Is true that $H^{s,p}(\mathbb{R}^n)\subset C^k(\mathbb{R}^n)$ when $s-k>n/p$?.

Question

In Functional Analysis by J. Cerda, the theorem 7.28 says:

theorem 7.8. If $s-k>n/2$ then $H^{s}(\mathbb{R}^n)\subset C^{k}(\mathbb{R}^n)$ (functions $k$ times differentiable) with $H^{s}(\mathbb{R}^n):=\left\{u\in L^2: \mathcal{F}^{-1}(1+|\xi|^2)^{s/2}\mathcal{F}(u))\in L^2\right\}$

Let $H^{s,p}(\mathbb{R}^n):=\left\{u\in L^p: \mathcal{F}^{-1}(1+|\xi|^2)^{s/2}\mathcal{F}(u)\in L^p\right\}$ Sobolev space.

I know that if $s>n/p$ then $H^{s,p}(\mathbb{R}^n)\subset C(\mathbb{R}^n)$ [Taylor,Partial Differential equations III, prop. 6.3, p.26]

Is there a similar result in the context of $L^p$? e.g.

If $s-k>n/p$, the Sobolev space $H^{s,p}(\mathbb{R}^n)\subset C^k(\mathbb{R}^n)$?

Answer 1

Yes, and it's part of the Sobolev embedding theorem. Wikipedia has a pretty good article on Sobolev spaces https://en.wikipedia.org/wiki/Sobolev_inequality which contains it. Here is the relevant quote from that article.

The second part of the Sobolev embedding theorem applies to embeddings in Hölder spaces $C^{ r,α}({\mathbf R}^n)$. If $n < pk$ and

$$ {\displaystyle {\frac {1}{p}}-{\frac {k}{n}}=-{\frac {r+\alpha }{n}}} $$

with $\alpha \in (0, 1]$ then one has the embedding $${\displaystyle W^{k,p}(\mathbf {R}^{n})\subset C^{r,\alpha }(\mathbf {R}^{n}})$$

Answer 2

Yes,actually $H^{s,p} = F^{s}_{p,2} ⊂ B^s_{p,\infty}$ where $F^{s}_{p,q}$ are the Triebel-Lizorkin spaces and $B^{s}_{p,q}$ are the Besov spaces. By Sobolev's embeddings for Besov spaces, you have $$ B^s_{p,\infty} ⊂ B^{s-n/p}_{\infty,\infty} = \mathcal{C}^{s-n/p} ⊂ C^k $$ where $\mathcal{C}^{s-n/p}$ are the Hölder spaces when $s-n/p$ is not an integer (and Hölder-Zygmund spaces if $s-n/p$ is an integer).

These results can be found for example in the book Theory of Function Spaces II by H. Triebel.