New York Journal of MathematicsNew York J. Math. 24 (2018) 21–42.

Off-diagonal sharp two-weight estimatesfor sparse operators

Stephan Fackler and Tuomas P. Hytonen

Abstract. For a class of sparse operators including majorants of sin-gular integral, square function, and fractional integral operators in auniform manner, we prove off-diagonal two-weight estimates of mixedtype in the two-weight and A∞-characteristics. These bounds are knownto be sharp in many cases; as a new result, we prove their sharpness forfractional square functions.

Contents

1. Introduction 21

Acknowledgement 24

2. Preliminaries 24

3. Testing constant type estimates for the operator norm 26

4. Sharp estimates for the testing constants 29

5. The fractional square function 37

References 40

1. Introduction

We prove two-weight Lpσ → Lqω estimates for sparse operators

(1.1) Ar,αS (f) =

(∑Q∈S

(|Q|−α

∫Qf

)r1Q

)1/r

,

Received November 22, 2017.2010 Mathematics Subject Classification. Primary 42B20. Secondary 42B25, 47G10,

47G40.Key words and phrases. Ap-A∞ estimates, sparse operators, off-diagonal estimates.The first author was supported by the DFG grant AR 134/4-1 “Regularitat evolu-

tionarer Probleme mittels Harmonischer Analyse und Operatortheorie”. The second au-thor was supported by the ERC Starting Grant “Analytic-probabilistic methods for bor-derline singular integrals” (grant agreement No. 278558), and by the Academy of Finlandvia the Finnish Centre of Excellence in Analysis and Dynamics Research (project Nos.271983 and 307333). Parts of the work were done during a research stay of the first authorat the University of Helsinki. He thanks the members of the harmonic analysis group fortheir hospitality.

ISSN 1076-9803/2018

21

22 STEPHAN FACKLER AND TUOMAS P. HYTONEN

where S is any sparse collection of dyadic cubes as defined below. By nowit is known that such Ar,αS dominate many classical operators T in the senseof pointwise estimates of the type

(1.2) |Tf(x)| .N∑j=1

Ar,αSj |f | (x),

where the collections Sj depend on the function f , but the implied constantsdo not, and so the norm estimates for Ar,αS , uniform over the sparse collectionS, imply similar estimates for the corresponding T .

Our main result reads as follows:

Theorem 1.1. Let 1 < p ≤ q < ∞, 0 < r < ∞, and 0 < α ≤ 1. Letω, σ ∈ A∞ be two weights. Then Ar,αS (·σ) maps Lpσ → Lqω if and only if thetwo-weight Aαpq-characteristic

[ω, σ]Aαpq(S) := supQ∈S|Q|−α ω(Q)1/qσ(Q)1/p′

is finite, and in this case

1 ≤

∥∥Ar,αS (·σ)∥∥Lpσ→Lqω

[ω, σ]Aαpq(S)(1.3)

.

[σ]1q

A∞+ [ω]

( 1r− 1p

)+

A∞, unless p = q > r and α < 1,

[ω]1r

(1− rp

)2

A∞[σ]

1r

(1−(1− rp

)2)

A∞+ [ω]

1r

(1−( rp

)2)

A∞[σ]

1r

( rp

)2

A∞,

where x+ := max(x, 0) in the exponent.

The necessity of the Aαpq-condition, and the lower bound in (1.3), followssimply by substituting f = 1Q for any Q ∈ S and estimating

Ar,αS (fσ) ≥ Ar,αQ(fσ) = |Q|−α σ(Q)1Q,

so the main point of the theorem is the other estimate.Theorem 1.1 includes several known cases: (The “Sobolev” case

1/p− 1/q = 1− α

of these results, together with multilinear extensions, can also be recoveredfrom the recent general framework of [ZK16].)

