**Context:** I am working on a PDE problem where I have an approximate sequence of measured value functions and I have to incorporate it compactly in a negative Sobolev space $ W ^ {- m, q} $ over the bounded interval $ mathbb {R} $. I am especially interested in spaces where $ q = $ 2. I only found such integration in the only theorem of the article:

**Evans – Weak convergence methods for nonlinear partial differential equations, 1990**.

**Theorem 6** (Compactness of measurements, page 7): Suppose the sequence $ { mu_k } _ {k = 1} ^ { infty} $ is delimited by $ mathcal {M} (U) $, $ U subset mathbb {R} ^ n $. so $ { mu_k } _ {k = 1} ^ { infty} $ is precompressed in $ W ^ {- 1, q} (U) $ for each $ 1 leq q <1 ^ * $.

Here $ mathcal {M} (U) $ represents the space of the Radon measurements signed on $ U $ of finite mass, $ 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 Sobolev conjugate.

The identical theorem (Lemma 2.55, page 38) is given in the book: **Malek, Necas, Rokyta, Ruzicka – Weak and measured solutions to evolutionary PDE, 1996**, with the difference that instead of $ 1 leq q <1 ^ * $, there is written $ 1 leq q < frac {n} {n-1} $ (Here it is not explicitly written that $ n geq 2 $).

**My question:** Theorem 6 works in one dimension ($ n = $ 1)? Simply put, do we have a compact footprint of space $ mathcal {M} (U) $ in space $ W ^ {- 1, q} (U) $, or $ U subset mathbb {R} $?

And in addition:

- I guess if we have a compact integration in $ W ^ {- 1, q} (U) $, 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} _ + $, probability measure space with finite first moment $ Pr_1 $, etc.) which are compactly integrated into certain negative spaces of Sobolev $ W ^ {- m, q} (U) $?

I think if we use the definition of the Sobolev conjugate: $ frac {1} {p ^ *} = frac {1} {p} – frac {1} {n} $, we get for $ p = 1, n = 1 $ the $ frac {1} {1 ^ *} = frac {1} {1} – frac {1} {1} Rightarrow 1 ^ * = infty $. So we would have Theorem 6 (maybe) to work for each $ 1 leq q < infty $ (then for $ q = $ 2 as well)? If we use $ p ^ * = frac {np} {n-p} $ we would have for $ n = 1, $ $ p ^ * = frac {p} {1-p} $ and here we couldn't take $ p = $ 1 and get $ p ^ * $.

I don't usually take care of measured and negative Sobolev spaces, so I don't know much about them. Help would be great and I really need it. And any additional references in addition to the two mentioned above would be nice. Thanks in advance.