Is it possible to control the oscillation of a BV vector field $u:\mathbb R^N \to \mathbb R^N$ at a point $x_0$ by the total variation of $u$?

$\begingroup$ When $N=1$ and you use the classical definition of total variation $$V^a_b(f)=\sup_{\mathcal{P}} \sum_{i=0}^{n_P1}  f(x_{i+1})f(x_i) $$ where $\mathcal{P} =\left\{P=\{ x_0, \dots , x_{n_P}\}P\text{ is a partition of } [a,b] \right\}$, then this is certainly true since $u$ is bounded. In the case $N>1$ you should recur to a concept of essential (in the sense of measure theory) oscillation. $\endgroup$ – Daniele Tampieri Jun 4 '19 at 12:45

$\begingroup$ @DanieleTampieri Thank you. How would the proof go in the general case? $\endgroup$ – Zyl Jun 4 '19 at 14:14

$\begingroup$ I am not sure on how to proceed, but intuitively I would try to show that $u\in BV$ (or $BV_\mathrm{loc}$) implies $u$ is essentially bounded, i.e. $u\in L^\infty_\mathrm{loc}$: then I would show that seestial bondedness imply a control of the (essential) oscillation in terms of the $BV$ norm. I do not post a full answer since I am not yet skilled enough to work quickly with the extension of those concepts in dimension $N>1$. $\endgroup$ – Daniele Tampieri Jun 4 '19 at 14:32
When $N>1$ you cannot control the oscillation by the total variation, because it would mean that $BV$ functions are locally bounded. However, a $BV$ function can have essential supremum equal $+\infty$ and essential infimum equal $\infty$ on every open set. In the case of vector field this phenomenon may apply to each component of the vector field. You can find an example here https://mathoverflow.net/a/321502/121665. This example is for Sobolev functions, but Sobolev functions are in $BV$.