Extrapolation of vector-valued rearrangement operators II

We determine the extrapolation law of rearrangement operators acting on the Haar system in vector valued H spaces: If 0 < q ≤ p < 2, then, ‖Tτ,q ⊗ IdX‖ q 2−q q ≤ A(p, q)‖Tτ,p ⊗ IdX‖ p 2−p p . For a fixed Banach space X, the extrapolation range 0 < q ≤ p < 2 is optimal. If, however, there exists 1 < p0 < ∞, so that ‖Tτ,p0 ⊗ IdE‖L0 E < ∞, for each UMD space E, then for any 1 < p < ∞, ‖Tτ,p ⊗ IdE‖Lp E < ∞, for any UMD space E. (The value p0 = 2 is not excluded.)