The One-Third-Trick and Shift Operators

TL;DR

This paper provides a martingale transform representation for the rearrangement and shift operators introduced by Figiel, enabling explicit combinatorial construction of the underlying sigma algebras and deriving known norm estimates.

Contribution

It introduces a novel martingale transform representation for Figiel's rearrangement and shift operators, with explicit combinatorial construction of sigma algebras.

Findings

Representation as martingale transform operators

Explicit combinatorial construction of sigma algebras

Derivation of known norm estimates

Abstract

In this paper we obtain a representation as martingale transform operators for the rearrangement and shift operators introduced by T. Figiel. The martingale transforms and the underlying sigma algebras are obtained explicitly by combinatorial means. The known norm estimates for those operators are a direct consequence of our representation.