu \frac{\varphi_{1,d-1}(\omega)}{\cosh(\alpha s)}$, where $s = \log|x|$ and $\omega = \frac{x}{|x|}$, 
with $(\lambda, \gamma, \nu) \in \mathbb{R}^3$ and some eigenfunction $\varphi_{1,d-1}$ associated to the first nonzero eigenvalue $\lambda_1 = d-1$ of $-\Delta_{\mathbb{S}^{d-1}}$.
**Lemma 1.9** (Comparison of the two regions).

$$ \Theta_{FS} = \left\{ (a, b) \in \Theta, \mathfrak{B}_{DGZ} + \frac{n-d}{n-1} \ge 0 \right\} = \left\{ (a, b) \in \Theta, \alpha^2 \le \frac{d-1}{n-1} \right\} = \{(a, b) \in \Theta, \alpha \in (0, 1]\}. $$

Hence, $\Theta_{DGZ} \subsetneq \Theta_{FS}$. Moreover, for any $(a, b) \in \Theta$, $\alpha = 1$ if and only if $a = b = 0$ (that is the standard Euclidean case).

The two regions are represented in Figure 1 with $d = 4$. The condition $\theta \in \Theta_{FS}$ is known in the literature as

<figure><img src="image_3.png" /><figcaption>Figure 1: $b_{FS}(a)$ (dashed style in red) and the curve $\mathfrak{B}_{DGZ} = 0$ (dotted style in blue) with $d = 4$.</figcaption></figure>

the Felli-Schneider condition. Felli and Schneider [FS03], building on the work of Catrina and Wang [CW01] initially proved that extremal functions for the optimal CKN inequality (1) cannot be radial whenever (16) fails. Conversely, in their work [DEL14], Dolbeault, Esteban and Loss computed the optimal constant in (1) and proved that extremals for the optimal CKN inequality (1) are radial and explicit whenever (16) holds. Combining Theorem 1.7 with Theorem 1.4 gives an immediate alternative proof of these latter facts. Our point of view may further clarify why the Felli-Schneider condition is optimal. Indeed, it is well-known that a tight Sobolev inequality implies a Poincaré inequality: precisely, applying (17) with $v = 1 + \epsilon f$ and letting $\epsilon \to 0$ leads to

$$ \int f^2 d\mu - \left( \int f d\mu \right)^2 \le \frac{1}{n\alpha^2} \int \Gamma_S(f) d\mu. \quad (20) $$

We prove the following.

**Proposition 1.10** (Poincaré inequality for the spherical CKN space). Let $(a, b) \in \Theta$. The Poincaré inequality (20) holds with optimal constant $C = \frac{1}{n\alpha^2}$ if and only if the Felli-Schneider condition (16) holds. In addition, equality holds in (20) if and only if $f$ is an eigenfunction associated to the first nonzero eigenvalue of $L_S$,

* if $0 < \alpha < \frac{d-1}{n-1}$,
  $$ f(x) = \lambda + \gamma \tanh(\alpha s), \quad \text{where } s = \log|x|, $$
  for some constants $(\lambda, \gamma) \in \mathbb{R}^2$.

* otherwise, if $\alpha = \frac{d-1}{n-1}$,
  $$ f(x) = \lambda + \gamma \tanh(\alpha s) + \nu \frac{\varphi_{1,d-1}(\omega)}{\cosh(\alpha s)}, \quad \text{where } s = \log|x| \text{ and } \omega = \frac{x}{|x|}, $$

with $(\lambda, \gamma, \nu) \in \mathbb{R}^3$ and some eigenfunction $\varphi_{1,d-1}$ associated to the first nonzero eigenvalue $\lambda_1 = d-1$ of $-\Delta_S^{d-1}$.
So, the Felli-Schneider condition cannot be improved in the statement of Theorem 1.7 and it is in fact equivalent to both Sobolev's and Poincaré's inequality with the given optimal constants on the spherical CKN space. In addition, Poincaré's inequality with constant $C = \frac{n-1}{\rho n}$ is in fact equivalent to the following integrated $CD(\rho, n)$ condition, where $\rho, n > 0$:

$$ \int \left( \Gamma_{\mathbf{s}}^{\mathbf{s}}(f) - \rho \Gamma_{\mathbf{s}}(f) - \frac{1}{n} (L_{\mathbf{s}} f)^2 \right) d\mu \geq 0, $$

see Proposition 4.8.3, Theorem 4.8.4 and their proofs in [BGL14]. Hence, the Felli-Schneider condition (16) can be interpreted as a curvature-dimension condition in integral form.

Let us also point out that if Sobolev's inequality (10) holds on a $d$-dimensional smooth manifold $(M, \mathfrak{g})$ without weight with optimal constant $C = \frac{4}{d(d-2)|\mathbf{s}^d|^{2/d}}$, then $(M, \mathfrak{g})$ must be isometric to the round sphere, as very recently demonstrated in [NV21].

## 1.4 The $n$-conformal invariants

In this last introductory paragraph, we expand on the conformal invariance of Sobolev's inequality in the setting of weighted manifolds and provide a deeper reason for why the three CKN model spaces satisfy equivalent conformal forms of the Sobolev inequality.

For the inequality (8) without weights, it turns out that $S_{\mathfrak{g}} = \frac{d-2}{4(d-1)}sc_{\mathfrak{g}}$ is a constant multiple of the scalar curvature of $\mathfrak{g}$ (see Propositions 3.6.20, 3.6.21 and 6.2.2, as well as the second displayed formula on p. 63 in [Heb97] or [BGL14, Sec. 6.9.2] for proofs of this classical result). In other words, inequality (8) is valid on the whole conformal class of the round sphere, including the Euclidean (where $S_{\mathfrak{g}} = 0$) and hyperbolic (where $S_{\mathfrak{g}} = -\frac{d-2}{4(d-1)}sc_{\mathfrak{g}}$) spaces.

For weighted manifolds, the notion of scalar curvature can be generalized as follows. As proposed in [BGL14, Sec. 6.9] (see also [CGY06] and [Cas12] for earlier perspectives$^{10}$), given a $d$-dimensional ($d \geq 2$) weighted Riemannian manifold $(M, \mathfrak{g}, \mu)$ with reference measure

$$ d\mu = e^{-W} dV_{\mathfrak{g}}, $$

where $W: M \to \mathbf{R}$ is a given weight and $dV_{\mathfrak{g}}$ the Riemannian volume, let

$$ \Gamma(f) = |\nabla_{\mathfrak{g}} f|^2_{\mathfrak{g}} $$

denote the associated carré du champ operator, so that $(M, \mu, \Gamma)$ is a Markov triple.

**Definition 1.11.** Take a real number $n \in [d, +\infty]$, which is not necessarily an integer. The $n$-conformal class of the triple $(M, \mu, \Gamma)$ is the set of all Markov triples $(M, c^{-n}\mu, c^2\Gamma)$, where $c: M \to (0, \infty)$ is any smooth and positive function. An $n$-conformal invariant is a map $S$ defined on the $n$-conformal class of $(M, \mu, \Gamma)$ with values in the set of functions over $M$, such that for any positive smooth function $c = e^{\tau}$,

$$ S(c^{-n}\mu, c^2\Gamma) = c^2 \left[ S(\mu, \Gamma) + \frac{n-2}{2} \left( L\tau - \frac{n-2}{2}\Gamma(\tau) \right) \right], \quad (21) $$

where

$$ L = \Delta_{\mathfrak{g}} - \Gamma(W, \cdot). $$

It is important to notice that the operator $L$ is uniquely determined by the carré du champ operator $\Gamma$ and the measure $\mu$ only. This is indeed the case since the operator $\Delta_{\mathfrak{g}}$ depends on the metric $\mathfrak{g}$ (which itself is uniquely determined by $\Gamma$) and since $W$ is related to the measure $\mu$ and the metric $\mathfrak{g}$ through the Riemannian measure $dV_{\mathfrak{g}}$. Also observe that setting $u = c^{-\frac{n-2}{2}}$, $s = S(\mu, \Gamma)$, $\tilde{s} = S(c^{-n}\mu, c^2\Gamma)$, then (21) can be reformulated as the following Yamabe-type equation:

$$ -Lu + su = \tilde{s}u^{\frac{n+2}{n-2}} \quad \text{in } M. $$

Note that the case where $n = d$, $L = \Delta_{\mathfrak{g}}$, $s = \frac{d-2}{4(d-1)}sc_{\mathfrak{g}}$ and $\tilde{s}$ constant, is the standard Yamabe equation.

<footnote>$^{10}$which correspond to the special case $\gamma = -2$ in Proposition 1.12 below.</footnote>
So, the Felli-Schneider condition cannot be improved in the statement of Theorem 1.7 and it is in fact equivalent to both Sobolev's and Poincaré's inequality with the given optimal constants on the spherical CKN space. In addition, Poincaré's inequality with constant $C = \frac{n-1}{\rho n}$ is in fact equivalent to the following integrated CD($\rho$, $n$) condition, where $\rho, n > 0$:

$$ \int \left( \Gamma_{\mathbf{s}}^{\mathbf{s}}(f) - \rho \Gamma_{\mathbf{s}}(f) - \frac{1}{n} (L_{\mathbf{s}} f)^2 \right) d\mu \geq 0, $$

see Proposition 4.8.3, Theorem 4.8.4 and their proofs in [BGL14]. Hence, the Felli-Schneider condition (16) can be interpreted as a curvature-dimension condition in integral form.

Let us also point out that if Sobolev's inequality (10) holds on a $d$-dimensional smooth manifold $(M, \mathfrak{g})$ without weight with optimal constant $C = \frac{4}{d(d-2)|\mathbf{s}^d|^{2/d}}$, then $(M, \mathfrak{g})$ must be isometric to the round sphere, as very recently demonstrated in [NV21].

## 1.4 The $n$-conformal invariants

In this last introductory paragraph, we expand on the conformal invariance of Sobolev's inequality in the setting of weighted manifolds and provide a deeper reason for why the three CKN model spaces satisfy equivalent conformal forms of the Sobolev inequality.

For the inequality (8) without weights, it turns out that $S_{\mathfrak{g}} = \frac{d-2}{4(d-1)}sc_{\mathfrak{g}}$ is a constant multiple of the scalar curvature of $\mathfrak{g}$ (see Propositions 3.6.20, 3.6.21 and 6.2.2, as well as the second displayed formula on p. 63 in [Heb97] or [BGL14, Sec. 6.9.2] for proofs of this classical result). In other words, inequality (8) is valid on the whole conformal class of the round sphere, including the Euclidean (where $S_{\mathfrak{g}} = 0$) and hyperbolic (where $S_{\mathfrak{g}} = -\frac{d-2}{4(d-1)}sc_{\mathfrak{g}}$) spaces.

For weighted manifolds, the notion of scalar curvature can be generalized as follows. As proposed in [BGL14, Sec. 6.9] (see also [CGY06] and [Cas12] for earlier perspectives$^{10}$), given a $d$-dimensional ($d \geq 2$) weighted Riemannian manifold $(M, \mathfrak{g}, \mu)$ with reference measure

$$ d\mu = e^{-W} dV_{\mathfrak{g}}, $$

where $W: M \to \mathbf{R}$ is a given weight and $dV_{\mathfrak{g}}$ the Riemannian volume, let

$$ \Gamma(f) = |\nabla_{\mathfrak{g}} f|^2_{\mathfrak{g}} $$

denote the associated carré du champ operator, so that $(M, \mu, \Gamma)$ is a Markov triple.

**Definition 1.11.** Take a real number $n \in [d, +\infty]$, which is not necessarily an integer. The $n$-conformal class of the triple $(M, \mu, \Gamma)$ is the set of all Markov triples $(M, c^{-n}\mu, c^2\Gamma)$, where $c: M \to (0, \infty)$ is any smooth and positive function. An $n$-conformal invariant is a map $S$ defined on the $n$-conformal class of $(M, \mu, \Gamma)$ with values in the set of functions over $M$, such that for any positive smooth function $c = e^{\tau}$,

$$ S(c^{-n}\mu, c^2\Gamma) = c^2 \left[ S(\mu, \Gamma) + \frac{n-2}{2} \left( L\tau - \frac{n-2}{2}\Gamma(\tau) \right) \right], \quad (21) $$

where

$$ L = \Delta_{\mathfrak{g}} - \Gamma(W, \cdot). $$

It is important to notice that the operator $L$ is uniquely determined by the carré du champ operator $\Gamma$ and the measure $\mu$ only. This is indeed the case since the operator $\Delta_{\mathfrak{g}}$ depends on the metric $\mathfrak{g}$ (which itself is uniquely determined by $\Gamma$) and since $W$ is related to the measure $\mu$ and the metric $\mathfrak{g}$ through the Riemannian measure $dV_{\mathfrak{g}}$. Also observe that setting $u = c^{-\frac{n-2}{2}}$, $s = S(\mu, \Gamma)$, $\tilde{s} = S(c^{-n}\mu, c^2\Gamma)$, then (21) can be reformulated as the following Yamabe-type equation:

$$ -Lu + su = \tilde{s}u^{\frac{n+2}{n-2}} \quad \text{in } M. $$

Note that the case where $n = d$, $L = \Delta_{\mathfrak{g}}$, $s = \frac{d-2}{4(d-1)}sc_{\mathfrak{g}}$ and $\tilde{s}$ constant, is the standard Yamabe equation.

<footnote>$^{10}$which correspond to the special case $\gamma = -2$ in Proposition 1.12 below.</footnote>
By a rather direct computation, see [BGL14, Prop. 6.9.2], whenever $S = S(\mu, \Gamma)$ is an $n$-conformal invariant, the Sobolev inequality

$$ \left( \int |v|^p d\mu \right)^{2/p} \le C \left( \int \Gamma(v) d\mu + \int S v^2 d\mu \right), \qquad (22) $$

(with given constant $C > 0$ and $p = \frac{2n}{n-2}$) is invariant in the $n$-conformal class of the triple $(M, \mu, \Gamma)$. In other words, if the Sobolev inequality (22) holds for some constant $C$, then it also holds with the same constant $C$ for all triples $(M, c^{-n}\mu, c^2\Gamma)$ where $c$ is any smooth and positive function.

Rephrasing what we said earlier, in the absence of weight, $S_g = \frac{d-2}{4(d-1)}sc_g$ is an example of a $d$-conformal invariant (where in this case $n = d$). The case of weighted Riemannian manifolds is a little bit more complicated and contains interesting examples. Let us recall Proposition 6.9.6 of [BGL14] (we will also provide a proof since the one in [BGL14] contains some mistakes as well as the statement).

**Proposition 1.12** ($n$-conformal invariant in a weighted manifold). Let $\gamma \in \mathbf{R}$ and $n > d$. Then,

$$ S_{\gamma}(\mu, \Gamma) = \theta_{n}(\gamma)[sc_{g} - \gamma\Delta_{g}W + \beta_{n}(\gamma)\Gamma(W)] \qquad (23) $$

is an $n$-conformal invariant if

$$ \beta_n(\gamma) = \frac{\gamma(n - 2d + 2) - 2(d - 1)}{2(n - d)} \quad \text{and} \quad \theta_n(\gamma) = \frac{n - 2}{4(d - 1) - 2\gamma(n - d)}. $$

This being recalled, a natural question arises in the context of the Euclidean CKN space we introduced in Definition 1.1: does there exist a (unique) real number $\gamma_0 \in \mathbf{R}$ such that this space satisfies

$$ S_{\gamma_0}(\mu_{\mathbf{E}}, \Gamma_{\mathbf{E}}) = 0 \ ? $$

This is indeed the case, as we are about to see. By Theorem 1.4, without any further computation, we deduce that for the same value of the parameter $\gamma = \gamma_0$, $S_{\gamma_0}(\mu_{\mathbf{S}}, \Gamma_{\mathbf{S}}) = \frac{n(n-2)}{4}\alpha^2 > 0$ is constant for the CKN sphere and $S_{\gamma_0}(\mu_{\mathbf{H}}, \Gamma_{\mathbf{H}}) = -\frac{n(n-2)}{4}\alpha^2$ for the CKN hyperbolic space.

**Proposition 1.13.** Let $n > d$ and

$$ \gamma_0 = - \frac{2(d-1)}{\alpha^2(n-d)(n-2)} \mathfrak{B}_{DGZ}, \qquad (24) $$

where $\mathfrak{B}_{DGZ}$ has been defined in (19).

$$ \text{Then, } S_{\gamma_0}(\mu_{\mathbf{E}}, \Gamma_{\mathbf{E}}) = 0, S_{\gamma_0}(\mu_{\mathbf{S}}, \Gamma_{\mathbf{S}}) = \frac{n(n-2)}{4}\alpha^2 \text{ and } S_{\gamma_0}(\mu_{\mathbf{H}}, \Gamma_{\mathbf{H}}) = -\frac{n(n-2)}{4}\alpha^2. $$

As an immediate collorary of the CKN inequality (1) and the above lemma, we recover the validity of Sobolev's inequality on our three model spaces, stated in Theorem 1.4 above.

**Remark 1.14.** In a forthcoming report, we will further explain how a weighted version of Otto's calculus can be introduced in order to prove a wider class of optimal CKN inequalities, by working directly on the Euclidean CKN space, rather than the CKN sphere.

The rest of the paper is organized as follow. In Section 2 below, we prove the conformal invariance of Sobolev's inequality in the CKN spaces (Theorem 1.4). Section 3 is dedicated to the characterization of the region of parameter $\Theta_{DGZ}$ (resp. $\Theta_{FS}$) for which the classical curvature-dimension condition (resp. the integrated form (15)) holds, from which Sobolev's inequality follows (Proposition 1.6 and Theorem 1.7). In Section 4, we prove all results pertaining to $n$-conformal invariance for general weighted manifolds (Propositions 1.12 and 1.13). At last, an appendix contains lists of known formulas and constants, proofs of the numerology relating them as well as rigorous justification of the integrations by parts implicitly used in the proof of Sobolev's inequality.

# 2 Conformal invariance of Sobolev type inequalities for CKN models

## Proof of Theorem 1.4

### △ As in the unweighted case, the proof reduces to a simple change of unknown, once the proper notion of
By a rather direct computation, see [BGL14, Prop. 6.9.2], whenever $S = S(\mu, \Gamma)$ is an $n$-conformal invariant, the Sobolev inequality

$$ \left( \int |v|^p d\mu \right)^{2/p} \le C \left( \int \Gamma(v) d\mu + \int S v^2 d\mu \right), \qquad (22) $$

(with given constant $C > 0$ and $p = \frac{2n}{n-2}$) is invariant in the $n$-conformal class of the triple $(M, \mu, \Gamma)$. In other words, if the Sobolev inequality (22) holds for some constant $C$, then it also holds with the same constant $C$ for all triples $(M, c^{-n}\mu, c^2\Gamma)$ where $c$ is any smooth and positive function.

Rephrasing what we said earlier, in the absence of weight, $S_g = \frac{d-2}{4(d-1)}sc_g$ is an example of a $d$-conformal invariant (where in this case $n = d$). The case of weighted Riemannian manifolds is a little bit more complicated and contains interesting examples. Let us recall Proposition 6.9.6 of [BGL14] (we will also provide a proof since the one in [BGL14] contains some mistakes as well as the statement).

**Proposition 1.12** ($n$-conformal invariant in a weighted manifold). Let $\gamma \in \mathbf{R}$ and $n > d$. Then,

$$ S_{\gamma}(\mu, \Gamma) = \theta_{n}(\gamma)[sc_{g} - \gamma\Delta_{g}W + \beta_{n}(\gamma)\Gamma(W)] \qquad (23) $$

is an $n$-conformal invariant if

$$ \beta_n(\gamma) = \frac{\gamma(n - 2d + 2) - 2(d - 1)}{2(n - d)} \quad \text{and} \quad \theta_n(\gamma) = \frac{n - 2}{4(d - 1) - 2\gamma(n - d)} $$

This being recalled, a natural question arises in the context of the Euclidean CKN space we introduced in Definition 1.1: does there exist a (unique) real number $\gamma_0 \in \mathbf{R}$ such that this space satisfies

$$ S_{\gamma_0}(\mu_{\mathbf{E}}, \Gamma_{\mathbf{E}}) = 0 \ ? $$

This is indeed the case, as we are about to see. By Theorem 1.4, without any further computation, we deduce that for the same value of the parameter $\gamma = \gamma_0$, $S_{\gamma_0}(\mu_S, \Gamma_S) = \frac{n(n-2)}{4}\alpha^2 > 0$ is constant for the CKN sphere and $S_{\gamma_0}(\mu_H, \Gamma_H) = -\frac{n(n-2)}{4}\alpha^2$ for the CKN hyperbolic space.

**Proposition 1.13.** Let $n > d$ and

$$ \gamma_0 = -\frac{2(d-1)}{\alpha^2(n-d)(n-2)} \mathfrak{B}_{DGZ}, \qquad (24) $$

where $\mathfrak{B}_{DGZ}$ has been defined in (19).

$$ \text{Then, } S_{\gamma_0}(\mu_{\mathbf{E}}, \Gamma_{\mathbf{E}}) = 0, S_{\gamma_0}(\mu_{\mathbf{S}}, \Gamma_{\mathbf{S}}) = \frac{n(n-2)}{4}\alpha^2 \text{ and } S_{\gamma_0}(\mu_{\mathbf{H}}, \Gamma_{\mathbf{H}}) = -\frac{n(n-2)}{4}\alpha^2. $$

As an immediate collorary of the CKN inequality (1) and the above lemma, we recover the validity of Sobolev's inequality on our three model spaces, stated in Theorem 1.4 above.

**Remark 1.14.** In a forthcoming report, we will further explain how a weighted version of Otto's calculus can be introduced in order to prove a wider class of optimal CKN inequalities, by working directly on the Euclidean CKN space, rather than the CKN sphere.

The rest of the paper is organized as follow. In Section 2 below, we prove the conformal invariance of Sobolev's inequality in the CKN spaces (Theorem 1.4). Section 3 is dedicated to the characterization of the region of parameter $\Theta_{DGZ}$ (resp. $\Theta_{FS}$) for which the classical curvature-dimension condition (resp. the integrated form (15)) holds, from which Sobolev's inequality follows (Proposition 1.6 and Theorem 1.7). In Section 4, we prove all results pertaining to $n$-conformal invariance for general weighted manifolds (Propositions 1.12 and 1.13). At last, an appendix contains lists of known formulas and constants, proofs of the numerology relating them as well as rigorous justification of the integrations by parts implicitly used in the proof of Sobolev's inequality.

# 2 Conformal invariance of Sobolev type inequalities for CKN models

## Proof of Theorem 1.4

### ▷ As in the unweighted case, the proof reduces to a simple change of unknown, once the proper notion of
conformal invariance has been introduced. First we prove that the Sobolev inequality in the CKN Euclidean space is equivalent to the Sobolev inequality in the spherical CKN space.

Recall that for

$$ \varphi(x) = \frac{1 + |x|^{2\alpha}}{2}, \quad x \in \mathbf{R}^d, \qquad (25) $$

and that

$$ \Gamma_{\mathbf{S}} = \varphi^2 \Gamma_{\mathbf{E}} \quad \text{and} \quad \mu_{\mathbf{S}} = \varphi^{-n} \mu_{\mathbf{E}}. $$

Apply (1) to the function $f = \varphi^{\frac{2-n}{2}} g$. On the one hand, we have

$$ \int f^p d\mu_{\mathbf{E}} = \int f^{\frac{2n}{n-2}} d\mu_{\mathbf{E}} = \int g^p d\mu_{\mathbf{S}}. $$

On the other hand, letting $V = \log \varphi$,

$$ \begin{aligned} \Gamma_{\mathbf{E}}(f) &= \Gamma_{\mathbf{E}}(\varphi^{\frac{2-n}{2}} g) = \varphi^{2-n} \Gamma_{\mathbf{E}}(g) + 2\varphi^{\frac{2-n}{2}} g \Gamma_{\mathbf{E}}(\varphi^{\frac{2-n}{2}}, g) + \Gamma_{\mathbf{E}}(\varphi^{\frac{2-n}{2}}) g^2 \\ &= \varphi^{2-n} \left( \Gamma_{\mathbf{E}}(g) - \frac{n-2}{2} \Gamma_{\mathbf{E}}(g^2, V) + \frac{(n-2)^2}{4} \Gamma_{\mathbf{E}}(V) g^2 \right). \end{aligned} $$

An integration by parts with respect to $\mu_{\mathbf{E}}$ yields

$$ \int \Gamma_{\mathbf{E}}(g^2, V) \varphi^{2-n} d\mu_{\mathbf{E}} + \int \Gamma_{\mathbf{E}}(\varphi^{2-n}, V) g^2 d\mu_{\mathbf{E}} = - \int L_{\mathbf{E}} V g^2 \varphi^{2-n} d\mu_{\mathbf{E}}, $$

so that we get

$$ \begin{aligned} \int \Gamma_{\mathbf{E}}(f) d\mu_{\mathbf{E}} &= \int \Gamma_{\mathbf{S}}(g) d\mu_{\mathbf{S}} - \frac{n-2}{2} \int \Gamma_{\mathbf{E}}(g^2, V) \varphi^{2-n} d\mu_{\mathbf{E}} + \frac{(n-2)^2}{4} \int \Gamma_{\mathbf{E}}(V) g^2 \varphi^2 d\mu_{\mathbf{S}} \\ &= \int \Gamma_{\mathbf{S}}(g) d\mu_{\mathbf{S}} + \frac{n-2}{2} \int \left( L_{\mathbf{E}}(V) - \frac{n-2}{2} \Gamma_{\mathbf{E}}(V) \right) g^2 \varphi^2 d\mu_{\mathbf{S}}, \end{aligned} $$

The CKN inequality (1) becomes

$$ \left( \int g^p d\mu_{\mathbf{S}} \right)^{2/p} \le C \left( \int \Gamma_{\mathbf{S}}(g) d\mu_{\mathbf{S}} + \frac{(n-2)}{2} \int \left( L_{\mathbf{E}}(V) - \frac{n-2}{2} \Gamma_{\mathbf{E}}(V) \right) \varphi^2 g^2 d\mu_{\mathbf{S}} \right), $$

and it is enough to compute the quantity

$$ L_{\mathbf{E}}(V) - \frac{n-2}{2} \Gamma_{\mathbf{E}}(V). $$

To that end, recall that $L_{\mathbf{E}}$ is given by

$$ L_{\mathbf{E}} = |x|^{2(1-\alpha)} [\Delta - a \nabla \log |x|^2 \cdot \nabla]. \qquad (26) $$

Since $V = \log \varphi$, we have

$$ L_{\mathbf{E}}(V) - \frac{n-2}{2} \Gamma_{\mathbf{E}}(V) = \frac{L_{\mathbf{E}}(\varphi)}{\varphi} - \frac{\Gamma_{\mathbf{E}}(\varphi)}{\varphi^2} - \frac{n-2}{2} \frac{\Gamma_{\mathbf{E}}(\varphi)}{\varphi^2} = \frac{L_{\mathbf{E}}(\varphi)}{\varphi} - \frac{n}{2} \frac{\Gamma_{\mathbf{E}}(\varphi)}{\varphi^2}. $$

Recalling the definition of $\varphi$ given in (25), we find that

$$ \frac{\Gamma_{\mathbf{E}}(\varphi)}{\varphi^2} = \frac{\alpha^2 |x|^{2\alpha}}{\varphi^2}, $$

and

$$ \frac{L_{\mathbf{E}}(\varphi)}{\varphi} = \frac{|x|^{2(1-\alpha)} [\Delta\varphi - a\nabla \log|x|^2 \cdot \nabla\varphi]}{\varphi} = \alpha \frac{d + 2(\alpha - 1) - 2a}{\varphi} = \frac{\alpha^2 n}{\varphi}. $$
conformal invariance has been introduced. First we prove that the Sobolev inequality in the CKN Euclidean space is equivalent to the Sobolev inequality in the spherical CKN space.