• For r = α = 1, (1.2) holds for all Calderon–Zygmund operators.The most general version is due to [Lac17], with a simplified proofin [Ler16], but its variants go back to [Ler13]. The bound (1.3) inthis case was obtained in [HP13] for p = q = 2 and in [HLa12] forgeneral p = q ∈ (1,∞). These improved the A2 theorem of [H12]by replacing a part of the A2 or Ap constant by the smaller A∞constant.

TWO-WEIGHT ESTIMATES FOR SPARSE OPERATORS 23

• For r = 2 and α = 1, (1.2) holds for several square function operatorsof Littlewood–Paley type [Ler11]. For p = q, the mixed bound (1.3),even for general r > 0, is from [LacL16]; this improves the pure Apbound of [Ler11].• For r = 1 and 0 < α < 1, (1.2) holds for the fractional integral

operator

Iγf(x) =

∫Rd

f(y)

|x− y|n−γdy,

when α = 1 − γ/n [LacMPT10]. When also p < q, (1.3) is due to[CrUM13a]. The “Sobolev” case with 1/p − 1/q = γ/n = 1 − α ∈(0, 1) was obtained by the same authors in [CrUM13b], elaboratingon the pure Apq bound of [LacMPT10]. Additional complicationswith p = q, which lead to the weaker version of our bound (1.3),have been observed and addressed in different ways in [CrUM13a,CrUM13b]; see also the discussion in [CrU17, Section 7].• For 0 < r < 1 and α = 1, the operators (1.1) can be related to

certain “rough” singular integral operators. Namely, several recentworks starting with [BFP16] have established sparse domination forever bigger classes of operators T in the form

|〈Tf, g〉| .∑Q∈S

(|Q|−1

∫Q|f |s)1/s(

|Q|−1∫Q|g|t)1/t

|Q| ,

for some s, t ≥ 1; this is weaker than (1.2), but still very powerfulfor many purposes. For s > 1 = t, the above domination can bewritten as

(1.4) |〈Tf, g〉| .⟨(

A1/s,1S |f |s

)1/s, |g|⟩,

reducing Lp bounds for T to Lp/s bounds for Ar,1S , where r = 1/s ∈(0, 1). In particular, [CoACDPO17] have proved the bound (1.4)when T = TΩ is a homogeneous singular integral with symbol Ω ∈L∞0 (Sn−1); in this case one can take any s > 1 with implied constants′ = s/(s− 1) in (1.4). Thus

‖TΩ‖Lpω→Lpω . s′ supS‖A1/s,1S (·σp/s)‖

1/s

Lp/sσp/s→Lp/sω

,

where σp/s = ω1−(p/s)′ is the Lp/s dual weight of ω. From the sharpreverse Holder inequality for A∞ weights [HP13] one checks, for

s = 1 + εd/[σ]A∞ ,

24 STEPHAN FACKLER AND TUOMAS P. HYTONEN

that [σp/s]A∞ . [σp]A∞ and [ω, σp/s]A1p/s,p/s

. [ω, σp]sA1pp

= [ω]s/pAp

.

With a similar estimate for the adjoint T ∗Ω : Lp′σp → Lp

′σp , this repro-

duces the bound

‖TΩ‖Lpω→Lpω . [ω]1/pAp

([ω]

1/p′

A∞+ [σp]

1/pA∞

)min([ω]A∞ , [σp]A∞) . [ω]p

′

Ap

from [LiPRR17], an elaboration of earlier results for the same oper-ators by [CoACDPO17, HRT17]. (Since we only reconsider knownresults here, we leave the details for the reader.)

Certainly, there are also important developments not covered by Theo-rem 1.1. We have used the A∞ assumption on the weights to bootstrap thegenerally insufficient two-weight condition [ω, σ]Aαpq(S) < ∞. Other promi-

nent assumptions found in the literature include:

• Orlicz norm bumps, studied in [CrUMP12, Ler13, CrUM13a],[CrUM13b, CrU17] and others, with early history in [Neu83, Per94a,Per94b];• testing conditions, pioneered in [Saw82, Saw88], with a recent cul-

mination in [LacSaSUT14, Lac14]; and• entropy bumps, introduced in [TV16] and studied in[LacSp15, RS16].

