## Abstract

A multi(sub)linear maximal operator that acts on the product of m Lebesgue spaces and is smaller than the m-fold product of the Hardy-Littlewood maximal function is studied. The operator is used to obtain a precise control on multilinear singular integral operators of Calderón-Zygmund type and to build a theory of weights adapted to the multilinear setting. A natural variant of the operator which is useful to control certain commutators of multilinear Calderón-Zygmund operators with BMO functions is then considered. The optimal range of strong type estimates, a sharp end-point estimate, and weighted norm inequalities involving both the classical Muckenhoupt weights and the new multilinear ones are also obtained for the commutators.

Original language | English |
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Pages (from-to) | 1222-1264 |

Number of pages | 43 |

Journal | Advances in Mathematics |

Volume | 220 |

Issue number | 4 |

DOIs | |

State | Published - 1 Mar 2009 |

Externally published | Yes |

### Bibliographical note

Funding Information:✩ A.K. Lerner’s research supported by the Spanish Ministry of Education under the program “Programa Ramón y Cajal, 2006.” S. Ombrosi’s research supported by a fellowship from the same institution. These two authors and C. Pérez are also supported by the same institution under research grant MTM2006-05622. R. Trujillo-González is also supported by the same institution under grant MTM2005-07347. R.H. Torres’ research supported in part by the National Science Foundation under grants OISE 0126272 and DMS 0400423. * Corresponding author.

### Funding

✩ A.K. Lerner’s research supported by the Spanish Ministry of Education under the program “Programa Ramón y Cajal, 2006.” S. Ombrosi’s research supported by a fellowship from the same institution. These two authors and C. Pérez are also supported by the same institution under research grant MTM2006-05622. R. Trujillo-González is also supported by the same institution under grant MTM2005-07347. R.H. Torres’ research supported in part by the National Science Foundation under grants OISE 0126272 and DMS 0400423. * Corresponding author.

Funders | Funder number |
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National Science Foundation | DMS 0400423, OISE 0126272 |

Ministerio de Educación, Cultura y Deporte | MTM2006-05622, MTM2005-07347 |

## Keywords

- Calderón-Zygmund theory
- Commutators
- Maximal operators
- Multilinear singular integrals
- Weighted norm inequalities