Recall that for

$$ \varphi(x) = \frac{1 + |x|^{2\alpha}}{2}, \quad x \in \mathbf{R}^d, \qquad (25) $$

and that

$$ \Gamma_S = \varphi^2 \Gamma_E \quad \text{and} \quad \mu_S = \varphi^{-n} \mu_E. $$

Apply (1) to the function $f = \varphi^{\frac{2-n}{2}} g$. On the one hand, we have

$$ \int f^p d\mu_E = \int f^{\frac{2n}{n-2}} d\mu_E = \int g^p d\mu_S. $$

On the other hand, letting $V = \log \varphi$,

$$ \begin{aligned} \Gamma_E(f) &= \Gamma_E(\varphi^{\frac{2-n}{2}} g) = \varphi^{2-n} \Gamma_E(g) + 2\varphi^{\frac{2-n}{2}} g \Gamma_E(\varphi^{\frac{2-n}{2}}, g) + \Gamma_E(\varphi^{\frac{2-n}{2}})g^2 \\ &= \varphi^{2-n} \left( \Gamma_E(g) - \frac{n-2}{2} \Gamma_E(g^2, V) + \frac{(n-2)^2}{4} \Gamma_E(V)g^2 \right). \end{aligned} $$

An integration by parts with respect to $\mu_E$ yields

$$ \int \Gamma_E(g^2, V) \varphi^{2-n} d\mu_E + \int \Gamma_E(\varphi^{2-n}, V) g^2 d\mu_E = - \int L_E V g^2 \varphi^{2-n} d\mu_E, $$

so that we get

$$ \begin{aligned} \int \Gamma_E(f) d\mu_E &= \int \Gamma_S(g) d\mu_S - \frac{n-2}{2} \int \Gamma_E(g^2, V) \varphi^{2-n} d\mu_E + \frac{(n-2)^2}{4} \int \Gamma_E(V) g^2 \varphi^2 d\mu_S \\ &= \int \Gamma_S(g) d\mu_S + \frac{n-2}{2} \int \left( L_E(V) - \frac{n-2}{2} \Gamma_E(V) \right) g^2 \varphi^2 d\mu_S, \end{aligned} $$

The CKN inequality (1) becomes

$$ \left( \int g^p d\mu_S \right)^{2/p} \le C \left( \int \Gamma_S(g) d\mu_S + \frac{(n-2)}{2} \int \left( L_E(V) - \frac{n-2}{2} \Gamma_E(V) \right) \varphi^2 g^2 d\mu_S \right), $$

and it is enough to compute the quantity

$$ L_E(V) - \frac{n-2}{2} \Gamma_E(V). $$

To that end, recall that $L_E$ is given by

$$ L_E = |x|^{2(1-\alpha)} [\Delta - a \nabla \log |x|^2 \cdot \nabla]. \qquad (26) $$

Since $V = \log \varphi$, we have

$$ L_E(V) - \frac{n-2}{2} \Gamma_E(V) = \frac{L_E(\varphi)}{\varphi} - \frac{\Gamma_E(\varphi)}{\varphi^2} - \frac{n-2}{2} \frac{\Gamma_E(\varphi)}{\varphi^2} = \frac{L_E(\varphi)}{\varphi} - \frac{n}{2} \frac{\Gamma_E(\varphi)}{\varphi^2}. $$

Recalling the definition of $\varphi$ given in (25), we find that

$$ \frac{\Gamma_E(\varphi)}{\varphi^2} = \frac{\alpha^2 |x|^{2\alpha}}{\varphi^2}, $$

and

$$ \frac{L_E(\varphi)}{\varphi} = \frac{|x|^{2(1-\alpha)} [\Delta\varphi - a\nabla \log|x|^2 \cdot \nabla\varphi]}{\varphi} = \alpha \frac{d + 2(\alpha - 1) - 2a}{\varphi} = \frac{\alpha^2 n}{\varphi}. $$
So finally,

$$L_{\mathbf{E}}(V) - \frac{n-2}{2} \Gamma_{\mathbf{E}}(V) = \frac{\alpha^2 n}{2 \varphi^2},$$

and the CKN inequality (1) takes the form (10), as announced.

Next, we prove that Sobolev's inequality in the CKN Euclidean space implies the Sobolev inequality on the CKN hyperbolic space. We mimic the previous proof. Define the function $\psi$ on the punctured open unit ball $\mathbf{B} \setminus \{0\} \subset \mathbf{R}^d$ by

$$\psi(x) = \frac{1 - |x|^{2\alpha}}{2}, \quad x \in \mathbf{B} \setminus \{0\}.$$

Then, on $\mathbf{B} \setminus \{0\}$,

$$\Gamma_{\mathbf{H}} = \psi^2 \Gamma_{\mathbf{E}} \quad \text{and} \quad \mu_{\mathbf{H}} = \psi^{-n} \mu_{\mathbf{E}}.$$

Apply the CKN inequality (1) to the function $f = \psi^{\frac{2-n}{2}} h$, where $h \in \mathcal{C}_c^{\infty}(\mathbf{B} \setminus \{0\})$. Again, we get

$$\int f^p d\mu_{\mathbf{E}} = \int f^{\frac{2n}{n-2}} d\mu_{\mathbf{E}} = \int h^p d\mu_{\mathbf{H}}.$$

and (1) becomes, with $U = \log \psi$,

$$\left(\int h^p d\mu_{\mathbf{H}}\right)^{2/p} \le C \left(\int \Gamma_{\mathbf{H}}(h) d\mu_{\mathbf{H}} + \frac{n-2}{2} \int \left(L_{\mathbf{E}}(U) - \frac{n-2}{2} \Gamma_{\mathbf{E}}(U)\right) \psi^2 h^2 d\mu_{\mathbf{H}}\right).$$

We obtain

$$L_{\mathbf{E}}(U) - \frac{n-2}{2} \Gamma_{\mathbf{E}}(U) = - \frac{\alpha^2 n}{2 \psi^2},$$

and so (11), as claimed. ▷

### 3 Sobolev's inequality for the spherical CKN model

This section is devoted to the proof of the optimal Sobolev inequality for the spherical CKN space (Theorem 1.7) under the Felli-Schneider condition (16). It is convenient to introduce spherical coordinates $\mathbf{R}^d \setminus \{0\} \ni x = r\theta$ with $r > 0$ and $\theta \in \mathbf{S}^{d-1}$. The Sobolev inequality on the CKN sphere (17) then takes the form

$$\left( \int_{(0,\infty) \times \mathbf{S}^{d-1}} |v|^p \frac{r^{d-1-pb}}{(1+r^{2\alpha})^n} dr dV_{\mathbf{S}^{d-1}} \right)^{2/p} \le C \int_{(0,\infty) \times \mathbf{S}^{d-1}} \left[ (\partial_r v)^2 + \frac{1}{r^2} |\nabla_{\theta} v|^2 \right] \frac{r^{2(\alpha-1)+d-1-pb}}{(1+r^{2\alpha})^{n-2}} dr dV_{\mathbf{S}^{d-1}} \\ + 4Z^{-\frac{2}{n}} \int_{(0,\infty) \times \mathbf{S}^{d-1}} v^2 \frac{r^{d-1-pb}}{(1+r^{2\alpha})^n} dr dV_{\mathbf{S}^{d-1}},$$

where $|\nabla_{\theta} v|$ is the Riemannian length of the Riemannian gradient $\nabla_{\theta} v$ on $\mathbf{S}^{d-1}$ and $dV_{\mathbf{S}^{d-1}}$ is the associated Riemannian volume. Using the change of variable $(0, \infty) \ni r = e^s$, with $s \in \mathbf{R}$, the inequality becomes (with a different constant $C$),

$$\left( \int_{\mathbf{R} \times \mathbf{S}^{d-1}} |v|^p \cosh(\alpha s)^{-n} ds dV_{\mathbf{S}^{d-1}} \right)^{2/p} \le C \int_{\mathbf{R} \times \mathbf{S}^{d-1}} \left[ (\partial_s v)^2 + |\nabla_{\theta} v|^2 \right] \cosh(\alpha s)^{2-n} ds dV_{\mathbf{S}^{d-1}} + Z^{-\frac{2}{n}} \int_{\mathbf{R} \times \mathbf{S}^{d-1}} v^2 \cosh(\alpha s)^{-n} ds dV_{\mathbf{S}^{d-1}}, \quad (27)$$

where we used the fact that $d - n\alpha - pb = 0$, see (75). This new chart is often called the Emden-Fowler transformation, as suggested in [CW01, DEL16]. In other words, in the cylindrical chart $(s, \theta) \in \mathbf{R} \times \mathbf{S}^{d-1}$, the
So finally,

$$L_{\mathbf{E}}(V) - \frac{n-2}{2} \Gamma_{\mathbf{E}}(V) = \frac{\alpha^2 n}{2 \varphi^2},$$

and the CKN inequality (1) takes the form (10), as announced.

Next, we prove that Sobolev's inequality in the CKN Euclidean space implies the Sobolev inequality on the CKN hyperbolic space. We mimic the previous proof. Define the function $\psi$ on the punctured open unit ball $\mathbf{B} \setminus \{0\} \subset \mathbf{R}^d$ by

$$\psi(x) = \frac{1 - |x|^{2\alpha}}{2}, \quad x \in \mathbf{B} \setminus \{0\}.$$

Then, on $\mathbf{B} \setminus \{0\}$,

$$\Gamma_{\mathbf{H}} = \psi^2 \Gamma_{\mathbf{E}} \quad \text{and} \quad \mu_{\mathbf{H}} = \psi^{-n} \mu_{\mathbf{E}}.$$

Apply the CKN inequality (1) to the function $f = \psi^{\frac{2-n}{2}}h$, where $h \in \mathcal{C}_c^{\infty}(\mathbf{B} \setminus \{0\})$. Again, we get

$$\int f^p d\mu_{\mathbf{E}} = \int f^{\frac{2n}{n-2}} d\mu_{\mathbf{E}} = \int h^p d\mu_{\mathbf{H}}.$$

and (1) becomes, with $U = \log \psi$,

$$\left(\int h^p d\mu_{\mathbf{H}}\right)^{2/p} \le C \left(\int \Gamma_{\mathbf{H}}(h) d\mu_{\mathbf{H}} + \frac{n-2}{2} \int \left(L_{\mathbf{E}}(U) - \frac{n-2}{2} \Gamma_{\mathbf{E}}(U)\right) \psi^2 h^2 d\mu_{\mathbf{H}}\right).$$

We obtain

$$L_{\mathbf{E}}(U) - \frac{n-2}{2} \Gamma_{\mathbf{E}}(U) = - \frac{\alpha^2 n}{2 \psi^2},$$

and so (11), as claimed. ▷

### 3 Sobolev's inequality for the spherical CKN model

This section is devoted to the proof of the optimal Sobolev inequality for the spherical CKN space (Theorem 1.7) under the Felli-Schneider condition (16). It is convenient to introduce spherical coordinates $\mathbf{R}^d \setminus \{0\} \ni x = r\theta$ with $r > 0$ and $\theta \in \mathbf{S}^{d-1}$. The Sobolev inequality on the CKN sphere (17) then takes the form

$$\left(\int_{(0,\infty) \times \mathbf{S}^{d-1}} |v|^p \frac{r^{d-1-pb}}{(1+r^{2\alpha})^n} dr dV_{\mathbf{S}^{d-1}}\right)^{2/p} \le C \int_{(0,\infty) \times \mathbf{S}^{d-1}} \left[ (\partial_r v)^2 + \frac{1}{r^2} |\nabla_{\theta} v|^2 \right] \frac{r^{2(\alpha-1)+d-1-pb}}{(1+r^{2\alpha})^{n-2}} dr dV_{\mathbf{S}^{d-1}} \\ + 4Z^{-\frac{2}{n}} \int_{(0,\infty) \times \mathbf{S}^{d-1}} v^2 \frac{r^{d-1-pb}}{(1+r^{2\alpha})^n} dr dV_{\mathbf{S}^{d-1}},$$

where $|\nabla_{\theta} v|$ is the Riemannian length of the Riemannian gradient $\nabla_{\theta} v$ on $\mathbf{S}^{d-1}$ and $dV_{\mathbf{S}^{d-1}}$ is the associated Riemannian volume. Using the change of variable $(0, \infty) \ni r = e^s$, with $s \in \mathbf{R}$, the inequality becomes (with a different constant $C$),

$$\left(\int_{\mathbf{R} \times \mathbf{S}^{d-1}} |v|^p \cosh(\alpha s)^{-n} ds dV_{\mathbf{S}^{d-1}}\right)^{2/p} \le C \int_{\mathbf{R} \times \mathbf{S}^{d-1}} \left[ (\partial_s v)^2 + |\nabla_{\theta} v|^2 \right] \cosh(\alpha s)^{2-n} ds dV_{\mathbf{S}^{d-1}} + Z^{-\frac{2}{n}} \int_{\mathbf{R} \times \mathbf{S}^{d-1}} v^2 \cosh(\alpha s)^{-n} ds dV_{\mathbf{S}^{d-1}}, \quad (27)$$

where we used the fact that $d - n\alpha - pb = 0$, see (75). This new chart is often called the Emden-Fowler transformation, as suggested in [CW01, DEL16]. In other words, in the cylindrical chart $(s, \theta) \in \mathbf{R} \times \mathbf{S}^{d-1}$, the
spherical CKN space takes a new and nice form. Notice that the space remains the same, it is only written in a new chart. More precisely, letting

$$ \varphi(s) = \cosh(\alpha s), \quad s \in \mathbf{R}, \qquad (28) $$

the metric becomes (with the upper indices)

$$ \mathfrak{g}_{\mathbf{s}} = \varphi^2 \mathfrak{h} = e^{2\tau_{\mathbf{s}} \mathfrak{h}}, \qquad (29) $$

where $\tau_{\mathbf{s}} = \log \varphi$ and $\mathfrak{h}$ is the standard product metric<fnref n="11" /> on $(0, \infty) \times \mathbf{S}^{d-1}$, represented by the $d$-dimensional matrix

$$ \begin{pmatrix} 1 & 0 \\ 0 & G_{\theta} \end{pmatrix}, $$

where $G_{\theta}$ is the matrix of $\mathfrak{g}_{\theta}$ in the chart $(s, \theta)$, and $\mathfrak{g}_{\theta}$ is the round metric of $\mathbf{S}^{d-1}$. For convenience, in Lemma 3.2 and its proof, as well as the proof of Proposition 1.6, we will abuse the notations and identify the tensors with their coordinates in the chart $(s, \theta)$, since it will be the only chart used in all the calculations. The carré du champ operator takes the form

$$ \Gamma_{\mathbf{s}}(f) = \varphi^2[(\partial_s f)^2 + |\nabla_{\theta} f|^2] = \varphi^2[(\partial_s f)^2 + \Gamma^{\theta}(f)], $$

where $\Gamma^{\theta}(f) = |\nabla_{\theta} f|^2$ is the carré du champ operator associated to the Laplace-Beltrami operator $\Delta_{\theta}$ on $\mathbf{S}^{d-1}$. The Riemannian volume becomes $dV_{\mathfrak{g}_{\theta}} = \varphi^{-d} dsdV_{\mathbf{S}^{d-1}}$ and the reference measure (not normalized measure)

$$ d\mu_{\mathbf{s}} = \varphi^{-n} dsdV_{\mathbf{S}^{d-1}}. $$

The corresponding weight $W_{\mathbf{s}}$ is defined by $d\mu_{\mathbf{s}} = e^{-W_{\mathbf{s}}} dV_{\mathfrak{g}_{\theta}}$, so that

$$ W_{\mathbf{s}} = (n - d) \log \varphi. $$

Finally, the associated generator takes the pleasant form

$$ L_{\mathbf{s}}(f) = \varphi^2 \left[ \partial_{ss} f + (2-n) \frac{\varphi'}{\varphi} \partial_s f + \Delta_{\theta} f \right]. $$

Taking advantage of this chart, let us begin by proving that the spherical CKN space satisfies the $CD(\rho, n)$ condition whenever condition (14) holds:

**Proof of Proposition 1.6**

◇ From [BGL14, Sec. C6], the generator $L_{\mathbf{s}}$ satisfies a $CD(\rho, n)$ condition (with $n > d$) if and only if, as a symmetric tensor (with lower indices),

$$ Ric(L_{\mathbf{s}}) - \rho g_{\mathbf{s}} \geq \frac{1}{n-d} \nabla^{\mathfrak{g}_{\mathbf{s}}} W_{\mathbf{s}} \otimes \nabla^{\mathfrak{g}_{\mathbf{s}}} W_{\mathbf{s}}. $$

Let us remark that, since $\mathfrak{g}_{\mathbf{s}} = \varphi^2 \mathfrak{h}$ (with upper indices), the corresponding metric tensors (with lower indices) satisfy

$$ g_{\mathbf{s}} = \frac{h}{\varphi^2}. $$

Compute first the r.h.s. of the above inequality. From the definition of $W_{\mathbf{s}}$, we have

$$ \frac{\nabla^{\mathfrak{g}_{\mathbf{s}}} W_{\mathbf{s}} \otimes \nabla^{\mathfrak{g}_{\mathbf{s}}} W_{\mathbf{s}}}{n-d} = (n-d) \left( \frac{\varphi'}{\varphi} \right)^2 J, $$

where $J$ is the $d$-dimensional matrix with all entries equal to zero but the first i.e. $J_{ij} = \delta_{i1}\delta_{j1}$ or more visually, letting $H$ (resp. $G_{\theta}$) be the matrix representing the standard product metric $h$ (resp. $g_{\theta}$) in the coordinates $(s, \theta)$ (resp. $\theta$),

$$ J = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} = H - \begin{pmatrix} 0 & 0 \\ 0 & G_{\theta} \end{pmatrix}. \qquad (30) $$

<footnote n="11">¹¹on the cotangent space of $(0, \infty) \times \mathbf{S}^{d-1}$, where $\mathbf{S}^{d-1}$ is viewed in a given chart</footnote>
spherical CKN space takes a new and nice form. Notice that the space remains the same, it is only written in a new chart. More precisely, letting

$$ \varphi(s) = \cosh(\alpha s), \quad s \in \mathbf{R}, \qquad (28) $$

the metric becomes (with the upper indices)

$$ \mathfrak{g}_{\mathbf{s}} = \varphi^2 \mathfrak{h} = e^{2\tau_{\mathbf{s}}} \mathfrak{h}, \qquad (29) $$

where $\tau_{\mathbf{s}} = \log \varphi$ and $\mathfrak{h}$ is the standard product metric<fnref n="11" /> on $(0, \infty) \times \mathbf{S}^{d-1}$, represented by the $d$-dimensional matrix

$$ \begin{pmatrix} 1 & 0 \\ 0 & G_{\theta} \end{pmatrix}, $$

where $G_{\theta}$ is the matrix of $\mathfrak{g}_{\theta}$ in the chart $(s, \theta)$, and $\mathfrak{g}_{\theta}$ is the round metric of $\mathbf{S}^{d-1}$. For convenience, in Lemma 3.2 and its proof, as well as the proof of Proposition 1.6, we will abuse the notations and identify the tensors with their coordinates in the chart $(s, \theta)$, since it will be the only chart used in all the calculations. The carré du champ operator takes the form

$$ \Gamma_{\mathbf{s}}(f) = \varphi^2[(\partial_s f)^2 + |\nabla_{\theta} f|^2] = \varphi^2[(\partial_s f)^2 + \Gamma^{\theta}(f)], $$

where $\Gamma^{\theta}(f) = |\nabla_{\theta} f|^2$ is the carré du champ operator associated to the Laplace-Beltrami operator $\Delta_{\theta}$ on $\mathbf{S}^{d-1}$. The Riemannian volume becomes $dV_{\mathfrak{g}_{\theta}} = \varphi^{-d} dsdV_{\mathbf{S}^{d-1}}$ and the reference measure (not normalized measure)

$$ d\mu_{\mathbf{s}} = \varphi^{-n} dsdV_{\mathbf{S}^{d-1}}. $$

The corresponding weight $W_{\mathbf{s}}$ is defined by $d\mu_{\mathbf{s}} = e^{-W_{\mathbf{s}}} dV_{\mathfrak{g}_{\theta}}$, so that

$$ W_{\mathbf{s}} = (n - d) \log \varphi. $$

Finally, the associated generator takes the pleasant form

$$ L_{\mathbf{s}}(f) = \varphi^2 \left[ \partial_{ss} f + (2-n) \frac{\varphi'}{\varphi} \partial_s f + \Delta_{\theta} f \right]. $$

Taking advantage of this chart, let us begin by proving that the spherical CKN space satisfies the $CD(\rho, n)$ condition whenever condition (14) holds:

**Proof of Proposition 1.6**

◇ From [BGL14, Sec. C6], the generator $L_{\mathbf{s}}$ satisfies a $CD(\rho, n)$ condition (with $n > d$) if and only if, as a symmetric tensor (with lower indices),

$$ Ric(L_{\mathbf{s}}) - \rho g_{\mathbf{s}} \geq \frac{1}{n-d} \nabla^{\mathfrak{g}_{\mathbf{s}}} W_{\mathbf{s}} \otimes \nabla^{\mathfrak{g}_{\mathbf{s}}} W_{\mathbf{s}}. $$

Let us remark that, since $\mathfrak{g}_{\mathbf{s}} = \varphi^2 \mathfrak{h}$ (with upper indices), the corresponding metric tensors (with lower indices) satisfy

$$ g_{\mathbf{s}} = \frac{h}{\varphi^2}. $$

Compute first the r.h.s. of the above inequality. From the definition of $W_{\mathbf{s}}$, we have

$$ \frac{\nabla^{\mathfrak{g}_{\mathbf{s}}} W_{\mathbf{s}} \otimes \nabla^{\mathfrak{g}_{\mathbf{s}}} W_{\mathbf{s}}}{n-d} = (n-d) \left( \frac{\varphi'}{\varphi} \right)^2 J, $$

where $J$ is the $d$-dimensional matrix with all entries equal to zero but the first i.e. $J_{ij} = \delta_{i1}\delta_{j1}$ or more visually, letting $H$ (resp. $G_{\theta}$) be the matrix representing the standard product metric $h$ (resp. $g_{\theta}$) in the coordinates $(s, \theta)$ (resp. $\theta$),

$$ J = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} = H - \begin{pmatrix} 0 & 0 \\ 0 & G_{\theta} \end{pmatrix}. \qquad (30) $$

<footnote n="11">¹¹on the cotangent space of $(0, \infty) \times \mathbf{S}^{d-1}$, where $\mathbf{S}^{d-1}$ is viewed in a given chart</footnote>
Now, applying formula (33) in Lemma 3.2 below, we get

$$Ric(L_S) - \rho g_S - \frac{1}{n-d} \nabla^{\mathfrak{g}s} W_S \otimes \nabla^{\mathfrak{g}s} W_S = -(n-d)\alpha^2 H + (d-2)(1-\alpha^2)(H-J) + (n-d)\alpha^2 J,$$

where, again, we conflate tensors and their matrices in the chart $(s, \theta)$. Hence,

$$Ric(L_S) - \rho g_S - \frac{1}{n-d} \nabla^{\mathfrak{g}s} W_S \otimes \nabla^{\mathfrak{g}s} W_S = \mathfrak{B}_{DGZ} \begin{pmatrix} 0 & 0 \\ 0 & G_{\theta} \end{pmatrix}$$

with the constant $\mathfrak{B}_{DGZ}$ is defined given in (19), and so, $L_S$ satisfies the curvature-dimension condition $CD(\rho, n)$ if and only if $\mathfrak{B}_{DGZ} \ge 0$. ▷

**Remark 3.1.** Since the matrix $H - J$ depends only on the variable $\theta$, when we restrict to functions depending on the variable $s$ only, the corresponding model always satisfies the $CD(\rho, n)$ condition, regardless of the sign of $\mathfrak{B}_{DGZ}$.

In the above proof, we made strong use of the following lemma.

