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

TL;DR

This paper investigates the growth of UMD constants in a class of iterated $L_p(L_q)$ spaces, providing bounds that depend exponentially on the iteration number, and offers a new elementary construction of super-reflexive non-UMD Banach lattices.

Contribution

It establishes exponential bounds for UMD constants in iterated $L_p(L_q)$ spaces and introduces a new elementary construction of super-reflexive non-UMD Banach lattices.

Findings

UMD constants grow exponentially with iteration number

Bounds depend only on parameters $p, q, s$

Constructs super-reflexive non-UMD Banach lattices via iteration

Abstract

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 $c_1=c_1(p, q)>1$ 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 $c_1^n \leq C_s(X_n) \leq c_2^n$ 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 $L_p(L_q)$.