Sawyer-type inequalities for Lorentz spaces

The Hardy-Littlewood maximal operator satisfies the classical Sawyer-type estimate $$ \left \Vert \frac{Mf}{v}\right \Vert_{L^{1,\infty}(uv)} \leq C_{u,v} \Vert f \Vert_{L^{1}(u)}, $$ where $u\in A_1$ and $uv\in A_{\infty}$. We prove a novel extension of this result to the general restricted weak type case. That is, for $p>1$, $u\in A_p^{\mathcal R}$, and $uv^p \in A_\infty$, $$ \left \Vert \frac{Mf}{v}\right \Vert_{L^{p,\infty}(uv^p)} \leq C_{u,v} \Vert f \Vert_{L^{p,1}(u)}. $$ From these estimates, we deduce new weighted restricted weak type bounds and Sawyer-type inequalities for the $m$-fold product of Hardy-Littlewood maximal operators. We also present an innovative technique that allows us to transfer such estimates to a large class of multi-variable operators, including $m$-linear Calder\'on-Zygmund operators, avoiding the $A_\infty$ extrapolation theorem and producing many estimates that have not appeared in the literature before. In particular, we obtain a new characterization of $A_p^{\mathcal R}$. Furthermore, we introduce the class of weights that characterizes the restricted weak type bounds for the multi(sub)linear maximal operator $\mathcal M$, denoted by $A_{\vec P}^{\mathcal R}$, establish analogous bounds for sparse operators and m-linear Calder\'on-Zygmund operators, and study the corresponding multi-variable Sawyer-type inequalities for such operators and weights. Our results combine mixed restricted weak type norm inequalities, $A_p^{\mathcal R}$ and $A_{\vec P}^{\mathcal R}$ weights, and Lorentz spaces.


Introduction
"Sawyer-type inequalities" is a terminology coined in the paper [CUMP2], where the authors prove that if u ∈ A 1 , and v ∈ A 1 or uv ∈ A ∞ , then where T is either the Hardy-Littlewood maximal operator or a linear Calderón-Zygmund operator. This result extends some questions previously considered by B. Muckenhoupt and R. Wheeden in [MW], and solves in the affirmative a conjecture formulated by E. Sawyer in [S] concerning the Hilbert transform. These problems were advertised by B. Muckenhoupt in [M3], where the terminology "mixed type norm inequalities" was introduced and was also used since then in other papers like [AM] or [MROS]. In general, this terminology refers to certain weighted estimates for some classical operators T , where a weight v is included in their level sets; that is, The structure of such sets makes it impossible, or very difficult, to use classical tools to measure them, such as Vitali's covering lemma or interpolation theorems. In this paper, we consider mixed restricted weak type norm inequalities, or Sawyer-type inequalities for Lorentz spaces; that is, we study estimates of the form where p ≥ 1, T is a classical operator, and u, v, w are weights. We also consider extensions of such inequalities to the multi-variable setting. Our goal is to prove estimates like (1.3) for sub-linear and multi-sub-linear maximal operators, and multi-linear Calderón-Zygmund operators. Observe that in the classical situation, namely when u = w, v ≈ 1, and T is either the Hardy-Littlewood maximal operator or a linear Calderón-Zygmund operator, the inequality (1.3) holds if w ∈ A R p (some authors use the notation A p,1 for this class of weights, as in [CHK]). The case when v ≈ 1 is much more difficult, and in what follows, we will study it in great detail. The starting point of this paper and our primary motivation to consider Sawyer-type inequalities for Lorentz spaces comes from the study of the m-fold product of Hardy-Littlewood maximal operators, M. J. Carro and E. R. P. proved in [CRoPe] that for exponents 1 ≤ p 1 , . . . , p m < ∞ and 1 p = 1 p1 + · · · + 1 pm , and weights w 1 , . . . , w m in A ∞ and ν w := w M ⊗ : L p1,1 (w 1 ) × · · · × L pm,1 (w m ) −→ L p,∞ (ν w ) is that w i ∈ A R pi , for i = 1, . . . , m. They left as an open question to prove that this last condition is also sufficient for (1.4) to hold. It is reasonable to think that this may indeed be true since the endpoint case was proved in [LOPTTG]. That is, for weights w 1 , . . . , w m ∈ A 1 , we have that (1.5) M ⊗ : L 1 (w 1 ) × · · · × L 1 (w m ) −→ L 1/m,∞ (w 1/m 1 . . . w 1/m m ). To prove this result, one has to control the following quantity for t > 0, which is related to the level sets (1.2): , where ν w = w 1/m 1 . . . w 1/m m . This is achieved by applying the classical Sawyer-type inequality (1.1) for the Hardy-Littlewood maximal operator M in combination with the observation that for locally integrable functions h 1 , . . . , h k , k j=1 (M h j ) −1 ∈ RH ∞ , with constant depending only on k and the dimension n.
