A $B_p$ condition for the strong maximal function

A strong version of the Orlicz maximal operator is introduced and a natural $B_p$ condition for the rectangle case is defined to characterize its boundedness. This fact let us to describe a sufficient condition for the two weight inequalities of the strong maximal function in terms of power and logarithmic bumps. Results for the multilinear version of this operator and for others multi(sub)linear maximal functions associated with bases of open sets are also studied.


Introduction
As it is well-known, Sawyer ([28]) characterized the pair of weights (u, v) for which the Hardy-Littlewood maximal operator, M , is a bounded operator from L p (v) to L p (u) for 1 < p < ∞. He showed that M : L p (v p ) → L p (u p ) if and only if (u, v) satisfies the testing condition On the other hand, it is also known that the two weight Muckenhoupt condition A p , is necessary but not sufficient for the maximal operator to be strong type (p, p).
The fact that Sawyer's condition involves the maximal operator itself makes it often difficult to test in practice. Therefore, though this condition characterizes completely the two weight problem, it would be more useful to look for sufficient conditions close in form to the A p condition. The first step in this direction was due to Neugebauer ([22]): he noticed that if the pair of weights (u, v) is such that for r > 1, for all cubes, then

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for all nonnegative functions f and some positive constant C independent of f . If for a given cube Q we define the normalized L p norm by u p,Q := 1 |Q| Q u p dx 1/p , then Neugebauer's condition can be restated in terms of a normalized L p norm as follows: (1.3) u pr,Q v −1 p ′ r,Q < ∞. Notice that the A p condition can also be rewritten as This tell us that if we replace the average L p and L p ′ norms in (1.4) by some stronger ones, then we can get a condition that is sufficient for (1.2) to hold. At the same time, this new condition preserves the geometric structure of the classical A p . Conditions like (1.3) are known as power bump conditions because the norms involved in the two weight A p condition are "bumped up" in the Lebesgue scale. Motivated by [22], [9] and [10], Pérez ([25], [23]) generalized these last conditions replacing the localized norms in (1.4) by some other larger than the L p one, but not as big as the L pr . Indeed, he proved that it was enough to substitute only the norm associated to the weight v −1 by a stronger one defined in terms of certain Banach function spaces X with an appropriate boundedness property.
To be more precise, we let Φ be a Young function (cf. section 2) and define the normalized Luxemburg norm on a cube Q by u Φ,Q := inf λ > 0 : 1 for s ≥ 0. SuchΦ is also a Young function and it plays, together with the class B p , an important role in the generalization of these bump conditions. Recall that a Young function Φ ∈ B p if there is a positive constant c for which This growth condition was introduced in [25] where it was proved that if Φ is a Young function such thatΦ ∈ B p , and (u, v) is a pair of weights such that for every cube Q, then (1.2) holds. Moreover the B p condition is sharp in the sense that: if M is strong (p, p) and (u, v) satisfy (1.7), thenΦ ∈ B p . This result has been generalized very recently to the more general context of Banach function spaces by Pérez and Mastylo ([21]). For a more complete account of all this we refer to the recent book [6]. Besides its inherent significance for this problem, for many other operators conditions like (1.7) have resulted in optimal sufficient conditions for weak and strong type inequalities. In general, given any pair of weights (u, v) we will define the A p bump condition as where Ψ and Φ are Young functions such thatΨ ∈ B p ′ and/orΦ ∈ B p and Q is any cube R n . This or related conditions are being used in the study of operators more singular than the Hardy-Littlewood maximal function. The first case was considered by Pérez in [26] for fractional integral operators where a fractional version of (1.8) was used to obtain a two weight L p estimate. The same problem for the Hilbert transform was proved in [5] and by different methods in [7] for any Calderón-Zygmund operator with C 1 kernel. Very recently the solution was extended in [8] to the Lipschitz case, proved full in generality in [18] and further improved in [15] with a better control on the bounds. The main goal of this paper is to study the two weight norm inequalities for the boundedness of the strong maximal function using an appropriate B p condition. We define this operator as where f is a locally integrable function and the supremum is taken over all rectangles R with sides parallel to the coordinate axes. The corresponding two weight problem for the strong maximal function was characterized by Jawerth (see [16]) in terms of a testing condition that it is even harder to verify than Sawyer's condition (1.1). The problem was also solved in [24] with a more similar approach to the one that we study here. It was proved that if (u, v) is a couple of weights satisfying the power condition for some r > 1 for every rectangle R, and suppose that u p satisfies the condition (A): there are constants 0 < λ < 1 and 0 < c(λ) < ∞ such that for all measurable sets E, then M R : L p (v p ) → L p (u p ). In this case, the strong weighted estimate is obtained from weak type ones using interpolation and the fact that there is a reverse Hölder's property for the weights that verifies (1.10). However, this good property disappears if we substitute the L pr -norm associated to the weight v −1 by a weaker one. Therefore, we will need a different approach to solve the two weight problem with general bump conditions. To state the main result of this article, we define the class of Young functions that enable us to obtain bump conditions in the rectangle case.