However, an in-depth discussion of any of these topics would take us too farafield. (We will use certain testing conditions as a tool, though.)

While some parameter combinations seem to be new in Theorem 1.1,we do not insist too much on this. Our main contribution is the unifiedapproach that covers all these cases at once, without being much longer thanthe proofs of the existing special cases. On the level of technical details, webuild on the previous approach of [HLi15] to the case p = q and α = 1. Inthe particular case of fractional square functions (corresponding to r = 2and α ∈ (0, 1)) we also prove, in Section 5, the sharpness of our estimate for1/p−1/q = 1−α. The sharpness seems to be new for this class of operators,although the bound itself was already obtained in [ZK16].

In the next section we introduce the necessary background material. Af-terwards we prove step by step sharp weighted estimates for sparse operatorsand conclude with the sharpness in the fractional square function case in thefinal section.

Acknowledgement. We would like to thank Pavel Zorin-Kranich and ananonymous referee for their friendly suggestions that eliminated several se-rious omissions in our original overview of related works.

2. Preliminaries

The definition in (1.1) is given for sparse families S of dyadic cubes. Letus be precise about these notions. The standard dyadic grid in Rn is thecollection D of cubes 2−j([0, 1)n + m) : j ∈ Z,m ∈ Zn. For us, a dyadic

TWO-WEIGHT ESTIMATES FOR SPARSE OPERATORS 25

grid is any family of cubes with similar nestedness and covering properties.Such systems of cubes may be parametrised by (ωk)k∈Z ∈ (0, 1n)Z as

Dω :=

Q+∑

j:2−j<`(Q)

2−jωj : Q ∈ D

,

but we will not need to make use of this explicit representation.

Definition 2.1. A collection S of dyadic cubes in Rn is called sparse if forsome η > 0 there exist pairwise disjoint (EQ)Q∈S such that for every Q ∈ Sthe set EQ is a measurable subset of Q with |EQ| ≥ η |Q|.

A weight ω on Rn is a locally integrable function ω : Rn → R≥0. The classof all A∞-weights consists of all weights ω for which their A∞-characteristic

[ω]A∞(Rn) := supQ

1

ω(Q)

∫QM(1Qω)

is finite, where (Mf)(x) := supQ3x |Q|−1∫Q |f | is the Hardy–Littlewood

maximal function and where both suprema run over cubes of positive andfinite diameter whose sides are parallel to the coordinate axes.

We introduce some convenient notation. For a positive Borel measureσ : B(Rn)→ R≥0 with σ(O) > 0 for all nonempty open subsets O ⊂ Rn anda locally integrable function f : Rn → R we use the abbreviation

〈f〉σQ = σ(Q)−1

∫Qf dσ

for the mean of f over Q with respect to σ.We conclude this section with some remarks about our notion of Aαpq:

Remark 2.2. The usual two-weight Ap-characteristic is defined as

[ω, σ]Ap = supQ|Q|−p ω(Q)σ(Q)p−1.

Hence, the relationship to the characteristic used in Theorem 1.1 is

[ω, σ]A1pp

= [ω, σ]1/pAp.

Only a limited range of parameters contains nontrivial pairs of weights:

Remark 2.3. The class of weights ω, σ : Rn → R>0 satisfying [ω, σ]Aαpq <∞is only nonempty if −α + 1

q + 1p′ ≥ 0. In the diagonal case p = q, which

maximizes the left hand side because of our standing assumption p ≤ q, thisholds if and only if α ≤ 1. Hence, we only get examples for α ∈ (0, 1] and,if α > 1

p′ , for q ∈ [p, pp(α−1)+1 ]. In particular, for α = 1 we necessarily are in

the diagonal case p = q. All this follows from the Lebesgue differentiation

26 STEPHAN FACKLER AND TUOMAS P. HYTONEN

theorem by considering cubes centered at and shrinking to a fixed point inRn and the following identity:

|Q|−α(∫

Qω

)1/q(∫Qσ

)1/p′

= |Q|−α+ 1q

+ 1p′

(1

|Q|

∫Qω

)1/q( 1

|Q|

∫Qσ

)1/p′

.