**Lemma 3.2 (Computation of Ric($L_S$)).** We have the following formulae

$$Ric_{g_S} = \frac{(d-1)\alpha^2}{\varphi^2} \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + \frac{1}{\varphi^2} [(d-2)(1-\alpha^2)\varphi^2 + (d-1)\alpha^2] \begin{pmatrix} 0 & 0 \\ 0 & G_{\theta} \end{pmatrix}, \quad (31)$$

and

$$\nabla \nabla^{\mathfrak{g}s} W_S = (n-d)\alpha^2 \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + (n-d)\alpha^2 \frac{1-\varphi^2}{\varphi^2} \begin{pmatrix} 0 & 0 \\ 0 & G_{\theta} \end{pmatrix}, \quad (32)$$

where $G_{\theta}$ is the matrix of round metric on the sphere $S^{d-1}$. With the constant $\mathfrak{B}_{DGZ}$ (given in (19)), we obtain

$$Ric(L_S) = Ric_{g_S} + \nabla \nabla^{\mathfrak{g}s} W_S = \frac{\alpha^2}{\varphi^2} [d-1 + \varphi^2(n-d)] \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + \frac{1}{\varphi^2} [\alpha^2(n-1) + \varphi^2 \mathfrak{B}_{DGZ}] \begin{pmatrix} 0 & 0 \\ 0 & G_{\theta} \end{pmatrix}. \quad (33)$$

**Proof**

Let us start with $Ric_{g_S}$, which is simply the Ricci tensor of the metric $g_S$. Since $g_S = e^{2\tau_S}h$ is conformal to $h$, we may apply (63) in the appendix to get (with lower indices)

$$Ric_{g_S} = Ric_h + (\Delta_h \tau_S)h + (d-2)(\nabla \nabla^h \tau_S + \nabla^h \tau_S \odot_h \nabla^h \tau_S - \nabla^h \tau_S \cdot_h \nabla^h \tau_S h). \quad (34)$$

Since $Ric_{g_{\theta}} = (d-2)g_{\theta}$, we have

$$Ric_h = (d-2)(H-J) = (d-2) \begin{pmatrix} 0 & 0 \\ 0 & G_{\theta} \end{pmatrix}.$$

Since $\varphi$ depends only on the variable $s$, we have

$$\Delta_h(\tau_S) = \tau_S'' = \frac{\varphi''}{\varphi} - \left(\frac{\varphi'}{\varphi}\right)^2 = \alpha^2 - \left(\frac{\varphi'}{\varphi}\right)^2,$$

$$\nabla^h \tau_S \odot_h \nabla^h \tau_S = \left(\frac{\varphi'}{\varphi}\right)^2 J,$$

$$\nabla^h \tau_S = \frac{\varphi'}{\varphi} \begin{pmatrix} 1 \\ 0 \end{pmatrix},$$

and

$$(\nabla^h \tau_S \cdot_h \nabla^h \tau_S) h = \left(\frac{\varphi'}{\varphi}\right)^2 H.$$
Now, applying formula (33) in Lemma 3.2 below, we get

$$Ric(L_S) - \rho g_S - \frac{1}{n-d} \nabla^{\mathfrak{g}s} W_S \otimes \nabla^{\mathfrak{g}s} W_S = -(n-d)\alpha^2 H + (d-2)(1-\alpha^2)(H-J) + (n-d)\alpha^2 J,$$

where, again, we conflate tensors and their matrices in the chart $(s, \theta)$. Hence,

$$Ric(L_S) - \rho g_S - \frac{1}{n-d} \nabla^{\mathfrak{g}s} W_S \otimes \nabla^{\mathfrak{g}s} W_S = \mathfrak{B}_{DGZ} \begin{pmatrix} 0 & 0 \\ 0 & G_{\theta} \end{pmatrix}$$

with the constant $\mathfrak{B}_{DGZ}$ is defined given in (19), and so, $L_S$ satisfies the curvature-dimension condition $CD(\rho, n)$ if and only if $\mathfrak{B}_{DGZ} \ge 0$. ▷

**Remark 3.1.** Since the matrix $H - J$ depends only on the variable $\theta$, when we restrict to functions depending on the variable $s$ only, the corresponding model always satisfies the $CD(\rho, n)$ condition, regardless of the sign of $\mathfrak{B}_{DGZ}$.

In the above proof, we made strong use of the following lemma.

**Lemma 3.2 (Computation of Ric($L_S$)).** We have the following formulae

$$Ric_{g_S} = \frac{(d-1)\alpha^2}{\varphi^2} \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + \frac{1}{\varphi^2} [(d-2)(1-\alpha^2)\varphi^2 + (d-1)\alpha^2] \begin{pmatrix} 0 & 0 \\ 0 & G_{\theta} \end{pmatrix}, \quad (31)$$

and

$$\nabla \nabla^{\mathfrak{g}s} W_S = (n-d)\alpha^2 \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + (n-d)\alpha^2 \frac{1-\varphi^2}{\varphi^2} \begin{pmatrix} 0 & 0 \\ 0 & G_{\theta} \end{pmatrix}, \quad (32)$$

where $G_{\theta}$ is the matrix of round metric on the sphere $S^{d-1}$. With the constant $\mathfrak{B}_{DGZ}$ (given in (19)), we obtain

$$Ric(L_S) = Ric_{g_S} + \nabla \nabla^{\mathfrak{g}s} W_S = \frac{\alpha^2}{\varphi^2} [d-1 + \varphi^2(n-d)] \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + \frac{1}{\varphi^2} [\alpha^2(n-1) + \varphi^2 \mathfrak{B}_{DGZ}] \begin{pmatrix} 0 & 0 \\ 0 & G_{\theta} \end{pmatrix}. \quad (33)$$

**Proof**

Let us start with $Ric_{g_S}$, which is simply the Ricci tensor of the metric $g_S$. Since $g_S = e^{2\tau_S}h$ is conformal to $h$, we may apply (63) in the appendix to get (with lower indices)

$$Ric_{g_S} = Ric_h + (\Delta_h \tau_S)h + (d-2)(\nabla \nabla^h \tau_S + \nabla^h \tau_S \odot_h \nabla^h \tau_S - \nabla^h \tau_S \cdot_h \nabla^h \tau_S h). \quad (34)$$

Since $Ric_{g_{\theta}} = (d-2)g_{\theta}$, we have

$$Ric_h = (d-2)(H-J) = (d-2) \begin{pmatrix} 0 & 0 \\ 0 & G_{\theta} \end{pmatrix}.$$

Since $\varphi$ depends only on the variable $s$, we have

$$\Delta_h(\tau_S) = \tau_S'' = \frac{\varphi''}{\varphi} - \left(\frac{\varphi'}{\varphi}\right)^2 = \alpha^2 - \left(\frac{\varphi'}{\varphi}\right)^2,$$

$$\nabla^h \tau_S \odot_h \nabla^h \tau_S = \left( \frac{\varphi'}{\varphi} \right)^2 J,$$

$$\nabla^h \tau_S = \frac{\varphi'}{\varphi} \begin{pmatrix} 1 \\ 0 \end{pmatrix},$$

and

$$(\nabla^h \tau_S \cdot_h \nabla^h \tau_S) h = \left( \frac{\varphi'}{\varphi} \right)^2 H.$$
Collecting the four terms and using (34), we get

$$Ric_g = H \left[ d - 2 + \alpha^2 - (d-1) \left( \frac{\varphi'}{\varphi} \right)^2 \right] + J(d-2)(\alpha^2 - 1).$$

Since $\varphi'^2 = \alpha^2(\varphi^2 - 1)$, the equation can be written,

$$Ric_g = H \frac{(d-1)\alpha^2}{\varphi^2} + (H - J)(d-2)(1 - \alpha^2),$$

which is the desired result.

Let us now compute $\nabla\nabla^{\text{gs}} W_S$, the Hessian with respect to the metric $g_S$. We have (see (61)),

$$\nabla\nabla^{\text{gs}} W_S = \nabla\nabla^h W_S + 2\nabla W_S \odot_h \nabla^h \tau_S - (\nabla^h W_S \cdot_h \nabla^h \tau_S) h.$$

Since $W_S$ depends only on the variable $s$, we easily get that

$$\nabla\nabla^{\text{gs}} W_S = J(n-d) \left[ \alpha^2 - \left( \frac{\varphi'}{\varphi} \right)^2 \right] + 2J(n-d) \left( \frac{\varphi'}{\varphi} \right)^2 - H(n-d) \left( \frac{\varphi'}{\varphi} \right)^2,$$

which is the expected result. ▷

**Remark 3.3.** As an immediate consequence of Proposition 1.6, the fact that $L_S$ is essentially self-adjoint when $n \ge 3$ (see [Ket15, Thm. 3.12]) and Theorem A, we see that Sobolev's inequality (17) holds (and so Poincaré's inequality (20) too), as soon as (14) holds. Also note that $f = \frac{\varphi'}{\varphi}$, seen as a function of the first of the cylindrical coordinates $(s, \theta)$, solves

$$-L_S f = n\alpha^2 f$$

and so equality in Poincaré's inequality (20) is achieved by $f$. In particular, the constant in Sobolev's inequality (17) is optimal.

In fact, one can do better and prove optimal inequalities in the optimal range of parameters given by the Felli-Schneider condition, as we describe next. The first crucial step consists in proving the following weaker integrated forms of the curvature-dimension condition (15).

**Proposition 3.4.** Let $(a, b) \in \Theta_{FS}$. In cylindrical coordinates $(s, \theta)$, for any $s \in (0, \infty)$ and any smooth positive function $f$ on $(0, \infty) \times \mathbf{S}^{d-1}$, there holds

$$\int \left( \Gamma_2^{\mathbf{S}}(f) - \rho \Gamma_{\mathbf{S}}(f) - \frac{1}{n} (L_{\mathbf{S}}f)^2 \right) f^{1-n} dV_{\mathbf{S}^{d-1}} \ge 0 \quad (35)$$

and

$$\int \left( \Gamma_2^{\mathbf{S}}(f) - \rho \Gamma_{\mathbf{S}}(f) - \frac{1}{n} (L_{\mathbf{S}}f)^2 \right) dV_{\mathbf{S}^{d-1}} \ge 0, \quad (36)$$

where $dV_{\mathbf{S}^{d-1}}$ is the standard volume on the sphere $\mathbf{S}^{d-1}$.

We establish Proposition 3.4 through a series of lemmas. First,

**Lemma 3.5** ($\Gamma_2^{\mathbf{S}}$ in the cylindrical chart). Let $(a, b) \in \Theta$. In cylindrical coordinates, we have for any smooth function $f$ on $(0, \infty) \times \mathbf{S}^{d-1}$

$$\frac{\Gamma_2^{\mathbf{S}}(f)}{\varphi^4} = (\partial_{ss}f)^2 + ||\nabla\nabla_{\theta}f||^2 + 2\Gamma^{\theta}(\partial_s f) + 2\frac{\varphi'}{\varphi}\partial_{ss}f\partial_s f + 4\frac{\varphi'}{\varphi}\Gamma^{\theta}(\partial_s f, f) - 2\frac{\varphi'}{\varphi}\partial_s f\Delta_{\theta}f \\ (\partial_s f)^2 \left[ d\left(\frac{\varphi'}{\varphi}\right)^2 + \alpha^2\left(\frac{d-1}{\varphi^2} + n - d\right) \right] + \Gamma^{\theta}(f)\left(2\left(\frac{\varphi'}{\varphi}\right)^2 + \alpha^2\frac{n-1}{\varphi^2} + \mathfrak{B}_{DGZ}\right), \quad (37)$$

where $||\nabla\nabla_{\theta}f||^2$ is the Hilbert-Schmidt norm with respect to the variable $\theta$, $\Gamma^{\theta}(f) = |\nabla_{\theta}f|^2$ the carré du champ operator associated to $\Delta_{\theta}$ and the function $\varphi$ has been defined in (28).
Collecting the four terms and using (34), we get

$$Ric_g = H \left[ d - 2 + \alpha^2 - (d-1) \left( \frac{\varphi'}{\varphi} \right)^2 \right] + J(d-2)(\alpha^2 - 1).$$

Since $\varphi'^2 = \alpha^2(\varphi^2 - 1)$, the equation can be written,

$$Ric_g = H \frac{(d-1)\alpha^2}{\varphi^2} + (H - J)(d-2)(1 - \alpha^2),$$

which is the desired result.

Let us now compute $\nabla\nabla^{\text{gs}} W_S$, the Hessian with respect to the metric $g_S$. We have (see (61)),

$$\nabla\nabla^{\text{gs}} W_S = \nabla\nabla^h W_S + 2\nabla W_S \odot_h \nabla^h \tau_S - (\nabla^h W_S \cdot_h \nabla^h \tau_S) h.$$

Since $W_S$ depends only on the variable $s$, we easily get that

$$\nabla\nabla^{\text{gs}} W_S = J(n-d) \left[ \alpha^2 - \left( \frac{\varphi'}{\varphi} \right)^2 \right] + 2J(n-d) \left( \frac{\varphi'}{\varphi} \right)^2 - H(n-d) \left( \frac{\varphi'}{\varphi} \right)^2,$$

which is the expected result. ▷

**Remark 3.3.** As an immediate consequence of Proposition 1.6, the fact that $L_S$ is essentially self-adjoint when $n \ge 3$ (see [Ket15, Thm. 3.12]) and Theorem A, we see that Sobolev's inequality (17) holds (and so Poincaré's inequality (20) too), as soon as (14) holds. Also note that $f = \frac{\varphi'}{\varphi}$, seen as a function of the first of the cylindrical coordinates $(s, \theta)$, solves

$$-L_S f = n\alpha^2 f$$

and so equality in Poincaré's inequality (20) is achieved by $f$. In particular, the constant in Sobolev's inequality (17) is optimal.

In fact, one can do better and prove optimal inequalities in the optimal range of parameters given by the Felli-Schneider condition, as we describe next. The first crucial step consists in proving the following weaker integrated forms of the curvature-dimension condition (15).

**Proposition 3.4.** Let $(a, b) \in \Theta_{FS}$. In cylindrical coordinates $(s, \theta)$, for any $s \in (0, \infty)$ and any smooth positive function $f$ on $(0, \infty) \times \mathbf{S}^{d-1}$, there holds

$$\int \left( \Gamma_2^{\mathbf{S}}(f) - \rho \Gamma_{\mathbf{S}}(f) - \frac{1}{n} (L_{\mathbf{S}}f)^2 \right) f^{1-n} dV_{\mathbf{S}^{d-1}} \ge 0 \quad (35)$$

and

$$\int \left( \Gamma_2^{\mathbf{S}}(f) - \rho \Gamma_{\mathbf{S}}(f) - \frac{1}{n} (L_{\mathbf{S}}f)^2 \right) dV_{\mathbf{S}^{d-1}} \ge 0, \quad (36)$$

where $dV_{\mathbf{S}^{d-1}}$ is the standard volume on the sphere $\mathbf{S}^{d-1}$.

We establish Proposition 3.4 through a series of lemmas. First,

**Lemma 3.5** ($\Gamma_2^{\mathbf{S}}$ in the cylindrical chart). Let $(a, b) \in \Theta$. In cylindrical coordinates, we have for any smooth function $f$ on $(0, \infty) \times \mathbf{S}^{d-1}$

$$\frac{\Gamma_2^{\mathbf{S}}(f)}{\varphi^4} = (\partial_{ss}f)^2 + ||\nabla\nabla_{\theta}f||^2 + 2\Gamma^{\theta}(\partial_s f) + 2\frac{\varphi'}{\varphi}\partial_{ss}f\partial_s f + 4\frac{\varphi'}{\varphi}\Gamma^{\theta}(\partial_s f, f) - 2\frac{\varphi'}{\varphi}\partial_s f\Delta_{\theta}f \\ (\partial_s f)^2 \left[ d\left(\frac{\varphi'}{\varphi}\right)^2 + \alpha^2\left(\frac{d-1}{\varphi^2} + n - d\right) \right] + \Gamma^{\theta}(f)\left(2\left(\frac{\varphi'}{\varphi}\right)^2 + \alpha^2\frac{n-1}{\varphi^2} + \mathfrak{B}_{DGZ}\right), \quad (37)$$

where $||\nabla\nabla_{\theta}f||^2$ is the Hilbert-Schmidt norm with respect to the variable $\theta$, $\Gamma^{\theta}(f) = |\nabla_{\theta}f|^2$ the carré du champ operator associated to $\Delta_{\theta}$ and the function $\varphi$ has been defined in (28).
### Proof

◇ We can use the definition of the $\Gamma_2$ operator to prove (37). But, since the Ricci curvature of $L_S$ has been computed in Lemma 3.2, we prefer to use the following Bochner-Lichnerowicz formula,

$$ \Gamma_2^S(f) = Ric(L_S)(\nabla f, \nabla f) + ||\nabla \nabla^{\mathfrak{g}s} f||^2, \quad (38) $$

where $||\nabla \nabla^{\mathfrak{g}s} f||^2$ is the Hilbert-Schmidt norm of the Hessian of $f$ with respect to the metric $\mathfrak{g}_S$ (see for instance [BGL14, P. 71]). From Lemma 3.2, equation (33), we have first

$$ \frac{Ric(L_S)(\nabla f, \nabla f)}{\varphi^4} = \frac{(\partial_s f)^2}{\varphi^2} \alpha^2 [d - 1 + \varphi^2(n - d)] + \frac{\Gamma^\theta(f)}{\varphi^2} [\alpha^2(n - 1) + \varphi^2 \mathfrak{B}_{DGZ}]. $$

It remains to compute $||\nabla \nabla^{\mathfrak{g}s} f||^2$. From (29) we have $\mathfrak{g}_S = \varphi^2 \mathfrak{h} = e^{2\tau_S} \mathfrak{h}$ and so we may apply formula (62) to get

$$ \frac{||\nabla \nabla^{\mathfrak{g}s} f||^2}{\varphi^4} = ||\nabla \nabla^{\mathfrak{h}} f||^2 + 2\Gamma^{\mathfrak{h}}(\tau_S, \Gamma^{\mathfrak{g}}(f)) + 2\Gamma^{\mathfrak{h}}(f)\Gamma^{\mathfrak{h}}(\tau_S) + (d-2)\Gamma^{\mathfrak{h}}(f, \tau_S)^2 - 2\Delta_{\mathfrak{h}}f\Gamma^{\mathfrak{h}}(f, \tau_S). $$

Since $\mathfrak{h}$ is the standard metric product and $\tau$ depends only on the variable $s$, we have

$$ ||\nabla \nabla^{\mathfrak{h}} f||^2 = (\partial_{ss} f)^2 + ||\nabla \nabla_{\theta} f||^2 + 2\Gamma^{\theta}(\partial_s f), $$

$$ \Gamma^{\mathfrak{h}}(\tau_S, \Gamma^{\mathfrak{g}}(f)) = 2 \frac{\varphi'}{\varphi} \partial_{ss} f \partial_s f + 2 \frac{\varphi'}{\varphi} \Gamma^{\theta}(\partial_s f, f), $$

$$ \Gamma^{\mathfrak{h}}(f) = (\partial_s f)^2 + \Gamma^{\theta}(f), \quad \Gamma^{\mathfrak{h}}(\tau_S) = \left(\frac{\varphi'}{\varphi}\right)^2, \quad \Gamma^{\mathfrak{h}}(f, \tau_S) = \frac{\varphi'}{\varphi} \partial_s f \quad \text{and} \quad \Delta_{\mathfrak{h}} f = \partial_{ss} f + \Delta_{\theta} f. $$

Collecting all the terms, we get

$$ \frac{||\nabla \nabla^{\mathfrak{g}s} f||^2}{\varphi^4} = (\partial_{ss} f)^2 + ||\nabla \nabla_{\theta} f||^2 + 2\Gamma^{\theta}(\partial_s f) + 4 \frac{\varphi'}{\varphi} \partial_{ss} f \partial_s f + 4 \frac{\varphi'}{\varphi} \Gamma^{\theta}(\partial_s f, f) \\ + 2 \left(\frac{\varphi'}{\varphi}\right)^2 [(\partial_s f)^2 + \Gamma^{\theta}(f)] + (d-2) \left(\frac{\varphi'}{\varphi}\right)^2 (\partial_s f)^2 - 2(\partial_{ss} f + \Delta_{\theta} f) \frac{\varphi'}{\varphi} \partial_s f, $$

that is

$$ \frac{||\nabla \nabla^{\mathfrak{g}s} f||^2}{\varphi^4} = (\partial_{ss} f)^2 + ||\nabla \nabla_{\theta} f||^2 + 2\Gamma^{\theta}(\partial_s f) + 2 \frac{\varphi'}{\varphi} \partial_{ss} f \partial_s f + 4 \frac{\varphi'}{\varphi} \Gamma^{\theta}(\partial_s f, f) \\ + 2 \left(\frac{\varphi'}{\varphi}\right)^2 \Gamma^{\theta}(f) + d \left(\frac{\varphi'}{\varphi}\right)^2 (\partial_s f)^2 - 2\Delta_{\theta} f \frac{\varphi'}{\varphi} \partial_s f. $$

Finally, by using (38), we get the expected formula (37). ◇

We restate the above lemma in the following more compact formulation.

**Lemma 3.6.** In the cylindrical chart, for any smooth function $f$ on $(0, \infty) \times \mathbf{S}^{d-1}$,

$$ \frac{1}{\varphi^4} \left( \Gamma_2^S(f) - \rho \Gamma_S(f) - \frac{1}{n} (L_S f)^2 \right) = \frac{n-1}{n} \left( \partial_{ss} f + 2 \frac{\varphi'}{\varphi} \partial_s f - \frac{1}{n-1} \Delta_{\theta} f \right)^2 + ||\nabla \nabla^{\theta} f||^2 - \frac{1}{n-1} (\Delta_{\theta} f)^2 \\ + 2\Gamma^{\theta}(\partial_s f + \frac{\varphi'}{\varphi} f) + \Gamma^{\theta}(f) \mathfrak{B}_{DGZ}. \quad (39) $$

### Proof

◇ In the cylindrical chart, the generator takes the following form, for a smooth function $f$:

$$ L_S(f) = \varphi^2 \left[ \partial_{ss} f + (2-n) \frac{\varphi'}{\varphi} \partial_s f + \Delta_{\theta} f \right], $$
### Proof

◇ We can use the definition of the $\Gamma_2$ operator to prove (37). But, since the Ricci curvature of $L_S$ has been computed in Lemma 3.2, we prefer to use the following Bochner-Lichnerowicz formula,

$$ \Gamma_2^S(f) = Ric(L_S)(\nabla f, \nabla f) + ||\nabla \nabla^{\mathfrak{g}s} f||^2, \quad (38) $$

where $||\nabla \nabla^{\mathfrak{g}s} f||^2$ is the Hilbert-Schmidt norm of the Hessian of $f$ with respect to the metric $\mathfrak{g}_S$ (see for instance [BGL14, P. 71]). From Lemma 3.2, equation (33), we have first

$$ \frac{Ric(L_S)(\nabla f, \nabla f)}{\varphi^4} = \frac{(\partial_s f)^2}{\varphi^2} \alpha^2 [d - 1 + \varphi^2(n - d)] + \frac{\Gamma^\theta(f)}{\varphi^2} [\alpha^2(n - 1) + \varphi^2 \mathfrak{B}_{DGZ}]. $$

It remains to compute $||\nabla \nabla^{\mathfrak{g}s} f||^2$. From (29) we have $\mathfrak{g}_S = \varphi^2 \mathfrak{h} = e^{2\tau \mathfrak{h}}$ and so we may apply formula (62) to get

$$ \frac{||\nabla \nabla^{\mathfrak{g}s} f||^2}{\varphi^4} = ||\nabla \nabla^{\mathfrak{h}} f||^2 + 2\Gamma^{\mathfrak{h}}(\tau_S, \Gamma^{\mathfrak{g}}(f)) + 2\Gamma^{\mathfrak{h}}(f)\Gamma^{\mathfrak{h}}(\tau_S) + (d-2)\Gamma^{\mathfrak{h}}(f, \tau_S)^2 - 2\Delta_{\mathfrak{h}}f\Gamma^{\mathfrak{h}}(f, \tau_S). $$

Since $\mathfrak{h}$ is the standard metric product and $\tau$ depends only on the variable $s$, we have

$$ ||\nabla \nabla^{\mathfrak{h}} f||^2 = (\partial_{ss} f)^2 + ||\nabla \nabla_{\theta} f||^2 + 2\Gamma^{\theta}(\partial_s f), $$

$$ \Gamma^{\mathfrak{h}}(\tau_S, \Gamma^{\mathfrak{g}}(f)) = 2 \frac{\varphi'}{\varphi} \partial_{ss} f \partial_s f + 2 \frac{\varphi'}{\varphi} \Gamma^{\theta}(\partial_s f, f), $$

$$ \Gamma^{\mathfrak{h}}(f) = (\partial_s f)^2 + \Gamma^{\theta}(f), \quad \Gamma^{\mathfrak{h}}(\tau_S) = \left(\frac{\varphi'}{\varphi}\right)^2, \quad \Gamma^{\mathfrak{h}}(f, \tau_S) = \frac{\varphi'}{\varphi} \partial_s f \quad \text{and} \quad \Delta_{\mathfrak{h}} f = \partial_{ss} f + \Delta_{\theta} f. $$

Collecting all the terms, we get

$$ \frac{||\nabla \nabla^{\mathfrak{g}s} f||^2}{\varphi^4} = (\partial_{ss} f)^2 + ||\nabla \nabla_{\theta} f||^2 + 2\Gamma^{\theta}(\partial_s f) + 4 \frac{\varphi'}{\varphi} \partial_{ss} f \partial_s f + 4 \frac{\varphi'}{\varphi} \Gamma^{\theta}(\partial_s f, f) \\ + 2 \left(\frac{\varphi'}{\varphi}\right)^2 [(\partial_s f)^2 + \Gamma^{\theta}(f)] + (d-2) \left(\frac{\varphi'}{\varphi}\right)^2 (\partial_s f)^2 - 2(\partial_{ss} f + \Delta_{\theta} f) \frac{\varphi'}{\varphi} \partial_s f, $$

that is

$$ \frac{||\nabla \nabla^{\mathfrak{g}s} f||^2}{\varphi^4} = (\partial_{ss} f)^2 + ||\nabla \nabla_{\theta} f||^2 + 2\Gamma^{\theta}(\partial_s f) + 2 \frac{\varphi'}{\varphi} \partial_{ss} f \partial_s f + 4 \frac{\varphi'}{\varphi} \Gamma^{\theta}(\partial_s f, f) \\ + 2 \left(\frac{\varphi'}{\varphi}\right)^2 \Gamma^{\theta}(f) + d \left(\frac{\varphi'}{\varphi}\right)^2 (\partial_s f)^2 - 2\Delta_{\theta} f \frac{\varphi'}{\varphi} \partial_s f. $$

Finally, by using (38), we get the expected formula (37). ◇

We restate the above lemma in the following more compact formulation.

