# Bounded Sobolev sequence together with convergence in Lebesgue space implies convergence in intermediate Sobolev spaces

I am trying to understand the proof of a theorem, but I just can't seem to wrap my head around this one step:

We have a sequence of functions $$(f_i)_{i \in \mathbb{N}}$$ that are bounded in the Sobolev space $$W^{k,p}$$, where $$k \in \mathbb{N}$$, $$p \geq 2$$, i.e. $$\|f_i\|_{W^{k,p}} \leq C$$ for some constant $$C$$ and all $$i$$. We also know that $$f_i$$ converges to some $$g$$ in $$L^p(\mathbb{R})$$ and this $$g$$ is also an element of $$W^{k,p}$$. The proof now just claims, that using standard properties of Sobolev spaces, this bound together with the convergence in $$L^p$$ implies the convergence $$f_i \to g$$ in all intermediate spaces $$W^{k',p}$$, where $$k' \in (0,k)$$.

I am rather new to Sobolev spaces, so I am quite confused as to what these standard properties might be. I tried to read up in the book on Sobolev spaces by Adams, but I could not really find anything helpful. To be honest, I have yet to understand what norms these intermediate spaces use. I also tried to first understand the statement asuming $$k'$$ is an integer, because in this case I am more familiar with the spaces, but I was also unsuccessful.

Do you have ideas on how to approach this task? Any help is appreciated

Here is the rough idea: typically, you have a compact embedding $$W^{k,p} \hookrightarrow W^{k',p}$$ for all $$k' \in (0,k)$$ [and I would say that in most cases, $$k'$$ is an integer].

Since $$f_i$$ is bounded in $$W^{k,p}$$, there is a subsequence $$f_{k_i}$$ with $$f_{k_i} \to h$$ in $$W^{k',p}$$ for some $$h \in W^{k',p}$$. However, this gives $$f_{k_i} \to g$$ in $$L^p$$, i.e., $$g = h$$.

Finally, you can use a subsequence-subsequence argument to show that the entire sequence $$f_i$$ converges in $$W^{k',p}$$.

• I have read up a bit on the compact embeddings, and have the impression that they usually require the domain to be at least bounded. In my case the domain is $\mathbb{R}$. To be very precise, the functions I am working with are all of the form $\varphi^{\frac{1}{p}}$, where $\varphi: \mathbb{R} \to \mathbb{R}$ is some density. Will the compact embeddings still work in this case? Furthermore, can you recommend me some literature where I can read up on these embeddings? Jun 3, 2021 at 9:39
• On $\mathbb R$, compactness seems to be difficult. In particular, you can consider a sequence of translates, i.e., for each $n \in \mathbb N$ set $u_n(x) = u(x - n)$. Then, $u_n$ will be bounded but will not contain a convergent subsequnce in typical function spaces.
– gerw
Jun 3, 2021 at 9:46

The most direct way to justify this would be to use the Gagliardo-Nirenberg interpolation inequality. There are certain technical assumptions (on the exponents and the domain), but for simplicity, let us fix $$0 \le k' < k$$ and take for granted the following: $$\| u \|_{W^{k',p}} \le A \| u \|^{\alpha}_{L^p} \cdot \| u \|^{1-\alpha}_{W^{k,p}} \qquad \text{for all } u \in W^{k,p},$$ where the constants $$0 < \alpha \le 1$$ and $$A > 0$$ may depend on everything (in particular the choice of $$k'$$) except $$u$$.

If $$\| f_i \|_{W^{k,p}} \le C$$ for all $$i$$, then $$\| f_i-f_j \|_{W^{k',p}} \le A \| f_i-f_j \|^\alpha_{L^p} \cdot \| f_i-f_j \|^{1-\alpha}_{W^{k,p}} \le 2C^{1-\alpha}A \| f_i-f_j \|^\alpha_{L^p}.$$ If additionally $$f_i$$ converges in $$L^p$$, then by the above, it's a Cauchy sequence in $$W^{k',p}$$, which implies convergence in this space.

• From the Wikipedia article you linked I can understand how you reach your first equation in the case that $k'$ is an integer. But will this also work, if $k'$ is an arbitrary positive real number? In that case, the norm of $W^{k', p}$ has the additional term of the Slobodeckij seminorm, and I dont understand what happens to this term. Jun 3, 2021 at 11:41