As we will show in Theorem 4.1, it turns out that the bound (1.4) holds if w i ∈ A R pi , for i = 1, . . . , m, solving in the affirmative the open question in [CRoPe] and completing the characterization of the restricted weak type bounds of M ⊗ for A ∞ weights. The strategy that we follow is similar to the one in [LOPTTG] for the endpoint case (1.5), but we have to replace the classical Sawyer-type inequality (1.1) by the estimate obtained in Theorem 3.8, which is a new restricted weak Sawyer-type inequality involving the class of weights A R p . That is, for p > 1, u ∈ A R p , and uv p ∈ A ∞ . The A R p condition on the weight u is a natural assumption since it is necessary when v ≈ 1. In Lemma 3.10 we also manage to track the dependence of the constant C u,v on the weights u and uv p , even in the endpoint case p = 1, refining the bound (1.1) in [CUMP2].
There is no reason to restrict ourselves to the study of one-variable Sawyer-type inequalities. Quite recently, the bound (1.1) has been extended to the multi-variable setting in [LiOPi]. More precisely, for weights w 1 , . . . , w m ∈ A 1 , and v ∈ A ∞ , Inspired by this result, we follow a similar approach to extend our Sawyer-type inequality (1.6) to the multi-variable setting, obtaining a generalization of (1.7) in Theorem 4.2. That is, for weights w 1 , . . . , w m and v such that for i = 1, . . . , m, w i ∈ A R pi and w i v pi ∈ A ∞ , Observe that this result is an extension of (1.4). To our knowledge, this multi-variable mixed restricted weak type inequalities for maximal operators involving the A R p condition on the weights have not been previously studied, and we found no record of them being conjectured in the literature. Motivated by the conjecture of E. Sawyer in [S], we can ask ourselves if it is possible to obtain bounds like (1.8) for multi-linear Calderón-Zygmund operators T . Once again, the endpoint case p 1 = · · · = p m = 1 has already been considered and extensively investigated in [LiOPi]. There, it was shown that for weights w 1 , . . . , w m ∈ A 1 , and as a corollary of (1.7), combined with a result in [OP], that allows replacing M by T using an extrapolation type argument based on the A ∞ extrapolation theorem obtained in [CUMP1,CGCMP]. We succeed in our goal and manage to get an extension of (1.9) to the general restricted weak setting. In Theorem 4.6 we prove, among other things, that for weights w 1 , . . . , w m and v such that for i = 1, . . . , m, w i ∈ A R pi and w i v pi ∈ A ∞ , and some other technical hypotheses on the weights, To achieve this, we build upon (1.8), but unlike in [LiOPi], we manage to avoid the use of extrapolation arguments like the ones in [OP]. Instead, we present in Theorem 4.4 a novel technique that allows us to replace M by T exploiting the fine structure of the Lorentz space L p,∞ , the A R p condition, and the recent advances in sparse domination. One can even go further and consider inequalities like (1.10) assuming multi-variable conditions on the weights involved, as it was done in [LiOPi] with the endpoint case p 1 = · · · = p m = 1 and weights in A 1 . In Section 5, we discuss our findings on this matter. There, we introduce the class of weights that characterizes the restricted weak type bounds of M, denoted by A R P , study some of its properties, deduce the corresponding restricted weak type bounds for sparse operators and multi-linear Calderón-Zygmund operators, and conjecture the main results on Sawyer-type inequalities with weights in A R P . It is worth mentioning that we couldn't find in the literature any trace of results like (1.10) involving M or multi-linear Calderón-Zygmund operators, A R p or A R P weights, and mixed restricted weak type inequalities. Curiously, we didn't find much about Sawyer-type inequalities for Lorentz spaces apart from the endpoint results studied in [AM,CUMP2,EKoMe,LiOP,LiOPi,MROS,M3,MW,OP,S], and some endpoint estimates for multi-variable fractional operators (see [Pi]), multi-linear pseudodifferential operators (see [CXY]), and the Hardy averaging operator (see [LoMR, MRO]). As we have seen before, these inequalities are fundamental to understand the behavior of the operator M ⊗ , but they appear naturally in the study of other classical operators, even in the one-variable case. Consider, for example, the case of the Hilbert transform H. Indeed, if p > 1 and w ∈ A R p , it is well-known that H : L p,1 (w) −→ L p,∞ (w). Hence, duality, linearity, and self-adjointness of H yield This is an example of an estimate like (1.3) involving the A R p condition on the weights and obtained almost without effort. The same inequality holds for the Hardy-Littlewood maximal operator M , but we cannot use the same argument, as shown in [CS]. In Theorem 4.8 we will generalize such a result for M , obtaining as a particular case, a new characterization of A R p and an alternative proof of the result in [CS]. In [HP, LOP], one can find similar endpoint estimates for Calderón-Zygmund operators, with p ′ = 1 and w ∈ A 1 (see also [CRR, OPRe, OsRa, St, WWZ]).
