elliptic regularity theory (so-called *Schauder theory*), namely that if $w\in C^{k,a}$ and $Lv=w$ then $v\in C^{k+l,a}$ and there is a universal constant $C$ so that $$\|v\|_{C^{k+l,a}}\leq C(\|Lv\|_{C^{k,a}}+\|v\|_{C^0})$$ (and we can drop the $\|v\|_{C^0}$ term if $v$ is orthogonal to $\operatorname{Ker}L$);

-   the adjoint operator $L^*$ and that $\sigma_{L^*}=(-1)^l\sigma_L^*$ so that $L^*$ is elliptic if and only if $L$ is elliptic; and

-   the Fredholm theory of $L$, namely that $\operatorname{Ker}L$ (and hence $\operatorname{Ker}L^*$) is finite-dimensional, and we can solve $Lv=w$ if and only if $w\in (\operatorname{Ker}L^*)^{\perp}$.

We shall discuss this in a model example which we shall use throughout this section.

The Laplacian on functions is given by $\Delta f=\d^*\d f$ which in normal coordinates at a point is given by $f\mapsto -\sum_i \frac{\partial^2f}{\partial x_i^2}$, so it is a linear second order differential operator. We see that its principal symbol is $\sigma_{\Delta}(x,\xi)f= -|\xi|^2f$ which is an isomorphism for $\xi\in T^*_xN\setminus\{0\}$, so $\Delta$ is elliptic. We therefore have that if $h\in C^{k,a}(N)$ and $\Delta f=h$ then $f\in C^{k+2,a}(N)$, and we have an estimate $$\|f\|_{C^{k+2,a}}\leq C(\|\Delta f\|_{C^{k,a}}+\|f\|_{C^0}).$$ We also know that $\Delta^*=\Delta$ and $\operatorname{Ker}\Delta$ is given by the constant functions (since if $f\in\operatorname{Ker}\Delta$ then $$0=\langle f,\Delta f\rangle_{L^2}=\langle f,\d^*\d f\rangle_{L^2}=\|\d f\|_{L^2}^2$$ so $\d 