Localization and Projections on Bi--Parameter BMO

Richard Lechner, Paul F. X. Müller

Published 2014-10-31Version 1

We prove that for any operator T on bi--parameter BMO the identity factors through T or Id - T. As a consequence, bi--parameter BMO is a primary Banach space. Bourgain's localization method provides the conceptual framework of our proof. It consists in replacing the factorization problem on the non--separable Banach space bi--parameter BMO by its localized, finite dimensional counterpart. We solve the resulting finite dimensional factorization problems by combinatorics of colored dyadic rectangles.