We study mixed weak type inequalities for the commutator [b, T ], where b is a BMO function and T is a Calderón-Zygmund operator. More precisely, we prove that for everyand v ∈ A∞(u). Our technique involves the classical Calderón-Zygmund decomposition, which allow us to give a direct proof. We use this result to prove an analogous inequality for higher order commutators. We also obtain a mixed estimation for a wide class of maximal operators associated to certain Young functions of L log L type which are in intimate relation with the commutators. This last estimate involves an arbitrary weight u and a radial function v which is not even locally integrable.2010 Mathematics Subject Classification. 42B20, 42B25.

In this paper we prove mixed inequalities for the maximal operator Φ , for general Young functions Φ with certain additional properties, improving and generalizing some previous estimates for the Hardy-Littlewood maximal operator proved by E. Sawyer. We show that given ≥ 1, if , are weights belonging to the 1-Muckenhoupt class and Φ is a Young function as above, then the inequality ({ ∈ ℝ ∶ Φ ()() () > }) ≤ ∫ ℝ Φ (| ()|) () () dx holds for every positive. A motivation for studying these type of estimates is to find an alternative way to prove the boundedness properties of Φ. Moreover, it is well-known that for the particular case Φ() = (1 + log +) with ∈ ℕ these maximal functions control, in some sense, certain operators in harmonic analysis.

We study mixed weak estimates of Sawyer type for maximal operators associated to the family of Young functions Φ(t) = t r (1 + log + t) δ , where r ≥ 1 and δ ≥ 0. More precisely, if u and v r are A1 weights, and w is defined as w = 1/Φ(v −1 ) then the following estimateholds for every positive t. This extends mixed estimates to a wider class of maximal operators, since when we put r = 1 and δ = 0 we recover a previous result for the classical Hardy-Littlewood maximal operator. This inequality generalizes the result proved by Sawyer in Proc. Amer. Math. Soc. 93 (1985), no. 4, 610-614. Moreover, it includes estimates for some maximal operators related with commutators of Calderón-Zygmund operators.2010 Mathematics Subject Classification. 42B20, 42B25.

We devote this note to correct an estimate concerning mixed inequalities for the generalized maximal function MΦ, when certain properties of the associated Young function Φ are assumed.Although the obtained estimates turn out to be slightly different, they are good extensions of mixed inequalities for the classical Hardy-Littlewood maximal functions Mr, with r ≥ 1. They also allow us to obtain mixed estimates for the generalized fractional maximal operator Mγ,Φ, when 0 < γ < n and Φ is an L log L type function.

We prove mixed weak estimates of Sawyer type for fractional operators. More precisely, let T be either the maximal fractional function Mγ or the fractional integral operator Iγ, 0 < γ < n, 1 ≤ p < n/γ and, holds for every positive t and every bounded function with compact support.As an important application of the results above we further more exhibe mixed weak estimates for commutators of Calderón-Zygmund singular integral and fractional integral operators when the symbol b is in the class Lipschitz-δ, 0 < δ ≤ 1.2010 Mathematics Subject Classification. 42B20, 42B25.