LIGUANG LIU AND TERESA LUQUE
Then we have the following result.
Theorem 1.2. Let 1 < p < ∞, and let Φ be a Young function such that the complementary Young function Φ satisfies the condition (1.11).
(i) Let (u, v) be a couple of weights such that u p satisfies the condition (A) and for some positive constant K and for all rectangles R. Then there is a constant C such that for all non-negative functions f .
for some positive constant K and for all rectangles R; thenΦ ∈ B * p . If this result is compared with the analogous one for the Hardy-Littlewood operator ([25, Theorem 1.5]), then there is a key difference between them. For the former not only do we need a more restrictive class of young functions (the class B * p ), but also it is necessary to ask for an extra condition (A) on the weight u. To understand the role of this extra condition (A), we should keep in mind the next relevant fact. The study of the boundedness properties of a maximal operator with respect to a family of bounded measurable sets is closely connected to studying the covering properties of that family (cf. [12]). But since the geometry of rectangles in R n is much more intricate than that of cubes in R n , the classical covering lemmas don't work in the rectangle case. Particularly, the Calderón-Zygmund decomposition that is strongly used in the cube case cannot be used here. In this sense, the condition (A) is necessary to deal with rectangles and with their covering properties (see Lemma 4.2 below). The problem with this condition is that, as happened with Sawyer's condition, it involves itself the operator and therefore it would be useful to have a replacement that did not. The A ∞ condition would be a good candidate since it is simpler than the (A) condition; however, it is also stronger (see for example [24, p. 1123]). Unfortunately, repeated efforts to get a weaker condition and a simpler covering argument have failed, but we believe that such a result would be very interesting and would provide new insights about the study of the strong maximal operator.
In this article, we also address similar questions involving the multilinear version of the strong maximal function and some other more general maximal functions. We define the strong multilinear maximal function as where f = (f 1 , · · · , f m ) is an m-dimensional vector of locally integrable functions and where the supremum is taken over all rectangles with sides parallel to the coordinate axes. This multilinear maximal operator was defined for first time by Lerner et al. in [19] with the usual cubes instead of rectangles. In that paper it is shown that this operator plays a central role in the theory of multilinear Calderón-Zygmund operators. The operator (1.13) as well as its version for a general basis B (cf. section 3) was introduced and studied in [14]. In particular, it was shown the weak boundedness of M R whenever the weights satisfy a certain power bump variant of the multilinear version of the A p condition. That is, for 1 < p 1 , · · · , p m < ∞ and 0 < p < ∞ such that 1 p = m j=1 1 pj , the multilinear strong maximal function maps provided that (ν, w) = (ν, w 1 , . . . , w m ) are weights that satisfy the power bump condition for some r > 1. In the case that ν = m j=1 w p/pj j the strong boundedness of M R is also characterized; see [14, Corollary 2.4 and Theorem 2.5]. Multiple weight theory that adapted to the basis B = Q and to its multilinear operator associated, M Q , has been fully developed by Lerner et al. [19] and generalized very recently by Moen [20].
Inspired by these previous works, we will introduce the multilinear version of (1.7) and (1.12) for weights (ν, w) associated with general basis. Then, the L p1 (w 1 )× · · ·×L pm (w m ) → L p (ν) boundedness of M B (cf. section 3) will be proved whenever ν is any arbitrary weight such that ν p satisfies condition (A). This result is given in Theorem 3.1. As an application of this theorem, we will obtain the strong version of [14, Theorem 2.3] and we will deduce the analogous result of [20, Theorem 6.6] for the strong multilinear maximal function. See Corollary 3.3 and Corollary 3.4.
The general organization of this paper is as follows. Section 2 contains some preliminaries about Orlicz spaces and a characterization of the strong B * p condition. Section 3 presents some definitions about general basis and the statement of the main strong weight result for a general multilinear operator (Theorem 3.1). Also, we give the proofs of Corollaries 3.3 and 3.4, and we will deduce the proof of Theorem 1.2 by applying Corollary 3.4. Finally, the last section shows the proof of Theorem 3.1.

