Questions on B.M.O and Gagliardo–Nirenberg–Sobolev inequality

Question

As I was reading a paper I came across with the following inequality:

Let $U \subset \mathbb R^3$ be an open, bounded and compact set with $C^2-$regular boundary $\Gamma=\partial U$. Consider $f\in L^2(0,T;H^1(\Gamma))\cap H^1(0,T;H^1(\Gamma)^{\ast})$ and let $\bar f:=\frac{1}{|\Gamma|}\int_{U}f$. Then using the Gagliardo-Nirenberg theorem one obtains:

${\vert \vert f-\bar f \vert \vert}_{L^4(\Gamma)} \le C {\vert \vert Df \vert \vert}_{L^2(\Gamma)} (*)$

So I 've only studied the Gagliardo–Nirenberg–Sobolev inequality in the case where $p \lt n$ and even in this case, no $\bar f$ appears. After searching about the limit case where $p=n$, I found the following:

1. If $u\in W^{1,n}(\mathbb {R}^{n})$ , then $u$ is a function of bounded mean oscillation and ${\vert \vert u \vert \vert}_{BMO} \le C {\vert \vert Du \vert \vert}_{L^n(\mathbb{R}^n)}$ for some constant $C$ depending only on $n$.
2. ${\vert \vert u \vert \vert}_{BMO} \simeq \sup_{Q}(\frac{1}{|Q|} \int_{Q} |u-\bar u|^p dx)^{1/p}$ for some hypercube $Q$ in $\mathbb R^n$

Although it seems that my problem solved, I have some questions:

1. Can I "identify" ${\vert \vert f-\bar f \vert \vert}_{L^4(\Gamma)}$ with ${\vert \vert f \vert \vert}_{BMO}$ and if yes why? Because the $L^4-$norm here confuses me and what is more I don't have a cear definition of ${\vert \vert \cdot \vert \vert}_{BMO}$
2. If the inequality with BMO holds (in order to deduce $(*)$) then how from $L^4-$norm in $(*)$ we go to $L^2-$norm? In the case I know(i.e when $p\lt n$) there is a relation between $p^{\ast}$ and $p$ but in the limit case of Gagliardo-Nirenberg inequality there is no $p^{\ast}$..

It's the first time I see B.M.O and I would really appreciate if somebody could help me figure out the connections between the above.

Thanks a lot in advance!