**Lemma A.9.** Assume that $n > 4$ and $(a, b) \in \Theta_{FS}$. Let $f \in D(L_S)$, $f \ge 0$. Then, in the variables $(t, \theta) \in (0, \pi) \times S^{d-1}$ introduced in (66),

$$ \int_0^\pi \left| \int_{S^{d-1}} \Gamma_2^S(f) dV_{S^{d-1}} \right| \sin^{n-1}(t) dt < +\infty $$

**Proof**

Let $(\zeta_k)_{k \in \mathbb{N}} \in C_c^\infty(0, \pi)$ 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 $|\zeta_k'| \le 2k$, $|\zeta_k''| \le 2k^2$ and $f_k = f\zeta_k$, so that $f_k \to f$ in $D(L_S)$. For $h \in C_c^\infty(\mathbf{R}^d \setminus \{0\})$ and $t \in (0, \pi)$, set $\gamma_2(h)(t) = \int_{S^{d-1}} \Gamma_S^2(h) dV_{S^{d-1}}$. Since $(a, b) \in \Theta_{FS}$, $\gamma_2$ is a nonnegative quadratic form (see Proposition 3.4) and so the Cauchy-Schwarz inequality holds:

$$ |\gamma_2(f_k)^{1/2} - \gamma_2(f_l)^{1/2}|^2 \le \gamma_2(f_k - f_l). $$

Thus, letting $d\mu_t = \frac{1}{Z} \sin^{n-1}(t)dt$,

$$ \int_0^\pi |\gamma_2(f_k)^{1/2} - \gamma_2(f_l)^{1/2}|^2 d\mu_t \le \int_0^\pi \gamma_2(f_k - f_l) d\mu_t = \int \Gamma_S^2(f_k - f_l) d\mu_S = \int (L_S(f_k - f_l))^2 d\mu_S. $$

Hence, $(\gamma_2(f_k)^{1/2})$ is a Cauchy sequence in $L^2(d\mu_x)$ and so $(\gamma_2(f_k))$ converges to some function $\gamma$ in $L^1(d\mu_t)$. In addition, for fixed $t$, there exists $K = K_t$ such that for all $k \ge K$ and $\theta \in S^{d-1}$, $f_k(t, \theta) = f(t, \theta)$, whence $\gamma_2(f_k)(t) \to \gamma_2(f)(t)$ for all $t \in (0, \pi)$. Hence, $\gamma = \gamma_2(f)$ and the lemma follows. ▷

## A.3 List of constants and regions of parameters

We recall in this section the definition of the parameters and also some useful properties. Recall that $d \in \mathbb{N}$ is the topological dimension of the considered spaces, and that we assume that $d \ge 3$. Recall from the introduction the definition of the parameter range

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

where $a_c = (d - 2)/2$. This is the set of parameters $(a, b)$ where the CKN inequality (1) holds for all test functions $v \in C_c^\infty(\mathbf{R}^d)$ which need not vanish near the origin (recall that the limit case $b = a + 1$ has been removed for simplicity). We also defined the number $\alpha = 1 + a - \frac{pb}{2}$, that is

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

Clearly $\alpha \ge 0$, for any $(a, b) \in \Theta$, including the limiting case $a = b = 0$ for which $\alpha = 1$. For any $(a, b) \in \Theta$, the exponent $p$ is given by

$$ p = \frac{d}{a_c - a + b} < 2^* = \frac{2d}{d - 2} $$

and

$$ p = \frac{2n}{n - 2}, $$

that is

$$ n = d + \frac{d(b - a)}{1 + a - b} = \frac{d}{1 + a - b}. $$

We always have $n \ge d$, and we call $n$ the intrinsic dimension of the considered model spaces. From a straight-forward computation, we have

$$ d - n\alpha - pb = 0 \qquad (75) $$

The constant $\mathfrak{B}_{DGZ} = \alpha^2(2 - n) + d - 2$ which appears throughout the paper takes the following form with respect to $a$ and $b$:

$$ \mathfrak{B}_{DGZ} = -2 \frac{(a_c - a)^2(1 + a - b)}{a_c + b - a} + 2a_c $$
**Lemma A.9.** Assume that $n > 4$ and $(a, b) \in \Theta_{FS}$. Let $f \in D(L_S)$, $f \ge 0$. Then, in the variables $(t, \theta) \in (0, \pi) \times S^{d-1}$ introduced in (66),

$$ \int_0^\pi \left| \int_{S^{d-1}} \Gamma_2^S(f) dV_{S^{d-1}} \right| \sin^{n-1}(t) dt < +\infty $$