LIGUANG LIU AND TERESA LUQUE
for all t ≥ 0. The normalized Φ-norm of a function f over a set E with finite measure is defined by The complementary of the Young function (1.5) has properties for all s, t ≥ 0. Also theΦ-norms are related to the L Φ -norms via the the generalized Hölder's inequality, namely Consider the Orlicz maximal operator where the supremum is taken over all cubes containing x. Pérez [25, Theorem 1.7] proved the following key observation: when 1 < p < ∞ and Φ is a doubling Young function, then Here we remark that the hypothesis of Φ being doubling was only used to prove the necessity of the B p condition but we show now that can be removed. Indeed, if we assume that for any non-negative function f the operator M Q Φ is bounded on L p (R n ) and we take f = χ [0,1] n , we have Now, it is easy to see that there exists a positive dimensional constant b such that whenever |y| > 1 .
where c n is a positive constant depending only on n. Since Φ is increasing and Φ(t) → ∞ as t → ∞, we can choose some t 0 > 0 such that for every t ≤ t 0 , Motivated by Pérez [25,Theorem 1.7] and the previous observation, in this section we consider the Orlicz maximal operator M R Φ associated with rectangles rather than cubes. Precisely, for each locally integrable function f and Young function Φ we define the Orlicz maximal operator where the supremum is taken over all rectangles with sides parallel to the coordinate axes containing x. In particular, when Φ(t) = t the maximal operator M R Φ is exactly the classical strong maximal function (1.9).
The next characterization shows that the boundedness of M R Φ is closely connected with the class B * p . Theorem 2.1. Let 1 < p < ∞. Suppose that Φ is a Young function. Then the following statements are equivalent: for all non-negative functions f and u; (iv) there exists a positive constant C such that for all non-negative functions f and all w satisfying the condition (A), As particular examples of Young functions Φ ∈ B * p , one can easily check that a Young function Φ satisfies the condition (1.11) if Proof of Theorem 2.1. We assume that (i) holds and show (ii). To this end, for each t > 0, we split the function f into f = f t + f t with f t := f χ |f |>t/2 and f t := f χ |f |≤t/2 . Then,

Then it follows that
where we have used the fact that Φ ′ (t) ≥ Φ(t) t for any Young function Φ. Now we take the integral in the variable y n−1 and we obtain We iterate this process by integrating over the next variables y n−3 , · · · , y 1 in turn and we obtain which proves (i).
To conclude the proof of this theorem, note that (iv) implies (ii) by choosing w = 1 in the right side of (2.4). In order to prove that (ii) implies (iv) we will proceed using an argument very similar to the one presented in the proof of Theorem 3.1. For this reason, we will give the details to complete this proof in the fourth section, Remark 4.3. Here we point out that the proof of Theorem 3.1 below is independent of Theorem 2.1.
It should be remarked that the classical B p condition (1.6) is not sufficient for the L p (R n )-boundedness of M R Φ . For this, we consider for example the function f = χ [0,1] n , and for all t ∈ (0, ∞) with 0 < δ < 1. It is easy to verify that such a function Φ satisfies (1.6) but fails for (1.11). For simplicity, we consider only the case when n = 2. If |x i | >> 1 (we can take |x i | > 4) for i = 1, 2, then Hence, M R Φ is not bounded on L p (R 2 ). The general case for n > 2 is similar and we omit the details.
Though the B p condition (1.6) is not sufficient for the L p (R n )-boundedness of M R Φ , we can remedy this situation if we restrict to those Young functions that are submultiplicative; see Proposition 2.2 below. We say that a Young function Φ is submultiplicative if for each t, s > 0, Proof. This is a simple consequence of the fact that for a submultiplicative Young function such that Φ ∈ B p , there exits ǫ > 0 for which Φ ∈ B p−ǫ ( [25,Lemma 4.3]). Indeed, using the previous theorem we only need to prove that Φ ∈ Φ * p . Note that

LIGUANG LIU AND TERESA LUQUE
It is clear that the first term in the right hand of (2.6) is bounded. On the other hand, (log + Φ(s)) n−1 s ǫ ≤ Φ(s) δ(n−1) s ǫ δ (n−1) , with δ > 0. Since Φ is in the class B p , it follows that Φ(t) t p for t ≥ 1 and hence for δ = ǫ p(n−1) the above term is bounded. This further implies that the second term of (2.6) is bounded by a constant multiple of which together with the aforementioned fact that Φ ∈ B p−ǫ gives the boundedness of the second term of (2.6).
Remark 2.3. We observe that a typical Young function that belongs to the class B p and that it is also submultiplicative is Φ(t) = t r with 1 ≤ r < p. Another more interesting example is the function Φ given by Φ(t) = t r (1 + log + t) α with 1 ≤ r < p and α > 0. It is not difficult to see that such functions are submultiplicative and they are in the B p class. See Cruz-Uribe and Fiorenza [4] for related discussions on the topic of submultiplicative Young functions.

