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

2008; Elsevier BV; Volume: 220; Issue: 4 Linguagem: Inglês

10.1016/j.aim.2008.10.014

1090-2082

Andrei K. Lerner, Sheldy Ombrosi, Carlos Pérez, Rodolfo H. Torres, Rodrigo Trujillo‐González,

Differential Equations and Boundary Problems

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.