Weighted mixed weak-type inequalities for multilinear fractional operators

The aim of this paper is to obtain mixed weak-type inequalities for multilinear fractional operators, extending results by F. Berra, M. Carena and G. Pradolini \cite{BCP}. We prove that, under certain conditions on the weights, there exists a constant $C$ such that $$\Bigg\| \frac{\mathcal G_{\alpha}(\vec f \,)}{v}\Bigg\|_{L^{q, \infty}(\nu v^q)} \leq C \ \prod_{i=1}^m{\|f_i\|_{L^1(u_i)}},$$ where $\mathcal G_{\alpha}(\vec f \,)$ is the multilinear maximal function $\mathcal M_{\alpha}(\vec f\,)$ that was introduced by K. Moen in \cite{M} or the multilineal fractional integral $\mathcal I_{\alpha}(\vec f \,)$. As an application a vector-valued weighted mixed inequality for $\mathcal I_{\alpha}(\vec f \,)$ will be provided as well.

1. Introduction E. Sawyer proved in 1985 the following mixed weak-type inequality.
Theorem 1.1 ([18]).If u, v ∈ A 1 , then there is a constant C such that for all t > 0, This estimate is a highly non-trivial extension of the classical weak type (1, 1) inequality for the maximal operator due to the presence of the weight function v inside the distribution set.Note that if v = 1, this result is a well-known estimate due to C. Fefferman and E. Stein [6].The inequality (1.1) also holds if u ∈ A 1 when v ∈ A 1 ; see [9].
In 2005, D. Cruz-Uribe, J. M. Martell and C. Pérez [5] extended (1.1) to R n .Furthermore, they settled that estimate for Calderón-Zygmund operators, answering affirmatively and extending a conjecture raised by E. Sawyer for the Hilbert transform [18].The precise statement of their result is the following.

M. BEL ÉN PICARDI
Theorem 1.2 ([5]).If u, v ∈ A 1 , or u ∈ A 1 and uv ∈ A ∞ , then there is a constant C such that, for all t > 0, where T is a Calderón-Zygmund operator with some regularity.
Quantitative versions of the previous result were obtained in [17] and also some counterparts for commutators in [1].
In [5], D. Cruz-Uribe, J. M. Martell and C. Pérez conjectured that (1.2) and (1.1) should hold for v ∈ A ∞ .This result is the most singular case, due to the fact that the A ∞ condition is the weakest possible asumption within the A p classes.
where T can be the Hardy-Littlewood maximal function, any Calderón-Zygmund operator or any rough singular integral.
In 2009, Lerner et al. [10] introduced the multi(sub)linear maximal function M defined by where ⃗ f = (f 1 , . . ., f m ) and the supremum is taken over all cubes Q containing x.This maximal operator is smaller than the product m i=1 M f i , which was the auxiliar operator used previously to estimate multilinear singular integral operators.
There is a connection between multilinear operators and mixed weak-type inequalities (see [10] or [12]).In fact, in a recent joint work with K. Li and S. Ombrosi we proved the following theorem.m .Suppose that ⃗ w ∈ A (1,...1) and νv We refer the reader to Section 2 for the definition of A (1,...,1) and more details about A ⃗ P weights.
The study of fractional integrals and associated maximal functions is important in harmonic analysis.We recall that the fractional integral operator or Riesz potential is defined by and the fractional maximal function by where the supremum is taken over all cubes Q containing x.Note that in the case α = 0 we recover the Hardy-Littlewood maximal operator.Properties of these operators can be found in the books by Stein [19] and Grafakos [7].
, then there exists a positive constant C such that for every t > 0 , where I α is the fractional integral or the fractional maximal function.
In the multilinear setting, a natural way to extend fractional integrals is the following.Definition 1.6.Let α be a number such that 0 < α < mn and let ⃗ f = (f 1 , . . ., f m ) be a collection of functions on R n .We define the multilinear fractional integral as [14] introduced the multi(sub)linear maximal operator M α asociated to the multilinear fractional integral I α .Definition 1.7.For 0 ≤ α < mn and ⃗ f = (f 1 , . . ., f m ) as above, we define the multi(sub)linear maximal operator M α by Observe that the case α = 0 corresponds to the multi(sub)linear maximal function M studied in [10].
At this point we present our contribution.Our first result is a counterpart of Theorem 1.5 for multilinear fractional maximal operators.
Note that if α = 0 then q = 1 m and we obtain Theorem 1.4 for the multi(sub)linear maximal operator M.
Remark.If in Theorem 1.8 we take m = 1 we get that 1 q = 1 − α n and the hypothesis on the weights reduces to u q ∈ A 1 and v ∈ A ∞ .Then we recover Theorem 1.5 in the case p = 1 for a more general class of weights v.The weight u q in Theorem 1.8 plays the role of the weight u in Theorem 1.5.
By extrapolation arguments, we can extend this result to multilinear fractional integrals.The theorem below was essentially obtained in [16]; however, for the sake of completeness, we will give a complete proof in Appendix A. Theorem 1.9 ([16]).Let 0 < α < mn.Let q = n mn−α , ⃗ u mq = (u mq 1 , . . ., u mq m ) ∈ A (1,...,1) , v q ∈ A ∞ and set ν = m i=1 u q i .Then there exists a constant C such that Finally, as a consequence of Theorem 1.8 and Theorem 1.9, we obtain the main result of this paper.
Theorem 1.10.Let 0 < α < mn.Let q = n mn−α , ⃗ u mq = (u mq 1 , . . ., u mq m ) and ν = m i=1 u q i .Suppose that ⃗ u mq ∈ A (1,...,1) and νv q ∈ A ∞ , or u mq 1 , . . ., u mq m ∈ A 1 and v mq ∈ A ∞ .Then there exists a constant C such that The rest of the article is organized as follows.In Section 2 we recall the definition of the A p and A ⃗ P classes of weights.Section 3 is devoted to the proof of Theorem 1.8.In Section 4, as an application of Theorem 1.10, we obtain a vector-valued extension of the mixed weighted inequalities for multilinear fractional integrals.We end this paper with an appendix, in which we give a proof of Theorem 1.9.

