**Background:** I work on a PDE problem where I have a rough sequence of functions with measure values and I need to compactly embed it in some negative space of Sobolev $ W ^ – m, q} $ in the limited interval in $ mathbb {R} $. I am mainly interested in spaces where $ q = 2 $. I found just one of those inlays in the article's only theorem:

**Evans – Weak Convergence Methods for Nonlinear Partial Differential Equations, 1990**.

**Theorem 6** (Compactness for measurements, page 7): suppose the sequence $ { mu_k } _ {k = 1} ^ { infty} $ is limited in $ mathcal {M} (U) $, $ U subset mathbb {R} ^ n $. So $ { mu_k } _ {k = 1} ^ { infty} $ it is pre-compact in $ W ^ – 1, q} (U) $ for each $ 1 leq q <1 ^ * $.

here $ mathcal {M} (U) $ represents the space of signed radon measurements in $ U $ with finite dough, $ U subset mathbb {R} ^ n $ is an open, bounded, and smooth subset of $ mathbb {R} ^ n, n geq 2 $ and $ 1 ^ * = frac {n} {n-1} $ represents a conjugate of Sobolev.

The identical theorem (Lemma 2.55, page 38) is given in the book: **Malek, Necas, Rokyta, Ruzicka – Weak and Measured Value Solutions for Evolutionary PDE, 1996**, with a difference that instead of $ 1 leq q <1 ^ * $, there it is written $ 1 leq q < frac {n} {n-1} $ (here it is not explicitly written that $ n geq 2 $)

**My question:** How Theorem 6 works in one dimension ($ n = 1 $)? I mean we have a compact space inlay $ mathcal {M} (U) $ in the space $ W ^ – 1, q} (U) $, where $ U subset mathbb {R} $?

And additionally:

- I guess if we have compact inlay on $ W ^ – 1, q} (U) $, then we also have it in the $ W ^ {- m, q} (U), m geq 1 $?
- Are there other measurement spaces (for example, finite positive measurement space) $ mathcal {M} _ + $, measures of probability space with finite first moment $ Pr_1 $etc.) which are compactly integrated into some negative Sobolev spaces $ W ^ – m, q} (U) $?

I think if we use Sobolev's conjugate definition: $ frac {1} {p ^ *} = frac {1} {p} – frac {1} {n} $, we arrived by $ p = 1, n = 1 $ the $ frac {1} {1 ^ *} = frac {1} {1} – frac {1} {1} Rightarrow 1 ^ * = infty $. So we would have that theorem 6 (maybe) works for every $ 1 leq q < infty $ (and then for $ q = 2 $ also)? If we use $ p ^ * = frac {np} {n-p} $ we would have for $ n = 1, $ $ p ^ * = frac {p} {1-p} $ and here we couldn't drink $ p = 1 $ and get $ p ^ * $.

I don't usually deal with Sobolev spaces with measure values and negatives, so I don't know much about them. Help with this would be great and I definitely need it. And any additional reference in addition to the two mentioned above would be good. Thanks in advance.