On the UMD constants for a class of iterated Lp(Lq) spaces

Let 1 < p = q < ∞ and (D, μ) = ({±1}, 12 δ−1 + 12 δ1). Define by recursion: X0 = C and Xn+1 = Lp(μ;Lq(μ;Xn)). In this paper, we show that there exist c1 = c1(p, q) > 1 depending only on p, q and c2 = c2(p, q, s) depending on p, q, s, such that the UMDs constants of Xn’s satisfy cn 1 Cs(Xn) cn 2 for all 1 < s <∞. Similar results will be showed for the analytic UMD constants. We mention that the first super-reflexive non-UMD Banach lattices were constructed by Bourgain. Our results yield another elementary construction of super-reflexive non-UMD Banach lattices, i.e. the inductive limit of Xn, which can be viewed as iterating infinitely many times Lp(Lq ). © 2012 Elsevier Inc. All rights reserved