Sobolev spaces types and cotypes

Question

Given $1\le p < \infty$, I am aware that $L^p$ is of type $\min\{2,p\}$ and of cotype $\max\{2,p\}$. It seems reasonable to assume that the Sobolev space $W^{k,p}$ ($k$ times weakly differentiable, $L^p$-integrability) is thus also of type $\min\{2,p\}$ and cotype $\max\{2,p\}$, yet I have not been able to find a reference for that.

Also, I am not quite sure if there are multiple kinds of types (Rademacher type, Fourier type,...?), and what their relationship is. So just to be sure, here is the definition of type I need.

Let $1\le p\le\infty$ and let $Z_j$ be a sequence of Bernoulli-Rademacher random variables. A Banach space $E$ is said to be of type $p$ if there is a type constant $C>0$ such that for all finite sequences $(x_n)_{n=1}^N$ in $E$, $$\left\|\sum_{n=1}^N Z_n x_n\right\| \le C\left(\sum_{n=1}^N \|x_n\|^p\right)^{1/p}.$$

The question is then what the type of $W^{k,p}$ is. I would be very glad if someone could point me to a reference where this question is treated, or provide a proof themselves. If this is relevant, you may assume that $W^{k,p} = W^{k,p}(\Omega)$ for some very nice set $\Omega$, say the unit ball in $\mathbb{R}^d$. You may also assume $k=1$.

Answer 1

According to Pelczynski and Senator (http://matwbn.icm.edu.pl/ksiazki/sm/sm84/sm84113.pdf ) the answer is that the Sobolev spaces have the same type and cotype as the parent $L^p$ space provided that the domain you're looking at is sufficiently nice. The unit ball in $\mathbb R^2$ would fall into the category of "sufficiently nice".

Wojtaszczyk provides a proof for the case of $W^{1,p}(\mathbb T^2)$ in his book Banach spaces for analysts, and Albiac and Kalton (Topics in Banach space theory) hint at it as well in section 6.2 without outright saying so.

There doesn't seem to be (that I can find, or think of) a simple proof of this fact though.

Answer 2

Here is an "indirect" and partial result: $W^{k,p}(\Omega)$, where $\infty>p\ge 2$ and $k\in\mathbb{Z}_+$ and $\Omega$ could be open sets $\mathbb{R}^d$ and torus, has type 2.

To see this, firstly notice that we have $W^{k,p}(\Omega)$ has martingale type 2, which is shown, for example, in

BRZEZNIAK 1995 https://link.springer.com/article/10.1007/BF01048965.

Then, since $W^{k,p}(\Omega)$ is UMD, and for UMD spaces, the notion of martingale type $p$ and type $p$ are equivalent, which is shown in

Analysis in Banach spaces Vol I, https://link.springer.com/book/10.1007/978-3-319-48520-1

then we arrive at this result.