**Lemma 3.6.** In the cylindrical chart, for any smooth function $f$ on $(0, \infty) \times S^{d-1}$,

$$ \frac{1}{\varphi^4} \left( \Gamma_2^S(f) - \rho \Gamma_S(f) - \frac{1}{n} (L_S f)^2 \right) = \frac{n-1}{n} \left( \partial_{ss} f + 2 \frac{\varphi'}{\varphi} \partial_s f - \frac{1}{n-1} \Delta_{\theta} f \right)^2 + ||\nabla \nabla^{\theta} f||^2 - \frac{1}{n-1} (\Delta_{\theta} f)^2 \\ + 2\Gamma^{\theta}(\partial_s f + \frac{\varphi'}{\varphi} f) + \Gamma^{\theta}(f) \mathfrak{B}_{DGZ}. \quad (39) $$

### Proof

◇ In the cylindrical chart, the generator takes the following form, for a smooth function $f$:

$$ L_S(f) = \varphi^2 \left[ \partial_{ss} f + (2-n) \frac{\varphi'}{\varphi} \partial_s f + \Delta_{\theta} f \right], $$
and from Lemma 3.5 (formula (37)), we obtain

$$ \begin{aligned} \frac{1}{\varphi^4} \left[ \Gamma_2^S(f) - \rho \Gamma_S(f) - \frac{1}{n} (L_S f)^2 \right] = & \frac{n-1}{n} (\partial_{ss} f)^2 + \|\nabla \nabla_{\theta} f\|^2 - \frac{1}{n} (\Delta_{\theta} f)^2 + 4 \frac{\varphi'}{\varphi} \frac{n-1}{n} \partial_{ss} f \partial_{s} f - \frac{4}{n} \frac{\varphi'}{\varphi} \partial_{s} f \Delta_{\theta} f \\ & - \frac{2}{n} \partial_{ss} f \Delta_{\theta} f + 2 \Gamma^{\theta}(\partial_{s} f) + 4 \frac{\varphi'}{\varphi} \Gamma^{\theta}(\partial_{s} f, f) \\ & + (\partial_{s} f)^2 \left[ \left( \frac{\varphi'}{\varphi} \right)^2 \left( d - \frac{(n-2)^2}{n} \right) + (n-d) \alpha^2 \frac{\varphi^2 - 1}{\varphi^2} \right] + \Gamma^{\theta}(f) \left( 2 \left( \frac{\varphi'}{\varphi} \right)^2 + \mathfrak{B}_{DGZ} \right). \end{aligned} $$

Since $\varphi'^2 = \alpha^2(\varphi^2 - 1)$, we get

$$ \begin{aligned} \frac{1}{\varphi^4} \left[ \Gamma_2^S(f) - \rho \Gamma_S(f) - \frac{1}{n} (L_S f)^2 \right] = & \frac{n-1}{n} (\partial_{ss} f)^2 + \|\nabla \nabla_{\theta} f\|^2 - \frac{1}{n} (\Delta_{\theta} f)^2 + 4 \frac{\varphi'}{\varphi} \frac{n-1}{n} \partial_{ss} f \partial_{s} f - \frac{4}{n} \frac{\varphi'}{\varphi} \partial_{s} f \Delta_{\theta} f \\ & + 4 \frac{n-1}{n} \left( \frac{\varphi'}{\varphi} \right)^2 (\partial_{s} f)^2 - \frac{2}{n} \partial_{ss} f \Delta_{\theta} f + \Gamma^{\theta}(f) \left( 2 \left( \frac{\varphi'}{\varphi} \right)^2 + \mathfrak{B}_{DGZ} \right) + 2 \Gamma^{\theta}(\partial_{s} f) + 4 \frac{\varphi'}{\varphi} \Gamma^{\theta}(\partial_{s} f, f). \end{aligned} $$

Formula (39) follows then easily since $\varphi$ depends only on the variable $s$. ▷

Remark 3.7. By the Cauchy-Schwarz inequality,

$$ \|\nabla \nabla_{\theta} f\|^2 \ge \frac{1}{d-1} (\Delta_{\theta} f)^2, $$

and so

$$ \|\nabla \nabla_{\theta} f\|^2 - \frac{1}{n-1} (\Delta_{\theta} f)^2 \ge \frac{n-d}{(d-1)(n-1)} (\Delta_{\theta} f)^2 \ge 0 $$

since $n \ge d$. We recover from (39) that, under the condition $\mathfrak{B}_{DGZ} \ge 0$, the generator $L_S$ satisfies the $CD(\rho, n)$ curvature-dimension condition.

The next ingredient is the following inequality valid on the sphere $S^{d-1}$ (or any smooth weighted manifold satisfying the $CD(d-2, d-1)$ condition).

**Lemma 3.8.** For any smooth positive function $f$ on $S^{d-1}$,

$$ \int \Gamma_2^{\theta}(f) f^{1-n} dV_{S^{d-1}} \ge (d-1) \int \Gamma^{\theta}(f) f^{1-n} dV_{S^{d-1}} + A \int \frac{\Gamma^{\theta}(f)^2}{f^2} f^{1-n} dV_{S^{d-1}}, \quad (40) $$

where

$$ A = \frac{n-1}{4(d+1)^2} (n(4d-5) + 3(4d+7)). \quad (41) $$

In particular,

$$ \int \Gamma_2^{\theta}(f) f^{1-n} dV_{S^{d-1}} \ge (d-1) \int \Gamma^{\theta}(f) f^{1-n} dV_{S^{d-1}}. \quad (42) $$

**Proof**

□ The operator $\Delta_{\theta}$ is the Laplace-Beltrami operator on the $(d-1)$-dimensional sphere, therefore, it satisfies the $CD(d-2, d-1)$ condition. Moreover, $Ric_{g_{\theta}}$ (the Ricci tensor of $\Delta_{\theta}$) satisfies

$$ Ric_{g_{\theta}}(\nabla_{\theta} f, \nabla_{\theta} f) = (d-2)\Gamma^{\theta}(f). \quad (43) $$

In [GZ21, p. 767], it is proved that under the $CD(K, m)$ condition, for a general operator $L$ associated to the measure $\mu$, and the operators $\Gamma$ and $\Gamma_2$, one has for any real parameters $q, \chi$,

$$ \int h^q \Gamma_2(h) d\mu \ge \frac{Km}{m-1} \int h^q \Gamma(h) d\mu + \int [Ah^{q-2} \Gamma(h)^2 + Bh^{q-1} \Gamma(h, \Gamma(h))] d\mu $$
and from Lemma 3.5 (formula (37)), we obtain

$$ 
\begin{aligned}
\frac{1}{\varphi^4} \left[ \Gamma_2^S(f) - \rho \Gamma_S(f) - \frac{1}{n} (L_S f)^2 \right] &= \frac{n-1}{n} (\partial_{ss} f)^2 + \|\nabla \nabla_{\theta} f\|^2 - \frac{1}{n} (\Delta_{\theta} f)^2 + 4 \frac{\varphi'}{\varphi} \frac{n-1}{n} \partial_{ss} f \partial_{s} f - \frac{4}{n} \frac{\varphi'}{\varphi} \partial_{s} f \Delta_{\theta} f \\
&\quad - \frac{2}{n} \partial_{ss} f \Delta_{\theta} f + 2 \Gamma^{\theta}(\partial_{s} f) + 4 \frac{\varphi'}{\varphi} \Gamma^{\theta}(\partial_{s} f, f) \\
&\quad + (\partial_{s} f)^2 \left[ \left( \frac{\varphi'}{\varphi} \right)^2 \left( d - \frac{(n-2)^2}{n} \right) + (n-d) \alpha^2 \frac{\varphi^2 - 1}{\varphi^2} \right] + \Gamma^{\theta}(f) \left( 2 \left( \frac{\varphi'}{\varphi} \right)^2 + \mathfrak{B}_{DGZ} \right).
\end{aligned}
$$

Since $\varphi'^2 = \alpha^2(\varphi^2 - 1)$, we get

$$ 
\begin{aligned}
\frac{1}{\varphi^4} \left[ \Gamma_2^S(f) - \rho \Gamma_S(f) - \frac{1}{n} (L_S f)^2 \right] &= \frac{n-1}{n} (\partial_{ss} f)^2 + \|\nabla \nabla_{\theta} f\|^2 - \frac{1}{n} (\Delta_{\theta} f)^2 + 4 \frac{\varphi'}{\varphi} \frac{n-1}{n} \partial_{ss} f \partial_{s} f - \frac{4}{n} \frac{\varphi'}{\varphi} \partial_{s} f \Delta_{\theta} f \\
&\quad + 4 \frac{n-1}{n} \left( \frac{\varphi'}{\varphi} \right)^2 (\partial_{s} f)^2 - \frac{2}{n} \partial_{ss} f \Delta_{\theta} f + \Gamma^{\theta}(f) \left( 2 \left( \frac{\varphi'}{\varphi} \right)^2 + \mathfrak{B}_{DGZ} \right) + 2 \Gamma^{\theta}(\partial_{s} f) + 4 \frac{\varphi'}{\varphi} \Gamma^{\theta}(\partial_{s} f, f).
\end{aligned}
$$

Formula (39) follows then easily since $\varphi$ depends only on the variable $s$. ▷

*Remark 3.7.* By the Cauchy-Schwarz inequality,

$$ \|\nabla \nabla_{\theta} f\|^2 \geq \frac{1}{d-1} (\Delta_{\theta} f)^2, $$

and so

$$ \|\nabla \nabla_{\theta} f\|^2 - \frac{1}{n-1} (\Delta_{\theta} f)^2 \geq \frac{n-d}{(d-1)(n-1)} (\Delta_{\theta} f)^2 \geq 0 $$

since $n \geq d$. We recover from (39) that, under the condition $\mathfrak{B}_{DGZ} \geq 0$, the generator $L_S$ satisfies the $CD(\rho, n)$ curvature-dimension condition.

The next ingredient is the following inequality valid on the sphere $S^{d-1}$ (or any smooth weighted manifold satisfying the $CD(d-2, d-1)$ condition).

**Lemma 3.8.** For any smooth positive function $f$ on $S^{d-1}$,

$$ \int \Gamma_2^{\theta}(f) f^{1-n} dV_{S^{d-1}} \geq (d-1) \int \Gamma^{\theta}(f) f^{1-n} dV_{S^{d-1}} + A \int \frac{\Gamma^{\theta}(f)^2}{f^2} f^{1-n} dV_{S^{d-1}}, \quad (40) $$

where

$$ A = \frac{n-1}{4(d+1)^2} (n(4d-5) + 3(4d+7)). \quad (41) $$

In particular,

$$ \int \Gamma_2^{\theta}(f) f^{1-n} dV_{S^{d-1}} \geq (d-1) \int \Gamma^{\theta}(f) f^{1-n} dV_{S^{d-1}}. \quad (42) $$

**Proof**

⊂ The operator $\Delta_{\theta}$ is the Laplace-Beltrami operator on the $(d-1)$-dimensional sphere, therefore, it satisfies the $CD(d-2, d-1)$ condition. Moreover, $Ric_{g_{\theta}}$ (the Ricci tensor of $\Delta_{\theta}$) satisfies

$$ Ric_{g_{\theta}}(\nabla_{\theta} f, \nabla_{\theta} f) = (d-2)\Gamma^{\theta}(f). \quad (43) $$

In [GZ21, p. 767], it is proved that under the $CD(K, m)$ condition, for a general operator $L$ associated to the measure $\mu$, and the operators $\Gamma$ and $\Gamma_2$, one has for any real parameters $q, \chi$,

$$ \int h^q \Gamma_2(h) d\mu \geq \frac{Km}{m-1} \int h^q \Gamma(h) d\mu + \int [Ah^{q-2} \Gamma(h)^2 + Bh^{q-1} \Gamma(h, \Gamma(h))] d\mu $$
where

$$
\begin{cases}
A = \frac{q(q-1)}{m-1} - \chi^2 - 2\chi \frac{q-1}{m-1}, \\
B = \frac{1}{m-1} \left( \frac{3q}{2} - \chi(m+2) \right).
\end{cases}
$$

Apply the previous inequality to our operator $\Delta_{\theta}$ with parameters $q = 1 - n$, $K = d - 2$, $m = d - 1$ and
$\chi = \frac{3q}{2(m+2)}$ so that $B = 0$. We obtain

$$
\int \Gamma_2^\theta(f) f^{1-n} dV_{S^{d-1}} \geq (d-1) \int_{S^{d-1}} \Gamma^\theta(f) f^{1-n} dV_{S^{d-1}} + A \int_{S^{d-1}} \frac{\Gamma^\theta(f)^2}{f^2} f^{1-n} dV_{S^{d-1}}
$$

where $A$ is given by (41) after a straightforward computation. In particular, $A \ge 0$ and the estimate (42)
follows. ▷

We can now turn to the **Proof of Proposition 3.4**

◇ From Lemma 3.6, we have

$$
\int \left( \Gamma_2^S(f) - \rho \Gamma_S(f) - \frac{1}{n} (L_S f)^2 \right) f^{1-n} dV_{S^{d-1}} \ge \varphi^4 \int \left( \|\nabla \nabla^\theta f\|^2 - \frac{1}{n-1} (\Delta_\theta f)^2 + \Gamma^\theta(f) \mathfrak{B}_{DGZ} \right) f^{1-n} dV_{S^{d-1}}.
$$

Now,

$$
\|\nabla \nabla_{\theta} f\|^2 - \frac{1}{n-1}(\Delta_{\theta} f)^2 = \frac{n-d}{n-1}\|\nabla \nabla_{\theta} f\|^2 + \frac{d-1}{n-1}\|\nabla \nabla_{\theta} f\|^2 - \frac{1}{n-1}(\Delta_{\theta} f)^2 \geq \frac{n-d}{n-1}\|\nabla \nabla_{\theta} f\|^2,
$$

where we used the Cauchy-Schwarz inequality to infer that

$$
\|\nabla \nabla_{\theta} f\|^2 \geq \frac{1}{d-1} (\Delta_{\theta} f)^2. \quad (44)
$$

So,

$$
\int \left( \Gamma_2^S(f) - \rho \Gamma_S(f) - \frac{1}{n} (L_S f)^2 \right) f^{1-n} dV_{S^{d-1}} \geq \varphi^4 \int \left( \frac{n-d}{n-1} \|\nabla \nabla^\theta f\|^2 + \Gamma^\theta(f) \mathfrak{B}_{DGZ} \right) f^{1-n} dV_{S^{d-1}}.
$$

Since, from (38) and (43),

$$
\|\nabla \nabla^\theta f\|^2 = \Gamma_2^\theta(f) - Ric_{S^{d-1}}(\nabla_\theta f, \nabla_\theta f) = \Gamma_2^\theta(f) - (d-2)\Gamma^\theta(f),
$$

the inequality becomes

$$
\varphi^{-4} \int \left( \Gamma_2^S(f) - \rho \Gamma_S(f) - \frac{1}{n} (L_S f)^2 \right) f^{1-n} dV_{S^{d-1}} \geq \\
\frac{n-d}{n-1} \int \Gamma_2^\theta(f) f^{1-n} dV_S^{d-1} + \left[ \frac{n-d}{n-1}(d-2) + \mathfrak{B}_{DGZ} \right] \int \Gamma^\theta(f) f^{1-n} dV_{S^{d-1}}.
$$

Using the estimate (42) in Lemma 3.8, we get

$$
\varphi^{-4} \int \left( \Gamma_2^S(f) - \rho \Gamma_S(f) - \frac{1}{n} (L_S f)^2 \right) f^{1-n} dV_{S^{d-1}} \geq \left[ \frac{n-d}{n-1} + \mathfrak{B}_{DGZ} \right] \int \Gamma^\theta(f) f^{1-n} dV_{S^{d-1}}. \quad (45)
$$

That is,

$$
\frac{n-d}{n-1} + \mathfrak{B}_{DGZ} \geq 0 \qquad (46)
$$

implies (35). The proof of inequality (36) is almost identical, except that instead of (40), one uses

$$
\int \Gamma_2^\theta(f) dV_{S^{d-1}} \geq (d-1) \int \Gamma^\theta(f) dV_{S^{d-1}},
$$
where

$$
\begin{cases}
A = \frac{q(q-1)}{m-1} - \chi^2 - 2\chi \frac{q-1}{m-1}, \\
B = \frac{1}{m-1} \left( \frac{3q}{2} - \chi(m+2) \right).
\end{cases}
$$

Apply the previous inequality to our operator $\Delta_{\theta}$ with parameters $q = 1 - n$, $K = d - 2$, $m = d - 1$ and
$\chi = \frac{3q}{2(m+2)}$ so that $B = 0$. We obtain

$$
\int \Gamma_2^\theta(f) f^{1-n} dV_{S^{d-1}} \geq (d-1) \int_{S^{d-1}} \Gamma^\theta(f) f^{1-n} dV_{S^{d-1}} + A \int_{S^{d-1}} \frac{\Gamma^\theta(f)^2}{f^2} f^{1-n} dV_{S^{d-1}}
$$

where $A$ is given by (41) after a straightforward computation. In particular, $A \ge 0$ and the estimate (42)
follows. ▷

We can now turn to the **Proof of Proposition 3.4**

◇ From Lemma 3.6, we have

$$
\int \left( \Gamma_2^S(f) - \rho \Gamma_S(f) - \frac{1}{n} (L_S f)^2 \right) f^{1-n} dV_{S^{d-1}} \ge \varphi^4 \int \left( \|\nabla \nabla^\theta f\|^2 - \frac{1}{n-1} (\Delta_\theta f)^2 + \Gamma^\theta(f) \mathfrak{B}_{DGZ} \right) f^{1-n} dV_{S^{d-1}}.
$$

Now,

$$
\|\nabla \nabla_{\theta} f\|^2 - \frac{1}{n-1}(\Delta_{\theta} f)^2 = \frac{n-d}{n-1}\|\nabla \nabla_{\theta} f\|^2 + \frac{d-1}{n-1}\|\nabla \nabla_{\theta} f\|^2 - \frac{1}{n-1}(\Delta_{\theta} f)^2 \geq \frac{n-d}{n-1}\|\nabla \nabla_{\theta} f\|^2,
$$

where we used the Cauchy-Schwarz inequality to infer that

$$
\|\nabla \nabla_{\theta} f\|^2 \geq \frac{1}{d-1} (\Delta_{\theta} f)^2. \quad (44)
$$

So,

$$
\int \left( \Gamma_2^S(f) - \rho \Gamma_S(f) - \frac{1}{n} (L_S f)^2 \right) f^{1-n} dV_{S^{d-1}} \geq \varphi^4 \int \left( \frac{n-d}{n-1} \|\nabla \nabla^\theta f\|^2 + \Gamma^\theta(f) \mathfrak{B}_{DGZ} \right) f^{1-n} dV_{S^{d-1}}.
$$

Since, from (38) and (43),

$$
\|\nabla \nabla^\theta f\|^2 = \Gamma_2^\theta(f) - Ric_{S^{d-1}}(\nabla_\theta f, \nabla_\theta f) = \Gamma_2^\theta(f) - (d-2)\Gamma^\theta(f),
$$

the inequality becomes

$$
\varphi^{-4} \int \left( \Gamma_2^S(f) - \rho \Gamma_S(f) - \frac{1}{n} (L_S f)^2 \right) f^{1-n} dV_{S^{d-1}} \geq \\
\frac{n-d}{n-1} \int \Gamma_2^\theta(f) f^{1-n} dV_S^{d-1} + \left[ \frac{n-d}{n-1}(d-2) + \mathfrak{B}_{DGZ} \right] \int \Gamma^\theta(f) f^{1-n} dV_{S^{d-1}}.
$$

Using the estimate (42) in Lemma 3.8, we get

$$
\varphi^{-4} \int \left( \Gamma_2^S(f) - \rho \Gamma_S(f) - \frac{1}{n} (L_S f)^2 \right) f^{1-n} dV_{S^{d-1}} \geq \left[ \frac{n-d}{n-1} + \mathfrak{B}_{DGZ} \right] \int \Gamma^\theta(f) f^{1-n} dV_{S^{d-1}}. \quad (45)
$$

That is,

$$
\frac{n-d}{n-1} + \mathfrak{B}_{DGZ} \geq 0 \qquad (46)
$$

implies (35). The proof of inequality (36) is almost identical, except that instead of (40), one uses

$$
\int \Gamma_2^\theta(f) dV_{S^{d-1}} \geq (d-1) \int \Gamma^\theta(f) dV_{S^{d-1}},
$$
which itself holds thanks to the Cauchy-Schwarz inequality (44), Bochner's formula (38) and the identity
$$ \int \Gamma_2^\theta(f) dV_{S^{d-1}} = \int (\Delta_\theta f)^2 dV_{S^{d-1}}. \triangleright $$

Now that the integrated curvature-dimension is established, we can turn to the proof of Sobolev's inequality.

**Proof of Theorem 1.7**

Fix $q \in [1, p)$. By the Caffarelli-Kohn-Nirenberg inequality (1) and Theorem 1.4, Sobolev's inequality holds on the CKN spherical space in the form (10). By Proposition 6.2.2 in [BGL14], (10) and Poincaré's inequality (A.5) imply the following tight form of Sobolev's inequality:

$$ \left( \int |v|^q d\mu \right)^{2/q} \le A \int \Gamma_S(v) d\mu + \int v^2 d\mu, \quad (47) $$

for some $A \in \mathbf{R}_+^*$ where $\mu = \frac{1}{Z}\mu_S$ is the normalized measure and $v \in H_0^1(\mu_S)$. Given $A \in \mathbf{R}_+^*$, consider the minimization problem

$$ I(A) = \inf \left\{ A \int \Gamma_S(v) d\mu + \int v^2 d\mu : v \in H_0^1(\mu_S), \|v\|_{L^q(\mu)} = 1 \right\}. $$

Using $v=1$ as a test function, we see that $I(A) \le 1$. Thus, (47) holds if and only if $I(A) = 1$. Thanks to the Banach-Alaoglu-Bourbaki and Lemma A.4, there exists a minimizer $v \in H_0^1(\mu_S)$ s.t. $\|v\|_{L^q(\mu)} = 1$. By Stampacchia's theorem [Sta66], $|v|$ is also a minimizer, so we may assume that $v \ge 0$ a.e. In addition, a constant multiple of $v$ (abusively denoted the same below) is a weak solution to

$$ -AL_Sv + v = v^{q-1} \quad \text{in } \mathbf{R}^d \setminus \{0\}. \quad (48) $$

By standard elliptic regularity (see e.g. [Heb97], proof of Theorem 6.2.1, p. 248) $v \in C^3(\mathbf{R}^d \setminus \{0\})$ and by the strong maximum principle (see e.g. [Heb97], Theorem 5.7.2), $v > 0$ in $\mathbf{R}^d \setminus \{0\}$. In addition,

$$ C \geq v \geq c > 0 \quad \text{and} \quad \Gamma_S(v) \leq C \quad (49) $$

for some constants $C, c > 0$. The upper bound on $v$ is obtained by standard Moser iteration (i.e. by multiplying (48) by the test function $\min(v, k)^{2\alpha-1}$, where $k \in \mathbf{N}^*$ and $\alpha \ge 1$ and making use of Sobolev's inequality (69) inductively). For the lower bound on $v$, we apply Proposition 6.3.4 in [BGL14] and repeat the considerations of p. 312 in the same reference. The upper bound on $\Gamma_S(v)$ is more delicate and proved in Lemma A.7.
Define the pressure function $\Phi = v^{-\frac{q-2}{2}}$. Then, $\Phi$ solves

$$ \Phi L_S \Phi - \frac{\nu}{2} \Gamma_S(\Phi) = -\lambda(\Phi^2 - 1) \quad \text{in } \mathbf{R}^d \setminus \{0\}, \quad (50) $$

where $\nu = \frac{2q}{q-2}$ and $\lambda = \frac{q-2}{2A} = \frac{2}{(\nu-2)A}$. Since $v$ is bounded above and below by positive constants and since $\Gamma_S(v)$ is bounded, equation (50) implies that for every $a \in \mathbf{R}$

$$ \Phi^a \in D(L_S) \quad (51) $$

Multiply equation (50) by $L_S(\Phi^{1-\nu})$ and integrate. Thanks to the integration by parts formula (65), we find for the right-hand-side

$$ \begin{aligned} \int \lambda(\Phi^2 - 1)L_S(\Phi^{1-\nu}) &= \lambda \int \Phi^2 L_S(\Phi^{1-\nu}) = -\lambda \int \Gamma_S(\Phi^2, \Phi^{1-\nu}) \\ &= c \int \Gamma_S(\Phi)\Phi^{1-\nu} \end{aligned} $$

where $c = 2\lambda(\nu - 1) = 4\frac{\nu-1}{(\nu-2)A}$ and where integration is understood with respect to the reference measure $\mu = \frac{1}{Z}\mu_S$. For the left-hand side, integrations by parts must be dealt with more carefully. By Lemma A.2 (or since $L_S$ is essentially self-adjoint by [Ket15, Thm. 3.12]), there exists a sequence of radial functions $\zeta_k \in C_c^\infty(\mathbf{R}^d \setminus \{0\})$ such that $\zeta_k \to 1$ in $D(L_S)$. By equation (50), $\Phi L_S \Phi - \frac{\nu}{2} \Gamma_S(\Phi)$ is bounded. By (51), $L_S(\Phi^{1-\nu}) \in L^2(\mu_S)$. So, by dominated convergence, as $k \to +\infty$,

$$ \int (L_S(\Phi^{1-\nu}))(\Phi L_S \Phi - \frac{\nu}{2} \Gamma_S(\Phi)) = \int (L_S(\Phi^{1-\nu}))(\Phi L_S \Phi - \frac{\nu}{2} \Gamma_S(\Phi))\zeta_k + o(1) $$
which itself holds thanks to the Cauchy-Schwarz inequality (44), Bochner's formula (38) and the identity
$$ \int \Gamma_2^\theta(f) dV_{S^{d-1}} = \int (\Delta_\theta f)^2 dV_{S^{d-1}}. $$

Now that the integrated curvature-dimension is established, we can turn to the proof of Sobolev's inequality.

