An elementary question about Sobolev spaces:


Is there some explicit theorem about embedding relation between spaces $BV(\Omega)$ and $L^p(\Omega)$?


Formulated otherwise: is $BV$ a subset of $L^2$ (i.e. $BV$ possess regularities of $L^2$) ?



Embedding of $BV$ and $L^p$ spaces
**Edit:** The most general imbedding I know of about $L^p$ spaces is that $BV(\Omega) \subset \subset L^{n/n-1}(\Omega)$ where $\Omega \subset \mathbb{R}^n$ and $n > 1$ (replace $n/(n-1)$ with $1$ when $n=1$). This embedding is **compact** for any $p < n/n-1$. Hence for your $n=2$, $n=3$ interest we have $f \in L^{2}(\Omega)$ for $n=2$ and $f \in L^{3/2}(\Omega)$ for $n=3$. On bounded domains you have all *lower* $L^p$ norms as well by an application of Holder's inequality. Sorry for my initial mis-understanding of your question. 