Weighted theory for general bases and proof of Theorem 1.2
We start by introducing some notation that we will use through this section. Another interesting example is the set ℜ of all rectangles in R 3 with sides parallel to the coordinate axes whose side lengths are s, t, and st, for some t, s > 0.
Assume that B is a basis and that {Ψ j } m j=1 is a sequence of Young functions, we define the multi(sub)linear Orlicz maximal function by In particular, when Ψ j (t) = t for all t ∈ (0, ∞) and all j ∈ {1, · · · , m}, we simply write M B − → Ψ as M B ; that is, When m = 1, we use M B Ψ and M B to respectively denote M B − → Ψ and M B . We say that w is a weight associated with the basis B if w is a non-negative measurable function in R n such that w(B) = B w(y) dy < ∞ for each B in B. A weight w associated with B is said to satisfy the A p,B condition, 1 < p < ∞, if In the limiting case p = 1 we say that w satisfies the A 1,B if for all B ∈ B; this is equivalent to M B w(x) ≤ c w(x) for almost all x ∈ R n . It follows from these definitions and Hölder's inequality that Then it is natural to define the class A ∞,B by setting For a general basis B we obtain the following strong type result.
Theorem 3.1. Let 1 < p 1 , · · · , p m < ∞ and 0 < p < ∞ such that 1 p = m j=1 1 pj . Assume that B is a basis and that {Ψ j } m j=1 is a sequence of Young functions such that is bounded from L p1 (R n ) × · · · × L pm (R n ) to L p (R n ). Let (ν, w ) = (ν, w 1 , · · · , w m ) be weights such that ν p satisfies condition (A), and that Then M B is bounded from L p1 (w p1 1 ) × · · · × L pm (w pm m ) to L p (ν p ). Remark 3.2. We observe that for all x ∈ R n and for all non-negative functions f = (f 1 , . . . , f m ), Thus, if we assume that each M B Ψj is bounded on L pj (R n ), then M B − → Ψ is bounded from L p1 (R n ) × · · · × L pm (R n ) to L p (R n ), and consequently, the conclusion of Theorem 3.1 gives us that M B is bounded from L p1 (w p1 1 ) × · · · × L pm (w pm m ) to L p (ν p ) when (ν, w) satisfies (3.1).
Between these general bases, we are particularly interested in Muckenhoupt basis introduced in [23]. We say that B is a Muckenhoupt basis if for any 1 < p < ∞ and for any w ∈ A p,B , M B is bounded in L p (w). Most of the important bases are in this class and, in particular, those mentioned above: Q, D, R. The fact that R is a Muckenhoupt basis can be found in [13]. The basis ℜ is also a Muckenhoupt basis as shown by R. Fefferman [11].

LIGUANG LIU AND TERESA LUQUE
Proof. For each j ∈ {1, · · · , m}, we set w j := w 1/pj j and Ψ j (t) := t p ′ j r for all t ∈ (0, ∞). Set ν := ν 1/p . Then the power bump condition (3.2) can be rewritten as In this case, for all x ∈ R n , Since B is a Muckenhoupt basis and (p ′ j r) ′ < p j , every M B Ψj is bounded on L pj (R n ). By Remark 3.2 this implies that M B − → Ψ is bounded from L p1 (R n ) × · · · × L pm (R n ) to L p (R n ). Thus, by Theorem 3.1 , which completes the proof.
A result stronger than Corollary 3.3 is the following boundedness of the multilinear strong maximal function, where (ν, w) satisfy some logarithmic type condition.
pj . Let (ν, w ) = (ν, w 1 , · · · , w m ) such that ν and all the w j 's are weights, and ν p satisfies condition (A). If there is a positive constant K such that for all rectangles R, where every Ψ j is a Young function such thatΨ j ∈ B * pj . Then M R is bounded from L p1 (w p1 1 ) × · · · × L pm (w pm m ) to L p (ν p ).
Proof. From Theorem 2.1 and the assumption that eachΨ j is a Young function satisfying the condition (1.11), it follows that every M R Ψj is bounded on L pj (R n ). Then, applying Remark 3.2 and Theorem 3.1 with B = R, we obtain the desired conclusion.
We end this section with the proof of Theorem 1.2 that is a straight consequence of Corollary 3.4.
Proof of Theorem 1.2. We notice first that (i) is the linear case (m = 1) of Corollary 3.4. For the proof of (ii) we proceed as in [25,Proposition 3.2]. That is, consider any non-negative function g and define the couple of weights (u, v) = (M R Φ (g 1/p ) −1 , g −1/p ). Obviously, (u, v) satisfies condition (1.12) with constant K = 1. Hence, by hypothesis there is a constant C such that Finally, by Theorem 2.1, this inequality implies that Φ ∈ B * p , which completes the proof.

