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.

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.

We obtain a Bloom‐type characterization of the two‐weighted boundedness of iterated commutators of singular integrals. The necessity is established for a rather wide class of operators, providing a new result even in the unweighted setting for the first order commutators.

Abstract. We obtain an L p (w) bound for Calderón-Zygmund operators T when w ∈ A 1 . This bound is sharp both with respect to w A 1 and with respect to p. As a result, we get a new L 1,∞ (w) estimate for T related to a problem of Muckenhoupt and Wheeden.

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We obtain an improved version of the pointwise sparse domination principle established by the first author in [19]. This allows us to determine nearly minimal assumptions on a singular integral operator T for which it admits a sparse domination.where f p p,Q = 1 |Q| Q |f | p , and S is a sparse family of cubes of R n . Recall that the family of cubes S is called η-sparse, 0 < η ≤ 1, if for every cube Q ∈ S, there exists a measurable set E Q ⊂ Q such that |E Q | ≥ η|Q|, and the sets {E Q } Q∈S are pairwise disjoint.Localization and sparseness are two main ingredients which make sparse bounds especially effective in quantitative weighted norm inequalities.The literature about sparse bounds is too extensive to be given here in more or less adequate form. We mention only that sparse bounds for Calderón-Zygmund operators can be found in [7,15,16,18,19,20]. Also, there are several general sparse domination principles [5,6,8,19].In [19], a sparse domination principle was obtained in terms of the grand maximal truncated operatordefined for a given operator T .2010 Mathematics Subject Classification. 42B20, 42B25.

Abstract. The Calderón operator S is the sum of the the Hardy averaging operator and its adjoint. The weights w for which S is bounded on L p (w) are the Calderón weights of the class C p . We give a new characterization of the weights in C p by a single condition which allows us to see that C p is the class of Muckenhoupt weights associated to a maximal operator defined through a basis in (0, ∞). The same condition characterizes the weighted weaktype inequalities for 1 < p < ∞, but that the weights for the strong type and the weak type differ for p = 1. We also prove that the weights in C p do not behave like the usual A p weights with respect to some properties and, in particular, we answer an open question on extrapolation for Muckenhoupt bases without the openness property.

We discuss several characterizations of the A ∞ class of weights in the setting of general bases. Although they are equivalent for the usual Muckenhoupt weights, we show that they can give rise to different classes of weights for other bases. We also obtain new characterizations for the usual A ∞ weights.