**Proof of Theorem 1.7**

Fix $q \in [1, p)$. By the Caffarelli-Kohn-Nirenberg inequality (1) and Theorem 1.4, Sobolev's inequality holds on the CKN spherical space in the form (10). By Proposition 6.2.2 in [BGL14], (10) and Poincaré's inequality (A.5) imply the following tight form of Sobolev's inequality:

$$ \left( \int |v|^q d\mu \right)^{2/q} \le A \int \Gamma_S(v) d\mu + \int v^2 d\mu, \quad (47) $$

for some $A \in \mathbf{R}_+^*$ where $\mu = \frac{1}{Z}\mu_S$ is the normalized measure and $v \in H_0^1(\mu_S)$. Given $A \in \mathbf{R}_+^*$, consider the minimization problem

$$ I(A) = \inf \left\{ A \int \Gamma_S(v) d\mu + \int v^2 d\mu : v \in H_0^1(\mu_S), \|v\|_{L^q(\mu)} = 1 \right\}. $$

Using $v=1$ as a test function, we see that $I(A) \le 1$. Thus, (47) holds if and only if $I(A) = 1$. Thanks to the Banach-Alaoglu-Bourbaki and Lemma A.4, there exists a minimizer $v \in H_0^1(\mu_S)$ s.t. $\|v\|_{L^q(\mu)} = 1$. By Stampacchia's theorem [Sta66], $|v|$ is also a minimizer, so we may assume that $v \ge 0$ a.e. In addition, a constant multiple of $v$ (abusively denoted the same below) is a weak solution to

$$ -AL_Sv + v = v^{q-1} \quad \text{in } \mathbf{R}^d \setminus \{0\}. \quad (48) $$

By standard elliptic regularity (see e.g. [Heb97], proof of Theorem 6.2.1, p. 248) $v \in C^3(\mathbf{R}^d \setminus \{0\})$ and by the strong maximum principle (see e.g. [Heb97], Theorem 5.7.2), $v > 0$ in $\mathbf{R}^d \setminus \{0\}$. In addition,

$$ C \ge v \ge c > 0 \quad \text{and} \quad \Gamma_S(v) \le C \quad (49) $$

for some constants $C, c > 0$. The upper bound on $v$ is obtained by standard Moser iteration (i.e. by multiplying (48) by the test function $\min(v, k)^{2\alpha-1}$, where $k \in \mathbf{N}^*$ and $\alpha \ge 1$ and making use of Sobolev's inequality (69) inductively). For the lower bound on $v$, we apply Proposition 6.3.4 in [BGL14] and repeat the considerations of p. 312 in the same reference. The upper bound on $\Gamma_S(v)$ is more delicate and proved in Lemma A.7.
Define the pressure function $\Phi = v^{-\frac{q-2}{2}}$. Then, $\Phi$ solves

$$ \Phi L_S \Phi - \frac{\nu}{2} \Gamma_S(\Phi) = -\lambda(\Phi^2 - 1) \quad \text{in } \mathbf{R}^d \setminus \{0\}, \quad (50) $$

where $\nu = \frac{2q}{q-2}$ and $\lambda = \frac{q-2}{2A} = \frac{2}{(\nu-2)A}$. Since $v$ is bounded above and below by positive constants and since $\Gamma_S(v)$ is bounded, equation (50) implies that for every $a \in \mathbf{R}$

$$ \Phi^a \in D(L_S) \quad (51) $$

Multiply equation (50) by $L_S(\Phi^{1-\nu})$ and integrate. Thanks to the integration by parts formula (65), we find for the right-hand-side

$$ \begin{aligned} \int \lambda(\Phi^2 - 1)L_S(\Phi^{1-\nu}) &= \lambda \int \Phi^2 L_S(\Phi^{1-\nu}) = -\lambda \int \Gamma_S(\Phi^2, \Phi^{1-\nu}) \\ &= c \int \Gamma_S(\Phi)\Phi^{1-\nu} \end{aligned} $$

where $c = 2\lambda(\nu - 1) = 4\frac{\nu-1}{(\nu-2)A}$ and where integration is understood with respect to the reference measure $\mu = \frac{1}{Z}\mu_S$. For the left-hand side, integrations by parts must be dealt with more carefully. By Lemma A.2 (or since $L_S$ is essentially self-adjoint by [Ket15, Thm. 3.12]), there exists a sequence of radial functions $\zeta_k \in C_c^\infty(\mathbf{R}^d \setminus \{0\})$ such that $\zeta_k \to 1$ in $D(L_S)$. By equation (50), $\Phi L_S \Phi - \frac{\nu}{2} \Gamma_S(\Phi)$ is bounded. By (51), $L_S(\Phi^{1-\nu}) \in L^2(\mu_S)$. So, by dominated convergence, as $k \to +\infty$,

$$ \int (L_S(\Phi^{1-\nu}))(\Phi L_S \Phi - \frac{\nu}{2} \Gamma_S(\Phi)) = \int (L_S(\Phi^{1-\nu}))(\Phi L_S \Phi - \frac{\nu}{2} \Gamma_S(\Phi))\zeta_k + o(1) $$
Since $\zeta_k$ is compactly supported, we may integrate by parts and deduce that

$$ 
\begin{aligned}
\int (L_S(\Phi^{1-\nu}))(\Phi L_S\Phi - \frac{\nu}{2}\Gamma_S(\Phi))\zeta_k &= -\int \Gamma_S((\Phi L_S\Phi - \frac{\nu}{2}\Gamma_S(\Phi))\zeta_k, \Phi^{1-\nu}) + o(1) \\
&= -\int \Gamma_S((\Phi L_S\Phi)\zeta_k, \Phi^{1-\nu}) + \frac{\nu}{2}\int \Gamma_S(\Gamma_S(\Phi))\zeta_k, \Phi^{1-\nu}) + o(1) \\
&=: I + II + o(1)
\end{aligned}
\quad (52) $$

Using the product rule to expand derivatives in the first integral, we find

$$ I = -\int \zeta_k L_S \Phi \Gamma_S(\Phi, \Phi^{1-\nu}) - \int \zeta_k \Phi \Gamma_S(L_S \Phi, \Phi^{1-\nu}) - \int (\Phi L_S \Phi) \Gamma_S(\zeta_k, \Phi^{1-\nu}) =: I_1 + I_2 + I_3 $$

Using (50) and the boundedness of $\Phi$ and $\Gamma_S(\Phi)$, we find

$$ I_3 \le C \left( \int \Gamma_S(\zeta_k) \right)^{1/2} = o(1) \quad (53) $$

Next, we deal with $I_1$. Thanks to the product rule for derivatives, we find

$$ 
\begin{aligned}
I_1 &= -\int \Gamma_S(\Phi, \Phi^{1-\nu} L_S \Phi \zeta_k) + \int \Phi^{1-\nu} \zeta_k \Gamma_S(\Phi, L_S \Phi) + \int \Phi^{1-\nu} L_S \Phi \Gamma_S(\Phi, \zeta_k) \\
&= -\int \Gamma_S(\Phi, \Phi^{1-\nu} L_S \Phi \zeta_k) + \int \Phi^{1-\nu} \zeta_k \Gamma_S(\Phi, L_S \Phi) + o(1)
\end{aligned}
$$

Since $\zeta_k$ has compact support, we may integrate by parts to find that

$$ I_1 = - \int \Phi^{1-\nu} (L_S \Phi)^2 \zeta_k + \int \Phi^{1-\nu} \zeta_k \Gamma_S(\Phi, L_S \Phi) + o(1). \quad (54) $$

For $I_2$ at last, the chain rule simply implies that

$$ I_2 = (\nu - 1) \int \Phi^{1-\nu} \zeta_k \Gamma_S(\Phi, L_S \Phi). \quad (55) $$

Now we turn to $II$ and apply the product rule.

$$ \frac{2}{\nu} II = \int \zeta_k \Gamma_S(\Gamma_S(\Phi), \Phi^{1-\nu}) + \int \Gamma_S(\Phi) \Gamma_S(\zeta_k, \Phi^{1-\nu}) = \int \zeta_k \Gamma_S(\Gamma_S(\Phi), \Phi^{1-\nu}) + o(1). $$

Thanks to the product rule again and integration by parts, it follows that

$$ \frac{2}{\nu} II = \int \Gamma_S(\Gamma_S(\Phi), \zeta_k \Phi^{1-\nu}) + \int \Phi^{1-\nu} \Gamma_S(\Gamma_S(\Phi), \zeta_k) = \\ \int \Gamma_S(\Gamma_S(\Phi), \zeta_k \Phi^{1-\nu}) - \int \Gamma_S(\Phi) \Gamma_S(\Phi^{1-\nu}, \zeta_k) - \int \Gamma_S(\Phi) \Phi^{1-\nu} L_S \zeta_k. $$

And so, since $\zeta_k \to 1$ in $D(L_S)$ and $\Phi, \Gamma_S(\Phi)$ are bounded,

$$ \frac{2}{\nu} II = \int \Gamma_S(\Gamma_S(\Phi), \zeta_k \Phi^{1-\nu}) + o(1) = - \int L_S(\Gamma_S(\Phi)) \zeta_k \Phi^{1-\nu} + o(1). \quad (56) $$

Plugging (53), (54), (55), (56) in (52), we find

$$ \int \left( \Gamma_2^S(\Phi) - \frac{1}{\nu} (L_S \Phi)^2 - \frac{c}{\nu} \Gamma_S(\Phi) \right) \Phi^{1-\nu} \zeta_k = o(1). $$

Since $\Phi \in D(L_S)$, thanks to Lemma A.9, we may pass to the limit as $k \to +\infty$ and deduce that

$$ \int \left( \Gamma_2^S(\Phi) - \frac{1}{\nu} (L_S \Phi)^2 - \frac{c}{\nu} \Gamma_S(\Phi) \right) \Phi^{1-\nu} = 0. \quad (57) $$
Since $\zeta_k$ is compactly supported, we may integrate by parts and deduce that

$$ 
\begin{aligned}
\int (L_S(\Phi^{1-\nu}))(\Phi L_S\Phi - \frac{\nu}{2}\Gamma_S(\Phi))\zeta_k &= -\int \Gamma_S((\Phi L_S\Phi - \frac{\nu}{2}\Gamma_S(\Phi))\zeta_k, \Phi^{1-\nu}) + o(1) \\
&= -\int \Gamma_S((\Phi L_S\Phi)\zeta_k, \Phi^{1-\nu}) + \frac{\nu}{2}\int \Gamma_S(\Gamma_S(\Phi))\zeta_k, \Phi^{1-\nu}) + o(1) \\
&=: I + II + o(1)
\end{aligned} 
\quad (52) $$

Using the product rule to expand derivatives in the first integral, we find

$$ I = -\int \zeta_k L_S \Phi \Gamma_S(\Phi, \Phi^{1-\nu}) - \int \zeta_k \Phi \Gamma_S(L_S \Phi, \Phi^{1-\nu}) - \int (\Phi L_S \Phi) \Gamma_S(\zeta_k, \Phi^{1-\nu}) =: I_1 + I_2 + I_3 $$

Using (50) and the boundedness of $\Phi$ and $\Gamma_S(\Phi)$, we find

$$ I_3 \le C \left( \int \Gamma_S(\zeta_k) \right)^{1/2} = o(1) \quad (53) $$

Next, we deal with $I_1$. Thanks to the product rule for derivatives, we find

$$ 
\begin{aligned}
I_1 &= -\int \Gamma_S(\Phi, \Phi^{1-\nu} L_S \Phi \zeta_k) + \int \Phi^{1-\nu} \zeta_k \Gamma_S(\Phi, L_S \Phi) + \int \Phi^{1-\nu} L_S \Phi \Gamma_S(\Phi, \zeta_k) \\
&= -\int \Gamma_S(\Phi, \Phi^{1-\nu} L_S \Phi \zeta_k) + \int \Phi^{1-\nu} \zeta_k \Gamma_S(\Phi, L_S \Phi) + o(1)
\end{aligned} 
$$

Since $\zeta_k$ has compact support, we may integrate by parts to find that

$$ I_1 = - \int \Phi^{1-\nu} (L_S \Phi)^2 \zeta_k + \int \Phi^{1-\nu} \zeta_k \Gamma_S(\Phi, L_S \Phi) + o(1). \quad (54) $$

For $I_2$ at last, the chain rule simply implies that

$$ I_2 = (\nu - 1) \int \Phi^{1-\nu} \zeta_k \Gamma_S(\Phi, L_S \Phi). \quad (55) $$

Now we turn to $II$ and apply the product rule.

$$ \frac{2}{\nu}II = \int \zeta_k \Gamma_S(\Gamma_S(\Phi), \Phi^{1-\nu}) + \int \Gamma_S(\Phi)\Gamma_S(\zeta_k, \Phi^{1-\nu}) = \int \zeta_k \Gamma_S(\Gamma_S(\Phi), \Phi^{1-\nu}) + o(1). $$

Thanks to the product rule again and integration by parts, it follows that

$$ \frac{2}{\nu}II = \int \Gamma_S(\Gamma_S(\Phi), \zeta_k \Phi^{1-\nu}) + \int \Phi^{1-\nu} \Gamma_S(\Gamma_S(\Phi), \zeta_k) = \\ \int \Gamma_S(\Gamma_S(\Phi), \zeta_k \Phi^{1-\nu}) - \int \Gamma_S(\Phi)\Gamma_S(\Phi^{1-\nu}, \zeta_k) - \int \Gamma_S(\Phi)\Phi^{1-\nu} L_S \zeta_k. $$

And so, since $\zeta_k \to 1$ in $D(L_S)$ and $\Phi, \Gamma_S(\Phi)$ are bounded,

$$ \frac{2}{\nu}II = \int \Gamma_S(\Gamma_S(\Phi), \zeta_k \Phi^{1-\nu}) + o(1) = -\int L_S(\Gamma_S(\Phi))\zeta_k \Phi^{1-\nu} + o(1). \quad (56) $$

Plugging (53), (54), (55), (56) in (52), we find

$$ \int \left( \Gamma_2^S(\Phi) - \frac{1}{\nu} (L_S \Phi)^2 - \frac{c}{\nu} \Gamma_S(\Phi) \right) \Phi^{1-\nu} \zeta_k = o(1). $$

Since $\Phi \in D(L_S)$, thanks to Lemma A.9, we may pass to the limit as $k \to +\infty$ and deduce that

$$ \int \left( \Gamma_2^S(\Phi) - \frac{1}{\nu} (L_S \Phi)^2 - \frac{c}{\nu} \Gamma_S(\Phi) \right) \Phi^{1-\nu} = 0. \quad (57) $$
By the integrated curvature-dimension condition (35), we deduce that

$$ \left(\frac{1}{n} - \frac{1}{\nu}\right) \int (L_S \Phi)^2 \Phi^{1-\nu} + \left(\rho - \frac{c}{\nu}\right) \int \Gamma_S(\Phi) \Phi^{1-\nu} \le 0. $$

Since $q < p$, we have $n < \nu$ and so, if $\rho \ge \frac{c}{\nu}$ i.e.

$$ A \ge \frac{4(\nu - 1)}{\nu(\nu - 2)\rho}, $$

we deduce that $L_S \Phi = 0$. Integrating against $\Phi$, $\Phi$ is constant. Hence $v = 1$, $I(A) = 1$, and (47) holds for $A = \frac{4(\nu-1)}{\nu(\nu-2)\rho}$. Let $q \nearrow 2^*$. Then $\nu \searrow n$ and the sharp inequality (17) follows.

It remains to study the case of equality. If $v \in H_0^1(\mu_S)$ is an extremal function for (17), then repeating the above considerations, the function $f = v^{-\frac{p-2}{2}}$ satisfies

$$ \int \left( \Gamma_2^S(f) - \rho \Gamma_S(f) - \frac{1}{n} (L_S f)^2 \right) f^{1-n} d\mu_S = 0, $$

In particular, if the parameters are such that inequality (46) is strict, it follows from (45) that $f$ must be a function of $s$ only. If $\frac{n-d}{n-1} + \mathfrak{B}_{DGZ} = 0$, then the estimate (40) in Lemma 3.8 provides the following improvement of (45):

$$ \int \left( \Gamma_2^S(f) - \rho \Gamma_S(f) - \frac{1}{n} (L_S f)^2 \right) f^{1-n} d\mu_S \ge \left[ \frac{n-d}{n-1} + \mathfrak{B}_{DGZ} \right] \int \Gamma^\theta(f) \varphi^4 f^{1-n} d\mu_S \\ + \frac{n-d}{n-1} A \int \frac{\Gamma^\theta(f)^2}{f^2} \varphi^4 f^{1-n} d\mu_S. $$

And so again, $f$ is a function of $s$ only, provided $n > d$. Using this information in (39), we deduce that if $n > d$,

$$ \partial_{ss} f + 2 \frac{\varphi'}{\varphi} \partial_s f = 0, $$

while for $n = d$ there must exist some function $R: \mathbf{R} \to \mathbf{R}$ s.t.

$$ \partial_{ss} f + 2 \frac{\varphi'}{\varphi} \partial_s f - \frac{1}{d-1} \Delta_{\theta} f = 0 \quad \text{and} \quad \partial_s f + \frac{\varphi'}{\varphi} f = R(s). \qquad (58) $$

In the former case, this means that $f(s) = \lambda + \gamma \tanh(\alpha s)$, for some constants $\lambda, \gamma \in \mathbf{R}$ such that $\lambda > |\gamma|$, since $f$ is bounded below by a positive constant. In the latter case, the second equation in (58) implies that $f$ can be written as $f = \frac{f_1(\theta)}{\varphi(s)} + f_2(s)$. Plugging this in the first equation implies that $f_1 + \frac{\Delta_{\theta} f_1}{d-1}$ is constant i.e. $f_1 = A_1 + B_1 \psi_2(\theta)$, where $A_1, B_1$ are constants and $\psi_2$ is any eigenfunction of $-\Delta_{\theta}$ associated to the eigenvalue $d-1$. This implies in turn that $f_2$ takes the form $f_2 = -\frac{A_1}{\varphi(s)} + A_3 + A_4 \tanh(s)$. Summarizing, we have just proved that $f = \lambda + \gamma \varphi_{1,d}$ for some constants $\lambda, \gamma$ and some eigenfunction $\varphi_{1,d}$ of $-\Delta_{\theta}$ associated to the eigenvalue $d$ (and written in cylindrical coordinates). Again, we must have $\lambda > |\gamma||\varphi_{1,d}|_{\infty}$ since $f$ is bounded below by a positive constant.

Conversely, we need to check that $f^{-\frac{n-2}{2}}$ where $f(s) = \lambda + \gamma \tanh(\alpha s)$ with $\lambda > |\gamma|$ if $n > d$ (resp. $f = \lambda + \gamma \varphi_{1,d}$, $\lambda > |\gamma||\varphi_{1,d}|_{\infty}$ if $n = d$) is indeed an extremal function for Sobolev's inequality. Multiplying $f$ by a constant if necessary, we may assume that $\int f^{-n} d\mu = 1$, where $\mu$ is the normalized measure on the CKN sphere. By direct computation, recalling that $\tanh(\alpha s)$ if $n > d$ (resp. $\varphi_{1,d}$ if $n = d$) is an eigenfunction for the operator $-L_S$ associated to the eigenvalue $n\alpha^2$, we find that

$$ f L_S f - \frac{n}{2} \Gamma_S(f) = \frac{n\alpha^2}{2} (1 - f^2) $$

This implies in turn that $v = f^{-\frac{n-2}{2}}$ satisfies $\int v^p d\mu = 1$ and solves

$$ -\frac{4}{n(n-2)\alpha^2} L_S v + v = v^{p-1} $$
By the integrated curvature-dimension condition (35), we deduce that

$$ \left(\frac{1}{n} - \frac{1}{\nu}\right) \int (L_S \Phi)^2 \Phi^{1-\nu} + \left(\rho - \frac{c}{\nu}\right) \int \Gamma_S(\Phi) \Phi^{1-\nu} \le 0. $$

Since $q < p$, we have $n < \nu$ and so, if $\rho \ge \frac{c}{\nu}$ i.e.

$$ A \ge \frac{4(\nu - 1)}{\nu(\nu - 2)\rho}, $$

we deduce that $L_S \Phi = 0$. Integrating against $\Phi$, $\Phi$ is constant. Hence $v = 1$, $I(A) = 1$, and (47) holds for $A = \frac{4(\nu-1)}{\nu(\nu-2)\rho}$. Let $q \nearrow 2^*$. Then $\nu \searrow n$ and the sharp inequality (17) follows.

It remains to study the case of equality. If $v \in H_0^1(\mu_S)$ is an extremal function for (17), then repeating the above considerations, the function $f = v^{-\frac{p-2}{2}}$ satisfies

$$ \int \left( \Gamma_2^S(f) - \rho \Gamma_S(f) - \frac{1}{n} (L_S f)^2 \right) f^{1-n} d\mu_S = 0, $$

In particular, if the parameters are such that inequality (46) is strict, it follows from (45) that $f$ must be a function of $s$ only. If $\frac{n-d}{n-1} + \mathfrak{B}_{DGZ} = 0$, then the estimate (40) in Lemma 3.8 provides the following improvement of (45):

$$ \int \left( \Gamma_2^S(f) - \rho \Gamma_S(f) - \frac{1}{n} (L_S f)^2 \right) f^{1-n} d\mu_S \ge \left[ \frac{n-d}{n-1} + \mathfrak{B}_{DGZ} \right] \int \Gamma^\theta(f) \varphi^4 f^{1-n} d\mu_S \\ + \frac{n-d}{n-1} A \int \frac{\Gamma^\theta(f)^2}{f^2} \varphi^4 f^{1-n} d\mu_S. $$

And so again, $f$ is a function of $s$ only, provided $n > d$. Using this information in (39), we deduce that if $n > d$,

$$ \partial_{ss} f + 2 \frac{\varphi'}{\varphi} \partial_s f = 0, $$

while for $n = d$ there must exist some function $R: \mathbf{R} \to \mathbf{R}$ s.t.

$$ \partial_{ss} f + 2 \frac{\varphi'}{\varphi} \partial_s f - \frac{1}{d-1} \Delta_{\theta} f = 0 \quad \text{and} \quad \partial_s f + \frac{\varphi'}{\varphi} f = R(s). \qquad (58) $$

In the former case, this means that $f(s) = \lambda + \gamma \tanh(\alpha s)$, for some constants $\lambda, \gamma \in \mathbf{R}$ such that $\lambda > |\gamma|$, since $f$ is bounded below by a positive constant. In the latter case, the second equation in (58) implies that $f$ can be written as $f = \frac{f_1(\theta)}{\varphi(s)} + f_2(s)$. Plugging this in the first equation implies that $f_1 + \frac{\Delta_{\theta} f_1}{d-1}$ is constant i.e. $f_1 = A_1 + B_1 \psi_2(\theta)$, where $A_1, B_1$ are constants and $\psi_2$ is any eigenfunction of $-\Delta_{\theta}$ associated to the eigenvalue $d-1$. This implies in turn that $f_2$ takes the form $f_2 = -\frac{A_1}{\varphi(s)} + A_3 + A_4 \tanh(s)$. Summarizing, we have just proved that $f = \lambda + \gamma \varphi_{1,d}$ for some constants $\lambda, \gamma$ and some eigenfunction $\varphi_{1,d}$ of $-\Delta_{\theta}$ associated to the eigenvalue $d$ (and written in cylindrical coordinates). Again, we must have $\lambda > |\gamma| \|\varphi_{1,d}\|_{\infty}$ since $f$ is bounded below by a positive constant.

Conversely, we need to check that $f^{-\frac{n-2}{2}}$ where $f(s) = \lambda + \gamma \tanh(\alpha s)$ with $\lambda > |\gamma|$ if $n > d$ (resp. $f = \lambda + \gamma \varphi_{1,d}$, $\lambda > |\gamma| \|\varphi_{1,d}\|_{\infty}$ if $n = d$) is indeed an extremal function for Sobolev's inequality. Multiplying $f$ by a constant if necessary, we may assume that $\int f^{-n} d\mu = 1$, where $\mu$ is the normalized measure on the CKN sphere. By direct computation, recalling that $\tanh(\alpha s)$ if $n > d$ (resp. $\varphi_{1,d}$ if $n = d$) is an eigenfunction for the operator $-L_S$ associated to the eigenvalue $n\alpha^2$, we find that

$$ f L_S f - \frac{n}{2} \Gamma_S(f) = \frac{n\alpha^2}{2} (1 - f^2) $$

This implies in turn that $v = f^{-\frac{n-2}{2}}$ satisfies $\int v^p d\mu = 1$ and solves

$$ -\frac{4}{n(n-2)\alpha^2} L_S v + v = v^{p-1} $$
Multiplying by $v$ and integrating by parts, the result follows. ▷

**Proof of Proposition 1.10**

As explained in the introduction, Poincaré's inequality (with constant $C = \frac{n-1}{pn} = \frac{1}{n\alpha^2}$) follows from Sobolev's inequality by linearization i.e. by applying (17) with $v = 1 + \epsilon f$ and letting $\epsilon \to 0$. Also, Poincaré's inequality (with the same constant $C$) is equivalent to the following integrated curvature-dimension condition

$$ \int \left( \Gamma_{2}^{\mathbf{s}}(f) - \rho \Gamma_{\mathbf{s}}(f) - \frac{1}{n} (L_{\mathbf{s}} f)^2 \right) d\mu_{\mathbf{s}} \geq 0. $$

Equality holds in Poincaré's inequality for some function $f$ if and only if equality holds in the above inequality. So, extremals are characterized exactly as in the case of Sobolev's inequality except in the case $\alpha^2 = \frac{d-1}{n-1}$, in which we can no longer use (40) to deduce that $f$ is radial. Still, we deduce from (39) that

$$ \partial_{ss} f + 2 \frac{\varphi'}{\varphi} \partial_s f - \frac{1}{n-1} \Delta_{\theta} f = 0 \quad \text{and} \quad \partial_s f + \frac{\varphi'}{\varphi} f = R(s) $$

The second equation in (58) implies that $f$ can be written as $f = \frac{f_1(\theta)}{\varphi(s)} + f_2(s)$. Plugging this in the first equation implies that $\alpha^2 f_1 + \frac{\Delta_{\theta} f_1}{n-1}$ is constant i.e. $f_1 = A_1 + B_1 \varphi_{1,d-1}(\omega)$, where $A_1, B_1$ are constants and $\varphi_{1,d-1}$ is any eigenfunction of $-\Delta_{\theta}$ associated to the eigenvalue $\alpha^2(n-1) = d-1$. This implies in turn that $f_2$ takes the form $f_2 = -\frac{A_1}{\varphi(s)} + A_3 + A_4 \tanh(\alpha s)$. Summarizing, we have just proved that extremals of Poincaré's inequality take the form $f = \lambda + \gamma \tanh(\alpha s) + \nu \frac{\varphi_{1,d-1}(\omega)}{\cosh(\alpha s)}$ for some constants $\lambda, \gamma, \nu$ and some eigenfunction $\varphi_{1,d-1}$ of $-\Delta_{\mathbf{s}}^{d-1}$, as desired. ▷

**Remark 3.9.** Up to our knowledge, the CKN sphere is the first example where the optimal constants for both the Sobolev and the Poincaré inequalities are explicit functions of $(\rho, n)$ yet the usual curvature-dimension condition doesn't hold, although the integral version (15) remains true. Beware though that the integrated curvature-dimension needed for (and equivalent to) Poincaré's inequality, i.e. inequality (15) without the weight $f^{1-n}$, is in general much weaker, as evidenced by any space for which the Poincaré inequality holds but not the Sobolev inequality, such as, for instance, the Euclidean space equipped with the Gaussian measure.

