## Interpolation of OperatorsThis book presents interpolation theory from its classical roots beginning with Banach function spaces and equimeasurable rearrangements of functions, providing a thorough introduction to the theory of rearrangement-invariant Banach function spaces. At the same time, however, it clearly shows how the theory should be generalized in order to accommodate the more recent and powerful applications. Lebesgue, Lorentz, Zygmund, and Orlicz spaces receive detailed treatment, as do the classical interpolation theorems and their applications in harmonic analysis. The text includes a wide range of techniques and applications, and will serve as an amenable introduction and useful reference to the modern theory of interpolation of operators. |

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### Contents

1 | |

35 | |

Chapter 3 Interpolation of Operators on Rearrangement Invariant Spaces | 95 |

Chapter 4 The Classical Interpolation Theorems | 183 |

Chapter 5 The KMethod | 291 |

Appendix A | 441 |

443 | |

445 | |

461 | |

List of Notations | 467 |

Pure and Applied Mathematics | 470 |

### Common terms and phrases

a-finite measure space absolutely continuous absolutely continuous norm admissible operator analytic arbitrary associate space Banach function space Banach space bounded operator Calderón compatible couple completes the proof constant continuously embedded Corollary couple Xo cubes Q decreasing rearrangement defined Definition denote diam(Q disjoint equimeasurable establishes estimate f and g f belongs finite measure function f function norm fundamental function G. H. Hardy Hardy's inequality Hence Hilbert transform Hölder's inequality holds integrable function interpolation spaces interpolation theorem interval J. E. Littlewood joint weak type K-functional Lebesgue Lemma Let f Let Xo linear operator LlogL Lorentz spaces Marcinkiewicz interpolation theorem maximal operator measurable functions measure space monotone convergence theorem nonatomic nonnegative obtain Proposition rearrangement-invariant Banach function rearrangement-invariant spaces resonant measure space Riesz-Fischer satisfies sequence shows simple functions strong type subset subspace substochastic u-measurable weak type weak-type Young's function