Properties of local orthonormal systems Part I: Unconditionality in Lp, 1<p<∞

Assume that we are given a filtration (Formula presented.) on a probability space (Formula presented.) of the form that each (Formula presented.) is generated by the partition of one atom of (Formula presented.) into two atoms of (Formula presented.) having positive measure. Additionally, assume that we are given a finite-dimensional linear space S of (Formula presented.) -measurable, bounded functions on Ω so that on each atom A of any σ-algebra (Formula presented.), all (Formula presented.) -norms of functions in S are comparable independently of n or A. Denote by (Formula presented.) the space of functions that are given locally, on atoms of (Formula presented.), by functions in S and by (Formula presented.) the orthoprojector (with respect to the inner product in (Formula presented.)) onto (Formula presented.). Since (Formula presented.) satisfies the above assumption and (Formula presented.) is then the conditional expectation (Formula presented.) with respect to (Formula presented.), for such filtrations, martingales (Formula presented.) are special cases of our setting. We show in this article that certain convergence results that are known for martingales (or rather martingale differences) are also true in the general framework described above. More precisely, we show that the differences (Formula presented.) form an unconditionally convergent series and are democratic in (Formula presented.) for (Formula presented.). This implies that those differences form a greedy basis in (Formula presented.) -spaces for (Formula presented.). © 2023 Wiley-VCH GmbH.