Firstly, I would like to say that I know that there are many topics asking about this example (topic 1, topic 2 and topic 3), but I didn't find a topic with the same doubt that I have.
The following example:
I didn't understand how the author stated that $|\int_{\partial B(0,\varepsilon)} u \phi \nu^i dS| \leq ||\phi||_{L^{\infty}} \int_{\partial B(0,\varepsilon)} \varepsilon^{- \alpha} dS$
I think what happens here is the following:
$|\int_{\partial B(0,\varepsilon)} u \phi \nu^i dS| \leq \int_{\partial B(0,\varepsilon)} |u \phi \nu^i| dS = \int_{\partial B(0,\varepsilon)} |\varepsilon^{-\alpha} \phi \nu^i| dS = \int_{\partial B(0,\varepsilon)} |\varepsilon^{-\alpha}| \ |\phi| \ |\nu^i| dS \leq \int_{\partial B(0,\varepsilon)} |\varepsilon^{-\alpha}| \ |\phi| dS$,
but I don't know what to do now, I think that I need to use the fact that if $f \in L^1$ and $g \in L^{\infty}$, then $fg \in L^p$ and $||fg||_{L^1} \leq ||f||_{L^1} ||g||_{L^{\infty}}$, but I don't know if I can use this because I don't know if $\phi \in L^{\infty}$. Anyone can help me to understand this inequality? Thanks in advance!
$\textbf{P.S.:} \ C^{\infty}_c(U)$ is the space of all $\mathcal{C}^{\infty}$-functions $\phi: U \longrightarrow \mathbb{R}$ with compact support in $U$.
$\phi\in C_c^{\infty}$. Continuous functions on compact sets are bounded.