Unconditional martingale difference sequences in Banach spaces

Abstract

For a Banach space Y, the question of whether L p ( μ , Y ) has an unconditional basis if 1 < p < ∞ and Y has unconditional basis, stood unsolved for a long time and was answered in the negative by Aldous. In this work we prove a weaker, positive result related to this question. We show that if ( y j ) is a basis of Y and ( d i ) is a martingale difference sequence spanning L p ( μ ) then the sequence ( d i ⊗ y j ) is a basis of L p ( μ , Y ) for 1 ⩽ p < ∞ . Moreover, if 1 < p < ∞ and ( y j ) is unconditional then ( d i ⊗ y j ) is strictly dominated by an unconditional tensor product basis. In addition, for 1 < p < ∞ , we show that if ( d i ) ⊂ L p ( μ ) is a martingale difference sequence then there exists a constant K > 0 so that ‖ ∑ i , j ∈ N ( α i j y j ) d i ‖ L p ( μ , Y ) ⩽ K ‖ ∑ i , j ∈ N ‖ α i j y j ‖ d i ‖ L p ( μ ) holds for every sequence ( y j ) ⊂ Y and every choice of finitely supported scalars ( α i j ) .