Embedding of $BV$ and $L^p$ spaces 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 $L^2$) ?
 A: 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. 
A: An $L^1_{loc}$ function  on $\mathbb{R}^n$ is in $BV_{loc}$ iff its distributional derivatives $\partial_i f\in\mathcal{M}^1_{loc}$, i.e. they are all locally finite (Radon) measures. If $n=1$, the situation is well-known, and $BV_{loc}\subset L^\infty_{loc}$. So assume $n\geq 2$. Since $W^{s,p}_{c}(\mathbb{R}^n)\subset C^0_{c}(\mathbb{R}^n)$ if $s>n/p$, you have that $$BV_{loc}(\mathbb{R}^n)\subset W^{1-s,p'}_{loc}(\mathbb{R}^n)\subset L^{p'}_{loc}(\mathbb{R}^n)$$ if $s\leq 1$ and $1/p+1/p'=1$, that is if $p'<n/(n-1)$. On the other hand, when $n>1$,  $1/r^\alpha$ is in $BV_{loc}(\mathbb{R}^n)$ if $\alpha<n-1$, since partial derivatives are in $L^1_{loc}$, but it is in $L^q$ only for $q<n/\alpha$, so that $BV_{loc}(\mathbb{R}^n)\subset L^q_{loc}(\mathbb{R}^n)$ fails for any $q>n/(n-1)$. I wouldn't bet on the limiting case.