# 4 The $n$-conformal invariant

## 4.1 The $n$-conformal invariant on a weighted manifold

We begin this section by proving Proposition 1.12, which constructs a one-parameter family of $n$-conformal invariants on any given weighted manifold, thereby generalizing the notion of scalar curvature to this setting.

**Proof of Proposition 1.12**

We want to check that $S_{\gamma}(\mu, \Gamma)$ satisfies condition (21). Let $c$ be a positive and smooth function on $M$, $\tau = \log c$ and $\gamma \in \mathbb{R}$. We are looking for the expression of the two numbers $\theta_n(\gamma)$ and $\beta_n(\gamma)$ in the definition of $S_{\gamma}(\mu, \Gamma)$ which are such that

$$ S_{\gamma}(c^{-n}\mu, c^2\Gamma) = c^2 \left[ S_{\gamma}(\mu, \Gamma) + \frac{n-2}{2} \left( L\tau - \frac{n-2}{2}\Gamma(\tau) \right) \right]. $$

The measure $\mu$ is transformed into $\tilde{\mu} = c^{-n}\mu$, and the carré du champ $\Gamma$ into $\tilde{\Gamma} = c^2\Gamma$. From (64), $sc_{\mathfrak{g}}$ becomes

$$ \check{sc}_{\mathfrak{g}} = c^2[sc_{\mathfrak{g}} + (d-1)(2\Delta_{\mathfrak{g}}\tau - (d-2)\Gamma(\tau))], $$

the weight $W = -\log \frac{d\mu}{d\check{V}_{\mathfrak{g}}}$ becomes

$$ \check{W} = -\log \frac{d\tilde{\mu}}{d\check{V}_{\mathfrak{g}}} = -\log \frac{c^{-n}d\mu}{c^{-d}d\check{V}_{\mathfrak{g}}} = -\log \left( c^{d-n} \frac{d\mu}{d\check{V}_{\mathfrak{g}}} \right) = W + (n-d)\tau, $$
Multiplying by $v$ and integrating by parts, the result follows. ▷

**Proof of Proposition 1.10**

As explained in the introduction, Poincaré's inequality (with constant $C = \frac{n-1}{\rho n} = \frac{1}{n\alpha^2}$) follows from Sobolev's inequality by linearization i.e. by applying (17) with $v = 1 + \epsilon f$ and letting $\epsilon \to 0$. Also, Poincaré's inequality (with the same constant $C$) is equivalent to the following integrated curvature-dimension condition

$$ \int \left( \Gamma_{2}^{\mathbf{s}}(f) - \rho \Gamma_{\mathbf{s}}(f) - \frac{1}{n} (L_{\mathbf{s}} f)^2 \right) d\mu_{\mathbf{s}} \geq 0. $$

Equality holds in Poincaré's inequality for some function $f$ if and only if equality holds in the above inequality. So, extremals are characterized exactly as in the case of Sobolev's inequality except in the case $\alpha^2 = \frac{d-1}{n-1}$, in which we can no longer use (40) to deduce that $f$ is radial. Still, we deduce from (39) that

$$ \partial_{ss} f + 2 \frac{\varphi'}{\varphi} \partial_s f - \frac{1}{n-1} \Delta_{\theta} f = 0 \quad \text{and} \quad \partial_s f + \frac{\varphi'}{\varphi} f = R(s) $$

The second equation in (58) implies that $f$ can be written as $f = \frac{f_1(\theta)}{\varphi(s)} + f_2(s)$. Plugging this in the first equation implies that $\alpha^2 f_1 + \frac{\Delta_{\theta} f_1}{n-1}$ is constant i.e. $f_1 = A_1 + B_1 \varphi_{1,d-1}(\omega)$, where $A_1, B_1$ are constants and $\varphi_{1,d-1}$ is any eigenfunction of $-\Delta_{\theta}$ associated to the eigenvalue $\alpha^2(n-1) = d-1$. This implies in turn that $f_2$ takes the form $f_2 = -\frac{A_1}{\varphi(s)} + A_3 + A_4 \tanh(\alpha s)$. Summarizing, we have just proved that extremals of Poincaré's inequality take the form $f = \lambda + \gamma \tanh(\alpha s) + \nu \frac{\varphi_{1,d-1}(\omega)}{\cosh(\alpha s)}$ for some constants $\lambda, \gamma, \nu$ and some eigenfunction $\varphi_{1,d-1}$ of $-\Delta_{\mathbf{s}}^{d-1}$, as desired. ▷

**Remark 3.9.** Up to our knowledge, the CKN sphere is the first example where the optimal constants for both the Sobolev and the Poincaré inequalities are explicit functions of $(\rho, n)$ yet the usual curvature-dimension condition doesn't hold, although the integral version (15) remains true. Beware though that the integrated curvature-dimension needed for (and equivalent to) Poincaré's inequality, i.e. inequality (15) without the weight $f^{1-n}$, is in general much weaker, as evidenced by any space for which the Poincaré inequality holds but not the Sobolev inequality, such as, for instance, the Euclidean space equipped with the Gaussian measure.

# 4 The $n$-conformal invariant

## 4.1 The $n$-conformal invariant on a weighted manifold

We begin this section by proving Proposition 1.12, which constructs a one-parameter family of $n$-conformal invariants on any given weighted manifold, thereby generalizing the notion of scalar curvature to this setting.

**Proof of Proposition 1.12**

We want to check that $S_{\gamma}(\mu, \Gamma)$ satisfies condition (21). Let $c$ be a positive and smooth function on $M$, $\tau = \log c$ and $\gamma \in \mathbb{R}$. We are looking for the expression of the two numbers $\theta_n(\gamma)$ and $\beta_n(\gamma)$ in the definition of $S_{\gamma}(\mu, \Gamma)$ which are such that

$$ S_{\gamma}(c^{-n}\mu, c^2\Gamma) = c^2 \left[ S_{\gamma}(\mu, \Gamma) + \frac{n-2}{2} \left( L\tau - \frac{n-2}{2}\Gamma(\tau) \right) \right]. $$

The measure $\mu$ is transformed into $\tilde{\mu} = c^{-n}\mu$, and the carré du champ $\Gamma$ into $\tilde{\Gamma} = c^2\Gamma$. From (64), $sc_{\mathfrak{g}}$ becomes

$$ \check{sc}_{\mathfrak{g}} = c^2[sc_{\mathfrak{g}} + (d-1)(2\Delta_{\mathfrak{g}}\tau - (d-2)\Gamma(\tau))], $$

the weight $W = - \log \frac{d\mu}{d\check{V}_{\mathfrak{g}}}$ becomes

$$ \check{W} = - \log \frac{d\tilde{\mu}}{d\check{V}_{\mathfrak{g}}} = - \log \frac{c^{-n} d\mu}{c^{-d} d\check{V}_{\mathfrak{g}}} = - \log \left( c^{d-n} \frac{d\mu}{d\check{V}_{\mathfrak{g}}} \right) = W + (n-d)\tau, $$
and finally, from (60), $\Delta_g$ becomes

$$ \check{\Delta}_g = c^2[\Delta_g - (d-2)\Gamma(\tau, \cdot)]. $$

So,

$$ S_{\gamma}(c^{-n}\mu, c^2\Gamma) = c^2\theta_n(\gamma) \left[ \begin{aligned} &sc_g + (d-1)(2\Delta_g(\tau) - (d-2)\Gamma(\tau)) \\ & - \gamma[\Delta_g(W + (n-d)\tau) - (d-2)\Gamma(\tau, W + (n-d)\tau)] + \beta_n(\gamma)\Gamma(W + (n-d)\tau) \end{aligned} \right] $$

that is

$$ S_{\gamma}(c^{-n}\mu, c^2\Gamma) = c^2\theta_n(\gamma) \left[ \begin{aligned} &sc_g + [2(d-1) - \gamma(n-d)]\Delta_g(\tau) \\ & + [\beta_n(\gamma)(n-d)^2 - (d-1)(d-2) + \gamma(d-2)(n-d)]\Gamma(\tau) \\ & - \gamma\Delta_g(W) + [\gamma(d-2) + 2\beta_n(\gamma)(n-d)]\Gamma(\tau, W) + \beta_n(\gamma)\Gamma(W) \end{aligned} \right]. $$

It has to be equal to

$$ c^2 \left[ S_{\gamma}(\mu, \Gamma) + \frac{n-2}{2} \left( \Delta_g(\tau) - \Gamma(W, \tau) - \frac{n-2}{2} \Gamma(\tau) \right) \right] = c^2 \left[ \theta_n(\gamma) [sc_g - \gamma \Delta_g(W) + \beta_n(\gamma) \Gamma(W)] + \frac{n-2}{2} \left( \Delta_g(\tau) - \Gamma(W, \tau) - \frac{n-2}{2} \Gamma(\tau) \right) \right], $$

that is

$$ \left\{ \begin{array}{l} \theta_n(\gamma)[2(d-1) - \gamma(n-d)] = \frac{n-2}{2} \\ \\ \theta_n(\gamma)[\beta_n(\gamma)(n-d)^2 - (d-1)(d-2) + \gamma(d-2)(n-d)] = -\frac{(n-2)^2}{4} \\ \\ \theta_n(\gamma)[\gamma(d-2) + 2\beta_n(\gamma)(n-d)] = -\frac{n-2}{2} \end{array} \right. \qquad (59) $$

which implies that

$$ \left\{ \begin{array}{l} \theta_n(\gamma) = \frac{n-2}{4(d-1) - 2\gamma(n-d)} \\ \\ \beta_n(\gamma) = \frac{\gamma(n-2d+2) - 2(d-1)}{2(n-d)}. \end{array} \right. $$

Let us notice that the second equation in (59) is automatically valid for this choice of parameters $\theta_n(\gamma)$ and $\beta_n(\gamma)$ and so we are done. ▷

**Remark 4.1.** As explained in the introduction, when $W = 0$ the $d$-conformal invariant is, up to a multiplicative constant, the scalar curvature. In a weighted Riemannian manifold, the $n$-conformal invariant is given by (23) and is a way to extend the definition of the scalar curvature in the weighted case.

## 4.2 The $n$-conformal invariant for the CKN spaces

In this section, we would like to prove that the three CKN spaces enjoy, for some $\gamma \in \mathbb{R}$, a constant $n$-conformal invariant. By construction, the three CKN models (Euclidean, spherical and hyperbolic) belong to the same $n$-conformal class. So, in virtue of Theorem 1.4 and Proposition 1.12, it suffices to prove that there exists a unique $\gamma \in \mathbb{R}$ such that $S_{\gamma} = 0$ for the Euclidean CKN space in order to prove Proposition 1.13.

### Proof of Proposition 1.13

Let $\gamma \in \mathbb{R}$. Then,

$$ S_{\gamma}(\mu_{\mathbb{E}}, \Gamma_{\mathbb{E}}) = \theta_{n}(\gamma)(sc_{\mathbb{g}_{\mathbb{E}}}-\gamma\Delta_{\mathbb{g}_{\mathbb{E}}}W_{\mathbb{E}}+\beta_{n}(\gamma)\Gamma_{\mathbb{E}}(W_{\mathbb{E}})). $$

So, we need to find $\gamma$ such that $sc_{\mathbb{g}_{\mathbb{E}}} - \gamma\Delta_{\mathbb{g}_{\mathbb{E}}}W_{\mathbb{E}} + \beta_n(\gamma)\Gamma_{\mathbb{E}}(W_{\mathbb{E}}) = 0$.

* Computation of the scalar curvature. From the identity (64) with $\Gamma_{\mathbb{E}} = c_{\mathbb{E}}\Gamma$ and $\tau_{\mathbb{E}} = \log c_{\mathbb{E}}$,

$$ sc_{\mathbb{g}_{\mathbb{E}}} = |x|^{2(1-\alpha)}(0 + (d-1)(2\Delta\tau_{\mathbb{E}} - (d-2)|\nabla\tau_{\mathbb{E}}|^2)), $$
and finally, from (60), $\Delta_g$ becomes

$$ \check{\Delta}_g = c^2[\Delta_g - (d-2)\Gamma(\tau, \cdot)]. $$

So,

$$ S_{\gamma}(c^{-n}\mu, c^2\Gamma) = c^2\theta_n(\gamma) \left[ \begin{aligned} & sc_g + (d-1)(2\Delta_g(\tau) - (d-2)\Gamma(\tau)) \\ & - \gamma[\Delta_g(W + (n-d)\tau) - (d-2)\Gamma(\tau, W + (n-d)\tau)] + \beta_n(\gamma)\Gamma(W + (n-d)\tau) \end{aligned} \right] $$

that is

$$ S_{\gamma}(c^{-n}\mu, c^2\Gamma) = c^2\theta_n(\gamma) \left[ \begin{aligned} & sc_g + [2(d-1) - \gamma(n-d)]\Delta_g(\tau) \\ & + [\beta_n(\gamma)(n-d)^2 - (d-1)(d-2) + \gamma(d-2)(n-d)]\Gamma(\tau) \\ & - \gamma\Delta_g(W) + [\gamma(d-2) + 2\beta_n(\gamma)(n-d)]\Gamma(\tau, W) + \beta_n(\gamma)\Gamma(W) \end{aligned} \right]. $$

It has to be equal to

$$ c^2 \left[ S_{\gamma}(\mu, \Gamma) + \frac{n-2}{2} \left( \Delta_g(\tau) - \Gamma(W, \tau) - \frac{n-2}{2} \Gamma(\tau) \right) \right] = c^2 \left[ \theta_n(\gamma) [sc_g - \gamma \Delta_g(W) + \beta_n(\gamma) \Gamma(W)] + \frac{n-2}{2} \left( \Delta_g(\tau) - \Gamma(W, \tau) - \frac{n-2}{2} \Gamma(\tau) \right) \right], $$

that is

$$ \left\{ \begin{array}{l} \theta_n(\gamma)[2(d-1) - \gamma(n-d)] = \frac{n-2}{2} \\ \\ \theta_n(\gamma)[\beta_n(\gamma)(n-d)^2 - (d-1)(d-2) + \gamma(d-2)(n-d)] = -\frac{(n-2)^2}{4} \\ \\ \theta_n(\gamma)[\gamma(d-2) + 2\beta_n(\gamma)(n-d)] = -\frac{n-2}{2} \end{array} \right. \qquad (59) $$

which implies that

$$ \left\{ \begin{array}{l} \theta_n(\gamma) = \frac{n-2}{4(d-1) - 2\gamma(n-d)} \\ \\ \beta_n(\gamma) = \frac{\gamma(n-2d+2) - 2(d-1)}{2(n-d)}. \end{array} \right. $$

Let us notice that the second equation in (59) is automatically valid for this choice of parameters $\theta_n(\gamma)$ and $\beta_n(\gamma)$ and so we are done. ▷

**Remark 4.1.** As explained in the introduction, when $W = 0$ the $d$-conformal invariant is, up to a multiplicative constant, the scalar curvature. In a weighted Riemannian manifold, the $n$-conformal invariant is given by (23) and is a way to extend the definition of the scalar curvature in the weighted case.

## 4.2 The $n$-conformal invariant for the CKN spaces

In this section, we would like to prove that the three CKN spaces enjoy, for some $\gamma \in \mathbb{R}$, a constant $n$-conformal invariant. By construction, the three CKN models (Euclidean, spherical and hyperbolic) belong to the same $n$-conformal class. So, in virtue of Theorem 1.4 and Proposition 1.12, it suffices to prove that there exists a unique $\gamma \in \mathbb{R}$ such that $S_{\gamma} = 0$ for the Euclidean CKN space in order to prove Proposition 1.13.

**Proof of Proposition 1.13**

Let $\gamma \in \mathbb{R}$. Then,

$$ S_{\gamma}(\mu_{\mathbb{E}}, \Gamma_{\mathbb{E}}) = \theta_{n}(\gamma)(sc_{\mathbb{g}_{\mathbb{E}}}-\gamma\Delta_{\mathbb{g}_{\mathbb{E}}}W_{\mathbb{E}}+\beta_{n}(\gamma)\Gamma_{\mathbb{E}}(W_{\mathbb{E}})). $$

So, we need to find $\gamma$ such that $sc_{\mathbb{g}_{\mathbb{E}}} - \gamma\Delta_{\mathbb{g}_{\mathbb{E}}}W_{\mathbb{E}} + \beta_n(\gamma)\Gamma_{\mathbb{E}}(W_{\mathbb{E}}) = 0$.

* Computation of the scalar curvature. From the identity (64) with $\Gamma_{\mathbb{E}} = c_{\mathbb{E}}\Gamma$ and $\tau_{\mathbb{E}} = \log c_{\mathbb{E}}$,

$$ sc_{\mathbb{g}_{\mathbb{E}}} = |x|^{2(1-\alpha)}(0 + (d-1)(2\Delta\tau_{\mathbb{E}} - (d-2)|\nabla\tau_{\mathbb{E}}|^2)), $$
hence,

$$sc_{g_E} = |x|^{-2\alpha}(d-1)(d-2)(1-\alpha^2).$$

• Computation of $\Delta_{g_E} W_E$. First, from the identity (60),

$$\Delta_{g_E} W_E = |x|^{2(1-\alpha)\kappa} (\Delta W_E - (d-2)\nabla_{\tau_E} \cdot \nabla W_E),$$

so

$$\Delta_{g_E} W_E = |x|^{-2\alpha} (d-2)\alpha^2 (n-d).$$

• Computation of $\Gamma_E(W_E)$. We have

$$\Gamma_E(W_E) = |x|^{-2\alpha} \alpha^h a^2 (n-d)^2.$$

So, in the end,

$$S_{\gamma}(\mu, \Gamma) = \theta_{n}(\gamma) |x|^{-2\alpha} \left( (d-1)(d-2)(1-\alpha^2) + \gamma \alpha^2 (n-d)(d-2) + \beta_{n}(\gamma) \alpha^2 (n-d)^2 \right),$$

and we need to find $\gamma \in \mathbb{R}$ such that

$$(d-1)(d-2)(1-\alpha^2) + \gamma\alpha^2(n-d)(d-2) + \beta_n(\gamma)\alpha^2(n-d)^2 = 0.$$

Since

$$\beta_n(\gamma) = \frac{\gamma(n - 2d + 2) - 2(d - 1)}{2(n - d)}$$

we have

$$\gamma = 2 \frac{(d-1)(d-2 + \alpha^2(2-n))}{\alpha^2(n-d)(2-n)}.$$

or by using the constant $\mathfrak{B}_{DGZ}$,

$$\gamma = \frac{2(d-1)}{\alpha^2(n-d)(2-n)} \mathfrak{B}_{DGZ}.$$

△

Remark 4.2. It is interesting to notice that the $n$-conformal invariant for the CKN spaces does not depend on the sign of $\mathfrak{B}_{DGZ}$ or the Felli-Schneider region.

# A Appendix

## A.1 Some Riemannian formulas

We recall here some general formulas on conformal transformations of a $d$-dimensional Riemannian manifold $(M, \mathfrak{g})$. All formulas can be found for example in [BGL14, Sec. 6.9]$^{12}$. We transform the metric (with upper indices) $\mathfrak{g}^{ij}$ into the conformal metric $\mathfrak{h}^{ij} = c^2 \mathfrak{g}^{ij}$, where $c$ is any positive and smooth function. We let $\tau = \log c$. Then,

• The carré du champ operator is given by

$$\Gamma^{\mathfrak{h}} = c^2 \Gamma^{\mathfrak{g}}.$$

• The Laplace-Beltrami operator is given by

$$\Delta_{\mathfrak{h}} = c^2 (\Delta_{\mathfrak{g}} - (d-2)\Gamma^{\mathfrak{g}}(\tau, \cdot)). \qquad (60)$$

<footnote>$^{12}$and also here https://en.wikipedia.org/wiki/List_of formulas_in_Riemannian_geometry</footnote>
hence,

$$sc_{g_E} = |x|^{-2\alpha}(d-1)(d-2)(1-\alpha^2).$$

* Computation of $\Delta_{g_E} W_E$. First, from the identity (60),

$$\Delta_{g_E} W_E = |x|^{2(1-\alpha)\kappa} (\Delta W_E - (d-2)\nabla_{\tau_E} \cdot \nabla W_E),$$

so

$$\Delta_{g_E} W_E = |x|^{-2\alpha} (d-2)\alpha^2 (n-d).$$

* Computation of $\Gamma_E(W_E)$. We have

$$\Gamma_E(W_E) = |x|^{-2\alpha} \alpha^h a^2 (n-d)^2.$$

So, in the end,

$$S_{\gamma}(\mu, \Gamma) = \theta_n(\gamma) |x|^{-2\alpha} \left( (d-1)(d-2)(1-\alpha^2) + \gamma \alpha^2 (n-d)(d-2) + \beta_n(\gamma) \alpha^2 (n-d)^2 \right),$$

and we need to find $\gamma \in \mathbb{R}$ such that

$$(d-1)(d-2)(1-\alpha^2) + \gamma \alpha^2 (n-d)(d-2) + \beta_n(\gamma) \alpha^2 (n-d)^2 = 0.$$

Since

$$\beta_n(\gamma) = \frac{\gamma(n - 2d + 2) - 2(d - 1)}{2(n - d)}$$

we have

$$\gamma = 2 \frac{(d-1)(d-2+\alpha^2(2-n))}{\alpha^2(n-d)(2-n)}.$$

or by using the constant $\mathfrak{B}_{DGZ}$,

$$\gamma = \frac{2(d-1)}{\alpha^2(n-d)(2-n)} \mathfrak{B}_{DGZ}.$$

△

*Remark 4.2.* It is interesting to notice that the $n$-conformal invariant for the CKN spaces does not depend on the sign of $\mathfrak{B}_{DGZ}$ or the Felli-Schneider region.

# A Appendix

## A.1 Some Riemannian formulas

We recall here some general formulas on conformal transformations of a $d$-dimensional Riemannian manifold $(M, \mathfrak{g})$. All formulas can be found for example in [BGL14, Sec. 6.9]$^{12}$. We transform the metric (with upper indices) $\mathfrak{g}^{ij}$ into the conformal metric $\mathfrak{h}^{ij} = c^2 \mathfrak{g}^{ij}$, where $c$ is any positive and smooth function. We let $\tau = \log c$. Then,

* The carré du champ operator is given by

$$\Gamma^{\mathfrak{h}} = c^2 \Gamma^{\mathfrak{g}}.$$

* The Laplace-Beltrami operator is given by

$$\Delta_{\mathfrak{h}} = c^2 (\Delta_{\mathfrak{g}} - (d-2)\Gamma^{\mathfrak{g}}(\tau, \cdot)). \qquad (60)$$

<footnote>$^{12}$and also here https://en.wikipedia.org/wiki/List_of formulas_in_Riemannian_geometry</footnote>
• For any smooth function $\psi$, the Hessian of $\psi$ with respect to the metric $\mathfrak{h}$, denoted $\nabla\nabla^\mathfrak{h}\psi$ is given by

$$ \nabla\nabla^\mathfrak{h}\psi = \nabla\nabla^\mathfrak{g}\psi + 2\nabla^\mathfrak{g}\psi \odot_{\mathfrak{g}} \nabla^\mathfrak{g}\tau - \Gamma^\mathfrak{g}(\psi, \tau)_{\mathfrak{g}}, \quad (61) $$

Here and below, $\nabla\nabla^\mathfrak{g}\psi$ is the Hessian of $\psi$ with respect to $\mathfrak{g}$ and $\nabla^\mathfrak{g}\psi \odot_{\mathfrak{g}} \nabla^\mathfrak{g}\tau$ is the symmetric tensor product, that is for any functions $f, g$,

$$ (\nabla^\mathfrak{g}\psi \odot_{\mathfrak{g}} \nabla^\mathfrak{g}\tau)(\nabla^\mathfrak{g}f, \nabla^\mathfrak{g}g) = \frac{1}{2} [\Gamma^\mathfrak{g}(f, \psi)\Gamma^\mathfrak{g}(g, \tau) + \Gamma^\mathfrak{g}(f, \tau)\Gamma^\mathfrak{g}(g, \psi)]. $$

In particular, one can deduce the Hilbert-Schmidt norm of $\nabla\nabla^\mathfrak{h}\psi$ with respect to the new metric $\mathfrak{h}$:

$$ \|\nabla\nabla^\mathfrak{h}\psi\|^2 = c^4 \left[ \|\nabla\nabla^\mathfrak{g}\psi\|^2 + 2\Gamma^\mathfrak{g}(\tau, \Gamma^\mathfrak{g}(\psi)) + 2\Gamma^\mathfrak{g}(\psi)\Gamma^\mathfrak{g}(\tau) + (d-2)\Gamma^\mathfrak{g}(\psi, \tau)^2 - 2(\Delta_{\mathfrak{g}}\psi)\Gamma^\mathfrak{g}(\psi, \tau) \right]. \quad (62) $$

• The Ricci curvature reads

$$ Ric_{\mathfrak{h}} = Ric_{\mathfrak{g}} + (\Delta_{\mathfrak{g}}\tau)_{\mathfrak{g}} + (d-2)(\nabla\nabla^\mathfrak{g}\tau + \nabla\tau \odot_{\mathfrak{g}} \nabla\tau - \Gamma^\mathfrak{g}(\tau)_{\mathfrak{g}}) \quad (63) $$

• At last, the scalar curvature is given by

$$ sc_{\mathfrak{h}} = c^2 [sc_{\mathfrak{g}} + (d-1)(2\Delta_{\mathfrak{g}}\tau - (d-2)\Gamma^\mathfrak{g}(\tau))] \quad (64) $$

## A.2 Integration by parts and elliptic theory on the CKN spherical space

Let $H_0^1(\mu_S)$ denote the closure of $C_c^\infty(\mathbf{R}^d \setminus \{0\})$ with respect to the norm

$$ \|u\|_{H_0^1(\mu_S)}^2 = \int (\Gamma_S(u) + u^2) \, d\mu_S $$

Let $u \in L^2(\mu_S)$. Then, $|x|^{1-\alpha} \frac{1+|x|^2\alpha}{2} \nabla u$ and $L_S u$ are well-defined distributions on $\mathbf{R}^d \setminus \{0\}$ and we may ask whether they are actual functions in $L^2(\mu_S)$, that is, we may consider $L_S$ as an unbounded operator in $L^2(\mu_S)$ with domain

$$ D(L_S) = \{u \in H_0^1(\mu_S) : L_S u \in L^2(\mu_S)\}, $$

equipped with the norm

$$ \|u\|^2 = \|u\|_{L^2(\mu_S)}^2 + \|L_S u\|_{L^2(\mu_S)}^2 = \int (u^2 + (L_S u)^2) \, d\mu_S. $$