Note that [1,1]Aαpq < ∞ if and only if −α + 1q + 1

p′ = 0, or equivalently,

α = 1 + 1q −

1p . Further, in the case −α + 1

q + 1p′ > 0 one can verify that

there exist (β, γ) ∈ (0, n) × (0, n) such that ω(t) = |t|−β and σ(t) = |t|−γsatisfy [ω, σ]Aαpq <∞.

3. Testing constant type estimates for the operator norm

As a first central step we prove the following estimate for the operatornorm of Ar,αS between two weighted Lp-spaces in terms of two testing con-stants.

Proposition 3.1. Let 1 < p ≤ q < ∞, r ∈ (0,∞), α ∈ (0, 1], S a sparsecollection of dyadic cubes and let ω, σ : Rn → R≥0 be weights. Define thetesting constants

T = supR∈S

σ(R)−r/p

∥∥∥∥∥ ∑Q∈S:Q⊂R

|Q|−αr σ(Q)r1Q

∥∥∥∥∥Lq/rω

,

T ∗ = supR∈S

ω(R)−1/(q/r)′

∥∥∥∥∥ ∑Q∈S:Q⊂R

|Q|−αr σ(Q)r−1ω(Q)1Q

∥∥∥∥∥L(p/r)′σ

.

Then ∥∥Ar,αS (σ·)∥∥rLpσ→Lqω

.

T + T ∗ if r < p,

T if r ≥ p.

As preparatory steps for the proof of the above result we show someintermediate results. In a first step we reduce the norm estimate for Ar,αS toan estimate for a linear operator. This reduction does not make use of theconcrete form of Ar,αS . We therefore formulate it in a more general setting.

Lemma 3.2. Let 1 < r < p ≤ q < ∞, C a collection of dyadic cubes andω, σ : Rn → R≥0 be weights. Then for nonnegative real numbers (cQ)Q∈C,the expressions

I := supf≥0,‖f‖

Lpσ≤1

∥∥∥∥∥∑Q∈C

cQ

(∫Qfσ

)r1Q

∥∥∥∥∥Lq/rω

,

II := supg≥0,‖g‖

Lp/rσ≤1

∥∥∥∥∥∑Q∈C

cQσ(Q)r−1 |Q| 〈gσ〉Q1Q

∥∥∥∥∥Lq/rω

are comparable with constants independent of the choice of (cQ), ω and σ.

TWO-WEIGHT ESTIMATES FOR SPARSE OPERATORS 27

Proof. This follows from the estimates(∫Qfσ

)r≤ σ(Q)r−1

∫Qf rσ

= σ(Q)r−1 |Q| 〈f rσ〉Q

= σ(Q)r1

σ(Q)

∫Qf rσ

≤ σ(Q)r(

infx∈Q

Mσ,rf

)r, Mσ,rf := (Mσ |f |r)1/r,

≤(∫

Q(Mσ,rf)σ

)r,

observing that ‖f r‖1/rLp/rσ

= ‖f‖Lpσ h ‖Mσ,rf‖Lpσ when p > r.

For the complement parameter range for r, a second argument for thedomination of the operator norm by the testing constants is needed. Thiscan again be proven in a more general context.

Lemma 3.3. Let 1 < p ≤ q <∞, r ∈ [p,∞), S a sparse collection of dyadiccubes and let ω, σ : Rn → R≥0 be weights. For nonnegative real numbers(cQ)Q∈S and measurable f : Rn → R≥0 we have∥∥∥∥∥∑

Q∈ScQ

(∫Qfσ

)r1Q

∥∥∥∥∥Lq/rω

. supR∈S

σ(R)−r/p

∥∥∥∥∥ ∑Q∈S:Q⊂R

cQσ(Q)r1Q

∥∥∥∥∥Lq/rω

· ‖f‖rLpσ .

