Let us first see the prove of sobolev embedding of on the plane.

**Lemma:** Suppose with compact support. Then

*Proof:* Let us suppose , the general case can be proved by approximation. Since has compact support, then

Then

Suppose we have a function , where is the unit disc in , can blow up wildly near the origin. However if we know , then actually and .

*Proof:* Because the bad thing happened only at origin, we can suppose has spt inside or . Put a substantially large square box with length inside the left half of the disc whose distance to the origin is see the picture.

puncture disc

Then on the three sides, , , , . Using the proof of the above, one can prove

Letting , we get . The same proof works for the right part . Finally . Choose a cut off function . Then

So .

**Remark:** This is called the removable singularity. There is a more general theorem related to this. Assume , such that . Suppose and . Then .

I learned this from Prof. Brezis’s class. Also see his book: Sobolev maps with values into the circle.