In order to motivate my question, I'd like to remember that if $\Omega$ is a bounded domain and $f \in L^q(\Omega)$ for some $q>1$, by Hölder inequality $f \in L^p(\Omega)$ for $p \in (1,q]$ with
\begin{equation} \|f\|_{L^p(\Omega)} \le |\Omega|^{(1/p-1/q)} \|f\|_{L^q(\Omega)} \end{equation}
Also, we easily see that if $f \in L^\infty(\Omega)$ then $f \in BMO(\Omega)$ with \begin{equation} \|f\|_{BMO(\Omega)} \le 2 \|f\|_{L^\infty(\Omega)} \end{equation}
Where the BMO space can be seen here, where we also can see that if $f \in BMO(\Omega)$ then $f \in L^p_{Loc}(\Omega)$ for all $1<p<\infty$. But I can't see any proof of this fact. I'd like to see one. Better yet, I'd like to know if is it possible to bound the $L^p$ Loc norm of $f$ by the $BMO$ norm of f. That is, is there a constant $C$ such that \begin{equation} \|f\|_{L^p_{Loc}(\Omega)} \le C\|f\|_{BMO(\Omega)} ? \end{equation}
A reference is also valid. Thank you.
You cannot have a norm bound like the one you wanted, since adding a constant to a function doesn't change it's $BMO$-norm but does affect the $L^p$-norm.
As for proving the non-quantitative inclusion one has to somehow manage to control the $L^p$-norm of a function with essentially an $L^1$-integral (or supremum of). Therefore something nontrivial must happen.
One nontrivial fact one can use is the John - Nirenberg inequality, $$|\{x\in B: |u(x)-u_B|>\lambda\}|\le C_1|B|\exp(-c_2\lambda /\|u\|_{BMO}) $$mentioned in the wikipedia article you cited. Using that we may arrive at $$\int_B |u-u_B|^p \ dx=p\int_0^\infty t^{p-1}|\{x\in B: |u(x)-u_B|>t\}|\ dt \\ \le C|B|\int_0^\infty t^{p-1}\exp(-ct/\|u\|_{BMO})\ dt<\infty $$ for any ball $B\subset \Omega$. Note that proving $u\in L^p_{loc}$ is equivalent to proving $u-a\in L^p_{loc}$ for some $a\in \mathbb R$. For more information and a proof of the John - Nirenberg inequality you may want to check out, say, prop 3.6 here or thm 5.2.1 here