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This has been asked on Mathematics Stack Exchange but apparently received no attention. The question is very basic in nature:

Is it true that $W^{2,1}_{\text{loc}}$ functions (after possibly modifying them on a set of measure zero) allow classical second order partial derivatives almost everywhere?

The corresponding problem for the first order partial derivatives of functions in $W^{1,1}_{\text{loc}}$ has positive answer by the ACL characterization of Sobolev functions (Nikodym's theorem). It seems to me that the answer to my question will follow by iteration but all the textbooks that I have checked are suspiciously silent on the topic.

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    $\begingroup$ The answer is yes. Since the first order partial derivatives are in $W^{1,1}_{rm loc}$ so partial derivatives have partial derivatives i.e. second order partial derivatives exist a.e. However, the function is no not necessarily twice differentiable in the Frechet sense. $\endgroup$
    – Piotr Hajlasz
    Commented Mar 11, 2021 at 16:27

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