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In an answer to the question Weak Lebesgue spaces and an estimate for BV functions it was remarked that if $u\in BV(\mathbb R^N)$ then there exists a Lebesgue negligible set $F \subset \mathbb R^N$ such that $$ |u(x) - u(y)| \le c_N |x-y| (M_R Du(x) + M_R Du(y)) $$ for $x,y\in \mathbb R^N \setminus F$ with $|x-y|\le R$, and $$ M_R Du(x) = \sup_{r\in(0,R)} \frac{|Du|(B_r(x))}{|B_r(x)|}, $$ where $|B_r(x)|$ denotes the Lebesgue measure of $B_r(x)$.

Also, $x\mapsto M_R Du(x)$ belongs to the weak $L^1$ space.

  • Is it true that $\forall \epsilon >0$ there exist $F_1, F_2$ such that $$M_R Du = F_1 + F_2$$ and $$\| F_1 \|_{w-L^1} \le \epsilon \quad \| F_2 \|_{L^1} \le C_\epsilon ,$$ where $C_\epsilon$ is some constant that depends on $\epsilon$?

  • If not, under what additional assumptions is it true?

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  • $\begingroup$ Where does the question originate from? Why do you want such a decomposition of $M_R Du$? $\endgroup$
    – user111164
    Commented Apr 24, 2019 at 21:04
  • $\begingroup$ @user111164 To understand better the role of $L^1$ vs weak-$L^1$ in this estimate. $\endgroup$
    – Riku
    Commented Apr 24, 2019 at 21:06

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