I’m working through a proof of the Schauder estimates for Elliptic PDE. Basically, right now I’ve gotten to the point where I have an estimate of the form:
$$ |D^2 v|_{\alpha} \leq C |\sum_{ij} a_{ij} D^{ij}v|_{\alpha} $$
For all $v \in C^{2, \alpha} \left( \mathbb{R}^n \right)$, where $|D^2 v|_{\alpha} = \sum_{ij} |D^{ij}v|_{\alpha}$.
Next, it seems the thing I need to do is employ this estimate on a ball. I don’t see why I can use this estimate on a ball instead of the full space. I was thinking you prove that you can extend every $C^{2,\alpha}(B_\rho)$ to a function on the full space without increasing the Holder semi-norm by more than a constant. But that turns out to be harder than I thought it would be.
Can someone give me a little direction?
Here is an approach. Use a partition of unity, and a smooth diffeomorphism, to change the problem to $v$ is compactly supported and in $C^{2,\alpha}([0,\infty) \times \mathbb R^{n-1})$. Let $\theta_0, \theta_1, \theta_2:\mathbb R \to \mathbb R$ be smooth, compactly supported functions such that $\theta_i^{(j)}(0) = \delta_{i,j}$. Then for $(t,x) \in (-\infty,0] \times \mathbb R^{n-1}$, let $$ v(t,x) = v(0,x) \theta_0(t) + \frac{\partial v}{\partial t}(0,x) \theta_1(t) + \frac{\partial^2 v}{\partial t^2}(0,x) \theta_2(t) .$$