Proof. It suffices to show the estimate for sparse collections S of cubeswhose side lengths are bounded from above by a fixed constant. In thefollowing we use the principal cubes associated to f and σ: consider F =∪∞k=0Fk for F0 = maximal cubes in S and Fk+1 := ∪F∈Fk chF (F ), wherechF (F ) := S 3 Q ( F maximal with 〈f〉σQ > 2〈f〉σF . Further, for Q ∈ Swe write π(Q) for the minimal cube in F containing Q. Using the principalcubes, we obtain∥∥∥∥∥

(∑Q∈S

cQ

(∫Qfσ

)r1Q

)1/r∥∥∥∥∥Lqω

=

∥∥∥∥∥(∑Q∈S

cQ(〈f〉σQ)rσ(Q)r1Q

)1/r∥∥∥∥∥Lqω

≤ 2

∥∥∥∥∥(∑F∈F

(〈f〉σF )r∑

Q:π(Q)=F

σ(Q)rcQ1Q

)1/r∥∥∥∥∥Lqω

.

28 STEPHAN FACKLER AND TUOMAS P. HYTONEN

Here we use that for F = π(Q) one has 〈f〉σQ ≤ 2〈f〉σF . In fact, if 〈f〉σQ >

2〈f〉σF would hold, then F ) F ′ ⊇ Q for some F ′ ∈ chF (F ) by the maxi-mality property of chF (F ). But then F ′ ∈ F satisfies F ′ ⊇ Q and is strictlysmaller than F , which contradicts the minimality of F = π(Q). Because ofp ≤ r the norm on the right hand side is dominated by∥∥∥∥∥

(∑F∈F

(〈f〉σF )p

( ∑Q:π(Q)=F

σ(Q)rcQ1Q

)p/r)1/p∥∥∥∥∥Lqω

=

∥∥∥∥∥∑F∈F

(〈f〉σF )p

( ∑Q:π(Q)=F

σ(Q)rcQ1Q

)p/r∥∥∥∥∥1/p

Lq/pω

≤

(∑F∈F

(〈f〉σF )p

∥∥∥∥∥( ∑Q:π(Q)=F

σ(Q)rcQ1Q

)1/r∥∥∥∥∥p

Lqω

)1/p

.

Note that in the last step we can use the triangle inequality because ofthe assumption p ≤ q. Furthermore, the last expression to the power r isdominated by

supR∈S

σ(R)−r/p

∥∥∥∥∥ ∑Q∈S:Q⊂R

cQσ(Q)r1Q

∥∥∥∥∥Lq/rω

·

(∑F∈F

(〈f〉σF )pσ(F )

)r/p.

We estimate the last factor. Let x ∈ Rn be fixed. If x lies in some cube inF0, we let Q0 be the unique cube in F0 that contains x. Further, if x liesin some cube in chF (Q0), we let Q1 be the unique cube in chF (Q0) withx ∈ Q1. Continuing inductively, we obtain a possibly finite or even emptychain of cubes Q0, Q1, . . . , each of them containing x. Let N ∈ N withx ∈ QN . Then

N∑k=0

(〈f〉σQk)p ≤N∑k=0

2−Np(〈f〉σQN )p . (Mσf)p(x).

Since x ∈ Rn and N is arbitrary, we get the estimate∑F∈F

(〈f〉σF )p1F . (Mσf)p.

In particular, we have∑F∈F

(〈f〉σF )pσ(F ) =

∫Rn

∑F∈F

(〈f〉σF )p1F dσ .∫Rn

(Mσf)p dσ . ‖f‖pLp(σ) .

As a central blackbox result we need the following norm characterizationproved by Lacey, Sawyer and Uriarte-Tuero [LacSaU09, Theorem 1.11].