Proof of the strong type estimate in the (m + 1)-weight case
To prove Theorem 3.1, we use an argument that combines ideas from [14, Theorem 2.5], the second proof of Theorem 3.7 in [19], and some other tools from [16] and [25]. First, we will recall an additional definition for general bases and a special case of a lemma from [16] that we will need for the proof of Theorem 3.1. The following definition concerns the concept of α-scattered families, which was considered in the works [17] and [16] and implicitly in [2] and [3]. The proof of the following lemma is in [16, p. 370, Lemma 1.5]; see also [14].
Lemma 4.2. Let B be a basis and let w be a weight associated to this basis. Suppose further that w satisfies condition (A) for some 0 < λ < 1 and 0 < c(λ) < ∞. Then given any finite sequence Proof of Theorem 3.1. Let N > 0 be a large integer. We will prove the required estimate for the quantity with a bound independent of N . We claim that for each integer k with |k| ≤ N , there exist a compact set K k and a finite sequence The sequence of sets {∪ B∈b k B} N k=−N is decreasing. Moreover, To see the claim, for each k we choose a compact set For this K k , there exists a finite sequence b k = {B k α } α≥1 of sets B k α ∈ B such that every B k α satisfies (4.1) and such that Finally, to ensure that {∪ B∈b k B} N k=−N is decreasing, we begin the above selection from k = N and once a selection is done for k we do the selection for k − 1 with the next additional requirement K k−1 ⊃ K k . This proves the claim. Since {∪ B∈b k B} N k=−N is a sequence of decreasing sets, we set Observe that these sets are decreasing in k, i.e., Ω k+1 ⊂ Ω k when −N < k ≤ N . We now distribute the sets in k b k over µ sequences {A i (l)} i≥1 , 0 ≤ l ≤ µ − 1, where µ will be chosen momentarily to be an appropriately large natural number. Set i 0 (0) = 1. In the first i 1 (0) − i 0 (0) entries of {A i (0)} i≥1 , i.e., for we place the elements of the sequence b N = {B N α } α≥1 in the order indicated by the index α. For the next i 2 (0) − i 1 (0) entries of {A i (0)} i≥1 , i.e., for we place the elements of the sequence b N −µ . We continue in this way until we reach the first integer m 0 such that N − m 0 µ ≥ −N , when we stop. For indices i satisfying i m0 (0) ≤ i < i m0+1 (0), we place in the sequence {A i (0)} i≥1 the elements of b N −m0µ . The sequences {A i (l)} i≥1 , 1 ≤ l ≤ µ − 1, are defined similarly, starting from b N −l and using the families b N −l−sµ , s = 0, 1, · · · , m l , where m l is chosen to be the biggest integer such that N − l − m l µ ≥ −N .
Since ν p is a weight associated to B and it satisfies condition (A), we can apply Lemma 4.2 to each {A i (l)} i≥1 for some fixed 0 < λ < 1. Then we obtain sequences which are λ-scattered with respect to the Lebesgue measure. In view of the definition of the set Ω k and the construction of the families {A i (l)} i≥1 , we can use assertion (c) of Lemma 4.2 to obtain that for any k = N − l − sµ with 0 ≤ l ≤ µ − 1 and 1 ≤ s ≤ m l , Observe that the first term in the last equality of (4.3) is equal to If we choose µ so large that c 2 −µp ≤ 1 2 and since everything involved is finite the first term on the right hand side of (4.3) can be subtracted from the left hand side of (4.3). This yields where in the penultimate step we used the generalized Hölder's inequality (2.2). Finally, we will obtain the claimed conclusion using the fact that the operator M R Φ is bounded on L p (R n ). The details are left to the reader.