Weighted BMO estimates for singular integrals and endpoint extrapolation in Banach function spaces

In this paper we prove sharp weighted BMO estimates for singular integrals, and we show how such estimates can be extrapolated to Banach function spaces.


Introduction
In the 70's Muckenhoupt and Wheeden extended the known L ∞ → BMO estimates for the Hilbert transform to the weighted setting.In particular, they proved that where Hf I is the average of Hf over I, holds if and only if w −1 ∈ A ∞ ∩ B 2 .This is Theorem 1 in [MW76].
The class B 2 , which was introduced in [HMW73], is the collection of all locally integrable weights w satisfying In the same paper (Theorem 2), Muckenhoupt and Wheeden characterize w −1 ∈ A 1 as the class of weights w for which the following weighted BMO estimate holds for all intervals I and all functions f .This result was later used by Harboure, Macías, and Segovia in [HMS88] to prove a BMO-to-L p extrapolation result, saying that if a sublinear operator T satisfies the same bound as in (1.1) with implicit constant depending only on [w −1 ] A 1 , then for all weights v ∈ A p and 1 < p < ∞.
We should mention the result of [Nie19], where a multilinear extrapolation result is obtained which generalizes and sharpens the one by Harboure, Macías, and Segovia.See also [CPRR19], where sharp linear versions of these two results are obtained.
The purpose of this article is twofold: first, we extend the extrapolation result of Harboure, Macías, and Segovia, to the setting of Banach function spaces.Second, we extend the results of Muckenhoupt and Wheeden to sparse operators as well as to Calderón-Zygmund operators whose kernels satisfy a Dini smoothness condition.We fully recover the results of [CPRR19], and in fact our extrapolation result is sharper, see Remark 3.3.
Let us first state a simplified version of our extrapolation theorem.
Theorem A. Let T be a sublinear operator.Suppose there is an increasing function φ : [1, ∞) → (0, ∞) such that for all weights w with w −1 ∈ A 1 , and all compactly supported functions f with f w ∈ L ∞ (R d ) we have Then, for all Banach function spaces X over R d for which M is bounded on X and its associate space X , and all compactly supported functions f ∈ X T f X d M X →X φ(2 M X→X ) f X .
Our full result is Theorem 3.1 in section 3.As for our result for Calderón-Zygmund operators, suppose T is an operator represented by for all x outside of the support of f , and where K satisfies for all x, y, z satisfying |x−y| > 2|x−z| > 0, and where Ω : [0, ∞) → [0, ∞) is an increasing subadditive function with Ω(0) = 0.
Similar to the situation in [MW76], it is not sufficient that w ∈ A ∞ for T to have weighted BMO estimates.What we need is a generalization of the B 2 condition that takes into account the interaction between the weight and the smoothness of the kernel.This is quantified by Note that, in dimension one and when Ω(t With this definition, we have ) and let T be an operator as above, then and for all compactly supported functions This is Theorem 4.4 in section 4.These estimates are based on sparse domination techniques, so they offer a simplified argument to the results from [MW76] and [MW78].
Notation.We work in the setting of R d equipped with the Lebesgue measure dx.For a measurable set E ⊆ R d we denote its measure by |E|.For a measurable function f ∈ L 0 (R d ), a measurable set E ⊆ R d of finite positive measure, and a q ∈ (0, ∞) we write , and denote the essential supremum of f in E by f ∞,E .Moreover, we denote the linearized mean of f on E by By a cube in R d we mean a half-open cube whose sides are parallel to the coordinate axes.
We will write A a,b,... B to mean that there is a constant C depending only on the parameters a, b,

Preliminaries
2.1.Dyadic analysis.In this paper we will be working in R d , but our results are also valid in more general spaces of homogeneous type.We will often reduce our arguments to dyadic grids: a dyadic grid is a collection of cubes with the property For a collection of cubes E in a dyadic grid D and a cube Q 0 ∈ D, we let E (Q 0 ) denote the cubes in E that are contained in Q 0 .We refer the interested reader to the monograph [LN19].

