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.

We give a simple proof of the A 2 conjecture proved recently by T. Hytönen. Our proof avoids completely the notion of the Haar shift operator, and it is based only on the "local mean oscillation decomposition". Also our proof yields a simple proof of the "two-weight conjecture" as well.2010 Mathematics Subject Classification. 42B20,42B25.

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In recent years, it has been well understood that a Calderón-Zygmund operator T is pointwise controlled by a finite number of dyadic operators of a very simple structure (called the sparse operators). We obtain a similar pointwise estimate for the commutator [b, T ] with a locally integrable function b. This result is applied into two directions. If b ∈ BM O, we improve several weighted weak type bounds for [b, T ]. If b belongs to the weighted BM O, we obtain a quantitative form of the two-weighted bound for [b, T ] due to Bloom-Holmes-Lacey-Wick.2010 Mathematics Subject Classification. 42B20, 42B25.