I'm have difficultly nailing down the direction of some implications. For $2 \leq q < \infty$, there are (at least) two ways to say that a Banach space $B$ has "cotype $q$".


* $B$ has **cotype** q.
* $B$ is isomorphic to a $q$-**uniformly convex Banach space**, i.e. a uniformly convex Banach space with a "power type" modulus of uniform convexity $\delta(\varepsilon) = C \varepsilon^q$. ($B$ is said to have **martingale cotype** q. The name comes from a [characterization by Pisier involving martingales](http://link.springer.com/article/10.1007/BF02760337).)


I know the following:


1. Every $q$-uniformly convex Banach space (and any space isomorphic to it) has cotype $q$.
2. There are nonreflexive spaces, e.g. $L^1$ and $\ell^1$, with cotype $2$. Since they are nonreflexive they are not isomorphic to a uniformly convex space.
3. The super-reflexive spaces are exactly those isomorphic to uniformly convex spaces, which in turn are all isomorphic to $q$-uniformly convex spaces.


However, I can't seem to find the answer to the following.



> 
> If a space is **super-reflexive** and of cotype $q$, is it isomorphic to a $q$-uniformly convex space?
> 
> 
> 


If not, is there a nice class of spaces where these two notions of cotype agree?




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**Update 1:** I have some partial answers of spaces for which the notions agree (but not yet a general answer to my question).


* **UMD spaces**. (see Cédric's answer).
* **Banach lattices of type $p>1$** (which includes the super-reflexive Banach lattices). I found a pair of interdependence diagrams on pp. 100, 101 of Lindenstrass and Tzafriri's "Classical Banach Spaces II" (these are some of the best math diagrams I have seen). On a Banach lattice, the modulus of convexity is of power type $q$ for an equivalent norm if and only if is of cotype $q$ and there is an upper estimate $p>1$. Following the diagram, "upper estimate" can be replaced with "type".


**Update 2:** On the bottom of p. 78 of Lindenstrass and Tzafriri's "Classical Banach Spaces II" my main question is listed as an open problem. So I guess my question becomes, has it been solved yet?



Martingale-cotype vs cotype on super-reflexive spaces
I'll attempt to answer this with what I've found:




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(1) The answer to my main question is that it is not true. In this [document of Pisier's](http://students.mimuw.edu.pl/~tt249057/other/figel_fulltext.pdf), he states



> 
> It is possible to find a uniformly convex space $B$ for which the index of type $p(B)$ differs from the corresponding index for the $M$-type. Similarly for the cotype. See [P4] for details.
> 
> 
> 


The reference [P4] is 



> 
> G. PISIER, Un exempte concernant la super-réflexivité. Annexe no. 2. Séminaire Maurey-Schwartz 1974-75. Ecole Polytechnique. Paris.
> 
> 
> 


which I can't seem to find, so I'll have to take his word for it.




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(2) As I added to my original post, it seems the two best cases where they agree are UMD spaces (thanks Cédric) and Banach lattices of type $p>1$.


