On the $\text{UMD}$ constants for a class of iterated $L_p(L_q)$ spaces

Let $1<p\mathrm{\ne }q<\mathrm{\infty }$ and $(D, \mu) = ({\pm 1}, 1/2 \delta_{-1} + 1/2 \delta_1)$\mathrm{.}Definebyrecursion:$X_0 = \C$and$X_{n+1} = L_p(\mu; L_q(\mu; X_n))$\mathrm{.}Inthispaper,weshowthatthereexist$c_1=c_1(p, q)>1$dependingonlyon$p, q$and$ c_2 = c_2(p, q, s)$dependingon$p, q, s$, such that the ${\text{UMD}}_s$ constants of $X_n$’s satisfy $c_1^n \leq C_s(X_n) \leq c_2^n$forall$1 < s < \infty$. 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 $L_p(L_q)$.