New maximal functions and multiple weights for the multilinear Calderón–Zygmund theory

• Published in: Advances in mathematics (New York. 1965) Vol. 220; no. 4; pp. 1222 - 1264
• Main Authors: Lerner, Andrei K., Ombrosi, Sheldy, Pérez, Carlos, Torres, Rodolfo H., Trujillo-González, Rodrigo
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.03.2009
• Subjects: Calderón–Zygmund theory; Commutators; Maximal operators; Multilinear singular integrals; Weighted norm inequalities; Weighted norm inequalities; 42B25; 42B20; Calderón–Zygmund theory; Commutators; Multilinear singular integrals; Maximal operators
• ISSN: 0001-8708; 1090-2082
• DOI: 10.1016/j.aim.2008.10.014

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.