TWO-WEIGHT ESTIMATES FOR SPARSE OPERATORS 29

Lemma 3.4. Let 1 < p ≤ q < ∞ and let ω, σ : Rn → R≥0 be weights. Fora collection of dyadic cubes D and nonnegative real (τQ)Q∈D consider

T (f) =∑Q∈D

τQ〈f〉Q1Q.

Then

‖T (·σ)‖Lpσ→Lqω ' supR∈D

ω(R)−1/q′

∥∥∥∥∥ ∑Q∈D:Q⊂R

τQ〈ω〉Q1Q

∥∥∥∥∥Lp′σ

+ supR∈D

σ(R)−1/p

∥∥∥∥∥ ∑Q∈D:Q⊂R

τQ〈σ〉Q1Q

∥∥∥∥∥Lqω

.

Our preparations for the proof of Proposition 3.1 are now completed.

Proof of Proposition 3.1. Let us first consider the case r < p ≤ q < ∞.We apply Lemma 3.2 for the choice cQ = |Q|−αr which reduces the estimate

to an estimate for T (·σ) : Lp/rσ → L

q/rω , where

T : f 7→∑Q∈S|Q|1−αr σ(Q)r−1〈g〉Q1Q.

Now, with the choice τQ = |Q|1−αr σ(Q)r−1 we are in the setting of Lem-ma 3.4. Putting everything together, we get∥∥Ar,αS (·σ)

∥∥Lpσ→Lqω

' supR∈S

ω(R)−1/(q/r)′

∥∥∥∥∥ ∑Q∈S:Q⊂R

|Q|1−αr σ(Q)r−1 |Q|−1 ω(Q)1Q

∥∥∥∥∥L(p/r)′σ

+ supR∈S

σ(R)−r/p

∥∥∥∥∥ ∑Q∈S:Q⊂R

|Q|1−αr σ(Q)r−1 |Q|−1 σ(Q)1Q

∥∥∥∥∥Lq/rω

.

Secondly, the case r ≥ p follows from Lemma 3.3 with cQ = |Q|−αr.

4. Sharp estimates for the testing constants

In the last section we saw that it suffices to control the size of the asso-ciated testing constants T and T ∗ instead of the operator norm of Ar,αS . Inthis section we establish sharp estimates for these constants. We again needsome preparatory steps before coming to the main result. We borrow thefollowing lemma from [CaOV04, Proposition 2.2].

Lemma 4.1. Let p ∈ (1,∞) and σ : B(Rn) → R≥0 be a positive Borelmeasure with σ(O) > 0 for all nonempty open O ⊂ Rn. For a collection ofdyadic cubes D and nonnegative (αQ)Q∈D set

ϕ =∑Q∈D

αQ1Q, ϕQ =∑

Q′∈D:Q′⊂QαQ′1Q′ .

30 STEPHAN FACKLER AND TUOMAS P. HYTONEN

Then

‖ϕ‖Lpσ '

(∑Q∈D

αQ(〈ϕQ〉σQ)p−1σ(Q)

)1/p

.

The first part of the following lemma can be deduced from the specialcase shown in [H14, Lemma 5.2]. However, we prefer to give a direct proof.

Lemma 4.2. Let ω, σ : Rn → R≥0 be weights and α, β, γ ≥ 0 be numberswith α+β+γ ≥ 1. For a sparse family S of cubes and a cube R the followingholds.

(1) For α > 0 one has the universal estimate∑Q∈S:Q⊂R

|Q|α σ(Q)βω(Q)γ . |R|α σ(R)βω(R)γ .

(2) For α = 0 one still has the weaker inequality∑Q∈S:Q⊂R

σ(Q)βω(Q)γ . [σ]βA∞ [ω]γA∞σ(R)βω(R)γ .

