Extrapolation of vector-valued rearrangement operators II

Abstract

We determine the extrapolation law of rearrangement operators acting on the Haar system in vector-valued Hp spaces: if 0<q⩽p<2, then, ∥ T τ , q ⊗ I d X ∥ q q / ( 2 − q ) ⩽ A ( p , q ) ∥ T τ , p ⊗ I d X ∥ p p / ( 2 − p ) For a fixed Banach space X, the extrapolation range 0<q⩽p<2 is optimal. If, however, there exists 1<p0<∞, so that ∥ T τ , p 0 ⊗ I d E ∥ L E p 0 < ∞ f o r e a c h U M D s p a c e E , then, for any 1<p<∞, ∥ T τ , p ⊗ I d E ∥ L E p < ∞ for any UMD space E. (The value p0=2 is not excluded.) We characterize Hilbert spaces in terms of vector-valued rearrangement operators. If ∥ T τ , 2 ⊗ I d X ∥ L Y 2 < ∞ a n d ∥ T τ ) p ∥ L p = ∞ ( f o r 1 < p ≠ 2 < ∞ ) , then X is isomorphic to a Hilbert space.