**Proof**

Let $(\zeta_k)_{k \in \mathbb{N}} \in C_c^\infty(0, \pi)$ 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 $|\zeta_k'| \le 2k$, $|\zeta_k''| \le 2k^2$ and $f_k = f\zeta_k$, so that $f_k \to f$ in $D(L_S)$. For $h \in C_c^\infty(\mathbf{R}^d \setminus \{0\})$ and $t \in (0, \pi)$, set $\gamma_2(h)(t) = \int_{S^{d-1}} \Gamma_S^2(h) dV_{S^{d-1}}$. Since $(a, b) \in \Theta_{FS}$, $\gamma_2$ is a nonnegative quadratic form (see Proposition 3.4) and so the Cauchy-Schwarz inequality holds:

$$ |\gamma_2(f_k)^{1/2} - \gamma_2(f_l)^{1/2}|^2 \le \gamma_2(f_k - f_l). $$

Thus, letting $d\mu_t = \frac{1}{Z} \sin^{n-1}(t)dt$,

$$ \int_0^\pi |\gamma_2(f_k)^{1/2} - \gamma_2(f_l)^{1/2}|^2 d\mu_t \le \int_0^\pi \gamma_2(f_k - f_l) d\mu_t = \int \Gamma_S^2(f_k - f_l) d\mu_S = \int (L_S(f_k - f_l))^2 d\mu_S. $$

Hence, $(\gamma_2(f_k)^{1/2})$ is a Cauchy sequence in $L^2(d\mu_x)$ and so $(\gamma_2(f_k))$ converges to some function $\gamma$ in $L^1(d\mu_t)$. In addition, for fixed $t$, there exists $K = K_t$ such that for all $k \ge K$ and $\theta \in S^{d-1}$, $f_k(t, \theta) = f(t, \theta)$, whence $\gamma_2(f_k)(t) \to \gamma_2(f)(t)$ for all $t \in (0, \pi)$. Hence, $\gamma = \gamma_2(f)$ and the lemma follows. ▷

## A.3 List of constants and regions of parameters

We recall in this section the definition of the parameters and also some useful properties. Recall that $d \in \mathbb{N}$ is the topological dimension of the considered spaces, and that we assume that $d \ge 3$. Recall from the introduction the definition of the parameter range

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

where $a_c = (d-2)/2$. This is the set of parameters $(a, b)$ where the CKN inequality (1) holds for all test functions $v \in C_c^\infty(\mathbf{R}^d)$ which need not vanish near the origin (recall that the limit case $b = a+1$ has been removed for simplicity). We also defined the number $\alpha = 1 + a - \frac{pb}{2}$, that is

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

Clearly $\alpha \ge 0$, for any $(a, b) \in \Theta$, including the limiting case $a = b = 0$ for which $\alpha = 1$. For any $(a, b) \in \Theta$, the exponent $p$ is given by

$$ p = \frac{d}{a_c - a + b} < 2^* = \frac{2d}{d-2} $$

and

$$ p = \frac{2n}{n-2}, $$

that is

$$ n = d + \frac{d(b-a)}{1+a-b} = \frac{d}{1+a-b}. $$

We always have $n \ge d$, and we call $n$ the intrinsic dimension of the considered model spaces. From a straight-forward computation, we have

$$ d - n\alpha - pb = 0 \qquad (75) $$

The constant $\mathfrak{B}_{DGZ} = \alpha^2(2-n) + d-2$ which appears throughout the paper takes the following form with respect to $a$ and $b$:

$$ \mathfrak{B}_{DGZ} = -2 \frac{(a_c - a)^2 (1 + a - b)}{a_c + b - a} + 2a_c $$
Let us also recall the definition of the Felli-Schneider region: for $a \le 0$,

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

Let us prove Lemma 1.9, which simplifies the expression of the Felli-Schneider region and shows its relation to our region $\Theta_{DGZ} = \{(a, b) \in \Theta, \mathfrak{B}_{DGZ} \ge 0\}$.