Since $\int \Gamma_S(u) \, d\mu_S = -\int u L_S u \, d\mu_S$ for $u \in C_c^\infty(\mathbf{R}^d \setminus \{0\})$, it easily follows that $L_S$ is a closed operator. In addition, the integration by parts formula holds on its domain:

**Lemma A.1.** Let $u, v \in D(L_S)$. Then,

$$ \int -L_S u v \, d\mu_S = \int \Gamma_S(u, v) \, d\mu_S \quad (65) $$

**Proof**

□ Assume first that $u, v \in C_c^\infty(\mathbf{R}^d \setminus \{0\})$. Then, (65) follows by standard integration by parts. Next, if $u \in D(L_S)$ and $v \in C_c^\infty(\mathbf{R}^d \setminus \{0\})$, take $u_n \in C_c^\infty(\mathbf{R}^d \setminus \{0\})$ s.t. $u_n \to u$ in $H_0^1(\mu)$. Using successively the definition of distributional derivatives, the convergence $u_n \to u$ in $L^2(\mu_S)$, standard integration by parts and the convergence $u_n \to u$ in $H_0^1(\mu)$, we find

$$ \int -L_S u v \, d\mu_S = \int u (-L_S v) \, d\mu_S = \lim_{n \to +\infty} \int u_n (-L_S v) \, d\mu_S = \lim_{n \to +\infty} \int \Gamma_S(u_n, v) \, d\mu_S = \int \Gamma_S(u, v) \, d\mu_S $$

Finally, if $u \in D(L_S)$ and if $v \in D(L_S)$, take $v_n \in C_c^\infty(\mathbf{R}^d \setminus \{0\})$ s.t. $v_n \to v$ in $H_0^1(\mu)$. Then, according to what we just proved,

$$ \int -L_S u v \, d\mu_S = \lim_{n \to +\infty} \int -L_S u v_n \, d\mu_S = \lim_{n \to +\infty} \int \Gamma_S(u, v_n) \, d\mu_S = \int \Gamma_S(u, v) \, d\mu_S $$

▷

The following approximation lemma will be useful to integrate by parts in more delicate settings than the above lemma.
• For any smooth function $\psi$, the Hessian of $\psi$ with respect to the metric $\mathfrak{h}$, denoted $\nabla\nabla^\mathfrak{h}\psi$ is given by

$$ \nabla\nabla^\mathfrak{h}\psi = \nabla\nabla^\mathfrak{g}\psi + 2\nabla^\mathfrak{g}\psi \odot_{\mathfrak{g}} \nabla^\mathfrak{g}\tau - \Gamma^\mathfrak{g}(\psi, \tau)_{\mathfrak{g}}, \quad (61) $$

Here and below, $\nabla\nabla^\mathfrak{g}\psi$ is the Hessian of $\psi$ with respect to $\mathfrak{g}$ and $\nabla^\mathfrak{g}\psi \odot_{\mathfrak{g}} \nabla^\mathfrak{g}\tau$ is the symmetric tensor product, that is for any functions $f, g$,

$$ (\nabla^\mathfrak{g}\psi \odot_{\mathfrak{g}} \nabla^\mathfrak{g}\tau)(\nabla^\mathfrak{g}f, \nabla^\mathfrak{g}g) = \frac{1}{2} [\Gamma^\mathfrak{g}(f, \psi)\Gamma^\mathfrak{g}(g, \tau) + \Gamma^\mathfrak{g}(f, \tau)\Gamma^\mathfrak{g}(g, \psi)]. $$

In particular, one can deduce the Hilbert-Schmidt norm of $\nabla\nabla^\mathfrak{h}\psi$ with respect to the new metric $\mathfrak{h}$:

$$ \|\nabla\nabla^\mathfrak{h}\psi\|^2 = c^4 \left[ \|\nabla\nabla^\mathfrak{g}\psi\|^2 + 2\Gamma^\mathfrak{g}(\tau, \Gamma^\mathfrak{g}(\psi)) + 2\Gamma^\mathfrak{g}(\psi)\Gamma^\mathfrak{g}(\tau) + (d-2)\Gamma^\mathfrak{g}(\psi, \tau)^2 - 2(\Delta_{\mathfrak{g}}\psi)\Gamma^\mathfrak{g}(\psi, \tau) \right]. \quad (62) $$

• The Ricci curvature reads

$$ Ric_{\mathfrak{h}} = Ric_{\mathfrak{g}} + (\Delta_{\mathfrak{g}}\tau)_{\mathfrak{g}} + (d-2)(\nabla\nabla^\mathfrak{g}\tau + \nabla\tau \odot_{\mathfrak{g}} \nabla\tau - \Gamma^\mathfrak{g}(\tau)_{\mathfrak{g}}) \quad (63) $$

• At last, the scalar curvature is given by

$$ sc_{\mathfrak{h}} = c^2 [sc_{\mathfrak{g}} + (d-1)(2\Delta_{\mathfrak{g}}\tau - (d-2)\Gamma^\mathfrak{g}(\tau))] \quad (64) $$

## A.2 Integration by parts and elliptic theory on the CKN spherical space

Let $H_0^1(\mu_S)$ denote the closure of $C_c^\infty(\mathbf{R}^d \setminus \{0\})$ with respect to the norm

$$ \|u\|_{H_0^1(\mu_S)}^2 = \int (\Gamma_S(u) + u^2) \, d\mu_S $$

Let $u \in L^2(\mu_S)$. Then, $|x|^{1-\alpha} \frac{1+|x|^2\alpha}{2} \nabla u$ and $L_S u$ are well-defined distributions on $\mathbf{R}^d \setminus \{0\}$ and we may ask whether they are actual functions in $L^2(\mu_S)$, that is, we may consider $L_S$ as an unbounded operator in $L^2(\mu_S)$ with domain

$$ D(L_S) = \{u \in H_0^1(\mu_S) : L_S u \in L^2(\mu_S)\}, $$

equipped with the norm

$$ \|u\|^2 = \|u\|_{L^2(\mu_S)}^2 + \|L_S u\|_{L^2(\mu_S)}^2 = \int (u^2 + (L_S u)^2) \, d\mu_S. $$

Since $\int \Gamma_S(u) \, d\mu_S = - \int u L_S u \, d\mu_S$ for $u \in C_c^\infty(\mathbf{R}^d \setminus \{0\})$, it easily follows that $L_S$ is a closed operator. In addition, the integration by parts formula holds on its domain:

**Lemma A.1.** Let $u, v \in D(L_S)$. Then,

$$ \int -L_S u v \, d\mu_S = \int \Gamma_S(u, v) \, d\mu_S \quad (65) $$

**Proof**

□ Assume first that $u, v \in C_c^\infty(\mathbf{R}^d \setminus \{0\})$. Then, (65) follows by standard integration by parts. Next, if $u \in D(L_S)$ and $v \in C_c^\infty(\mathbf{R}^d \setminus \{0\})$, take $u_n \in C_c^\infty(\mathbf{R}^d \setminus \{0\})$ s.t. $u_n \to u$ in $H_0^1(\mu)$. Using successively the definition of distributional derivatives, the convergence $u_n \to u$ in $L^2(\mu_S)$, standard integration by parts and the convergence $u_n \to u$ in $H_0^1(\mu)$, we find

$$ \int -L_S u v \, d\mu_S = \int u (-L_S v) \, d\mu_S = \lim_{n \to +\infty} \int u_n (-L_S v) \, d\mu_S = \lim_{n \to +\infty} \int \Gamma_S(u_n, v) \, d\mu_S = \int \Gamma_S(u, v) \, d\mu_S $$

Finally, if $u \in D(L_S)$ and if $v \in D(L_S)$, take $v_n \in C_c^\infty(\mathbf{R}^d \setminus \{0\})$ s.t. $v_n \to v$ in $H_0^1(\mu)$. Then, according to what we just proved,

$$ \int -L_S u v \, d\mu_S = \lim_{n \to +\infty} \int -L_S u v_n \, d\mu_S = \lim_{n \to +\infty} \int \Gamma_S(u, v_n) \, d\mu_S = \int \Gamma_S(u, v) \, d\mu_S $$

▷ The following approximation lemma will be useful to integrate by parts in more delicate settings than the above lemma.
**Lemma A.2.** Assume $n > 4$. Let $u \in D(L_S)$ be such that $u, \Gamma_S(u)$ are bounded. Then, there exists $u_k \in C_c^\infty(\mathbb{R}^d \setminus \{0\})$ such that $u_k \to u$ in $D_{L_S}$.

**Remark A.3.** The assumptions $n > 4$ and $u, \Gamma_S(u)$ bounded can be removed and replaced by $n \ge 3$, but the proof is more involved (see [Ket15, Thm. 3.12]).

**Proof**

In Section 3, the model given in (27) is written with the variables $(s, \theta) \in \mathbb{R} \times \mathbb{S}^{d-1}$. Choosing now $s = \frac{1}{\alpha}\text{Argch}(1/\sin(t))$ the model becomes with the variables $(t, \theta) \in (0, \pi) \times \mathbb{S}^{d-1}$,

$$L_S(f) = \alpha^2(\partial_{tt}f + c'(t)\partial_t f) + \frac{1}{\sin^2(t)}\Delta_{\theta}f, \qquad (66)$$

for any smooth function $f$ defined in $(0, \pi) \times \mathbb{S}^{d-1}$, where $c(t) = (n-1)\log(\sin(t))$. The carré du champ operator becomes

$$\Gamma_S(f) = \alpha^2(\partial_t f)^2 + \frac{1}{\sin^2(t)}\Gamma^\theta(f) \qquad (67)$$

and invariant measure

$$d\mu_S(t, \theta) = \frac{1}{Z}e^{c(t)}dtdV_{S^{d-1}} = \frac{1}{Z}(\sin t)^{n-1}dtdV_{S^{d-1}}, \qquad (68)$$

where $Z$ is a normalization constant and $dV_{S^{d-1}}$ is the volume in $\mathbb{S}^{d-1}$.

Let now $\zeta_k \in C_c^\infty(0, \pi)$ denote a standard cut-off function such that $0 \le \zeta_k \le 1$, $\zeta_k = 0$ in $(0, 1/k) \cup (\pi - 1/k, \pi)$, $\zeta_k = 1$ in $(2/k, \pi - 2/k)$ and $|\zeta_k'| \le 2k$, $|\zeta_k''| \le 2k^2$. Setting $u_k = u\zeta_k$, we find

$$\int (L_S(u - u_k))^2 d\mu_S \le 2 \int (L_S u)^2 (\zeta_k - 1)^2 d\mu_S + 4 \int \Gamma_S(u, \zeta_k)^2 d\mu_S + 2 \int u^2 (L_S \zeta_k)^2 d\mu_S =: I_1 + I_2 + I_3$$

By dominated convergence, $I_1 \to 0$ as $k \to +\infty$. For $I_2$, thanks to (67) and (68),

$$I_2 \le C \|\Gamma_S(u)\|_{\infty}^2 k^{2-n} \to 0$$

Similarly, thanks to (66) and (68),

$$I_3 \le C \|u\|_{\infty}^2 k^{4-n} \to 0.$$

<figure><img src="image_4.png" /></figure>

Our next tool is the following version of the Rellich-Kondrachov compactness theorem.

**Lemma A.4.** Let $(M, g, \mu)$ be a smooth connected weighted $d$-dimensional Riemannian manifold s.t. $d \ge 3$, $\mu(M) < +\infty$ and Sobolev's inequality holds i.e. there exist constants $A, B \ge 0$, $p \in [2, \frac{2d}{d-2}]$ such that for every $v \in C_c^\infty(M)$,

$$(\int |v|^p d\mu)^{\frac{2}{p}} \le A \int \Gamma(v) d\mu + B \int v^2 d\mu \qquad (69)$$

Let $H_0^1(\mu)$ be the closure of $A_0 = C_c^\infty(M)$ for the norm $\|u\|_{H_0^1(\mu)}^2 = \int (u^2 + \Gamma(u))d\mu$ and let $q \in [1, p)$. Then, the embedding $H_0^1(\mu) \hookrightarrow L^q(\mu)$ is compact.

**Proof**

Cover $M$ by a countable increasing family of open sets $(\Omega_k)_{k \in \mathbb{N}}$ with compact closure and for each $k \in \mathbb{N}$, let $\eta_k \in C_c^\infty(\Omega_{k+1})$ be such that $\eta_k = 1$ in $\Omega_k$. Let $(u_m)$ be a bounded sequence in $H_0^1(\mu)$. Since $d\mu = e^{-W}dV_g$ and $W, g$ are smooth, the $H_0^1(\mu)$ and the standard $H_0^1$ norm are equivalent for functions compactly supported in a fixed $\Omega_k$. By the classical Rellich-Kondrachov theorem, we deduce that for fixed $k$, the sequence $(u_m\eta_k)_m$ is compact in $L^r(\Omega_{k+1}, d\mu)$ for $r \in (q, p)$. Since $(u_m)$ is bounded in the Hilbert space $H_0^1(\mu)$, by the Banach-Alaoglu theorem, $(u_m)$ is also compact in $H_0^1(\mu)$ for the weak topology. By a standard diagonal argument,
**Lemma A.2.** Assume $n > 4$. Let $u \in D(L_S)$ be such that $u, \Gamma_S(u)$ are bounded. Then, there exists $u_k \in C_c^\infty(\mathbb{R}^d \setminus \{0\})$ such that $u_k \to u$ in $D_{L_S}$.

**Remark A.3.** The assumptions $n > 4$ and $u, \Gamma_S(u)$ bounded can be removed and replaced by $n \ge 3$, but the proof is more involved (see [Ket15, Thm. 3.12]).

**Proof**

In Section 3, the model given in (27) is written with the variables $(s, \theta) \in \mathbb{R} \times \mathbb{S}^{d-1}$. Choosing now $s = \frac{1}{\alpha}\operatorname{Argch}(1/\sin(t))$ the model becomes with the variables $(t, \theta) \in (0, \pi) \times \mathbb{S}^{d-1}$,

$$L_S(f) = \alpha^2(\partial_{tt}f + c'(t)\partial_t f) + \frac{1}{\sin^2(t)}\Delta_{\theta}f, \qquad (66)$$

for any smooth function $f$ defined in $(0, \pi) \times \mathbb{S}^{d-1}$, where $c(t) = (n-1)\log(\sin(t))$. The carré du champ operator becomes

$$\Gamma_S(f) = \alpha^2(\partial_t f)^2 + \frac{1}{\sin^2(t)}\Gamma^\theta(f) \qquad (67)$$

and invariant measure

$$d\mu_S(t, \theta) = \frac{1}{Z}e^{c(t)}dtdV_{S^{d-1}} = \frac{1}{Z}(\sin t)^{n-1}dtdV_{S^{d-1}}, \qquad (68)$$

where $Z$ is a normalization constant and $dV_{S^{d-1}}$ is the volume in $\mathbb{S}^{d-1}$.

Let now $\zeta_k \in C_c^\infty(0, \pi)$ denote a standard cut-off function such that $0 \le \zeta_k \le 1$, $\zeta_k = 0$ in $(0, 1/k) \cup (\pi - 1/k, \pi)$, $\zeta_k = 1$ in $(2/k, \pi - 2/k)$ and $|\zeta_k'| \le 2k$, $|\zeta_k''| \le 2k^2$. Setting $u_k = u\zeta_k$, we find

$$\int (L_S(u - u_k))^2 d\mu_S \le 2 \int (L_S u)^2 (\zeta_k - 1)^2 d\mu_S + 4 \int \Gamma_S(u, \zeta_k)^2 d\mu_S + 2 \int u^2 (L_S \zeta_k)^2 d\mu_S =: I_1 + I_2 + I_3$$

By dominated convergence, $I_1 \to 0$ as $k \to +\infty$. For $I_2$, thanks to (67) and (68),

$$I_2 \le C\|\Gamma_S(u)\|_{\infty}^2 k^{2-n} \to 0$$

Similarly, thanks to (66) and (68),

$$I_3 \le C\|u\|_{\infty}^2 k^{4-n} \to 0.$$

<figure><img src="image_4.png" /></figure>

Our next tool is the following version of the Rellich-Kondrachov compactness theorem.

**Lemma A.4.** Let $(M, g, \mu)$ be a smooth connected weighted $d$-dimensional Riemannian manifold s.t. $d \ge 3$, $\mu(M) < +\infty$ and Sobolev's inequality holds i.e. there exist constants $A, B \ge 0$, $p \in [2, \frac{2d}{d-2}]$ such that for every $v \in C_c^\infty(M)$,

$$(\int |v|^p d\mu)^{\frac{2}{p}} \le A \int \Gamma(v) d\mu + B \int v^2 d\mu \qquad (69)$$

Let $H_0^1(\mu)$ be the closure of $A_0 = C_c^\infty(M)$ for the norm $\|u\|_{H_0^1(\mu)}^2 = \int (u^2 + \Gamma(u))d\mu$ and let $q \in [1, p)$. Then, the embedding $H_0^1(\mu) \hookrightarrow L^q(\mu)$ is compact.

**Proof**

Cover $M$ by a countable increasing family of open sets $(\Omega_k)_{k \in \mathbb{N}}$ with compact closure and for each $k \in \mathbb{N}$, let $\eta_k \in C_c^\infty(\Omega_{k+1})$ be such that $\eta_k = 1$ in $\Omega_k$. Let $(u_m)$ be a bounded sequence in $H_0^1(\mu)$. Since $d\mu = e^{-W}dV_g$ and $W, g$ are smooth, the $H_0^1(\mu)$ and the standard $H_0^1$ norm are equivalent for functions compactly supported in a fixed $\Omega_k$. By the classical Rellich-Kondrachov theorem, we deduce that for fixed $k$, the sequence $(u_m\eta_k)_m$ is compact in $L^r(\Omega_{k+1}, d\mu)$ for $r \in (q, p)$. Since $(u_m)$ is bounded in the Hilbert space $H_0^1(\mu)$, by the Banach-Alaoglu theorem, $(u_m)$ is also compact in $H_0^1(\mu)$ for the weak topology. By a standard diagonal argument,
a subsequence of $(u_m)$ (denoted the same) converges weakly in $H_0^1(\mu)$ to some function $u \in H_0^1(\mu)$ such that $(u_m \eta_k)_m$ converges to $u \eta_k$ in $L^r(\mu)$. Now, using Hölder's and Sobolev's inequality we find

$$ \begin{aligned} \|u_m - u\|_{L^q(\mu)} &\le \|(u_m - u)\eta_k\|_{L^q(\mu)} + \|(u_m - u)(1 - \eta_k)\|_{L^q(\mu)} \\ &\le \|(u_m - u)\eta_k\|_{L^r(\mu)} \mu(M)^{\frac{1}{q} - \frac{1}{r}} + \|u_m - u\|_{L^p(\mu)} \mu(M \setminus \Omega_k)^{\frac{1}{q} - \frac{1}{p}} \\ &\le C \left( \|(u_m - u)\eta_k\|_{L^r(\mu)} + \mu(M \setminus \Omega_k) \right)^{\frac{1}{q} - \frac{1}{p}} \end{aligned} $$

Hence,

$$ \limsup_{m \to +\infty} \|u_m - u\|_{L^q(\mu)} \le C\mu(M \setminus \Omega_k)^{\frac{1}{q} - \frac{1}{p}} $$

Letting $k \to +\infty$, the claim follows. ▷

As an immediate consequence of the above lemma (and a proof by contradiction), we have

**Corollary A.5.** Make the same assumptions as in Lemma A.4. Assume in addition that constants belong to $H_0^1(\mu)$. Then, Poincaré's inequality holds i.e. there exists a constant $C_P > 0$ such that

$$ \int v^2 d\mu - \left( \int v d\mu \right)^2 \le C_P \int \Gamma(v) d\mu \quad \text{for } v \in \mathcal{A}_0 = C_c^\infty(M) $$

Finally, we state and prove elliptic regularity estimates, which are useful to justify integrations by parts in our proof of Sobolev's inequality.

**Lemma A.6 (General elliptic estimates).** Assume that $(a, b) \in \Theta_{FS}$ (defined in (18)). Let also $h : (0, 1] \times S^{d-1} \mapsto \mathbf{R}$ be a smooth and bounded function satisfying $\int_{S^{d-1}} h dV_{S^{d-1}} = 0$ and solving the equation

$$ \partial_{tt}h - \frac{(n-1)(n-3)}{4} \frac{h}{t^2} + \frac{\Delta_\theta h}{\alpha^2 \sin^2(t)} = R, $$

where $R$ is a smooth and bounded function on $(0, 1] \times S^{d-1}$. We assume also that, uniformly on $(0, 1] \times S^{d-1}$

$$ |h(t, \theta)| \le C t^{\frac{n-1}{2}} \quad \text{and} \quad |R(t, \theta)| \le C t^{\frac{n-1}{2}}, $$

for some constant $C > 0$. Then, there exists a constant $C' > 0$ s.t. uniformly on $(0, 1] \times S^{d-1}$,

$$ |h(t, \theta)| \le C' t^{\frac{n+1}{2}}, \quad \Gamma^\theta(h)(t, \theta) = |\nabla_\theta h|^2(t, \theta) \le C' t^{n+1} \quad \text{and} \quad |\partial_t h(t, \theta)| \le C' t^{\frac{n-1}{2}}. $$

**Proof**

Let $(P_k)_{k \ge 0}$ be the orthonormal basis of eigenvectors of the operator $-\Delta_\theta$ on $S^{d-1}$ associated to the increasing sequence of eigenvalues $(\lambda_k)_{k \ge 0}$ (recall that $\lambda_k \ge \lambda_1 = d-1$, for $k \ge 1$ and $\lambda_0 = 0$). We decompose $h$ in the basis $(P_k)_{k \ge 0}$,

$$ h(t, \theta) = \sum_{k=1}^{\infty} h_k(t) P_k(\theta), \quad (t, \theta) \in (0, \pi) \times S^{d-1}, $$

where $h_k(t) = \int_{S^{d-1}} P_k(\theta) h(t, \theta) dV_{S^{d-1}}$ (noting that $\int h P_0 dV_{S^{d-1}} = \int h dV_{S^{d-1}} = 0$, whence $h_0 = 0$). For each $k \ge 1$, $h_k$ satisfies

$$ h_k'' - \frac{(n-1)(n-3)}{4} \frac{h_k}{t^2} - \frac{\lambda_k}{\alpha^2 \sin^2(t)} h_k = R_k, $$

where $R_k = \int_{S^{d-1}} R P_k dV_{S^{d-1}}$, which satisfies again $|R_k(t, \theta)| \le C_k t^{\frac{n-1}{2}}$. The equation can be replaced by the following one

$$ h_k'' - a \frac{h_k}{t^2} = R_{2,k}, $$

where $a = \frac{(n-1)(n-3)}{4} + \frac{\lambda_k}{\alpha^2} > 0$ and $R_{2,k}$ satisfies the same estimate as $R_k$.
a subsequence of $(u_m)$ (denoted the same) converges weakly in $H_0^1(\mu)$ to some function $u \in H_0^1(\mu)$ such that $(u_m \eta_k)_m$ converges to $u \eta_k$ in $L^r(\mu)$. Now, using Hölder's and Sobolev's inequality we find

$$ \begin{aligned} \|u_m - u\|_{L^q(\mu)} &\le \|(u_m - u)\eta_k\|_{L^q(\mu)} + \|(u_m - u)(1 - \eta_k)\|_{L^q(\mu)} \\ &\le \|(u_m - u)\eta_k\|_{L^r(\mu)} \mu(M)^{\frac{1}{q} - \frac{1}{r}} + \|u_m - u\|_{L^p(\mu)} \mu(M \setminus \Omega_k)^{\frac{1}{q} - \frac{1}{p}} \\ &\le C \left( \|(u_m - u)\eta_k\|_{L^r(\mu)} + \mu(M \setminus \Omega_k) \right)^{\frac{1}{q} - \frac{1}{p}} \end{aligned} $$

Hence,

$$ \limsup_{m \to +\infty} \|u_m - u\|_{L^q(\mu)} \le C\mu(M \setminus \Omega_k)^{\frac{1}{q} - \frac{1}{p}} $$

Letting $k \to +\infty$, the claim follows. ▷

As an immediate consequence of the above lemma (and a proof by contradiction), we have

**Corollary A.5.** Make the same assumptions as in Lemma A.4. Assume in addition that constants belong to $H_0^1(\mu)$. Then, Poincaré's inequality holds i.e. there exists a constant $C_P > 0$ such that

$$ \int v^2 d\mu - \left( \int v d\mu \right)^2 \le C_P \int \Gamma(v) d\mu \quad \text{for } v \in \mathcal{A}_0 = C_c^\infty(M) $$

Finally, we state and prove elliptic regularity estimates, which are useful to justify integrations by parts in our proof of Sobolev's inequality.

