A REMARKABLE REARRANGEMENT OF THE HAAR SYSTEM IN Lp

We introduce a non-standard but, to our opinion natural, order on the initial segments of the Haar system and investigate the isomorphic classiication of the linear span, in L p , of block bases, with respect to this order. 0. Introduction. In DS] it was proved that every unconditional basic sequence fx i g 1 i=1 in L p , 2 < p < 1, which is not equivalent to the natural basis of`p has the property that for some K 1 and every positive integer n there are n vectors of the form y i = P j2 i a j x j , i = 1; : : :; n, where the sets i are pairwise disjoint and the sequence fy i g is equivalent, with constant K, to the unit vector basis in`n 2. See also JMST] for a generalization of this fact for more general lattices. It was left open in these two papers (and speciically asked in Problem 3.A of DS]) whether fy i g can be chosen to be a block basis, i.e.,whether the sets i can be chosen to be successive, that is, maxfj; j 2 k g < minfj; j 2 l g for all k < l. The initial motivation of this paper was to solve this question in the negative. As is well known the Haar basis, fh n;i g 1 n=0; 2 n i=1 , in L p (0; 1), 1 < p < 1, in its common order, has a block basis (i.e., the Rademacher functions) equivalent to the unit vector basis of`2. It follows from the main results of this paper (Theorems 2.2 and 3.1) that the initial segments of the Haar system can be rearranged so that they will not have (the nite version of) this property anymore. We denote the new order by .