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The following result is Theorem 1.1 in:

P. Maheux, L. Saloff-Coste, Analyse sur les boules d'un opérateur sous-elliptique. Math. Ann. 303 (1995), 713–740.

Theorem. Let $(M,g)$ be a complete Riemannian $n$-manifold, such that $Ric\geq kg$ for some $k\in\mathbb{R}$. Let $1\leq p<n$ and $p^*=np/(n-p)$. Then there are constants $C=C(p)$ and $A=A(p)$ such that for every $r>0$ and $\phi\in C^\infty(B(x,r))$ we have $$ \left(\int_{B(x,r)}|\phi-\phi_{B(x,r)}|^{p^*}\right)^{1/p^*}\leq A e^{C(1+\sqrt{|k|}r)}r\, Vol_g(B(x,r))^{\frac{1}{p^*}-\frac{1}{p}} \left(\int_{B(x,r)}|\nabla\phi|^p\right)^{1/p}, $$ where $$ \phi_{B(x,r)}=\frac{1}{Vol_g(B(x,r))}\int_{B(x,r)}\phi. $$

It follows that under the above assumptions, if $\phi\in C_0^\infty(B(x,r))$, then $$ \left(\int_{B(x,r)}|\phi|^{p^*}\right)^{1/p^*}\leq A e^{C(1+\sqrt{|k|}r)}r\, Vol_g(B(x,r))^{\frac{1}{p^*}-\frac{1}{p}} \left(\int_{B(x,r)}|\nabla\phi|^p\right)^{1/p}. $$ (Perhaps with differnet costants $A$ and $C$.)

The key point is that the measure on manifolds with bounded Ricci curvature satisfies so called doubling condition and that such manifolds satisfy the Poincare inequality due to Buser. For more information about how the Poincaré inequality and doubling condition imply the Sobolev inequality in a very general setting see:

P. Hajłasz, P. Koskela, Pekka Sobolev met Poincaré. Mem. Amer. Math. Soc. 145 (2000), no. 688.

For a related question see; Sobolev and Poincaré inequalities on compact Riemannian manifolds

The following result is Theorem 1.1 in:

P. Maheux, L. Saloff-Coste, Analyse sur les boules d'un opérateur sous-elliptique. Math. Ann. 303 (1995), 713–740.

Theorem. Let $(M,g)$ be a complete Riemannian $n$-manifold, such that $Ric\geq kg$ for some $k\in\mathbb{R}$. Let $1\leq p<n$ and $p^*=np/(n-p)$. Then there are constants $C=C(p)$ and $A=A(p)$ such that for every $r>0$ and $\phi\in C^\infty(B(x,r))$ we have $$ \left(\int_{B(x,r)}|\phi-\phi_{B(x,r)}|^{p^*}\right)^{1/p^*}\leq A e^{C(1+\sqrt{|k|}r)}r\, Vol_g(B(x,r))^{\frac{1}{p^*}-\frac{1}{p}} \left(\int_{B(x,r)}|\nabla\phi|^p\right)^{1/p}, $$ where $$ \phi_{B(x,r)}=\frac{1}{Vol_g(B(x,r))}\int_{B(x,r)}\phi. $$

It follows that under the above assumptions, if $\phi\in C_0^\infty(B(x,r))$, then $$ \left(\int_{B(x,r)}|\phi|^{p^*}\right)^{1/p^*}\leq A e^{C(1+\sqrt{|k|}r)}r\, Vol_g(B(x,r))^{\frac{1}{p^*}-\frac{1}{p}} \left(\int_{B(x,r)}|\nabla\phi|^p\right)^{1/p}. $$ (Perhaps with differnet costants $A$ and $C$.)

The key point is that the measure on manifolds with bounded Ricci curvature satisfies so called doubling condition and that such manifolds satisfy the Poincare inequality due to Buser. For more information about how the Poincaré inequality and doubling condition imply the Sobolev inequality in a very general setting see:

P. Hajłasz, P. Koskela, Sobolev met Poincaré. Mem. Amer. Math. Soc. 145 (2000), no. 688.

For a related question see; Sobolev and Poincaré inequalities on compact Riemannian manifolds

The following result is Theorem 1.1 in:

P. Maheux, L. Saloff-Coste, Analyse sur les boules d'un opérateur sous-elliptique. Math. Ann. 303 (1995), 713–740.

Theorem. Let $(M,g)$ be a complete Riemannian $n$-manifold, such that $Ric\geq kg$ for some $k\in\mathbb{R}$. Let $1\leq p<n$ and $p^*=np/(n-p)$. Then there are constants $C=C(p)$ and $A=A(p)$ such that for every $r>0$ and $\phi\in C^\infty(B(x,r))$ we have $$ \left(\int_{B(x,r)}|\phi-\phi_{B(x,r)}|^{p^*}\right)^{1/p^*}\leq A e^{C(1+\sqrt{|k|}r)}r\, Vol_g(B(x,r))^{\frac{1}{p^*}-\frac{1}{p}} \left(\int_{B(x,r)}|\nabla\phi|^p\right)^{1/p}, $$ where $$ \phi_{B(x,r)}=\frac{1}{Vol_g(B(x,r))}\int_{B(x,r)}\phi. $$

It follows that under the above assumptions, if $\phi\in C_0^\infty(B(x,r))$, then $$ \left(\int_{B(x,r)}|\phi|^{p^*}\right)^{1/p^*}\leq A e^{C(1+\sqrt{|k|}r)}r\, Vol_g(B(x,r))^{\frac{1}{p^*}-\frac{1}{p}} \left(\int_{B(x,r)}|\nabla\phi|^p\right)^{1/p}. $$ (Perhaps with differnet costants $A$ and $C$.)

