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I just read in a paper by Martio and Zeimer$^1$ that smooth functions ($C^1$) of several real variables have the have the Lusin property (N). I have two questions.

  1. First, could someone give me a reference for this fact.
  2. Second, what other classes of functions of several variables (in fact I only care about two variables) have the Lusin property (N)? (Martio and Zeimer also mentioned locally Lipshitzian and continuous in the Sobolov space $W^{1,p}(G,\mathbb{R}^n)$, where $G$ is the domain of the function, and $p>n$.)
  1. Lusin's condition (N) and mappings with nonnegative Jacobians. Michigan Math. J. Volume 39, Issue 3 (1992), 495-508.
Trevor J Richards
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1 Answers1

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A $C^1$ map is locally Lipschitz. And since countable unions preserve null sets, we can cover the domain into countably many balls where the map is Lipschitz, and work with each separately. This reduces the problem to the Lipschitz case, which is treated here: Why does a Lipschitz function $f:\mathbb{R}^d\to\mathbb{R}^d$ map measure zero sets to measure zero sets?

If you want a book reference, see the Evans-Gariepy book.

You ask about other classes for which property (N) is known... as far as Sobolev spaces are concerned, the 1995 paper by Malý and Martio is still pretty close to the edge of what's known, but do check its citations on Google Scholar to get the latest improvements.

In a different direction, there are several versions of definition of $n$-dimensional absolute continuity, starting with Malý in 1999 and continuing with Hencl and Bongiorno. This paper states some of their results (which include property N) and gives references.

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