Sobolev spaces complement of Hausdorff codimension 2, restriction theorem

Question

Let $X$ be an open domain in $R^n$. Let $E$ be a subspace of $X$ with Hausdorff dimension $m$. Fix $k$ and $p$. What are the optimal assumptions on $m$ and $n$ so that the trivial map $W^k_p(X) \to W^k_p(X \setminus E)$ becomes an isomorphism?

I am mostly interested in the case $k = 1$ and $p = 2$, and in that situation, it seems to me that the optimal bound is $m \leq n - 2$...

Here the Sobolev space $W^k_p(X)$ is defined as completion of smooth functions on $X$ with respect to the Sobolev norm (and not by the restriction from $R^n$).

Answer 1

If $E$ is a closed set such that Hausdorff measure $H^{n-p}(E)$ is $\sigma$-finite, then its capacity satisfies $Cap_p(E)=0$ and it follows that $W^{1,p}(X)=W^{1,p}(X\setminus E)$. Sobolev $W^{1,p}$ functions simply `do not see' sets of capacity zero.

In particular if $E$ is a linear subspace of dimension $m\leq n-p$, then $W^{1,p}(X)=W^{1,p}(X\setminus E)$. You can find more about capacity in the following books:

L. C. Evans, R. F. Gariepy, Measure theory and fine properties of functions. Revised edition. Textbooks in Mathematics. CRC Press, Boca Raton, FL, 2015.

W. P. Ziemer, Weakly differentiable functions. Sobolev spaces and functions of bounded variation. Graduate Texts in Mathematics, 120. Springer-Verlag, New York, 1989.

J. Malý, W. P. Ziemer, Fine regularity of solutions of elliptic partial differential equations. Mathematical Surveys and Monographs, 51. American Mathematical Society, Providence, RI, 1997.