Let $u \in BV(\Omega \subset \mathbb R^N, \mathbb{R}^N)$. Is it true that there exists a function $f$ in the [weak $L^1$ space](https://en.wikipedia.org/wiki/Lp_space#Weak_Lp) such that 
$$|u(y)-u(x)| \le |x-y|\big|f(y) - f(x)\big|$$
holds for a.e. $x,y$?




---


This question is motivated by [Convergence of the difference quotient of a BV function](https://mathoverflow.net/questions/327985/convergence-of-the-difference-quotient-of-a-bv-function).



Weak Lebesgue spaces and an estimate for BV functions
A result of this kind can be found in [1] (Lemma A.3):



> 
> 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 $|x-y|\le R$.
> 
> 
> 


Here
$$
M\_R Du(x) = \sup\_{r\in(0,R)} \frac{|Du|(B\_r(x))}{|B\_r(x)|},
$$
where $|B\_r(x)|$ denotes the Lebesgue measure of $B\_r(x)$.


From Lemma A.2 (which is stated for $L^1$ functions but holds for measures as well) if follows that $x\mapsto M\_R Du(x)$ belongs to the weak $L^1$ space.


### References


[1] De Lellis C., Crippa G. [Estimates And Regularity Results For The Diperna–Lions Flow](https://www.math.uzh.ch/fileadmin/user/delellis/publikation/Estimates_ODEs.pdf). J. Reine Angew. Math. 616 (2008), 15–46.


