Upper endpoint estimates and extrapolation for commutators

In this note we revisit the upper endpoint estimates for commutators following the line by Harboure, Segovia and Torrea. Relying upon the suitable BMO subspace suited for the commutator that was introduced by Accomazzo, we obtain a counterpart for commutators of the upper endpoint extrapolation result by Harboure, Mac\'ias and Segovia. Multilinear counterparts are provided as well.


Introduction and main results
Extrapolation has been a fruitful area of research since the 80s.The first results in that direction were due to Rubio de Francia [14,13].We briefly discuss the general principle behind that kind of results in the following lines.
We say that w is a weight if it is a non-negative locally integrable function on R n .Recall that w ∈ A p for 1 < p < ∞ if where M stands for the Hardy-Littlewood maximal function where each Q is a cube of R n with its sides parallel to the axis.A fundamental property of the A p classes is that they characterize the weighted L p boundedness of the Hardy-Littlewood maximal operator and they are good 2020 Mathematics Subject Classification.Primary 42B20; Secondary 42B25.The first author was supported by the National Natural Science Foundation of China (Grant 12001400).The second and the third authors were partially supported by FONCyT PICT 2018-02501.The second author was partially supported by Spanish Government Ministry of Science and Innovation through grant PID2020-113048GB-I00.The third author was partially supported as well by FONCyT PICT 2019-00018 and by Junta de Andalucía UMA18FEDERJA002.214 KANGWEI LI, SHELDY OMBROSI, AND ISRAEL P. RIVERA-R ÍOS weights for a number of operators in the theory such as singular integrals, commutators and some further ones.
The Rubio de Francia extrapolation results say that if T is a sublinear operator such that for some 1 < p 0 < ∞ T f L p 0 (w) ≤ c w,T,p0 f L p 0 (w) (1.1) for every w ∈ A p0 , then for every w ∈ A p and every 1 < p < ∞.This approach has been extensively studied by a number of authors in a wide variety of settings.For instance, in the linear setting there are fundamental works due to Cruz-Uribe, Martell, and Pérez [9,4,5,8,7,6], Duoandikoetxea [11,12], Dragicevic, Grafakos, Petermichl, Pereyra [10], Harboure, Macías, Segovia [18,17].After a number of intermediate results in the multilinear setting (see for instance [16,2,7]) the question was succesfully solved in the last years, in works such as [26,25,28].
A useful development in the area since Rubio de Francia's pioneering works consisted in learning that the operator involved in (1.1) and (1.2) actually plays no role.To be more precise, it can be replaced by a condition on pairs of functions.Assume that F is a family of pairs of functions such that for some 1 < p 0 < ∞ f L p 0 (w) ≤ c w,T,p0 g L p 0 (w) for every (f, g) ∈ F and every w ∈ A p0 , then for every (f, g) ∈ F, for every w ∈ A p and every 1 < p < ∞.
Another line of research would consist in considering the endpoints, namely p 0 = ∞ or p 0 = 1 as a "departing" point for extrapolation.For instance, the following result was obtained in [15,9] Theorem 1.Let (f, g) be a pair of functions and suppose that holds for all w with w −1 ∈ A 1 , where c w depends only on [w −1 ] A1 .Then for all 1 < p < ∞ and all w ∈ A p , we have where cw depends only on [w] Ap .
There are a number of operators that do not map L ∞ into L ∞ such as the Hilbert transform.However for the Hilbert transform H itself and even for a larger class of operators, the Calderón-Zygmund operators, it is possible to show that they map L ∞ into BMO.Weighted versions of that result were studied first in [27].There it was shown that if w ∈ A 1 , then In view of this estimate, it seems natural to think about extending this result to Calderón-Zygmund operators, and also, within the framework of extrapolation whether it would be possible to extrapolate from that weighted L ∞ → BM O bound in order to obtain weighted L p estimates.Those questions were answered in the positive in the inspiring paper [18] by Harboure, Macías, and Segovia.In that work the following extrapolation result was settled.
Theorem 2. Let T be a sublinear operator defined on Then for every 1 < p < ∞ and every w ∈ A p we have that Quite recently in [3], a quantitative version of this result was obtained.In that paper it was shown that if δ ∈ (0, 1) and where σ = w − 1 p−1 .Note that since M L p (w) [w] Ap such an estimate yields In the same paper it is shown that for Calderón-Zygmund operators namely ϕ(t) = t and hence the sharp exponent for the A p constant max 1, for that class is not recovered.Such a fact is not surprising since the current best known extrapolation argument from the lower endpoint neither recovers the sharp estimate.At this point we would like to note that a way more general version of the aforementioned extrapolation result, replacing L p (w) spaces by function Banach spaces and the A p constant by suitable boundedness constants of the maximal function over those spaces, was obtained very recently in [29].Also a quantitative multilinear result in that direction was provided in [28,Corollary 4.14] Now we turn our attention to our contribution in this work.We recall that given b ∈ BM O, the Coifman-Rochberg-Weiss commutator is defined as It is well known that [b, T ] is bounded on L p (w) and that, as Pérez showed in [30], [b, T ] is not of weak type (1, 1) but it satisfies the following estimate instead: where Φ(t) = t log(e + t).The quantitative dependence was obtained in [23].
In view of (1. (1) For every ball B and every The function b satisfies the following condition.For any cube [19] the authors point out that if T is the Hilbert transform and any of the conditions in the preceding Theorem are satisfied, then necessarily b is constant and hence [b, H] = 0.This fact leads us to think about the possibility of considering a "smaller" oscillation in the left hand side of (1.5).
Aiming for a dual of the Hardy spaces for commutators studied by Pérez [30] and Ky [21], Accomazzo introduced in [1] the spaces BM O q b which are defined as follows.Given a function b, and .
Note that f BM O q b = 0 if and only if f = α + βb and hence in order to consider f BM O q b as a norm one needs to take quotient by the subspace 1, b (the space of linear combinations of 1 and b).It readily follows from the definition that BM O ⊂ BM O q b for every q.It is also easy Inspired by the definition of BM O q b we provide the following result for commutators.
The next natural question would be whether it is possible to extrapolate from the condition above.We show that that is the case under some additional conditions.

