Let $s$ be an integer and $L$ a periodic linear partial differential operator $L=\{L_{ij}\}$ of order $l$ on $\mathcal{P}$, the space of $2\pi$-periodic functions $R^n\longrightarrow C^m$. The sobolev $s$ norm that I am familiar with is defined on $\mathcal{P}$ by $$||\phi||_s=\sqrt{\sum_{\xi\in\mathbb{Z}^n}(1+|\xi|^2)^s|\phi_\xi|^2} $$ where $\phi_\xi$ are the Fourier coefficients of $\phi.$
In Frank Warner's Foundations of Differentiable Manifolds and Lie groups Proposition 6.25 states that
There are constants $c,k,c'>0$ such that $$||L\phi||_s\leq ck||\phi||_{s+l}+c'||\phi||_{s+l-1} $$ where $c$ depends only on $n,m,l$ and $s$, where $k$ is a bound on the absolute values of the coefficients of the highest order terms in $L$ and $c'$ depends only on $n,m,l,s$ and derivatives of coefficients of $L$ up to order $l.$
I understand the proof in the case $m=1$. Warner then claims that one may use the inequality $||L\phi||\leq const\sum_{ij}||L_{ij}\phi_j||_s$) where $const$ is a constant depending only on $m$, to prove the general case. I can't seem to get any constants in my calculation, so I was wondering whether it is correct. Here is my approach:
\begin{align*} ||L\phi||^2_s&=\sum_\xi(1+|\xi|^2)^s|(L\phi)_\xi|^2\\&=\sum_\xi(1+|\xi|^2)^s|((L\phi)_1)_\xi|^2+\sum_\xi(1+|\xi|^2)^s|((L\phi)_2)_\xi|^2+\ldots\sum_\xi(1+|\xi|^2)^s|((L\phi)_m)_\xi|^2\\&=||(L\phi)_1||^2_s+||(L\phi)_2||^2_s+\ldots+||(L\phi)_m||^2_s \\&=||\sum_j L_{1j}\phi_j||^2_s+||\sum_j L_{2j}\phi_j||^2_s+\ldots+||\sum_j L_{mj}\phi_j||^2_s\\&\leq \left(\sum_{ij}||L_{ij}\phi_j||_s\right)^2. \end{align*}
So far so good. Now, apply the Cauchy-Schwarz inequality to the constant $1$ vector $\mathbf{1}$ and the vector with entries $\|L_{i,j} \phi_j\|_s$ to get that
$$ \left( \sum_{ij} \|L_{i,j} \phi_j\|_s\right)^2\leq m^2 \sum_{ij} \| L_{i,j} \phi_j\|^2_s $$
Thus, $$ \| L \phi\|_s\leq m\sqrt{\sum_{i,j} \|L_{i,j} \phi_j\|^2_s}\leq m\sum_{i,j} \| L_{i,j} \phi_j\|_s, $$ since $\|x\|_1\geq \|x\|_2$ for any $x\in \mathbb{R}^{2m},$ where $\|\cdot\|_p$ denotes the usual $\ell^p$ norm.