Laplace equation inequalities and essential supremum I started a PDE course and got some issues with technical problems. This one centers around the proof for the solution of the laplacian equation. I am asked to verify the following inequality which one also finds on page 24 of L.C. Evans 'Partial Differential Equations'. 
$$ \Bigg\lvert \int\limits_{B(0,\epsilon)} \Phi(y)\Delta_x\,f(x-y)\, dy \Bigg\rvert \leq C\Vert D^2 f\Vert_{L^{\infty}(\mathbb{R}^n)}\int\limits_{B(0,\epsilon)}\lvert\Phi(y)\rvert \, dy \leq \begin{cases} C\epsilon^2 \lvert \log \epsilon \rvert & (n=2) \\ C\epsilon^2 & (n\geq3)\end{cases}$$
What I already did: I already estimated the RHS integral part for the case $n = 2$. This was quite easy since I can switch to polar coordinates after which it is more or less ordinary integration and results in $\frac{1}{2}\epsilon^2\log\lvert\epsilon\rvert$ as upper estimate from the integral part alone.
Where my problems are: I have two problems. First, I don't have a clue what the essential supremum norm on $D^2f$, that is the Hessian matrix, is. I know what the essential supremum norm means and could compute it for ordinary functions but how do I apply it to matrices? I searched online and in the appendix of Evan's book but nothing came of it. Second, how can I estimate the RHS integral for $n\geq 3$? I can no longer switch to polar coordinates, how do I integrate this not so easy function 
$$ \Phi(x) := \frac{1}{n(n-2)\alpha(n)}\frac{1}{\lvert x\rvert^{n-2}}$$ function over an $n$-dimensional ball or produce a plausible estimate of it? Evans himself writes those inequalities down without any comment. 
 A: First problem. The estimate for the integral is simply 
$$ \Bigg\lvert \int\limits_{B(0,\varepsilon)} \Phi(y)\Delta_x\,f(x-y)\, dy \Bigg\rvert \leq C\Vert \Delta_x\,f(x-y) \Vert_{L^\infty({B(0,\varepsilon))}}\int\limits_{B(0,\varepsilon)}\lvert\Phi(y)\rvert \, dy $$
followed by a pointwise estimate $|\Delta f| \le C |D^2 f|$, where $|\cdot|$ stands for any chosen matrix norm. Let us choose the $l^1$ norm $|A| := \sum_{i,j} |a_{ij}|$, then this inequality is trivially satisfied with $C=1$, as $|\Delta f| \le |f_{x_1 x_1}|+\ldots+|f_{x_n x_n}|$. Of course in general $C$ depends on the choice of the matrix norm. 
The $L^p$ spaces (including $p=\infty$) of matrix-valued functions $A$ are defined in just the same way as the $L^p$ spaces of real-valued functions $g$ - we just put $|A|$ in the place of $|g|$. 
Second problem. According to the spherical Fubini theorem, 
\begin{align*}
\int_{B(0,\varepsilon)} \frac{dx}{|x|^{n-2}} 
& = \int_0^\varepsilon r^{n-1} \int_{S(0,1)} \frac{dy}{|ry|^{n-2}} \, dr \\
& = |S(0,1)| \int_0^\varepsilon r \, dr \\
& = \frac 12 |S(0,1)| \varepsilon^2.
\end{align*}
