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 $R^n$).



Sobolev spaces complement of Hausdorff codimension 2, restriction theorem
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.


