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

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

2011-12-04

arxiv.org/abs/1112.0739v1

Mathematics

Functional Analysis

18 pages

Scientific paper

Let $1 < p \neq q < \infty$ 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 $c_1=c_1(p, q)>1$ 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 $L_p(L_q)$.

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