Use of cut-off functions and partitions of unity This is a simple problem, but I would still be very thankful if you could give me an advice on it.
I'm trying to show that in a compact M-dimensional manifold,
$$\int e^w \sqrt{g}\, dx \leq C \exp \Big( c \|Du\|_{L^n}^n + \|u\|_{L^n}^n \Big),$$ 
where we used the standard Einstein notation.
I know I should use the Trudinger inequality, so I see that
$$ |u(x)| = \frac{|u(x)|}{ \kappa \|D u\|_{L^n} } \cdot \kappa \|D u\|_{L^n}\leq \frac{n-1}{n} \left( \left( \frac{|u(x)|}{ \kappa \|DU\|_{L^n}} \right)^{\tfrac{n}{n-1}} + \Big( \kappa \| Du\|_{L^n} \Big)^{\tfrac{n}{n-1}} \right).$$
So then
\begin{align*} \int e^{|u(x)|} dx &\leq \int \exp\left(\frac{n-1}{n} \left( \left( \frac{|u(x)|}{ \kappa \|DU\|_{L^n}} \right)^{\tfrac{n}{n-1}} + \Big( \kappa \| Du\|_{L^n} \Big)^{\tfrac{n}{n-1}} \right)\right) dx \\
&\leq \int\left( \exp\left(\frac{n-1}{n}  \left( \frac{|u(x)|}{ \kappa \|DU\|_{L^n}} \right)^{\tfrac{n}{n-1}} \right) \exp\left(\frac{n-1}{n} \Big( \kappa \| Du\|_{L^n} \Big)^{\tfrac{n}{n-1} }\right)\right) dx \\
& = \exp\left(\frac{n-1}{n} \Big( \kappa \| Du\|_{L^n} \Big)^{\tfrac{n}{n-1} }\right) \int \exp\left(\frac{n-1}{n} \left(\frac{|u(x)|}{ \kappa \|DU\|_{L^n}}\right)^{\tfrac{n}{n-1}}\right)\,dx \\
&\leq \exp\left(\frac{n-1}{n} \Big( \kappa \| Du\|_{L^n} \Big)^{\tfrac{n}{n-1} }\right) \cdot C \\
&= C \exp\left(\frac{n-1}{n} \Big( \kappa \| Du\|_{L^n} \Big)^{\tfrac{n}{n-1}}\right),
\end{align*}
since by the Trudinger inequality, $\displaystyle \int \exp\left(\frac{n-1}{n} \left(\frac{|u(x)|}{ \kappa \|DU\|_{L^n}}\right)^{\tfrac{n}{n-1}}\right) dx \leq C.$
Now, I would like to try to use partitions of unity to transport this inequality from $\mathbb{R}^n$ to manifolds, and also take advantage of cutoff functions. I read about them, but I'm not sure how to implement them. Do you think you could tell me how to do it?
Thank you very much in advance!
 A: I'm going to use $w$ instead of $u$ in my answer:
Since $M$ is a compact manifold, let $U$ be an open set containing $M$.  Then $\exists$ a cutoff function $\varphi \in C_{0}^{\infty}(M)$ such that $0 \leq \varphi \leq 1$, $\phi \equiv 1$ on $M$, and $supp(\varphi) \subset U$.
To deal with the full manifold, cover it by $\displaystyle M = \cup_{\alpha =1}^{\infty}M_{\alpha}$, since $M$ is compact.
Then, we want to pick smooth functions $\chi$ such that $\displaystyle \chi_{\alpha} \in C_{0}^{\infty}(M_{\alpha})$, so each one lives inside the corresponding neighborhood with the property that $\sum_{\alpha=1}^{N}\chi_{\alpha}=1$.
Now, for any function $w$ on $M$, we write $\displaystyle w = \left( \sum_{\alpha = 1}^{N}\chi_{\alpha}\right)w = \sum_{\alpha=1}^{N}\left(\chi_{\alpha}w \right)$.
Notice that $\displaystyle w = \frac{w}{\kappa ||Dw||_{L^{n}}} \cdot \kappa ||Dw||_{L^{n}} \leq \frac{n-1}{n}\left( \left( \frac{w}{\kappa ||Dw ||_{L^{n}}}\right)^{\frac{n}{n-1}} + (\kappa ||Dw||_{L^{n}})^{\frac{n}{n-1}}\right)$.
Then, $\displaystyle \int e^{w} \leq \int \exp\left[ \frac{n-1}{n} \left(\frac{w}{\kappa ||Dw||_{L^{n}}}\right)^{\frac{n}{n-1}}+\frac{n-1}{n}\left( \kappa ||Dw||_{L^{n}}\right)^{\frac{n}{n-1}}\right]$
$\displaystyle = \int \left(\exp\left[ \frac{n-1}{n} \left( \frac{w}{\kappa ||Dw||_{L^{n}}}\right)^{\frac{n}{n-1}}\right] \exp \left[\frac{n-1}{n} \left( \kappa ||Dw||_{L^{n}}\right)^{\frac{n}{n-1}} \right]\right)$
$\displaystyle =\exp \left[\frac{n-1}{n} \left( \kappa ||Dw||_{L^{n}}\right)^{\frac{n}{n-1}} \right]  \int \exp\left[ \frac{n-1}{n} \left( \frac{w}{\kappa ||Dw||_{L^{n}}}\right)^{\frac{n}{n-1}}\right]$
$\displaystyle \leq C \exp\left[ \frac{n-1}{n}\left(\kappa ||Dw||_{L^{n}} \right)^{\frac{n}{n-1}}\right]$.