Sawyer-type inequalities also arise in the broadly studied topic involving commutators of linear operators T with a BM O function b, although we will not deal with them in this paper. The crucial initial observation is that we can write [b, T ] as a complex integral operator using Cauchy's integral theorem, obtaining that for ε > 0, where T z (f ) := e zb T f e zb , z ∈ C. This approach was introduced in the celebrated paper [CRW] and was further developed in [ABKP]. In the context of Lorentz spaces, for p > 1 and a weight w, and in virtue of Minkowski's inequality, we get that for any ε > 0, Since b ∈ BM O, as a consequence of the John-Nirenberg inequality, there is a constant s 0 > 0 such that for |z| ≤ s 0 , v −1 := |e zb | = e ℜ(z)b ∈ A p , and hence, it is possible to deduce weighted inequalities for commutators from estimates of the form for a norm or a quasi-norm · X , and v −1 ∈ A p . Further results for commutators involving Sawyer-type inequalities can be found in [B,BCP1] (see also [BCP2,BCP3]).
Recently, E. R. P. has shown in [RoPe] that Sawyer-type inequalities for Lorentz spaces play a fundamental role in the extension to the multi-variable setting of the restricted weak type Rubio de Francia's extrapolation presented in [CGS, CS]. His approach suggests that Conjecture 5.7 will be crucial for proving multi-variable extrapolation theorems involving weights in A R P .
Given f ∈ L 1 loc (R n ), the Hardy-Littlewood maximal operator M is defined by where the supremum is taken over all cubes Q ⊆ R n containing x. Muckenhoupt studied the boundedness of M on Lebesgue spaces L p (w) (see [M1]). Given a positive and locally integrable function w, called weight, and 1 < p < ∞, if, and only if w ∈ A p ; that is, if where we use the notation Buckley proved in [Bu] that for 1 ≤ p < ∞, Ap , and if p > 1, then The restricted weak type bounds of M were studied in [CHK, KT]. For 1 ≤ p < ∞, As usual, we write A B if there exists a positive constant C > 0, independent of A and B, such that A ≤ CB. If the implicit constant C depends on some parameter α, we may write α at our discretion. If A B and B A, then we write A ≈ B.
We now give the definitions of some other classes of weights that will appear later. For more information about them, see [CUMP2,CUN,DMRO,GCRF]. Define the class of weights This quantity is usually referred to as the Fujii-Wilson A ∞ constant (see [F]). More generally, given a weight u, and p > 1, we say that w ∈ A p (u) if and as before, we define If u is a doubling weight for cubes in R n , and w ∈ A ∞ (u), then We have that In [LOPTTG], the following multi-variable extension of the Hardy-Littlewood maximal operator was introduced in connection with the theory of multi-linear Calderón-Zygmund operators: Commonly, this operator is referred to as the curly operator. For 1 ≤ p 1 , . . . , p m < ∞, P = (p 1 , . . . , p m ), 1 p = 1 p1 + · · · + 1 pm , and weights w 1 , . . . , w m , with w = (w 1 , . . . , w m ), and ν w := w if, and only if w ∈ A P . We are using throughout the paper the standard notation T : X 1 × · · · × X m −→ X 0 to denote that T is a bounded operator from X 1 × · · · × X m to X 0 , where X i is an appropriate function space.
2.2. Dyadic grids and sparse collections of cubes. A general dyadic grid D is a collection of cubes in R n with the following properties: (a) For any Q ∈ D, its side length l Q is of the form 2 k , for some k ∈ Z.
The cubes of a fixed side length 2 k form a partition of R n .
The standard dyadic grid in R n consists of the cubes 2 −k ([0, 1) n + j), with k ∈ Z and j ∈ Z n . It is well-known (see [HP]) that if one considers the perturbed dyadic grids with α ∈ {0, 1 3 } n , then for any cube Q ⊆ R n , there exist α and a cube Q α ∈ D α such that Q ⊆ Q α and l Qα ≤ 6l Q .
A collection of cubes S is said to be η-sparse if there exists 0 < η < 1 such that for every cube Q ∈ S, there exists a set E Q ⊆ Q with η|Q| ≤ |E Q |, and for every For more information about these topics, see [LN].

Calderón-Zygmund operators.
We say that a function ω : [0, ∞) → [0, ∞) is a modulus of continuity if it is continuous, increasing, sub-additive and such that ω(0) = 0. We say that ω satisfies the Dini condition if We give the definition of the multi-linear ω-Calderón-Zygmund operators. We denote by S (R n ) the space of all Schwartz functions on R n and by S ′ (R n ) its dual space, the set of all tempered distributions on R n .
