# Operator Valued Hardy Spaces

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*Tao Mei*

The author gives a systematic study of the
Hardy spaces of functions with values in the noncommutative
\(L^p\)-spaces associated with a semifinite von Neumann algebra
\(\mathcal{M}.\) This is motivated by matrix valued Harmonic
Analysis (operator weighted norm inequalities, operator Hilbert
transform), as well as by the recent development of noncommutative
martingale inequalities. In this paper noncommutative Hardy spaces are defined
by noncommutative Lusin integral function, and it is proved
that they are equivalent to those defined by noncommutative
Littlewood-Paley G-functions. The main results of this paper
include:

(i) The analogue in the author's setting of the classical
Fefferman duality theorem between \(\mathcal{H}^1\) and
\(\mathrm{BMO}\).

(ii) The atomic decomposition of the author's noncommutative
\(\mathcal{H}^1.\)

(iii) The equivalence between the norms of the noncommutative
Hardy spaces and of the noncommutative \(L^p\)-spaces
\((1 < p < \infty )\).

(iv) The noncommutative Hardy-Littlewood maximal
inequality.

(v) A description of \(\mathrm{BMO}\) as an intersection of two dyadic
\(\mathrm{BMO}\).

(vi) The interpolation results on these Hardy
spaces.

#### Table of Contents

# Table of Contents

## Operator Valued Hardy Spaces

- Contents v6 free
- Introduction 18 free
- Chapter 1. Preliminaries 512 free
- Chapter 2. The Duality between H[sup(1)] and BMO 1522
- Chapter 3. The Maximal Inequality 2835
- Chapter 4. The Duality between H[sup(p)] and BMO[sup(q)],1 < p < 2 3643
- Chapter 5. Reduction of BMO to dyadic BMO 5259
- Chapter 6. Interpolation 5663
- Bibliography 6370