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 convergence $\delta(\varepsilon) = C \varepsilon^q$. ($B$ is said to have martingale cotype q. The name comes from a characterization by Pisier involving martingales.)
I know the following:
- Every $q$-uniformly convex Banach space (and any space isomorphic to it) has cotype $q$.
- 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.
- 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?
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?