Paul F. X. Müller

LOCALIZATION AND PROJECTIONS ON BI-PARAMETER BMO

We prove that for any operator

A decomposition theorem for singular integral operators on spaces of homogeneous type

T

A local version of a result of gamlen and gaudet

on bi--parameter BMO the identity factors through

A Decomposition for Hardy Martingales III

T

Classification of the isomorphic types of martingale-H 1 spaces

or

A family of complemented subspaces in VMO and its isomorphic classification

I - T

On the span of some three valued martingale difference sequences inL p (1<p<∞) andH 1

. Bourgains 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.

Rearrangements of the Haar system that preserve BMO

Let (X,d,μ) be a space of homogeneous type. Under the assumption μ({x})=0 for all x∈X, we prove a decomposition theorem for singular integral operators on (X,d,μ). Isotropic Haar expansion gives a representation of the integral operator as a series of simple shifts and rearrangements plus two paraproducts. This yields a UMD-valued T(1) theorem on spaces of homogeneous type.

Rearrangements with supporting trees, isomorphisms and shift operators

LetXDp be the span of the Haar function {hj: J ε D} inLp (1 <p < ∞) endowed withLp norm. Then for any finite setD, the spacesXDp andl*Dp areKp-isomorphic whereKp depends onp only.

Factorization of the identity through operators with large diagonal

We prove Davis decompositions for vector valued Hardy martingales and illustrate their use. This paper continues [17] and [18] on Davis and Garsia Inequalities. AMS Subject Classification 2000: 60G42 , 60G46, 32A35 Key-words: Hardy Martingales, Martingale Inequalities, Embedding.We prove Davis decompositions for vector valued Hardy martingales and illustrate their use. This paper continues the work in [ 18 ] and [ 19 ] on Davis and Garsia Inequalities.