Definition 2.1. An m-linear ω-Calderón-Zygmund operator is an m-linear and continuous oper- for which there exists a locally integrable function K(y 0 , y 1 , . . . , y m ), defined away from the diagonal y 0 = y 1 = · · · = y m in (R n ) m+1 , satisfying, for some constant C K > 0, the size estimate for all (y 0 , y 1 , . . . , y m ) ∈ (R n ) m+1 with y 0 = y j for some j ∈ {1, . . . , m}, and the smoothness estimate qm . If we take ω(t) = t ε for some ε > 0, we recover the classical multi-linear Calderón-Zygmund operators. In general, an m-linear ω-Calderón-Zygmund operator with ω satisfying the Dini condition can be extended to a bounded operator from L 1 (R n ) × · · · × L 1 (R n ) to L 1/m,∞ (R n ). The multilinear Calderón-Zygmund theory has been investigated by many authors. For more information on this matter, see [GT, LOPTTG, LZ] and the publications cited there.

Sawyer-type inequalities for maximal operators
We devote this section to the study of a novel restricted weak type inequality that extends the classical Sawyer-type inequality (1.1) for the Hardy-Littlewood maximal operator. To this end, we will need some previous results.
The following lemma contains well-known results on weights (see [CUMP2,CUN,GCRF,LiOPi]), but we will give most of their proofs since we need to keep track of the constants of the weights involved.
Lemma 3.1. Let u and w be weights.
Proof. To prove (a), fix a cube Q ⊆ R n . By Hölder's inequality, we have that and the desired result follows taking the supremum over all cubes Q. The property (b) follows from [CUN,Theorem 4.2]. Let q ≥ 1, and fix a cube Q ⊆ R n . Then, from which the desired result follows, as before.
To prove (c), fix a cube Q ⊆ R n , and a measurable set E ⊆ Q. Then, In particular, for every ε > 0, and δ := ε β , if |E| < δ|Q|, then u(E) < εu(Q), and the desired result follows from this fact applying the last theorem in [M2].
To prove (d), take q, r > 1 such that u ∈ A q and w ∈ A r . We will show that uw ∈ A s , for and in virtue of Hölder's inequality with exponent α : The property (e) follows immediately from Corollary 4.6 in [CUN].
To prove (f), observe that in virtue of [G,Theorem 7.2.7], we have that for 0 To prove (g), we already know by (f) that v p ∈ RH ∞ , with constant bounded by c m,n,p , so by (c), there exists r > 1, depending only on m, n, p, such that [v p ] Ar ≤ C m,n,p . By (d), for q > 1 such that u ∈ A q , and s = s(m, n, p, q) = q + r − 1, [uv The next lemma gives a result on weights that will be handy later on.
Lemma 3.2. Let u and v be weights, and suppose that u ∈ A ∞ . Then, uv ∈ A ∞ if, and only if v ∈ A ∞ (u).
Proof. Let us first assume that uv ∈ A ∞ . Since u ∈ A ∞ , there exists s > 1 such that u ∈ RH s , and since uv ∈ A ∞ , there exists r > 1 such that uv ∈ A r . Take q := rs s−1 > 1. We will show that v ∈ A q (u). Fix a cube Q. Then, Hence, For the converse, let us assume that v ∈ A ∞ (u). It follows from Theorem 3.1 in [DMRO] that there exist δ, C > 0 such that for every cube Q ⊆ R n and every measurable set E ⊆ Q, Similarly, since u ∈ A ∞ , there exist ε, c > 0 such that for every cube Q ⊆ R n and every measurable set so for every cube Q ⊆ R n and every measurable set E ⊆ Q, Remark 3.3. This result is an extension of Lemma 2.1 in [CUMP2], where it is shown that if u ∈ A 1 and v ∈ A ∞ (u), then uv ∈ A ∞ .
We introduce a weighted version of the dyadic Hardy-Littlewood maximal operator.
Definition 3.4. Let D be a general dyadic grid in R n , and let u be a weight. For a measurable function f , we consider the function where the supremum is taken over all cubes Q ∈ D that contain x. If u = 1, we simply write M D f .
The following bound for the operator M D u is essential. Theorem 3.5. Let D be a general dyadic grid in R n , and let u and v be weights. If u ∈ A ∞ and uv ∈ A ∞ , then there exists a constant C u,v , independent of D, such that for every measurable function f , Proof. In virtue of Lemma 3.2, v ∈ A ∞ (u) and hence, this theorem follows from the proof of Theorem 1.4 in [CUMP2].
Remark 3.6. If we examine the proof of Theorem 1.4 in [CUMP2], and we combine it with Appendix A in [CUMP3], we can take , where r, q > 1 are such that uv ∈ A R r and v ∈ A q ′ (u). Remark 3.7. The bound of Theorem 3.5 also holds for the weighted Hardy-Littlewood maximal operator M u , with constant C := 2 n 6 np p p We can now state and prove the main result of this section.
