Consider a measurable, real function $f$ defined on $\mathbb R^n$ which belongs to $L^p(\mathbb R^n)\cap \operatorname{BMO}(\mathbb R^n)$, for some $1\leq p<\infty$. An interpolation inequality exists (see e.g. this paper) that reads $$ \|f\|_{L^q(\mathbb R^n)}\leq C \|f\|^{p/q}_{L^p(\mathbb R^n)}\|f\|^{1-p/q}_{\operatorname{BMO}(\mathbb R^n)} $$ for any $p<q<\infty$, where the constant $C$ does not depend on the choice of $f$.
Now, it seems to me that the constant $C$ has to depend on $q$ and goes to infinity as $q\to\infty^-$. My thoughts are: if $C$ did not depend on $q$, it would follow that $$ \limsup_{q\to\infty^-}\|f\|_{L^q(\mathbb R^n)}<\infty, $$ which would imply that $f\in L^\infty(\mathbb R^n)$. But this is definitely not true, since for instance it is well-known that the Sobolev space $H^{n/2}(\mathbb R^n)=W^{n/2,2}(\mathbb R^n)$ (defined using the Fourier transform) is contained in $L^2(\mathbb R^n)\cap\operatorname{BMO}(\mathbb R^n)$ but not in $L^\infty(\mathbb R^n)$ (please correct me if I have overlooked something).
My question is then, do you know a reference discussing the above inequality and the dependence of the constant $C$ on $p$ and $q$?