Proof. For both parts it suffices to treat the case α+ β + γ = 1. In fact, ifα + β + γ ≥ 1, let δ = (α + β + γ)−1 ≤ 1. Then, for example for part (1),one has ∑

Q∈S:Q⊂R|Q|α σ(Q)βω(Q)γ ≤

( ∑Q∈S:Q⊂R

|Q|δα σ(Q)δβω(Q)δγ

)1/δ

.(|R|αδ σ(R)βδω(R)γδ

)1/δ= |R|α σ(R)βω(R)γ .

Now let α+ β+ γ = 1. Let us start with the proof of part (1). We have theestimate∑Q∈S:Q⊂R

|Q|α σ(Q)βω(Q)γ =∑

Q∈S:Q⊂R|Q|α+β+γ

(1

|Q|

∫Qσ

)β( 1

|Q|

∫Qω

)γ≤

∑Q∈S:Q⊂R

|Q| infQM(σ1R)β inf

QM(ω1R)γ .

Since β + γ < 1, there exists some p ∈ (1,∞) with pβ < 1 and p′γ < 1.Using Holder’s inequality, the above term is dominated by( ∑

Q∈S:Q⊂R|Q| inf

QM(σ1R)pβ

)1/p( ∑Q∈S:Q⊂R

|Q| infQM(ω1R)p

′γ

)1/p′

.

We now estimate the first term in brackets. Using the sparseness of S and

‖f‖L1,∞ ' sup|E|∈(0,∞) |E|1− 1

pβ (∫E |f |

pβ)1/(pβ), this term is dominated up to

TWO-WEIGHT ESTIMATES FOR SPARSE OPERATORS 31

a universal constant by( ∑Q∈S:Q⊂R

|E(Q)| infQM(σ1R)pβ

)1/p

≤(∫

RM(σ1R)pβ

)1/p

. ‖M(σ1R)‖βL1,∞ |R|1/p−β

. ‖σ1R‖βL1 |R|1/p−β

= σ(R)β |R|1/p−β .

Putting all the estimates together, we obtain the asserted inequality. Forpart (2) observe that by Holder’s inequality

∑Q∈S:Q⊂R

σ(Q)βω(Q)γ ≤

( ∑Q∈S:Q⊂R

σ(Q)

)β( ∑Q∈S:Q⊂R

ω(Q)

)γ. [σ]βA∞ [ω]γA∞σ(R)βω(R)γ .

The inequalities used in the last estimate can be seen as follows (for ω):∑Q∈S:Q⊂R

∫Qω ≤

∑Q∈S:Q⊂R

|Q| infE(Q)

M(1Qω)

.∫RM(1Qω) ≤ [ω]A∞ω(R).

The next proposition is the heart of the article. Here we prove the re-quired estimates on the testing constants T and T ∗. Note the emergence ofadditional factors in the diagonal case for fractional sparse operators.

Theorem 4.3. Let ω, σ : Rn → R≥0 be A∞-weights, S a sparse family ofdyadic cubes, r ∈ (0,∞) and α ∈ (0, 1]. For

1 < p ≤ q <∞ with − α+1

q+

1

p′≥ 0

one has

T .

[ω, σ]rAαpq [σ]1−(1−r/p)2A∞

[ω](1−r/p)2A∞

if p = q and α < 1 and p > r,

[ω, σ]rAαpq [σ]r/qA∞

else.

Further, if p > r, the second testing constant satisfies

T ∗ .

[ω, σ]rAαpq [ω]1−(r/p)2

A∞[σ]

(r/p)2

A∞if p = q and α < 1,

[ω, σ]rAαpq [ω]1−r/pA∞

else.

Proof. Throughout the proof, we write [ω, σ] := [ω, σ]Aαpq for brevity.

