It is well-known that if $\Omega\in \mathbb{R}^n$ is a bounded domain, then we have the embedding $$ L^4({\Omega})\subset L^2({\Omega}) $$ since $||f||_{L^2(\Omega)}\leq C(\Omega) ||f||_{L^4(\Omega)}$ by Hölder's inequality. However, the above inequality no longer holds when $\Omega$ is unbounded.

Now, for general domain $\Omega$, let us consider the set $\mathcal{F}=\{f:\Omega\to\mathbb{R}:||f||_{L^2(\Omega)}\leq C||f||_{L^4(\Omega)}< \infty\}$, and $C$ is an universal constant independent of $f$. I am wondering if there exists a function space $X$ such that $X\subset\mathcal{F}$, where $X$ contains functions with non-compact support? It is okay to assume $f$ is smooth.