Theorem 3.8. Fix p ≥ 1, and let u and v be weights such that u ∈ A R p and uv p ∈ A ∞ . Then, there exists a constant C > 0 such that for every measurable function f , Proof. It is known (see [HP,L1]) that there exists a collection {D α } α of 2 n general dyadic grids in R n such that Hence, , and it suffices to establish the result for the operator M D , with D a general dyadic grid in R n . We first discuss the case p = 1, which was proved in [CUMP2]. We reproduce the proof here keeping track of the constants. By the definition of the A 1 condition, and hence, the desired result follows, with C = 24 n [u] A1 C u,v . Now, we discuss the case p > 1. Let us take f = χ E , with E a measurable set in R n , and fix a cube Q ∈ D. As before, by the definition of the A R p condition, .
We can now apply Theorem 3.5 to conclude that Combining all the previous estimates, we have that Since p > 1, L p,∞ (uv p ) is a Banach space, and by standard arguments (see [G,Exercise 1.4.7]), we can extend the previous estimate to arbitrary measurable functions f , gaining a factor of 4p ′ in the constant. Hence, the desired result follows, with C = 4 · 24 n p ′ Remark 3.9. For p = 1 and u ∈ A 1 , a more general version of Theorem 3.8 was established in [LiOP], replacing the hypothesis that uv ∈ A ∞ by the weaker assumption that v ∈ A ∞ . It is unknown to us whether the hypothesis that uv p ∈ A ∞ can be replaced by v ∈ A ∞ when p > 1.
In virtue of Lemma 3.1, if u ∈ A ∞ and v ∈ RH ∞ , then for every p ≥ 1, uv p ∈ A ∞ , and we have a whole class of non-trivial examples of weights that satisfy the hypotheses of Theorem 3.8.
Observe that the conclusion of Theorem 3.8 is completely elementary if p > 1 and u ∈ A p , since However, this argument doesn't work in the general case, because the inequality may fail for some measurable functions h on R n , and arbitrary weights u and v, as can be seen by choosing h( To provide applications of Theorem 3.8 we need to give a more precise estimate of the constant C that appears there in terms of the corresponding constants of the weights involved. We achieve this in the following lemma.
Lemma 3.10. In Theorem 3.8, if r ≥ 1 is such that uv p ∈ A R r , then one can take is a function that increases in each variable, and it depends only on r, p, and the dimension n.
Proof. We first discuss the case when r > 1. We already know that we can take . For convenience, we write V := v p . Let us first bound the factor M u q L q (uV 1−q ) . For the space of homogeneous type (R n , d ∞ , u(x)dx), it follows from the proof of Theorem 1.3 in [HPR] that . Now, given a cube Q ⊆ R n , and applying Hölder's inequality with exponent q, we have that and taking the supremum over all cubes Q, we get that [V ] A∞(u) ≤ M u L q (uV 1−q ) . Combining the previous estimates, we obtain that and applying Theorem 2.3 in [HPR], u ∈ RH s for s = 1 + 1 2 n+1 cp,n[u] 2p A R p −1 , and [u] RHs ≤ 2. Since uV ∈ A 2r , Lemma 3.2 tells us that if we choose q ′ = 2rs ′ , then Finally, observe that q ′ = 2 n+2 rc p,n [u] 2p A R p , and 1 < q ≤ 2, so and the desired result follows, with The case when r = 1 follows, for example, from the case when r = 2 and the fact that if 1/2 A1 . Remark 3.11. It would be interesting to obtain the sharp dependence of C on the constants of the weights involved; our results are most certainly far from optimal.

Applications
In this section, we will provide several applications of the Sawyer-type inequality established in Theorem 3.8, obtaining mixed restricted weak type estimates for multi-variable maximal operators, sparse operators and Calderón-Zygmund operators.
The first result that we present is the converse of Theorem 3.3 in [CRoPe], that was left as an open question. Combining both theorems, we obtain the complete characterization of the restricted weak type bounds of the operator M ⊗ for A ∞ weights.
Proof. The case when p 1 = · · · = p m = 1 was proved in [LOPTTG], and we build upon that proof to demonstrate the remaining cases.
For i = 1, . . . , m, and takingṽ i := j =i (M f j ) −1 , we have that Using the fact thatṽ i ∈ RH ∞ , with constant independent of f (see Lemma 3.1), Hölder's inequality, and Theorem 3.8, we obtain that Iterating this result, we get that for each t > 0 and every natural number N , and letting N tend to infinity, the last term vanishes, and we conclude that Observe that in virtue of Lemma 3.1, for i = 1, . . . , m, we have that w iṽ pi i ∈ A si , where s i > 1 depends only on m, n, p i , and , and hence, the desired result follows, with which depends on the constants of the weights w 1 , . . . , w m in an increasing way.