The key point is that the measure on manifolds with bounded Ricci curvature satisfies so called doubling condition and that such manifolds satisfy the Poincare inequality due to Buser. For more information about how the Poincaré inequality and doubling condition imply the Sobolev inequality in a very general setting see:

P. Hajłasz, P. Koskela, Pekka Sobolev met Poincaré. Mem. Amer. Math. Soc. 145 (2000), no. 688.

The following result is Theorem 1.1 in:

P. Maheux, L. Saloff-Coste, Analyse sur les boules d'un opérateur sous-elliptique. Math. Ann. 303 (1995), 713–740.

Theorem. Let $(M,g)$ be a complete Riemannian $n$-manifold, such that $Ric\geq kg$ for some $k\in\mathbb{R}$. Let $1\leq p<n$ and $p^*=np/(n-p)$. Then there are constants $C=C(p)$ and $A=A(p)$ such that for every $r>0$ and $\phi\in C^\infty(B(x,r))$ we have $$ \left(\int_{B(x,r)}|\phi-\phi_{B(x,r)}|^{p^*}\right)^{1/p^*}\leq A e^{C(1+\sqrt{|k|}r)}r\, Vol_g(B(x,r))^{\frac{1}{p^*}-\frac{1}{p}} \left(\int_{B(x,r)}|\nabla\phi|^p\right)^{1/p}, $$ where $$ \phi_{B(x,r)}=\frac{1}{Vol_g(B(x,r))}\int_{B(x,r)}\phi. $$

It follows that under the above assumptions, if $\phi\in C_0^\infty(B(x,r))$, then $$ \left(\int_{B(x,r)}|\phi|^{p^*}\right)^{1/p^*}\leq A e^{C(1+\sqrt{|k|}r)}r\, Vol_g(B(x,r))^{\frac{1}{p^*}-\frac{1}{p}} \left(\int_{B(x,r)}|\nabla\phi|^p\right)^{1/p}. $$ (Perhaps with differnet costants $A$ and $C$.)

The key point is that the measure on manifolds with bounded Ricci curvature satisfies so called doubling condition and that such manifolds satisfy the Poincare inequality due to Buser. For more information about how the Poincaré inequality and doubling condition imply the Sobolev inequality in a very general setting see:

P. Hajłasz, P. Koskela, Pekka Sobolev met Poincaré. Mem. Amer. Math. Soc. 145 (2000), no. 688.

For a related question see; Sobolev and Poincaré inequalities on compact Riemannian manifolds

You can find some references at: Sobolev and Poincaré inequalities on compact Riemannian manifolds but I will try to be more precise. Finding righr references for this particular result is not as simple as it seems.

The following result is Theorem 1.1 in:

P. Maheux, L. Saloff-Coste, Analyse sur les boules d'un opérateur sous-elliptique. Math. Ann. 303 (1995), 713–740.

Theorem. Let $(M,g)$ be a complete Riemannian $n$-manifold, such that $Ric\geq kg$ for some $k\in\mathbb{R}$. Let $1\leq p<n$ and $p^*=np/(n-p)$. Then there are constants $C=C(p)$ and $A=A(p)$ such that for every $r>0$ and $\phi\in C^\infty(B(x,r))$ we have $$ \left(\int_{B(x,r)}|\phi-\phi_{B(x,r)}|^{p^*}\right)^{1/p^*}\leq A e^{C(1+\sqrt{|k|}r)}r\, Vol_g(B(x,r))^{\frac{1}{p^*}-\frac{1}{p}} \left(\int_{B(x,r)}|\nabla\phi|^p\right)^{1/p}, $$ where $$ \phi_{B(x,r)}=\frac{1}{Vol_g(B(x,r))}\int_{B(x,r)}\phi. $$

It follows that under the above assumptions, if $\phi\in C_0^\infty(B(x,r))$, then $$ \left(\int_{B(x,r)}|\phi|^{p^*}\right)^{1/p^*}\leq A e^{C(1+\sqrt{|k|}r)}r\, Vol_g(B(x,r))^{\frac{1}{p^*}-\frac{1}{p}} \left(\int_{B(x,r)}|\nabla\phi|^p\right)^{1/p}. $$ (Perhaps with differnet costants $A$ and $C$.)

The key point is that the measure on manifolds with bounded Ricci curvature satisfies so called doubling condition and that such manifolds satisfy the Poincare inequality due to Buser. For more information about how the Poincaré inequality and doubling condition imply the Sobolev inequality in a very general setting see:

P. Hajłasz, P. Koskela, Pekka Sobolev met Poincaré. Mem. Amer. Math. Soc. 145 (2000), no. 688.