Muckenhoupt weights.
A weight w in R d can be associated with the measure through w(E) := E w dx.A classical result by Muckenhoupt is that the Hardy-Littlewood maximal operator where the supremum is taken over all cubes However, in the case p = ∞, we have L ∞ (R d , w) = L ∞ (R d ) for all weights w and hence, , then we do get a meaningful condition for p = ∞.Note that for p ∈ (1, ∞) we have For p = ∞, this expression yields the condition The assertion follows.
If D is a dyadic grid in R d , then we define Then, exactly as in the above result, we have Finiteness of this constant characterizes the class p≥1 A p (D).Note that in particular we have In the non-dyadic case, the Fujii-Wilson characteristic is defined as where the supremum is taken over all cubes.
We have the following well-known result (which, in fact, is a characterization of A ∞ (D)): Proposition 2.2.Let S ⊆ D be an η-sparse collection of cubes and let Q 0 ∈ D. Then for a weight w we have Proof.We have as desired.
2.3.Weighted BMO spaces.The space BMO(R d ) is defined as the space of function f ∈ L 0 (R d ) for which the sharp maximal function ).This is facilitated through the following definition.Definition 2.3.For a weight w we define the spaces BMO where functions are identified modulo constants.
We also define the weak analogues BMO 1,weak where functions are identified modulo constants.For a dyadic grid D in R d , the spaces BMO 1 w (D), BMO ∞ w (D), BMO 1,weak w (D), and BMO ∞,weak w (D) are defined analogously, this time taking the supremum over all cubes in D, and functions are identified modulo constants on cubes in D.
Thus, in this case we have For a dyadic grid D in R d , analogously defining the sharp maximal operators M ,D and M ,D 1,∞ with respect to this grid, we also have Proof.We only prove the first equality, the others being analogous.Since for all cubes Q, we have This proves the inequality Letting ε → 0, the assertion follows.Proposition 2.5.Let w be a weight.We have Thus, in particular, . This similarly holds for BMO 1 w (D), BMO ∞ w (D) for a dyadic grid D in R d .Proof.We only prove the second equivalence, the first one being analogous.The inequality ≥ follows from taking c = f Q .For the converse inequality, fix a cube Q and note that for any c ∈ R we have As the result follows analogously for BMO w (D), this proves the assertion.
In the unweighted case, it follows from the John-Strömberg characterization of BMO that the weak BMO space is equivalent to the usual one.More precisely, denoting the non-decreasing rearrangement of a function f by f * , for a cube Q and λ ∈ (0, 1) we define We can extend John-Strömberg's characterization of BMO through medians (see [Str79]) to the weighted setting using the following pointwise version by A. Lerner [Ler03, Theorem 1.3]: Proof.By Proposition 2.4, (2.3), and Proposition 2.1, we have The result now follows from the fact that M 1 2 f M weak f .2.4.Banach function spaces.We denote the positive measurable functions on R d by L 0 (R d ) + .
Definition 2.7.Suppose ρ : (iv) for every measurable E ⊆ R d of positive measure there exists a measurable F ⊆ E of positive measure with ρ(1 F ) < ∞.Then we call ρ a function norm.Moreover, we call the space Given a function norm ρ, we define a new function norm ρ through The Köte dual of X is defined as the Banach function space associated to ρ , i.e., Property (iv) is called the saturation property of ρ and is equivalent to the existence of a weak order unit, i.e., an f > 0 for which ρ(f ) < ∞.Property (iv) is also equivalent to the fact that ρ satisfies (i) and hence, is a function norm.Thus, X is a Banach function space over R d .
The Fatou property (iii) is equivalent to the assertion that ρ = ρ and hence, X = X isometrically.Thus, in particular we have We say that a Banach function space X over R d is order-continuous when for all (f n ) n∈N in X with f n ↓ 0 a.e.we have f n X → 0. We note that X is reflexive if and only if X and X are order-continuous.For proofs of the above claims we refer the reader to [Zaa67].
By carefully tracking the constants in the proof of [Ler10b, Corollary 4.3], we arrive at the following quantitative version: Theorem 2.8.Let X be a Banach function space over R d and suppose that M : X → X is bounded.Then the following assertions are equivalent: (i) There is a constant C X > 0 such that Moreover, if C X and C X,weak are the smallest possible constant in (i) and (ii), then Proof.Our strategy will be to prove that (ii)⇒(i)⇒(iii)⇒(ii).The assertion (ii)⇒(i) with C X ≤ C X,weak follows from the fact that M weak f ≤ M f .For (i)⇒(iii) we use that it was shown in [Ler10b, Theorem 1.1] that (i) is equivalent to the assertion (iv) there is a C > 0 such that ) and g ∈ X .Moreover, if C is the smallest possible constant in this estimate, then Thus, we have