The next application that we provide is an extension of Theorem 3.8 to the multi-variable setting, which in turn, extends Theorem 4.1. The proof is based on the previous one, and is similar to that of Theorem 1.4 in [LiOPi].
Proof. The first inequality follows from the fact that M( f ) ≤ M ⊗ ( f ). For the second one, we can assume, without loss of generality, that f i ∈ L ∞ c (R n ), i = 1, . . . , m. Fix y, R > 0 and define with constant independent of f (see Lemma 3.1). In virtue of Hölder's inequality and Theorem 3.8, we get that Iterating this result, we deduce that for each y > 0 and every natural number N , and letting first N tend to infinity, and then R, the last term vanishes, and we conclude that Remark 4.3. In the case when p 1 = · · · = p m = 1, the previous result is a corollary of Theorem 1.4 in [LiOPi].
Observe that if we take weights w i ∈ A R pi , i = 1, . . . , m, and v ∈ RH ∞ , then the hypotheses of Theorem 4.2 are satisfied.
The next result will be crucial to work with Calderón-Zygmund operators in the mixed restricted weak setting.
Theorem 4.4. Let 0 < p < ∞, let S be an η-sparse collection of cubes, and let v, w be weights. Suppose that there exists 0 < ε ≤ 1 such that ε < p, wv −ε ∈ A ∞ , and Then, there exists a constant C > 0, independent of S, such that the inequality holds for every vector of measurable functions f = (f 1 , . . . , f m ).

Proof. In virtue of Kolmogorov's inequalities, we obtain that
where the supremum is taken over all measurable sets F with 0 < w(F ) < ∞. For one of such sets F , and W := wv −ε , we have that By hypothesis, S is η-sparse, so for each Q ∈ S, W (3Q) ≤ ( 3 n η W A R r ) r W (E Q ). Using this, we get that The sides of an n-dimensional cube have Lebesgue measure 0 in R n , so we can assume that the cubes in S are open. For Q ∈ S and z ∈ E Q , we define Q z := Q(z, l Q ), the open cube of center z and side length twice the side length of Q. We have that Since the sets {E Q } Q∈S are pairwise disjoint, and using Hölder's inequality with exponent p ε > 1, .
Observe that for every measurable function g, M c W (g) L ∞ (w) ≤ g L ∞ (w) , and by standard arguments (see [G,Theorem 7.1.9]), it is easy to show that . In particular, and applying Marcinkiewicz's interpolation theorem (see [BS,Theorem 4.13

Combining the previous estimates, we obtain that
, and the desired result follows, with Remark 4.5. For 0 < p ≤ 1, if we take v such that v δ ∈ A ∞ for some δ > 0, and w = uv p , with u ∈ A 1 , then the previous result can be established via an extrapolation argument (see [OP,Theorem 1.1]).
Under the conditions that 0 < p ≤ 1, and w = uv p , we can find weights u and v that satisfy the hypotheses of Theorem 1.1 in [OP] but not the ones of Theorem 4.4, and vice versa. If we take a non-constant weight u ∈ A 1 , and v = u −1/p , then v ∈ RH ∞ ⊆ A ∞ , and uv p = 1, but for every 0 < ε ≤ 1 such that ε < p, we have that v −ε = u ε/p ∈ A 1 , and since u is non-constant, v −ε ∈ RH ∞ . Similarly, if we take a non-constant weight v ∈ A 1 , and u = v −p , then uv p = 1, and for every ε > 0, uv p−ε = v −ε ∈ RH ∞ ⊆ A ∞ , but u ∈ RH ∞ and is non-constant, so u ∈ A 1 (see Lemma 3.1).
The previous examples show that, sometimes, some of the hypotheses of Theorem 4.4 may be redundant. Let us be more precise on this fact. If w ∈ A ∞ , and wv −ε is a weight, then [v −ε ] RH∞(w) < ∞ implies that wv −ε ∈ A ∞ . Indeed, given a cube Q ⊆ R n , and a measurable set E ⊆ Q, we have that and since w ∈ A ∞ , there exist δ, C > 0 such that and hence, wv −ε ∈ A ∞ (see [DMRO]). The next application of Theorem 3.8 follows from the combination of Theorems 4.2 and 4.4, and gives us mixed restricted weak type bounds for multi-variable sparse operators that can also be deduced for other operators, such as multi-linear Calderón-Zygmund operators, using sparse domination techniques (see [Li]).
