This is an announcement for the paper "On the \text{UMD} constants for a class of iterated $L_p(L_q)$ spaces" by Yanqi Qiu.
Abstract: Let $1 < p \neq q < \infty$ and $(D, \mu) = ({\pm 1}, \frac{1}{2} \delta_{-1} + \frac{1}{2}\delta_1)$. Define by recursion: $X_0 = \C$ and $X_{n+1} = L_p(\mu; L_q(\mu; X_n))$. In this paper, we show that there exist $c_1=c_1(p, q)>1$ and $ c_2 = c_2(p, q, s) > 1$, such that the $\text{UMD}_s$ constants of $X_n$'s satisfy $c_1^n \leq C_s(X_n) \leq c_2^n$ for all $1 < s < \infty$. Similar results will be showed for the analytic $\text{UMD}$ constants. We mention that the first super-reflexive non-$\text{UMD}$ Banach lattices were constructed by Bourgain. Our results yield another elementary construction of super-reflexive non-$\text{UMD}$ Banach lattices, i.e. the inductive limit of $X_n$, which can be viewed as iterating infinitely many times $L_p(L_q)$.
Archive classification: math.FA
Remarks: 18 pages
Submitted from: yqi.qiu@gmail.com
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1112.0739
or