as desired
For (iii)⇒(ii), we use [Ler04, Theorem 1] which states that there is a dimensional constant λ ∈ (0, 1) such that for all f satisfying the property that |{x ∈ R d : |f (x)| > α}| < ∞ for all α > 0 and all g ∈ L 1 loc (R d ) we have This proves (ii) with C X,weak d M X →X and (2.4).The assertion follows.
3. Rubio de Francia extrapolation from weighted BMO Theorem 3.1.Let T be an operator in L 0 (R d ).Suppose there is an increasing function φ : [1, ∞) → (0, ∞) such that for all weights w −1 ∈ A 1 and all compactly supported Then, for all Banach function spaces X over R d for which M is bounded on X and X , and all compactly supported functions f ∈ X we have where C X is the constant from Theorem 2.8.If T is a (sub)linear operator and X is order-continuous, then T extends to a bounded operator X → X satisfying

An analogous assertion holds when you replace BMO
We need the following lemma which is based on the Rubio de Francia algorithm.
Lemma Let X be a Banach function space over R d for which M : X → X.Then for all f ∈ X, g ∈ X there exists a weight w −1 ∈ A 1 such that where we have recursively defined This proves (3.5), as desired.
Proof of Theorem 3.1.Let f ∈ X have compact support and let g ∈ X with g X = 1.By Lemma 3.2 we can pick a weight Hence, by Proposition 2.4 we have Hence, since f has compact support and thus proving the first result.
For the next assertion, since X is order continuous the functions in X of compact support are dense in X.Indeed, given f ∈ X, set as desired.Thus, the result follows from the fact that if T is (sub)linear, then it is is uniformly continuous and hence, extends to all of X with the same bound.The assertion for BMO ∞ w (R d ) replaced by BMO ∞,weak w (R d ) and C X replaced by C X,weak is proved analogously.
Remark 3.3.It was shown in [CPRR19] that the initial estimate (3.1) implies that for p ∈ (1, ∞) and w ∈ A p we have We recover this result in (3.3), since if X = L p (R d ; w), then X = L p (R d ; w 1−p ).As a matter of fact, we improve on this estimate in (3.2).Indeed, for X = L p (w; R d ) we obtain where X is the constant appearing in the Fefferman-Stein inequality is bounded if and only if w ∈ A p , the constant C X is bounded under much more general conditions.For example, it is bounded when w ∈ A ∞ .