Theorem 4.6. Let 1 ≤ p 1 , . . . , p m < ∞, and let 1 p = 1 p1 + · · · + 1 pm . Let w 1 , . . . , w m be weights, with w i ∈ A R pi , i = 1, . . . , m, and write ν w = w p/p1 1 Let v be a weight such that ν w v p is a weight, and w i v pi ∈ A ∞ , i = 1, . . . , m. Moreover, suppose that there exists 0 < ε ≤ 1 such that ε < p, ν w v p−ε ∈ A ∞ , and [v −ε ] RH∞(ν w v p ) < ∞. Then, there exists a constant C > 0 such that the inequality holds for every vector of measurable functions f = (f 1 , . . . , f m ), where T is either a sparse operator of the form where S is an η-sparse collection of dyadic cubes, or any operator that can be conveniently dominated by such sparse operators, like m-linear ω-Calderón-Zygmund operators with ω satisfying the Dini condition.
Remark 4.7. In the case when p 1 = · · · = p m = 1, and T is a multi-linear Calderón-Zygmund operator, the previous result follows from Theorem 1.9 in [LiOPi].
In general, there are examples of weights that satisfy the hypotheses of Theorem 4.6 apart from the constant weights. For instance, if 1 ≤ p 1 , . . . , p m ≤ m ′ , we can take Indeed, in virtue of Theorem 2.7 in [CGS], we have The last application that we provide of Theorem 3.8 can be interpreted as a dual version of it, and generalizes [CS,Proposition 2.10].
Theorem 4.8. Fix p > 1, and let u and v be weights such that u ∈ A R p , uv p ∈ A ∞ , and for some 0 < ε ≤ 1, uv p−ε is a weight and [v −ε ] RH∞(uv p ) < ∞. Then, there exists a constant C > 0 such that for every measurable function f , Proof. It is known (see [L1]) that there exist a collection {D α } α of 2 n general dyadic grids in R n , and a collection {S α } α of 1 2 -sparse families of cubes, with S α ⊆ D α , such that for every measurable function F , Hence, By duality, and self-adjointness of A Sα , and in virtue of Hölder's inequality, we have that and the desired result follows from Theorem 4.4 and Theorem 3.8.
Remark 4.9. It is clear from the previous proof that Theorem 4.8 is also true for operators that can be conveniently dominated by sparse operators A Sα . Even more, for a self-adjoint operator T , and by duality, an inequality like (4.2) follows immediately from an inequality like (1.6), with T in place of M .
Note that for p > 1, if u ∈ A p , and v is a weight, then for every measurable function f , Hence, we obtain the conclusion of Theorem 4.8 without assuming that for some 0 < ε ≤ 1, We would like to prove Theorem 4.8 without this technical hypothesis, but unfortunately, at the time of writing, we don't know how to do it. Observe that if v = 1, then in Theorem 4.8 we can take ε = 1, and C = C n,p [u] p+1 A R p , and the dependence on u of the constant C is explicit, although the exponent p + 1 may not be sharp. Also, by testing on characteristic functions and using Kolmogorov's inequalities, we see that the condition that u ∈ A R p is necessary. This argument yields a new characterization of A R p weights and refines [CS,Proposition 2.10].
Theorem 4.10. Fix p > 1, and let u be a weight. If u ∈ A R p , then for every measurable function f , Moreover, if such an inequality holds for some constant C > 0, then u ∈ A R p , and [u] A R p ≤ p ′ C.

Sawyer-type inequalities and multi-variable conditions on weights
In [LiOPi,Theorem 1.5], Li, Ombrosi, and Picardi obtained an endpoint Sawyer-type inequality for the operator M involving A 1 weights. It is natural to ask if something similar can be done in the general restricted weak setting, establishing a result for M like the one in Theorem 4.2, but assuming a multi-variable condition on the tuple of weights involved instead of imposing an individual condition on each weight. In this section we study this question.
In view of Theorem 4.2 for v = 1, it is reasonable to begin with the characterization of the weights for which the operator M and its centered version M c are bounded in the restricted weak setting. This will give us the appropriate multi-variable condition on the weights. We use ideas from [CHK,Section 3] and [G,Theorem 7.1.9].
Theorem 5.1. Let 1 ≤ p 1 , . . . , p m < ∞, and 1 p = 1 p1 + · · · + 1 pm . Let w 1 , . . . , w m , and ν be weights. The inequality holds for every vector of measurable functions f if, and only if Proof. First, recall that by [G,Theorem 1.4.16.(v)] (see also [BKS,Theorem 4.4]), we have that Now, fix a cube Q, and γ > 1, and for i = 1, . . . , m, choose a non-negative function f i such that f i L p i ,1 (wi) ≤ 1 and the hypothesis (5.1) and (5.3) imply that For the converse, suppose that the quantity [ w, ν] A R P < ∞. Observe that so it suffices to establish the result for the operator M c . If for some i = 1, . . . , m, f i L p i ,1 (wi) = ∞, then there is nothing to prove, so we may assume that f i L p i ,1 (wi) < ∞ for every i = 1, . . . , m. Fix λ > 0, and let E λ := {x ∈ R n : M c ( f )(x) > λ}.