**Proof of Lemma 1.9**

Since

$$\mathfrak{B}_{DGZ} + \frac{n-d}{n-1} = (n-2)\left(-\alpha^2 + \frac{d-1}{n-1}\right),$$

we have

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

The fact that

$$\Theta_{FS} = \left\{ (a, b) \in \Theta, \alpha^2 \le \frac{d-1}{n-1} \right\}$$

is more delicate and is proved in [DEL16, Sec. 3].

Finally, the identity $\Theta_{FS} = \{(a, b) \in \Theta, \alpha \in [0, 1]\}$ is a little trickier. Since $(a, b) \in \Theta$, then

$$0 < \alpha = \frac{(a_c - a)(a + 1 - b)}{a_c - a + b} \le 1 - \frac{a}{a_c}. \qquad (76)$$

In the case where $a \ge 0$ it follows that $0 < \alpha \le 1$. Assume now that $a < 0$. By definition of $\alpha$,

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

Then the inequality $b \ge \mathfrak{B}_{FS}$ is equivalent to

$$\frac{1 + a_c}{\alpha + a_c - a} \ge \frac{d}{2\sqrt{(a_c - a)^2 + d - 1}}.$$

Since $1 + a_c = d/2$, the previous condition becomes

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

If $\alpha > 1$, this inequality implies $a > 0$, contradicting our assumption. We just proved that

$$\Theta_{FS} \subset \{(a, b) \in \Theta, \alpha \in (0, 1]\}.$$

Since $\alpha = 0$ for $(a, b) = (0, 1)$, $\alpha = 1$ for $(a, b) = (0, 0)$, since $\Theta$ is connected and since $\alpha$ depends continuously on $(a, b)$, we deduce that $\Theta_{FS} = \{(a, b) \in \Theta, \alpha \in (0, 1]\}$. At last, if $\alpha = 1$, then, from the previous displayed inequality, $a \ge 0$ and from (76), $a \le 0$. That is, $a = b = 0$. ▷

**Remark A.10.** The normalizing constant $Z$ defined in Theorem 1.7 is finite. Indeed, using (75) and the change of variable $|x| = e^t$, we find

$$Z = \int d\mu_{\mathbf{S}} = \int_{\mathbb{R}^d \setminus \{0\}} \left( \frac{2}{1 + |x|^{2\alpha}} \right)^n |x|^{-bp} dx = \int_{\mathbb{R}^d \setminus \{0\}} \left( \frac{2}{|x|^{-\alpha} + |x|^{\alpha}} \right)^n |x|^{-d} dx = |\mathbf{S}^{d-1}| \int_{\mathbb{R}} \cosh(\alpha t)^{-n} dt < \infty.$$
Let us also recall the definition of the Felli-Schneider region: for $a \le 0$,

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

Let us prove Lemma 1.9, which simplifies the expression of the Felli-Schneider region and shows its relation to our region $\Theta_{DGZ} = \{(a, b) \in \Theta, \mathfrak{B}_{DGZ} \ge 0\}$.

**Proof of Lemma 1.9**

Since

$$\mathfrak{B}_{DGZ} + \frac{n-d}{n-1} = (n-2)\left(-\alpha^2 + \frac{d-1}{n-1}\right),$$

we have

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

The fact that

$$\Theta_{FS} = \left\{ (a, b) \in \Theta, \alpha^2 \le \frac{d-1}{n-1} \right\}$$

is more delicate and is proved in [DEL16, Sec. 3].

Finally, the identity $\Theta_{FS} = \{(a, b) \in \Theta, \alpha \in [0, 1]\}$ is a little trickier. Since $(a, b) \in \Theta$, then

$$0 < \alpha = \frac{(a_c - a)(a + 1 - b)}{a_c - a + b} \le 1 - \frac{a}{a_c}. \qquad (76)$$

In the case where $a \ge 0$ it follows that $0 < \alpha \le 1$. Assume now that $a < 0$. By definition of $\alpha$,

