Let $f: \Omega \subset \mathbb{R}^N \to \mathbb{R}^M$ be a Sobolev or BV vector field.

A heuristic that I've heard frequently is the following:

$f$ is almost Lipschitz on a large "good" set but there is a small "bad" set where $Df$ is very large.

What theorems make this heuristic rigorous?

  • 2
    $\begingroup$ In some sense, this follows from en.wikipedia.org/wiki/… applied to $Df$. $\endgroup$ – gerw Apr 15 at 18:01
  • $\begingroup$ @gerw Could you add some details about this? $\endgroup$ – Riku Apr 23 at 18:42
  • $\begingroup$ This brings to mind Federer Theorem 3.1.8. $\endgroup$ – Umberto P. Apr 23 at 18:51
  • $\begingroup$ @UmbertoP. I don't have access to the book these days. Could you write it down, please? $\endgroup$ – Riku Apr 23 at 18:58
  • $\begingroup$ Good and bad sets says Calderon Zygmund decomposition to me $\endgroup$ – George Dewhirst Apr 23 at 20:46

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