**Memoirs of the American Mathematical Society**

2001;
80 pp;
Softcover

MSC: Primary 46;

Print ISBN: 978-0-8218-2734-5

Product Code: MEMO/154/731

List Price: $52.00

Individual Member Price: $31.20

**Electronic ISBN: 978-1-4704-0324-9
Product Code: MEMO/154/731.E**

List Price: $52.00

Individual Member Price: $31.20

# On the Connection between Weighted Norm Inequalities, Commutators and Real Interpolation

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*Jesús Bastero; Mario Milman; Francisco J. Ruiz*

We show that the class of weights \(w\) for which the Calderón operator is bounded on \(L^p(w)\) can be used to develop a theory of real interpolation which is more general and exhibits new features when compared to the usual variants of the Lions-Peetre methods. In particular we obtain extrapolation theorems (in the sense of Rubio de Francia's theory) and reiteration theorems for these methods. We also consider interpolation methods associated with the classes of weights for which the Calderón operator is bounded on weighted Lorentz spaces and obtain similar results. We extend the commutator theorems associated with the real method of interpolation in several directions. We obtain weighted norm inequalities for higher order commutators as well as commutators of fractional order. One application of our results gives new weighted norm inequalities for higher order commutators of singular integrals with multiplications by BMO functions. We also introduce analogs of the space BMO in order to consider the relationship between commutators for Calderón type operators and their corresponding classes of weights.

#### Readership

Graduate students and research mathematicians interested in functional analysis.

#### Table of Contents

# Table of Contents

## On the Connection between Weighted Norm Inequalities, Commutators and Real Interpolation

- Contents vii8 free
- 1. Introduction 110 free
- 2. Calderón weights 716 free
- 3. Applications to real interpolation: reiteration and extrapolation 1726
- 4. Other classes of weights 2433
- 5. Extrapolation of weighted norm inequalities via extrapolation theory 2837
- 6. Applications to function spaces 3342
- 7. Commutators defined by the K-method 3746
- 8. Generalized commutators 4150
- 9. The quasi Banach case 5059
- 10. Applications to Harmonic Analysis 5463
- 11. BMO type spaces associated to Calderón weights 6473
- 12. Atomic decompositions and duality 7180
- References 7786