Weighted BMO estimates for singular integrals
We will begin this section discussing the action of sparse operators on L ∞ w .These serve as a way to gain intuition for the more complicated Calderón-Zygmund operators.We note, however, that unlike in the classical L p case, estimates for sparse operators do not immediately imply bounds for Calderón-Zygmund operators because BMO is not a lattice.4.1.Sparse operators.For η ∈ (0, 1), a collection of sets S in R d is called η-sparse if there exists a pairwise disjoint collection of sets {E S } S∈S such that E S ⊆ S and |E S | ≥ η|S| for all S ∈ S. For such a collection we define the sparse operator associated to S as In general, these operators do not map L ∞ to BM O.There are several obstacles that prevent this, which we now describe.
For any family S of sets in R d , let h S be its height function: First we note that A S f may not even be locally integrable for f ∈ L ∞ (R d ).In fact, the situation is much worse.Indeed, consider the sets In general, we need some boundedness of the maximal function associated to S for A S to behave well.In particular, if S consists of intervals (or generally cubes in any dimension), then A S is bounded on L 2 (R d ), so we have no problems with the local integrability of A S (f ).But even this is not enough to reasonably define A S as an operator from L ∞ (R d ) to BMO(R d ).
In general, we need S to have finite total measure, at least qualitatively.However, this is also not enough because there may still be issues similar to those that appear when comparing BMO and dyadic BMO.Indeed, consider This family is 1 2 -sparse and has finite total measure, thus A S (f ) is integrable for all f ∈ L ∞ (R d ).However, letting When the sparse collection consists of cubes belonging to the same dyadic family D, then we can obtain BMO(D) estimates.
For (4.3) we note that since , as desired.The inequality (4.4) then follows from (2.2). 4.2.Calderón-Zygmund operators.We now turn our study to BMO bounds for Calderón-Zygmund operators.In particular, we consider operators T which are weak-type (1, 1)bounded operator and are given by We will furthermore suppose that K satisfies the following smoothness condition: for all x, y, z satisfying |x − y| > 2|x − z| > 0, and where Ω : [0, ∞) → [0, ∞) is an increasing subadditive function with Ω(0) = 0. We can quantify the smoothness of the kernel in terms of the Dini condition of Ω: We be interested in L ∞ w → BM O w estimates for T , so we should be careful about whether T f is well-defined.Indeed, if f ∈ L ∞ w (R d ) has compact support, and w ), so we can use the representation (4.5).
To study the L ∞ → BMO boundedness of such operators we will employ the following pointwise domination theorem.Originally, a similar theorem was proved by A. Lerner in [Ler10a], but we will use the following version by T. Hytönen: Theorem 4.2 ([Hyt14, Theorem 2.3]).For any measurable function f on a cube Q 0 in R d there exists a sparse collection S ⊆ Q 0 such that The idea is to use this theorem, but applied to T f .One can estimate ω λ (T f ; Q) in terms of averages of f , in particular we have the following estimate which goes back to [JT85]: This, together with Theorem 4.2, yields the following: for every compatly supported function f ∈ L 1 (R d ) and every cube Q 0 there exists a sparse collection S of cubes in Q (4.7) We will be able to prove certain weighted estimates for such operators, but unless w −1 ∈ A 1 the estimates will depend on the interaction between Ω and the weight.We will quantify this interaction with a variant of the B 2 condition where the supremum is taken over all intervals I and c I is the center of I.
This class was introduced originally in [HMW73], and then in [MW76] where it was part of the characterization of the L ∞ w → BM O 1 w boundedness of the Hilbert transform.In our situation we need to adapt this condition to incorporate the Dini modulus: where the supremum is taken over all cubes Q.Note that in dimension one and when Ω(x) = x this reduces to the B 2 condition.We can prove a result analogous to Lemma 1 from [HMW73] which gives sufficient conditions for a weight to belong to B(Ω).In particular, if w ∈ A 1 then Proof. a cube Q, and set S m = {x :  Theorem 4.4.Let w −1 ∈ L 1 loc (R d ) and let T be an operator as above, then for all functions f ∈ L ∞ w (R d ) with compact support.Proof.Take f ∈ L ∞ with compact support and fix a cube Q 0 .Combining (4.6) and (4.7) we have for some sparse collection S ⊂ D(Q 0 ).
Assume without loss of generality that f w ≤ 1, then