real analysis - Embedding of BV and L^P spaces

An $L^1_{loc}$ function on $mathbb{R}^n$ is in $BV_{loc}$ iff its distributional derivatives $partial_i finmathcal{M}^1_{loc}$, 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 $ngeq 2$. 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 $sleq 1$ 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.

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