Preliminaries
By a weight we mean a non-negative locally integrable function defined on R n such that 0 < w(x) < ∞ almost everywhere.We recall that a weight w belongs to the class A p , introduced by B. Muckenhoupt [15] where p ′ is the conjugate exponent of p defined by the equation Since the A p classes are increasing with respect to p, it is natural to define the A ∞ class of weights by In 2009, Lerner et al. [10] showed that there is a way to define an analogue of the Muckenhoupt A p classes for multiple weights.We say that ⃗ w satisfies the When The multilinear A ⃗ P condition has the following characterization in terms of the linear A p classes.
Observe that in the particular case where every p i = 1 we have p = 1 m .By Theorem 2.3, given ⃗ w = (w 1 , . . ., w m ), we have that the following statements hold: does not imply that w i ∈ A 1 for every i = 1, . . ., m.We can see this with a simple counterexample.Let m = 2 and consider the weights 1 , w 3. Proof of Theorem 1.8 In order to prove Theorem 1.8 we need the following pointwise estimate for M α in terms of the multilinear maximal operator M.This is a multilinear version of Lemma 4 in [2], and to prove it, we follow a similar approach to the one that is used there.
Proof.Let us fix x ∈ R n and let Q be a cube containing x. Applying Hölder's inequality with 1 1− α mn and mn α we obtain .

□
Now we have all the tools that we need to prove Theorem 1.8. where A vector-valued extension of Theorem 1.10 Recently, D. Carando, M. Mazzitelli and S. Ombrosi [4] obtained a generalization of the Marcinkiewicz-Zygmund inequalities to the context of multilinear operators.We recall one of the results in that work that extends previously known results from [8] and [3].Theorem 4.1 ([4]).Let 0 < p, q 1 , . . ., q m < r < 2 or r = 2 and 0 < p, q 1 , . . ., q m < ∞ and, for each Let S be a multilinear operator such that S : .
As a consequence of this theorem and Theorem 1.10 we obtain the following mixed weighted vector valued inequality for a multilinear fractional operator I α .
Observe that under the hypothesis of Corollary 4.2, S satisfies S : (νv q ).So we are under the hypothesis of Theorem 4.1.
5. Appendix A. Proof of Theorem 1.9 In order to prove Theorem 1.9 we will need two known results.The first one is due to K. Moen.
Theorem 5.1 ([14, Theorem 3.1]).Suppose that 0 < α < mn; then for every w ∈ A ∞ and all 0 < s < ∞ we have for all functions ⃗ f with f i bounded with compact support.
The second result we will rely upon is due to D. Cruz-Uribe, J. M. Martell and C. Pérez.Theorem 5.2 ([5, Theorem 1.7]).Let F be a family of pairs of functions that satisfies that there exists a number p 0 , 0 < p 0 < ∞, such that, for all w ∈ A ∞ , R n f (x) p0 w(x) dx ≤ C R n g(x) p0 w(x) dx for all (f, g) ∈ F such that the left hand side is finite, and with C depending only on [w] A∞ .Then, for all weights u, v such that u ∈ A 1 and v ∈ A ∞ , we have that Having those results at our disposal we proceed as follows.First of all observe that if ⃗ u mq = (u mq 1 , . . ., u mq m ) ∈ A (1,...,1) , then ν = u q 1 . . .u q m ∈ A = sup λ>0 λ q νv q x ∈ R n :