Theorem 5. Let T be a linear operator such that for every b ∈ BM O and every
and such that Lerner's grand maximal operator Observe that the operator M T was introduced in [22] in order to study sparse domination.There it was shown that in the case of T being a Calderón-Zygmund operator where T * stands for the maximal Calderón-Zygmund operator.Since both M and T * are bounded on L p (w) for w ∈ A p , the result above combined with the estimate in Theorem 1.4 allows to provide an alternative proof of the weighted L p boundedness of the commutator [b, T ].
Here we just presented the results in the linear setting.However results in the multilinear setting are feasible as well and will be obtained in Section 4.
The remainder of the paper is organized as follows.In Section 2 we gather some preliminaries.
In Section 3 we settle Theorems 4 and 5. Finally in Section 4 we present and settle the multilinear counterparts of the main results.

Preliminaries
We recall that T is a Calderón-Zygmund operator if T is a linear operator that is bounded on L 2 and it admits a representation in terms of a kernel where K satisfies the following properties: where ω is a continuous subadditive function such that In the definition of commutators we used BMO functions.We recall that b A fundamental property of this space of functions is the well-known John-Nirenberg that says that the integrability of the oscillations self-improves to exponential integrability, namely, there exist constants λ, c > 0 such that for every ball B and every BMO function Note that this in turn implies that for every α > 0. Another fact that we will use in what follows is that if B is a ball then We remit the interested reader to [20] for more details on BMO.Quite related to the definition of BMO is that of the sharp maximal function.Given δ > 0, we define .
We would like to end this preliminaries section by gathering some basic facts about multilinear theory.We recall that a linear operator T is an m-linear Calderón-Zygmund operator if T : 1 pi and it admits the following representation: Note that in this context the commutator [b, T ] j f (x) is defined as Note that the definition is essentially equivalent whichever index we commute in.Hence throughout the remainder of this work we will consider just the case [b, T ] 1 .

Proof of Theorem 4.
Let B be a ball and c 2 , λ constants to be chosen.Let where . Then we begin arguing as follows: Note that for L 1 , choosing λ = b 2B we have that for δ < ε < 1, calling First we focus on L 11 .We argue as follows: Now we turn to L 12 .Choosing c 2 = T f 2 (c B ) we have that Rev. Un.Mat.Argentina, Vol.66, No. 1 (2023)

UPPER ENDPOINT ESTIMATES AND EXTRAPOLATION 221
From this point taking into account the smoothness condition of the kernel we may argue as follows: We continue bounding L 2 .Note that For L 21 by Kolmogorov's inequality, where in the last step we have used Hölder's inequality and (2.1).For L 22 , we have that, using the smoothness condition of the kernel, where in the last step we have used Hölder's inequality, (2.1) and (2.2).This ends the proof.
Note that this yields Observe that if we call by hypothesis we have that and hence by Theorem 1 we have that for all 1 < q < ∞ and every w ∈ A q g L q (w) ≤ cw b BM O f L q (w) .
Since by hypothesis as well we know that For I 1 , we have , where we have used the weak endpoint estimate of T .Now we turn to estimate I 2 .We argue as follows: The estimate of I 22 can be handled similarly as before, that is, we use Kolmogorov's inequality and then the weak type endpoint estimate where the last inequality holds since w −1 1 ∈ A ∞ .It remains to consider I 21 .Similarly as before, we have where we have used that This requires that the kernel satisfies the log-Dini condition.This completes the proof.
Having the Theorem above at our disposal we can obtain the following result.To deal with g we argue as in the linear setting.M T is bounded by hypothesis.Hence we are done.
Note that in the case of T being a Calderón-Zygmund operator, in the bilinear case, a careful calculus to bound M T was presented in [24].Such an estimate, that we recall in the following line, can be extended to the multilinear case directly.Therefore, we have for every 0 < s < 1 m .Of course, since M s is increasing with s, the inequality holds for all s > 0. In particular, we can choose s = 1 m , which in turn allows us to show that where we have used the fact that if (w 1 , . . ., w m ) ∈ A p then w p ∈ A mp (when p < ∞).This ends the argument and completes an alternative proof of the boundedness of [b, T ] i in the multilinear setting.

Theorem 3 .
3) and (1.4) one may wonder what can be said about commutators.In [19, Theorem A], Harboure, Segovia, and Torrea provided the following result.Let T be a Calderón-Zygmund operator and let b ∈ BM O. Then the following statements are equivalent:

Theorem 4 .
Let b ∈ BM O and T a Calderón-Zygmund operator satisfying a log-Dini regularity condition.Then for every ball B, if δ ∈ (0, 1) and r > 1 we have that

3. 2 .
Proof of Theorem 5. Let us fix a ball B and x ∈ B. Following the same notation as that in the proof of Theorem 4, if we choose