We first show that this set is open. If for some i = 1, . . . , m, f i ∈ L 1 loc (R n ), then E λ = R n . Otherwise, observe that for any r > 0, and x ∈ R n , the function Taking the supremum over all compact subsets K of E λ and using the inner regularity of ν(x)dx, we obtain (5.1) with constant Remark 5.2. In fact, we have proved that Remark 5.3. Observe that if M is bounded as in (5.1), then for every cube Q, if we choose f 1 = · · · = f m = χ Q , we get that In virtue of Theorem 5.1, we define the following class of weights.
Definition 5.4. Let 1 ≤ p 1 , . . . , p m < ∞, and 1 p = 1 p1 + · · · + 1 pm . Let w 1 , . . . , w m , and ν be weights. We say that (w 1 , . . . , w m , ν) belongs to the class The condition that defines the class of A R P weights depends on their behavior on cubes, and has been obtained following the ideas of Chung, Hunt, and Kurtz (see [CHK]). One can ask if it is possible to obtain a different condition, resembling the one obtained by Kerman and Torchinsky (see [KT]). Our next theorem gives a positive answer to this question, recovering their results in the case when m = 1 and w 1 = ν.
Proof. It is clear that (a) implies (b), and we have already proved in Theorem 5.1 that (a) and (d) are equivalent. Let us show that (b) implies (c). Fix a cube Q and measurable sets E i , for we apply (b) to conclude that and hence, w, ν A R P ≤ c < ∞. We now prove that (c) is equivalent to (d). First, observe that for every i = 1, . . . , m, where the first supremum is taken over all measurable sets E i such that 0 < w i (E i ) < ∞, and the second one is taken over all non-empty measurable sets E i ⊆ Q. Now, in virtue of [CHK,Lemma 2.8] and Kolmogorov's inequalities, we have that Note that a similar argument to the one in the proof of Theorem 4.4 shows that for 0 < ε ≤ 1 such that ε < p, and r ≥ 1 such that ν ∈ A R r , (5.5) M S ( f ) L p,∞ (ν) ≤ A S (| f |) L p,∞ (ν) ≤ C ε,η,n,p,r [ν] where and since S is a countable collection of dyadic cubes, the proof of Theorem 5.1 can be rewritten to show that M S : L p1,1 (w 1 ) × · · · × L pm,1 (w m ) −→ L p,∞ (ν) if, and only if , so (5.5) implies that Finally, in virtue of Theorem 1.2 and Proposition 3.1 in [Li] (see also [L2,Theorem 3.1]), if T is an m-linear ω-Calderón-Zygmund operator with ω satisfying the Dini condition, then there exists a dimensional constant 0 < η < 1 such that given compactly supported functions f i ∈ L 1 (R n ), i = 1, . . . , m, there exists an η-sparse collection of dyadic cubes S such that |T (f 1 , . . . , f m )| ≤ c n C T A S (| f |).
Hence, (5.4) follows from (5.6) and the standard density argument in [G,Exercise 1.4.17]. Moreover, Remark 5.6. Given weights w 1 , . . . , w m , and ν = m i=1 w p/pi i , the equivalence between (b) and (c) in Theorem 5.5 can be found in [BCMR]. Moreover, if p 1 = · · · = p m = 1, then the equivalence between (a) and (d) can be found in [LOPTTG]. Observe that if w ∈ A P , then (w 1 , . . . , w m , ν w ) ∈ A R P . In [LOPTTG], strong and weak type bounds for m-linear Calderón-Zygmund operators were established for the first time for tuples of weights in A P . In [LZ], these results were extended to m-linear ω-Calderón-Zygmund operators with ω Dini < ∞.
We can now state our main conjecture on Sawyer-type inequalities with A R P weights, a complete multi-variable version of Theorem 3.8 for M.
Remark 5.8. This conjecture is true in the case when p 1 = · · · = p m = 1 and ν = ν w , as shown in [LiOPi,Theorem 1.5]. We don't know if the hypothesis that νv p ∈ A ∞ can be replaced by v δ ∈ A ∞ for some δ > 0.
In virtue of Hölder's inequality, if w ∈ m i=1 A R pi , then (w 1 , . . . , w m , ν w ) ∈ A R P , so this conjecture extends the result for M presented in Theorem 4.2. Also, combining such conjecture with Theorem 4.4, we would get a generalization of Theorem 4.6 in the line of [LiOPi,Theorem 1.9].
As it happens in the one-dimensional case, the conclusion of Conjecture 5.7 is completely elementary if 1 < p 1 , . . . , p m < ∞, w ∈ A P , and ν = ν w , since = M( f ) L p (ν w ) [ w] In the general case, observe that for every θ > 0, with V := v θ . We suspect that a wise choice of θ (maybe θ = p or θ = mp) and the argument in the proof of Theorem 1.5 in [LiOPi] could lead to some advances towards our conjecture. This idea requires further investigation.