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

Abstract

Let $1<p\ne q<\infty$ and $(D,\mu )=(\{\pm 1\},1/2{\delta }_{-1}+1/2{\delta }_1)$. 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 $c_1=c_1(p,q)>1$ depending only on $p,q$ and $c_2=c_2(p,q,s)$ depending on $p,q,s$, such that the ${\text{UMD}}_s$ constants of $X_n$'s satisfy $c_1^n\le C_s(X_n)\le c_2^n$ 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 $L_p(L_q)$.

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