I'm trying to make my way through Griffiths and Harris's proof of the Hodge Theorem in Principles of Algebraic Geometry. I've gotten through most of it just fine, but I've been stuck on the estimates in the second half of their Regularity Lemma II:
Lemma. Let $P$ be a first order differential operator, where we write $Pu=Qu+Ru$ for
$$(Qu_i)=\sum_{k,j}a_{ij}^k(x)\frac{\partial u_j(x)}{\partial x_k}$$$$(Ru_i)=\sum_{j}b_{ij}(x)u_j(x).$$Further let $P$ satisfy the Gärding inequality:$$||Pu||_0 + ||u||_0 \geq ||u||_1$$for compactly supported $u$. If the equation $Pu=v$ holds for some compactlysupported $v$ in the Solbolev space $\mathcal H_s$, then $u\in \mathcal H_{s+1}$.
(This is stated far after most of the proof of the Hodge Theorem, in their Distributions and Currents section.) The proof reduces down to finding a constant bound for $Qu_\epsilon$, where $$u_\epsilon = \int_{\mathbb R^n}u \chi_\epsilon(x-y)dy$$and $\chi_\epsilon := 1/\epsilon^n\chi(x/\epsilon)$, and $\chi$ is any radially symmetric compactly supported function (this is standard smoothing, which I understand, I'm just including it for reference). We seek to do this by bounding the difference $Qu_\epsilon - (Qu)_\epsilon$, for a similar definition of $(Qu)_\epsilon$ as a smoothed $Qu$, for reasons I also understand (we can bound $(Qu)_\epsilon$), but I don't understand the estimates from their on out. They first argue that for the $s=0$ case the $i$th component of this expression is$$\frac{\partial}{\partial x^k}\left(\sum_{j, k}a^k_{ij}u_j\right) - \sum_{j,k} a^k_{ij}\frac{\partial}{\partial x^k}(u_j)_{\epsilon} - \left[\sum_{j,k}\frac{\partial a^k_{ij}}{\partial x^k}u_j\right]_\epsilon,$$which I don't particularly understand. They then proceed to bound the norm of the "last term" with a constant times the $L^2$ norm of $u$, and the "other term" as$$\frac{1}{\epsilon^{n+1}}\sum_{j,k}\int_{\mathbb R^n}(D_k\chi)\left(\frac y\epsilon\right)\left(a^k_{ij}(x-y)-a^k_{ij}(x)\right)u_j(x-y)dy$$which they in turn bound by$$\left(\frac{C}{\epsilon^{n+1}}\int_{||y||\leq \epsilon K}\left|(D_k\chi)\left(\frac{y}{\epsilon}\right) \right| |y|dy\right) ||u||_0\leq C'||u_0||$$
These four estimates are completely opaque to me; I've copied them over verbatim (as far as I can tell). I'm looking for, if possible, a self-contained justification for each of the last three centered expressions, including the inequality in the last one, and (ideally) some explanation as to what the notation means (for example, what are we summing over / what is $k$ in the first of those three centered expressions).