**Lemma A.6 (General elliptic estimates).** Assume that $(a, b) \in \Theta_{FS}$ (defined in (18)). Let also $h : (0, 1] \times \mathbf{S}^{d-1} \mapsto \mathbf{R}$ be a smooth and bounded function satisfying $\int_{\mathbf{S}^{d-1}} h dV_{\mathbf{S}^{d-1}} = 0$ and solving the equation

$$ \partial_{tt}h - \frac{(n-1)(n-3)}{4} \frac{h}{t^2} + \frac{\Delta_\theta h}{\alpha^2 \sin^2(t)} = R, $$

where $R$ is a smooth and bounded function on $(0, 1] \times \mathbf{S}^{d-1}$. We assume also that, uniformly on $(0, 1] \times \mathbf{S}^{d-1}$

$$ |h(t, \theta)| \le C t^{\frac{n-1}{2}} \quad \text{and} \quad |R(t, \theta)| \le C t^{\frac{n-1}{2}}, $$

for some constant $C > 0$. Then, there exists a constant $C' > 0$ s.t. uniformly on $(0, 1] \times \mathbf{S}^{d-1}$,

$$ |h(t, \theta)| \le C' t^{\frac{n+1}{2}}, \quad \Gamma^\theta(h)(t, \theta) = |\nabla_\theta h|^2(t, \theta) \le C' t^{n+1} \quad \text{and} \quad |\partial_t h(t, \theta)| \le C' t^{\frac{n-1}{2}}. $$

**Proof**

Let $(P_k)_{k \ge 0}$ be the orthonormal basis of eigenvectors of the operator $-\Delta_\theta$ on $\mathbf{S}^{d-1}$ associated to the increasing sequence of eigenvalues $(\lambda_k)_{k \ge 0}$ (recall that $\lambda_k \ge \lambda_1 = d-1$, for $k \ge 1$ and $\lambda_0 = 0$). We decompose $h$ in the basis $(P_k)_{k \ge 0}$,

$$ h(t, \theta) = \sum_{k=1}^{\infty} h_k(t) P_k(\theta), \quad (t, \theta) \in (0, \pi) \times \mathbf{S}^{d-1}, $$

where $h_k(t) = \int_{\mathbf{S}^{d-1}} P_k(\theta) h(t, \theta) dV_{\mathbf{S}^{d-1}}$ (noting that $\int h P_0 dV_{\mathbf{S}^{d-1}} = \int h dV_{\mathbf{S}^{d-1}} = 0$, whence $h_0 = 0$). For each $k \ge 1$, $h_k$ satisfies

$$ h_k'' - \frac{(n-1)(n-3)}{4} \frac{h_k}{t^2} - \frac{\lambda_k}{\alpha^2 \sin^2(t)} h_k = R_k, $$

where $R_k = \int_{\mathbf{S}^{d-1}} R P_k dV_{\mathbf{S}^{d-1}}$, which satisfies again $|R_k(t, \theta)| \le C_k t^{\frac{n-1}{2}}$. The equation can be replaced by the following one

$$ h_k'' - a \frac{h_k}{t^2} = R_{2,k}, $$

where $a = \frac{(n-1)(n-3)}{4} + \frac{\lambda_k}{\alpha^2} > 0$ and $R_{2,k}$ satisfies the same estimate as $R_k$.
We are now able to solve the ODE. The method of variation of constants gives the explicit solution:

$$h_k(t) = At^{\gamma_+} + Bt^{\gamma_-} + \frac{t^{\gamma_+}}{\gamma_+ - \gamma_-} \int_1^t R_{2,k}(y)y^{1-\gamma_+} dy - \frac{t^{\gamma_-}}{\gamma_+ - \gamma_-} \int_0^t R_{2,k}(y)y^{1-\gamma_-} dy,$$

where A, B are constants and

$$\gamma_{\pm} = \frac{1 \pm \sqrt{1 + 4a}}{2} = \frac{1}{2} \pm \frac{1}{2} \sqrt{(n-2)^2 + 4 \frac{\lambda_k}{\alpha^2}},$$

so that $\gamma_+ > 0$, $\gamma_- < 0$. Then, by the estimate satisfied by $R_{2,k}$ we have near 0,

$$\left| \frac{t^{\gamma_+}}{\gamma_+ - \gamma_-} \int_1^t R_{2,k}(y) y^{1-\gamma_+} dy \right| \le C t^{\gamma_+} \left| \int_1^t y^{\frac{n+1}{2}-\gamma_+} dy \right| \le C \left( t^{\gamma_+} + t^{\frac{n+3}{2}} \right)$$

and

$$\left| \frac{t^{\gamma_{-}}}{\gamma_{+} - \gamma_{-}} \int_{0}^{t} R_{2,k}(y) y^{1-\gamma_{-}} dy \right| \leq C t^{\gamma_{-}} \int_{0}^{t} y^{\frac{n+1}{2}-\gamma_{-}} dy \leq C t^{\frac{n+3}{2}}$$

Since $h_k$ is a bounded function, we deduce that $B = 0$ and

$$|h_k(t)| \leq C(t^{\gamma_+} + t^{\frac{n+3}{2}}).$$

We claim that for $k \ge 1$, $\gamma_+ \ge \frac{n+1}{2}$, whence $|h_k(t)| \le C t^{\frac{n+1}{2}}$. Indeed, by definition of $\gamma_+$, one can check that the inequality $\gamma_+ \ge (n+1)/2$ is equivalent to $\frac{\lambda_k}{n-1} \ge \alpha^2$. Since $\lambda_k \ge d-1$ for any $k \ge 1$ and since by Lemma 1.9, $(a,b) \in \Theta_{FS}$ if and only if $\alpha^2 \le \frac{d-1}{n-1}$, we indeed have $\gamma_+ \ge (n+1)/2$.

Next, we prove that the estimate remains valid for the function $h$. Define

$$H_K = \sum_{k=1}^{K} h_k P_k.$$

so that $h = \lim_{K \to \infty} H_K$ pointwise. From the previous computations, we know that, uniformly in $(0,1) \times \mathbf{S}^{d-1}$,

$$|H_K(t, \theta)| \le C_K t^{\frac{n+1}{2}}$$

We prove now that the inequality is uniform in the parameter $K$. Assume this is not the case i.e.

$$\sup_{K \ge 1} \|H_K\|_{(\frac{n+1}{2})} = +\infty, \qquad (70)$$

where

$$\|H_K\|_{(\frac{n+1}{2})} = \sup_{t \in (0,1], \theta \in \mathbf{S}^{d-1}} \frac{|H_K(t, \theta)|}{t^{\frac{n+1}{2}}}.$$

There exist a sequence $((t_K, \theta_K))_{K \ge 1}$ in $(0,1] \times \mathbf{S}^{d-1}$, such that

$$\lim_{K \to \infty} \frac{|H_K(t_K, \theta_K)|}{t_K^{\frac{n+1}{2}}} = \infty. \qquad (71)$$

By compactness, one can assume that $(t_K)$ (resp. $(\theta_K)$) converges to $t_\infty \in [0,1]$ (resp. $\theta_\infty \in \mathbf{S}^{d-1}$). There are two cases, either $t_\infty > 0$ or $t_\infty = 0$. The first case is not possible. Indeed, $h$ is bounded by assumption, whence $H_K$ is bounded by a constant independent of $K$ and so (71) contradicts $t_\infty > 0$. The remaining case $t_\infty = 0$ is more tricky. Let

$$G_K(z, \theta) = \frac{H_K(zt_K, \theta)}{t_K^{\frac{n+1}{2}} \|H_K\|_{(\frac{n+1}{2})}}$$
We are now able to solve the ODE. The method of variation of constants gives the explicit solution:

$$h_k(t) = At^{\gamma_+} + Bt^{\gamma_-} + \frac{t^{\gamma_+}}{\gamma_+ - \gamma_-} \int_1^t R_{2,k}(y)y^{1-\gamma_+} dy - \frac{t^{\gamma_-}}{\gamma_+ - \gamma_-} \int_0^t R_{2,k}(y)y^{1-\gamma_-} dy,$$

where A, B are constants and

$$\gamma_{\pm} = \frac{1 \pm \sqrt{1 + 4a}}{2} = \frac{1}{2} \pm \frac{1}{2} \sqrt{(n-2)^2 + 4 \frac{\lambda_k}{\alpha^2}},$$

so that $\gamma_+ > 0$, $\gamma_- < 0$. Then, by the estimate satisfied by $R_{2,k}$ we have near 0,

$$\left| \frac{t^{\gamma_+}}{\gamma_+ - \gamma_-} \int_1^t R_{2,k}(y) y^{1-\gamma_+} dy \right| \le C t^{\gamma_+} \left| \int_1^t y^{\frac{n+1}{2}-\gamma_+} dy \right| \le C \left( t^{\gamma_+} + t^{\frac{n+3}{2}} \right)$$

and

$$\left| \frac{t^{\gamma_{-}}}{\gamma_{+} - \gamma_{-}} \int_{0}^{t} R_{2,k}(y) y^{1-\gamma_{-}} dy \right| \leq C t^{\gamma_{-}} \int_{0}^{t} y^{\frac{n+1}{2}-\gamma_{-}} dy \leq C t^{\frac{n+3}{2}}$$

Since $h_k$ is a bounded function, we deduce that $B = 0$ and

$$|h_k(t)| \leq C(t^{\gamma_+} + t^{\frac{n+3}{2}}).$$

We claim that for $k \ge 1$, $\gamma_+ \ge \frac{n+1}{2}$, whence $|h_k(t)| \le C t^{\frac{n+1}{2}}$. Indeed, by definition of $\gamma_+$, one can check that the inequality $\gamma_+ \ge (n+1)/2$ is equivalent to $\frac{\lambda_k}{n-1} \ge \alpha^2$. Since $\lambda_k \ge d-1$ for any $k \ge 1$ and since by Lemma 1.9, $(a,b) \in \Theta_{FS}$ if and only if $\alpha^2 \le \frac{d-1}{n-1}$, we indeed have $\gamma_+ \ge (n+1)/2$. Next, we prove that the estimate remains valid for the function $h$. Define

$$H_K = \sum_{k=1}^{K} h_k P_k.$$

so that $h = \lim_{K \to \infty} H_K$ pointwise. From the previous computations, we know that, uniformly in $(0,1) \times \mathbf{S}^{d-1}$,

$$|H_K(t, \theta)| \le C_K t^{\frac{n+1}{2}}$$

We prove now that the inequality is uniform in the parameter K. Assume this is not the case i.e.

$$\sup_{K \ge 1} \|H_K\|_{(\frac{n+1}{2})} = +\infty, \qquad (70)$$

where

$$\|H_K\|_{(\frac{n+1}{2})} = \sup_{t \in (0,1], \theta \in \mathbf{S}^{d-1}} \frac{|H_K(t, \theta)|}{t^{\frac{n+1}{2}}}.$$

There exist a sequence $((t_K, \theta_K))_{K \ge 1}$ in $(0,1] \times \mathbf{S}^{d-1}$, such that

$$\lim_{K \to \infty} \frac{|H_K(t_K, \theta_K)|}{t_K^{\frac{n+1}{2}}} = \infty. \qquad (71)$$

By compactness, one can assume that $(t_K)$ (resp. $(\theta_K)$) converges to $t_\infty \in [0,1]$ (resp. $\theta_\infty \in \mathbf{S}^{d-1}$). There are two cases, either $t_\infty > 0$ or $t_\infty = 0$. The first case is not possible. Indeed, $h$ is bounded by assumption, whence $H_K$ is bounded by a constant independent of $K$ and so (71) contradicts $t_\infty > 0$. The remaining case $t_\infty = 0$ is more tricky. Let

$$G_K(z, \theta) = \frac{H_K(zt_K, \theta)}{t_K^{\frac{n+1}{2}} \|H_K\|_{(\frac{n+1}{2})}}$$
for any $z \in (0, 1/t_K]$. From the equation satisfied by each $h_k$, we have

$$ \partial_{zz} G_K(z, \theta) - \frac{(n-1)(n-3)}{4} \frac{G_K(z, \theta)}{z^2} + t_K^2 \frac{\Delta_\theta G_K(z, \theta)}{\alpha^2 \sin^2(zt_K)} = \frac{t_K^2}{t_K^{\frac{n+1}{2}} ||H_K||_{(\frac{n+1}{2})}} \sum_{k=1}^{K} R_k(zt_k) P_k(\theta). $$

By assumption on $R$, we have

$$ \left| \frac{t_K^2}{t_K^{\frac{n+1}{2}} ||H_K||_{(\frac{n+1}{2})}} \sum_{k=1}^{K} R_k(zt_k) P_k(\theta) \right| \le C \frac{t_K}{||H_K||_{(\frac{n+1}{2})}} \xrightarrow{K \to \infty} 0, $$

uniformly for $z$ in a compact subset of $\mathbf{R}^*$. By standard elliptic regularity, it follows that the sequence $(G_K)$ converges to $G$ solution on $(0, \infty) \times \mathbf{S}^{d-1}$ of the PDE

$$ \partial_{zz} G(z, \theta) - \frac{(n-1)(n-3)}{4} \frac{G(z, \theta)}{z^2} + \frac{\Delta_\theta G(z, \theta)}{\alpha^2 z^2} = 0. $$

Now, using the same argument we have

$$ G = \sum_{k=0}^{\infty} G_k P_k. $$

where again $G_k(z) = \int_{\mathbf{S}^{d-1}} G(z, \theta) P_k(\theta) dV_{\mathbf{S}^{d-1}}$. Then for each $k \ge 0$, we have $G_k(t) = At^{\gamma+} + Bt^{\gamma-}$ where $\gamma_\pm$ are the same constants as before. But, the function $G_k$, defined on $(0, \infty)$, is bounded. This implies that $A = B = 0$ and then $G = 0$. But by its definition, we know that

$$ G(1, \theta_\infty) = 1, $$

which gives a contradiction: the hypothesis (70) is not valid. We conclude that uniformly in $(0, 1] \times \mathbf{S}^{d-1}$, we have

$$ |h(t, \theta)| \le C t^{\frac{n+1}{2}}. $$

It remains to prove the gradient estimates. Fix $t > 0$ and for $z \in (1/4, 2)$, $\theta \in \mathbf{S}^{d-1}$, let this time $G(z, \theta) = h(tz, \theta)$ and $S(z, \theta) = R(tz, \theta)$ so that

$$ \partial_{zz} G - \frac{(n-1)(n-3)}{4} \frac{G}{z^2} + \frac{t^2}{\alpha^2 \sin^2(tz)} \Delta_\theta G = t^2 S \quad \text{in } (1/4, 2) \times \mathbf{S}^{d-1} $$

Note that the coefficients of the elliptic operator on the left-hand side are bounded in $C^2$-norm by a constant independent of $t$ so that, by standard elliptic regularity,

$$ |\partial_z G| + |\nabla_\theta G| \le C \left( t^2 \|S\|_{L^\infty((1/4,2)\times\mathbf{S}^{d-1})} + \|G\|_{L^\infty((1/4,2)\times\mathbf{S}^{d-1})} \right) \le C t^{\frac{n+1}{2}} \quad \text{in } (1/2, 3/2) \times \mathbf{S}^{d-1}, $$

for some constant $C > 0$ independent of $t$. The desired estimates on $h$ follow by applying the above estimate at $z = 1$. ▷

**Lemma A.7.** Whenever $(a, b) \in \Theta_{FS}$ (defined in (18)), the solution $v$ of the equation (48) has a bounded carré du champ operator

$$ ||\Gamma_{\mathbf{S}}(v)||_{\infty} < +\infty. $$

### Proof

We use the chart and notation introduced in (66). We have to prove that $v$, solution of (48) has a bounded carré du champ, that is $||\Gamma_{\mathbf{S}}(v)||_\infty < +\infty$. Letting $c(t) = (n-1)\log(\sin(t))$ and $h = e^{\frac{c}{2}}v$, equation (48) becomes

$$ \partial_{tt} h - \left( \frac{2c'' + c'^2}{4} \right) h + \frac{1}{\alpha^2 \sin^2(t)} \Delta_\theta h = R, \quad (72) $$
for any $z \in (0, 1/t_K]$. From the equation satisfied by each $h_k$, we have

$$ \partial_{zz} G_K(z, \theta) - \frac{(n-1)(n-3)}{4} \frac{G_K(z, \theta)}{z^2} + t_K^2 \frac{\Delta_\theta G_K(z, \theta)}{\alpha^2 \sin^2(zt_K)} = \frac{t_K^2}{t_K^{\frac{n+1}{2}} ||H_K||_{(\frac{n+1}{2})}} \sum_{k=1}^K R_k(zt_k)P_k(\theta). $$

By assumption on $R$, we have

$$ \left| \frac{t_K^2}{t_K^{\frac{n+1}{2}} ||H_K||_{(\frac{n+1}{2})}} \sum_{k=1}^K R_k(zt_k)P_k(\theta) \right| \le C \frac{t_K}{||H_K||_{(\frac{n+1}{2})}} \xrightarrow{K \to \infty} 0, $$

uniformly for $z$ in a compact subset of $\mathbf{R}^*$. By standard elliptic regularity, it follows that the sequence $(G_K)$ converges to $G$ solution on $(0, \infty) \times \mathbf{S}^{d-1}$ of the PDE

$$ \partial_{zz} G(z, \theta) - \frac{(n-1)(n-3)}{4} \frac{G(z, \theta)}{z^2} + \frac{\Delta_\theta G(z, \theta)}{\alpha^2 z^2} = 0. $$

Now, using the same argument we have

$$ G = \sum_{k=0}^{\infty} G_k P_k. $$

where again $G_k(z) = \int_{\mathbf{S}^{d-1}} G(z, \theta)P_k(\theta)dV_{\mathbf{S}^{d-1}}$. Then for each $k \ge 0$, we have $G_k(t) = At^{\gamma+} + Bt^{\gamma-}$ where $\gamma_\pm$ are the same constants as before. But, the function $G_k$, defined on $(0, \infty)$, is bounded. This implies that $A = B = 0$ and then $G = 0$. But by its definition, we know that

$$ G(1, \theta_\infty) = 1, $$

which gives a contradiction: the hypothesis (70) is not valid. We conclude that uniformly in $(0, 1] \times \mathbf{S}^{d-1}$, we have

$$ |h(t, \theta)| \le C t^{\frac{n+1}{2}}. $$

It remains to prove the gradient estimates. Fix $t > 0$ and for $z \in (1/4, 2)$, $\theta \in \mathbf{S}^{d-1}$, let this time $G(z, \theta) = h(tz, \theta)$ and $S(z, \theta) = R(tz, \theta)$ so that

$$ \partial_{zz} G - \frac{(n-1)(n-3)}{4} \frac{G}{z^2} + \frac{t^2}{\alpha^2 \sin^2(tz)} \Delta_\theta G = t^2 S \quad \text{in } (1/4, 2) \times \mathbf{S}^{d-1} $$

Note that the coefficients of the elliptic operator on the left-hand side are bounded in $C^2$-norm by a constant independent of $t$ so that, by standard elliptic regularity,

$$ |\partial_z G| + |\nabla_\theta G| \le C \left( t^2 \|S\|_{L^\infty((1/4,2)\times\mathbf{S}^{d-1})} + \|G\|_{L^\infty((1/4,2)\times\mathbf{S}^{d-1})} \right) \le C t^{\frac{n+1}{2}} \quad \text{in } (1/2, 3/2) \times \mathbf{S}^{d-1}, $$

for some constant $C > 0$ independent of $t$. The desired estimates on $h$ follow by applying the above estimate at $z = 1$. ▷

**Lemma A.7.** Whenever $(a, b) \in \Theta_{FS}$ (defined in (18)), the solution $v$ of the equation (48) has a bounded carré du champ operator

$$ ||\Gamma_{\mathbf{S}}(v)||_{\infty} < +\infty. $$

### Proof

We use the chart and notation introduced in (66). We have to prove that $v$, solution of (48) has a bounded carré du champ, that is $||\Gamma_{\mathbf{S}}(v)||_\infty < +\infty$. Letting $c(t) = (n-1)\log(\sin(t))$ and $h = e^{\frac{c}{2}}v$, equation (48) becomes

$$ \partial_{tt} h - \left( \frac{2c'' + c'^2}{4} \right) h + \frac{1}{\alpha^2 \sin^2(t)} \Delta_\theta h = R, \quad (72) $$
where

$$R = \frac{1}{\alpha^2 A} \left( h - e^{\frac{c(2-q)}{2}} h^{q-1} \right).$$

This transformation allows us to deal with a simpler PDE. We know that $v = e^{-\frac{c}{2}}h$ is bounded and positive. So, for some constant $C$ (the value of which is allowed to change from line to line),

$$0 \le h \le C e^{\frac{c}{2}} = C \sin(t)^{\frac{n-1}{2}}.$$

Thus $|R| \le C \sin(t)^{\frac{n-1}{2}}$. And then, from the definition of $h$, the following inequality,

$$\left(\partial_t h - \frac{c'}{2} h\right)^2 + \frac{\Gamma^\theta(h)}{\alpha^2 \sin^2(t)} \le C \sin(t)^{n-1} \quad (73)$$

is equivalent to $||\Gamma_S(v)||_\infty < +\infty$.

We know that $h$ is a smooth function on $(0, \pi) \times S^{d-1}$. So, to prove the previous inequality, it is enough to work around $t = 0$ and $t = \pi$. By symmetry, it suffices to treat the case $t = 0$.

By definition of $c$, equation (72) can be written as follow

$$\partial_{tt}h - \frac{(n-1)(n-3)}{4} \frac{h}{\sin^2(t)} + \frac{1}{\alpha^2 \sin^2(t)} \Delta_{\theta}h = R - \frac{(n-1)(n-3)}{4} h,$$

or, since we are working around $t = 0$, we have

$$\partial_{tt}h - \frac{(n-1)(n-3)}{4} \frac{h}{t^2} + \frac{1}{\alpha^2 \sin^2(t)} \Delta_{\theta}h = R_2, \quad (74)$$

with

$$R_2 = R - \frac{(n-1)(n-3)}{4} h + \frac{(n-1)(n-3)}{4} \left( \frac{1}{\sin^2(t)} - \frac{1}{t^2} \right) h,$$

which satisfies again $|R_2| \le C t^{\frac{n-1}{2}}$.

Let us write $h = h - h_0 + h_0$ where $h_0 = \int_{S^{d-1}} h dV_{S^{d-1}}$. Then, since $h_0$ doesn't depend on $\theta$,

$$\left(\partial_t h - \frac{c'}{2} h\right)^2 + \frac{\Gamma^\theta(h)}{\alpha^2 \sin^2(t)} \le 2\left(\partial_t(h - h_0) - \frac{c'}{2}(h - h_0)\right)^2 + 2\left(\partial_t h_0 - \frac{c'}{2}h_0\right)^2 + \frac{\Gamma^\theta(h - h_0)}{\alpha^2 \sin^2(t)}.$$

Then Lemma A.6 insures that $|h - h_0| \le C t^{\frac{n+1}{2}}$, $|h' - h'_0| \le C t^{\frac{n-1}{2}}$ and $\Gamma^\theta(h - h_0) = \Gamma^\theta(h) \le C t^{n+1}$. Hence,

$$2\left(\partial_t(h - h_0) - \frac{c'}{2}(h - h_0)\right)^2 + 2\frac{\Gamma^\theta(h - h_0)}{\alpha^2 \sin^2(t)} \le C t^{n-1}.$$

Now, using the same method as in the proof of Lemma A.6, one can check that $h_0(t) = At^{\frac{n-1}{2}} + O(t^{\frac{n+1}{2}})$ and $h'_0 = At^{\frac{n-3}{2}} + O(t^{\frac{n-1}{2}})$. Thus, we have

$$\partial_t h_0 - \frac{c'}{2} h_0 = O(t^{\frac{n-1}{2}})$$

that is

$$\left(\partial_t h_0 - \frac{c'}{2} h_0\right)^2 \le C t^{n-1}.$$

Finally, inequality (73) is satisfied, which concludes the proof. ▷

**Remark A.8.** It is interesting to see that with this method, one can check that the function $v$ has a bounded carré du champ if and only of $(a, b) \in \Theta_{FS}$.

We end this section with the following weaker estimate on higher derivatives of $v$.
where

$$R = \frac{1}{\alpha^2 A} \left( h - e^{\frac{c(2-q)}{2}} h^{q-1} \right).$$

This transformation allows us to deal with a simpler PDE. We know that $v = e^{-\frac{c}{2}}h$ is bounded and positive. So, for some constant $C$ (the value of which is allowed to change from line to line),

$$0 \le h \le C e^{\frac{c}{2}} = C \sin(t)^{\frac{n-1}{2}}.$$

Thus $|R| \le C \sin(t)^{\frac{n-1}{2}}$. And then, from the definition of $h$, the following inequality,

$$\left(\partial_t h - \frac{c'}{2} h\right)^2 + \frac{\Gamma^\theta(h)}{\alpha^2 \sin^2(t)} \le C \sin(t)^{n-1} \quad (73)$$

is equivalent to $||\Gamma_S(v)||_\infty < +\infty$.

We know that $h$ is a smooth function on $(0, \pi) \times S^{d-1}$. So, to prove the previous inequality, it is enough to work around $t=0$ and $t=\pi$. By symmetry, it suffices to treat the case $t=0$.

By definition of $c$, equation (72) can be written as follow

$$\partial_{tt}h - \frac{(n-1)(n-3)}{4} \frac{h}{\sin^2(t)} + \frac{1}{\alpha^2 \sin^2(t)} \Delta_{\theta}h = R - \frac{(n-1)(n-3)}{4}h,$$

or, since we are working around $t=0$, we have

$$\partial_{tt}h - \frac{(n-1)(n-3)}{4} \frac{h}{t^2} + \frac{1}{\alpha^2 \sin^2(t)} \Delta_{\theta}h = R_2, \quad (74)$$

with

$$R_2 = R - \frac{(n-1)(n-3)}{4}h + \frac{(n-1)(n-3)}{4} \left( \frac{1}{\sin^2(t)} - \frac{1}{t^2} \right)h,$$

which satisfies again $|R_2| \le Ct^{\frac{n-1}{2}}$.

Let us write $h = h - h_0 + h_0$ where $h_0 = \int_{S^{d-1}} h dV_{S^{d-1}}$. Then, since $h_0$ doesn't depend on $\theta$,

$$\left(\partial_t h - \frac{c'}{2} h\right)^2 + \frac{\Gamma^\theta(h)}{\alpha^2 \sin^2(t)} \le 2\left(\partial_t(h - h_0) - \frac{c'}{2}(h - h_0)\right)^2 + 2\left(\partial_t h_0 - \frac{c'}{2}h_0\right)^2 + \frac{\Gamma^\theta(h - h_0)}{\alpha^2 \sin^2(t)}.$$

Then Lemma A.6 insures that $|h - h_0| \le Ct^{\frac{n+1}{2}}$, $|h' - h'_0| \le Ct^{\frac{n-1}{2}}$ and $\Gamma^\theta(h - h_0) = \Gamma^\theta(h) \le Ct^{n+1}$. Hence,

$$2\left(\partial_t(h - h_0) - \frac{c'}{2}(h - h_0)\right)^2 + 2\frac{\Gamma^\theta(h - h_0)}{\alpha^2 \sin^2(t)} \le Ct^{n-1}.$$

Now, using the same method as in the proof of Lemma A.6, one can check that $h_0(t) = At^{\frac{n-1}{2}} + O(t^{\frac{n+1}{2}})$ and $h'_0 = At^{\frac{n-3}{2}} + O(t^{\frac{n-1}{2}})$. Thus, we have

$$\partial_t h_0 - \frac{c'}{2} h_0 = O(t^{\frac{n-1}{2}})$$

that is

$$\left(\partial_t h_0 - \frac{c'}{2} h_0\right)^2 \le Ct^{n-1}.$$

Finally, inequality (73) is satisfied, which concludes the proof. ▷

*Remark A.8.* It is interesting to see that with this method, one can check that the function $v$ has a bounded carré du champ if and only of $(a, b) \in \Theta_{FS}$.

We end this section with the following weaker estimate on higher derivatives of $v$.