Notice that $\frac{n-1}{n} < 1$ and that $\frac{n}{n-1} > 1$, so we have that this is 
$\displaystyle \leq C \exp \left[ \left( \kappa ||Dw||_{L^{n}} \right)^{\frac{n}{n-1}}\right]$
$\displaystyle = C \exp\left[ \kappa^{\frac{n}{n-1}} ||Dw||_{L^{n}}^{\frac{n}{n-1}}\right]$,
and recall that in the Trudinger Inequality, $\kappa$ is chosen to be large, so this becomes
$\displaystyle \leq C \exp\left[\kappa ||Dw||_{L^{n}}^{\frac{n}{n-1}}\right]$   (*)
Now, let's take a look at $||Dw||$.  Substituting in our expression for $w$, and applying Leibnitz' Rule, we obtain
$\displaystyle ||Dw|| = \left\Vert D \left(\sum_{\alpha = 1}^{N} \chi_{\alpha} w \right)\right\Vert_{L^{n}} = \left\Vert \sum_{\alpha=1}^{N} D \left( \chi_{\alpha} w\right)\right\Vert_{L^{n}} \leq \sum_{\alpha=1}^{N}\left\Vert D\left( \chi_{\alpha}w\right)\right\Vert_{L^{n}}$
$\displaystyle = \sum_{\alpha =1}^{N} \left\Vert \left(D\chi_{\alpha} \right)w + \chi_{\alpha} \left( Dw\right)\right\Vert_{L^{n}}$
$\displaystyle \leq \sum_{\alpha =1}^{N} \left\Vert \left(D\chi_{\alpha}\right)w \right\Vert_{L^{n}} + \sum_{\alpha =1}^{N}\left\Vert \chi_{\alpha}\left(Dw \right) \right\Vert_{L^{n}}$
$\displaystyle \leq b\left\Vert w \right\Vert_{L^{n}} + \left\Vert Dw \right\Vert_{L^{n}}$, where $b$ is a constant chosen to be sufficiently large.
Substituting this all back into (), we obtain that ()
$\displaystyle \leq C \exp \left[\kappa \left(b \left\Vert w \right\Vert_{L^{n}} + ||Dw||_{L^{n}}\right)^{\frac{n}{n-1}} \right]$.
And since $b$ is chosen to be sufficiently large, we have that this 
$\displaystyle \leq C \exp \left[\kappa \left( b ||w||_{L^{n}} + ||Dw||_{L^{n}}\right)^{n} \right]$
$\displaystyle \leq C \exp \left[\kappa\cdot 2^{n} \left( b^{n} ||w||_{L^{n}}^{n} + ||Dw||_{L^{n}}^{n} \right) \right]$
$\displaystyle = C \exp \left[ \kappa \cdot 2^{n} \cdot b^{n} ||w||_{L^{n}}^{n} + k \cdot 2^{n} ||Dw||_{L^{n}}^{n} \right]$.  (**)
Let $\kappa \cdot 2^{n} \cdot b^{n} ||w||_{L^{n}}^{n} + k \cdot 2^{n} ||Dw||_{L^{n}}^{n}   = ||w||_{L^{n}}^{n} + (\kappa \cdot 2^{n} \cdot b^{n} - 1)||w||_{L^{n}}^{n} + \kappa \cdot 2^{n} ||Dw||_{L^{n}}^{n}$
By Poincare, this is 
$\displaystyle \leq ||w||_{L^{n}}^{n}+(\kappa \cdot 2^{n}\cdot b^{n} - 1)\cdot \tilde{c}^{n}||Dw||_{L^{n}}^{n} + \kappa \cdot 2^{n} ||Dw||_{L^{n}}^{n}$
$\displaystyle ||w||_{L^{n}}^{n} + \left[(\kappa \cdot 2^{n} \cdot b^{n} - 1)\cdot \tilde{c}^{n} + \kappa \cdot 2^{n} \right]\cdot ||Dw||_{L^{n}}^{n}$
$\displaystyle ||w||_{L^{n}}^{n} + c ||Dw||_{L^{n}}^{n}$.
So, finally, (**) $\displaystyle \leq C \exp \left[ ||w||_{L^{n}}^{n} + c ||Dw||_{L^{n}}^{n} \right]$.
I'm not 100% sure this is correct.  As you can see, I had to get a bit creative with the constants, so if anybody is out there who can critique my math, that would be great! :)
