Can the derivative of a BV function $f:\mathbb{R}^n\to\mathbb{R}^n$ be controlled by the symmetric part of the derivative $\frac{1}{2}(Df+(Df)^T)$?



Control the derivative of a BV function by its symmetric part
This is not a full answer, but some comments that might put you on the right track. 


If a function $f=(f\_1,\ldots,f\_n):\mathbb{R}^n\to\mathbb{R}^n$ is sufficiently smooth and has compact support, then
$$
f\_k=\frac{2}{n\omega\_n}\sum\_{1\leq i\leq j\leq n}\left(\epsilon\_{jk}\*\frac{\partial K\_{ij}}{\partial x\_i}-\epsilon\_{ij}\*\frac{\partial K\_{ij}}{\partial x\_k}+\epsilon\_{ki}\*\frac{\partial K\_{ij}}{\partial x\_j}\right),
$$
where 
$$
\epsilon\_{ij}=\frac{1}{2}\left(\frac{\partial f\_i}{\partial x\_j}+\frac{\partial f\_j}{\partial x\_i}\right)
\quad
\text{are components of} 
\quad 
\epsilon(f)=\frac{1}{2}\left(Df+(Df)^T\right) 
$$
and $K\_{ij}(x)=x\_ix\_j/|x|^n$ see page 63 in [1]. Taking the derivative of the above formula we obtain the representation of the derivative of $f\_k$ as a singular integral applied to $\epsilon(f)$ (the derivative of $\partial K\_{ij}/\partial x\_i$ has nonintegrable singularity and the convolution needs to be interpreted as a singular integral, see [1])). Thus for sufficiently smooth functions we can represent $Df$ as a singular integral of $\epsilon(f)$. In this way one can prove the so called Korn inequality:
$\Vert Df\Vert\_p\leq C\Vert \epsilon(f)\Vert\_p$ for $1<p<\infty$. 


The problem is that in the case of $BV$ functions the derivative is a Radon measure and singular integrals do not act well on measures. However, from the fact that $\epsilon(f)$ is a measure (functions such that $\epsilon(f)$ is a Radon measure are called functions of bounded deformation) one can conclude that $f$ is approximately differentiable a.e. so some estimates are possible (see Corollary 1 page 66 in [1]).


See also [2] for a more recent development and related references.


**[1] P. Hajłasz,** [On approximate differentiability of functions with bounded deformation.](http://www.pitt.edu/~hajlasz/OriginalPublications/Hajlasz-OnApproximate-ManuscriptaMath-91-1996-61-72.pdf). *Manuscripta Math.* 91 (1996), 61–72. 


**[2] J. Verdera,**
[Capacitary differentiability of potentials of finite Radon measures.](https://arxiv.org/abs/1812.11419). arXiv:1812.11419.