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

Then the inequality $b \ge \mathfrak{B}_{FS}$ is equivalent to

$$\frac{1 + a_c}{\alpha + a_c - a} \ge \frac{d}{2\sqrt{(a_c - a)^2 + d - 1}}.$$

Since $1 + a_c = d/2$, the previous condition becomes

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

If $\alpha > 1$, this inequality implies $a > 0$, contradicting our assumption. We just proved that

$$\Theta_{FS} \subset \{(a, b) \in \Theta, \alpha \in (0, 1]\}.$$

Since $\alpha = 0$ for $(a, b) = (0, 1)$, $\alpha = 1$ for $(a, b) = (0, 0)$, since $\Theta$ is connected and since $\alpha$ depends continuously on $(a, b)$, we deduce that $\Theta_{FS} = \{(a, b) \in \Theta, \alpha \in (0, 1]\}$. At last, if $\alpha = 1$, then, from the previous displayed inequality, $a \ge 0$ and from (76), $a \le 0$. That is, $a = b = 0$. ▷

**Remark A.10.** The normalizing constant $Z$ defined in Theorem 1.7 is finite. Indeed, using (75) and the change of variable $|x| = e^t$, we find

$$Z = \int d\mu_S = \int_{\mathbb{R}^d \setminus \{0\}} \left( \frac{2}{1 + |x|^{2\alpha}} \right)^n |x|^{-bp} dx = \int_{\mathbb{R}^d \setminus \{0\}} \left( \frac{2}{|x|^{-\alpha} + |x|^{\alpha}} \right)^n |x|^{-d} dx = |\mathbf{S}^{d-1}| \int_{\mathbb{R}} \cosh(\alpha t)^{-n} dt < \infty.$$
References

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[BGL14] D. Bakry, I. Gentil, and M. Ledoux. *Analysis and geometry of Markov diffusion operators*. Springer, 2014.

[BL96] D. Bakry and M. Ledoux. Sobolev inequalities and Myers's diameter theorem for an abstract Markov generator. *Duke Math. J.*, 85(1):253–270, 1996.

[BV91] M.-F. Bidaut-Veron and L. Véron. Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations. *Invent. Math.*, 106(3):489–539, 1991.

[Cas12] J. S. Case. Smooth metric measure spaces, quasi-Einstein metrics, and tractors. *Cent. Eur. J. Math.*, 10(5):1733–1762, 2012.

[CGY06] S.-Y. A. Chang, M. J. Gursky, and P. Yang. Conformal invariants associated to a measure. *Proc. Natl. Acad. Sci. USA*, 103(8):2535–2540, 2006.

[CKN84] L. Caffarelli, R. Kohn, and L. Nirenberg. First order interpolation inequalities with weights. *Compos. Math.*, 53:259–275, 1984.

[CW01] F. Catrina and Z.-Q. Wang. On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions. *Commun. Pure Appl. Math.*, 54(2):229–258, 2001.

[DEL14] J. Dolbeault, M. J. Esteban, and M. Loss. Nonlinear flows and rigidity results on compact manifolds. *J. Funct. Anal.*, 267(5):1338–1363, 2014.

[DEL16] J. Dolbeault, M. J. Esteban, and M. Loss. Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces. *Invent. Math.*, 206(2):397–440, 2016.

[DGZ20] L. Dupaigne, I. Gentil, and S. Zugmeyer. Sobolev's inequality under a curvature-dimension condition, 2020. To appear in *Ann. Fac. Sci. Toulouse Math.*

[FS03] V. Felli and M. Schneider. Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type. *J. Differ. Equations*, 191(1):121–142, 2003.

[GZ21] I. Gentil and S. Zugmeyer. A family of Beckner inequalities under various curvature-dimension conditions. *Bernoulli*, 27(2):751–771, 2021.

[Heb97] E. Hebey. *Introduction à l'analyse non linéaire sur les variétés*. Paris: Diderot Editeur, 1997.

[Heb00] E. Hebey. *Nonlinear analysis on manifolds: Sobolev spaces and inequalities*, volume 5. Providence, RI: American Mathematical Society (AMS), 2000.

[Ket15] C. Ketterer. Cones over metric measure spaces and the maximal diameter theorem. *J. Math. Pures Appl. (9)*, 103(5):1228–1275, 2015.

[NV21] F. Nobili and I.Y. Violo. Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds. *Arxiv*, 2021.

[Pro15] A. Profeta. The sharp Sobolev inequality on metric measure spaces with lower Ricci curvature bounds. *Potential Anal.*, 43(3):513–529, 2015.

