eBook ISBN:  9781470404857 
Product Code:  MEMO/188/881.E 
List Price:  $57.00 
MAA Member Price:  $51.30 
AMS Member Price:  $34.20 
eBook ISBN:  9781470404857 
Product Code:  MEMO/188/881.E 
List Price:  $57.00 
MAA Member Price:  $51.30 
AMS Member Price:  $34.20 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 188; 2007; 64 ppMSC: Primary 46; 32
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 LittlewoodPaley Gfunctions. 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 HardyLittlewood 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

Chapters

Introduction

1. Preliminaries

2. The duality between $\mathcal {H}^1$ and BMO

3. The maximal inequality

4. The duality between $\mathcal {H}^p$ and $\operatorname {BMO}^q$, $1 < p < 2$

5. Reduction of BMO to dyadic BMO

6. Interpolation


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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 LittlewoodPaley Gfunctions. 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 HardyLittlewood maximal inequality.
(v) A description of \(\mathrm{BMO}\) as an intersection of two dyadic \(\mathrm{BMO}\).
(vi) The interpolation results on these Hardy spaces.

Chapters

Introduction

1. Preliminaries

2. The duality between $\mathcal {H}^1$ and BMO

3. The maximal inequality

4. The duality between $\mathcal {H}^p$ and $\operatorname {BMO}^q$, $1 < p < 2$

5. Reduction of BMO to dyadic BMO

6. Interpolation