© 2005

Isomorphisms between H1 Spaces

  • Study of dyadic H^1, its isomorphic invariants and its position within the two classes of martingale and atomic H^1 spaces, and simultaneously, a detailed analysis of the Haar system

  • Only basic knowledge in real, complex and functional analysis required, and some probability theory


Part of the Monografie Matematyczne book series (MONOGRAFIE, volume 66)

Table of contents

About this book


This book presents a thorough and self-contained presentation of H¹ and its known isomorphic invariants, such as the uniform approximation property, the dimension conjecture, and dichotomies for the complemented subspaces.
The necessary background is developed from scratch. This includes a detailed discussion of the Haar system, together with the operators that can be built from it (averaging projections, rearrangement operators, paraproducts, Calderon-Zygmund singular integrals). Complete proofs are given for the classical martingale inequalities of C. Fefferman, Burkholder, and Khinchine-Kahane, and for large deviation inequalities. Complex interpolation, analytic families of operators, and the Calderon product of Banach lattices are treated in the context of H^p spaces.
Througout the book, special attention is given to the combinatorial methods developed in the field, particularly J. Bourgain's proof of the dimension conjecture, L. Carleson's biorthogonal system in H¹, T. Figiel's integral representation, W.B. Johnson's factorization of operators, B. Maurey's isomorphism, and P. Jones' proof of the uniform approximation property. An entire chapter is devoted to the study of combinatorics of colored dyadic intervals.


Funktionalanalysis Harmonische Analysis Kombinatorik Martingale Singular integral Stochastische Analysis proof

Authors and affiliations

  1. 1.Institute of AnalysisJohannes Kepler University LinzAustria

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"Ein bedeutendes, in Inhalt und Form vorzügliches Standardwerk über faszinierende, tiefliegende mathematische Probleme vor, zu deren Lösung viele der besten Mathematiker beigetragen haben." (Monatshefte für Mathematik)