[Sta66] G. Stampacchia. Equations elliptiques du second ordre à coefficients discontinus. Séminaire de mathématiques supérieures (été 1965) 16. Les Presses de l'Université de Montréal, 1966.

This work was supported by the French ANR-17-CE40-0030 EFI project.

L. D., I. G. Institut Camille Jordan, Umr Cnrs 52065, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne cedex. dupaigne, gentil@math.univ-lyon1.fr

S. Z. ENS de Lyon, CNRS, UMPA UMR 5669, F-69364 Lyon Cedex 07. simon.zugmeyer@ens-lyon.fr
## References

[BFL08] R. D. Benguria, R. L. Frank, and M. Loss. The sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half-space. *Math. Res. Lett.*, 15(4):613–622, 2008.

[BGL14] D. Bakry, I. Gentil, and M. Ledoux. *Analysis and geometry of Markov diffusion operators*. Springer, 2014.

[BL96] D. Bakry and M. Ledoux. Sobolev inequalities and Myers's diameter theorem for an abstract Markov generator. *Duke Math. J.*, 85(1):253–270, 1996.

[BV91] M.-F. Bidaut-Veron and L. Véron. Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations. *Invent. Math.*, 106(3):489–539, 1991.

[Cas12] J. S. Case. Smooth metric measure spaces, quasi-Einstein metrics, and tractors. *Cent. Eur. J. Math.*, 10(5):1733–1762, 2012.

[CGY06] S.-Y. A. Chang, M. J. Gursky, and P. Yang. Conformal invariants associated to a measure. *Proc. Natl. Acad. Sci. USA*, 103(8):2535–2540, 2006.

[CKN84] L. Caffarelli, R. Kohn, and L. Nirenberg. First order interpolation inequalities with weights. *Compos. Math.*, 53:259–275, 1984.

[CW01] F. Catrina and Z.-Q. Wang. On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions. *Commun. Pure Appl. Math.*, 54(2):229–258, 2001.

[DEL14] J. Dolbeault, M. J. Esteban, and M. Loss. Nonlinear flows and rigidity results on compact manifolds. *J. Funct. Anal.*, 267(5):1338–1363, 2014.

[DEL16] J. Dolbeault, M. J. Esteban, and M. Loss. Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces. *Invent. Math.*, 206(2):397–440, 2016.

[DGZ20] L. Dupaigne, I. Gentil, and S. Zugmeyer. Sobolev's inequality under a curvature-dimension condition, 2020. To appear in *Ann. Fac. Sci. Toulouse Math.*.

[FS03] V. Felli and M. Schneider. Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type. *J. Differ. Equations*, 191(1):121–142, 2003.

[GZ21] I. Gentil and S. Zugmeyer. A family of Beckner inequalities under various curvature-dimension conditions. *Bernoulli*, 27(2):751–771, 2021.

[Heb97] E. Hebey. *Introduction à l'analyse non linéaire sur les variétés*. Paris: Diderot Editeur, 1997.

[Heb00] E. Hebey. *Nonlinear analysis on manifolds: Sobolev spaces and inequalities*, volume 5. Providence, RI: American Mathematical Society (AMS), 2000.

[Ket15] C. Ketterer. Cones over metric measure spaces and the maximal diameter theorem. *J. Math. Pures Appl. (9)*, 103(5):1228–1275, 2015.

[NV21] F. Nobili and I.Y. Violo. Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds. *Arxiv*, 2021.

[Pro15] A. Profeta. The sharp Sobolev inequality on metric measure spaces with lower Ricci curvature bounds. *Potential Anal.*, 43(3):513–529, 2015.

[Sta66] G. Stampacchia. Equations elliptiques du second ordre à coefficients discontinus. Séminaire de mathématiques supérieures (été 1965) 16. Les Presses de l'Université de Montréal, 1966.

This work was supported by the French ANR-17-CE40-0030 EFI project.

L. D., I. G. Institut Camille Jordan, Umr Cnrs 52065, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne cedex. dupaigne, gentil@math.univ-lyon1.fr

S. Z. ENS de Lyon, CNRS, UMPA UMR 5669, F-69364 Lyon Cedex 07. simon.zugmeyer@ens-lyon.fr