Localization and projections on bi-parameter BMO

TL;DR

This paper establishes a factorization property for operators on bi-parameter BMO spaces using Bourgain's localization method, which reduces the problem to finite-dimensional cases and employs combinatorial geometry of dyadic rectangles.

Contribution

It introduces a novel application of Bourgain's localization method to bi-parameter BMO, solving finite-dimensional factorization problems with geometric and combinatorial techniques.

Findings

Identity factors through any operator or its complement on bi-parameter BMO.

Localization reduces infinite-dimensional problems to finite-dimensional ones.

Utilizes geometry and combinatorics of dyadic rectangles for solutions.

Abstract

We prove that for any operator $T$ on bi--parameter BMO the identity factors through $T$ or $I - T$. Bourgain's localization method provides the conceptual framework of our proof. It consists in replacing the factorization problem on the non--separable bi--parameter BMO by its localized, finite dimensional counterpart. We solve the resulting finite dimensional factorization problems by exploiting the geometry and combinatorics of colored dyadic rectangles.