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

Abstract

Let 1 < p = q < ∞ and (D, μ) = ({±1}, 1 2 δ −1 + 1 2 δ 1 ). Define by recursion: X 0 = C and X n+1 = L p (μ; L q (μ; 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 UMD s constants of X n 's satisfy c n 1 C s (X n ) c n 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 X n , which can be viewed as iterating infinitely many times L p (L q ).