32 STEPHAN FACKLER AND TUOMAS P. HYTONEN

Part I. We begin with the estimate for the testing constant T . Here westart with the special case q > r. By Lemma 4.1 and q

r > 1 we rewrite thenorm as ∥∥∥∥∥ ∑

Q∈S:Q⊂R|Q|−αr σ(Q)r1Q

∥∥∥∥∥Lq/rω

'

( ∑Q∈S:Q⊂R

|Q|−αr σ(Q)rω(Q)

·

[ω(Q)−1

∑Q′∈S:Q′⊂Q

∣∣Q′∣∣−αr σ(Q′)rω(Q′)

]q/r−1)r/q.

As a first step we estimate the inner sum (from now on omitting Q′ ∈ S)∑Q′⊂Q

∣∣Q′∣∣−αr σ(Q′)rω(Q′).

For this let δ > 0 be arbitrary. For each summand we have the estimate∣∣Q′∣∣−αr σ(Q′)rω(Q′)(4.1)

= (∣∣Q′∣∣−α σ(Q′)1/p′ω(Q′)1/q)δ

∣∣Q′∣∣α(δ−r)σ(Q′)r−δ/p

′ω(Q′)1−δ/q

≤ [ω, σ]δ∣∣Q′∣∣α(δ−r)

σ(Q′)r−δ/p′ω(Q′)1−δ/q.

We want to use Lemma 4.2. Its assumptions imply restrictions on the pos-sible range of δ. In fact, the following four conditions must be satisfied:

α(δ − r) ≥ 0⇔ δ ≥ r(4.2)

r − δ

p′≥ 0⇔ δ ≤ rp′

1− δ

q≥ 0⇔ δ ≤ q

α(δ − r) + r − δ

p′+ 1− δ

q≥ 1⇔ δ

(α− 1

p′− 1

q

)+ r(1− α) ≥ 0.

The first three conditions imply that δ ∈ [r,min(rp′, q)] which has nonemptyinterior by our assumption q > r. The restriction imposed by the lastinequality depends on the parameters. First, if α − 1

p′ −1q = 0, the last

condition holds because of α ≤ 1. Secondly, if α < 1p′ + 1

q (this implies

α < 1), we have

δ ≤ r 1− α1p′ + 1

q − α.

Note that because of 1p′ + 1

q ≤ 1 the fraction on the right hand side is bigger

or equal to 1. Hence, if q > p, we can find a δ with δ > r satisfying allconditions in (4.2). However, in the case p = q the only possible choice is

TWO-WEIGHT ESTIMATES FOR SPARSE OPERATORS 33

δ = r. This is the only case where the strict inequality δ > r cannot beachieved. Summarizing our findings, Lemma 4.2 gives the estimate∑Q′⊂Q

∣∣Q′∣∣−αr σ(Q′)rω(Q′)

. [ω, σ]δ |Q|α(δ−r) σ(Q)r−δ/p′ω(Q)1−δ/q

[σ]

r/pA∞

[ω]1−r/pA∞

, p = q and α < 1,

1, else.

We now show estimates for arbitrary δ > 0 satisfying the inequalities in (4.2).In the following we will ignore the additional factors in the case p = q andα < 1. Using the estimate for the inner sum we have∥∥∥∥∥ ∑Q∈S:Q⊂R

|Q|−αr σ(Q)r1Q

∥∥∥∥∥Lq/rω

. [ω, σ]δ(1−r/q)

( ∑Q∈S:Q⊂R

|Q|−αr σ(Q)rω(Q)

·(ω(Q)−1σ(Q)r−δ/p

′ |Q|α(δ−r) ω(Q)1−δ/q)q/r−1

)r/q

= [ω, σ]δ(1−r/q)

∑Q∈S:Q⊂R

|Q|α(−q+δq/r−δ) ω(Q)1+δ/q−δ/rσ(Q)q−δ/p′(q/r−1)

r/q

.

We pull another power of the two-weight constant out of the sum usingexactly the power for which the sum becomes independent of the weight ω.Explicitly, this gives

|Q|−αq+δαq/r−δα ω(Q)1+δ/q−δ/rσ(Q)q−δ/p′(q/r−1)

=(|Q|−α ω(Q)1/qσ(Q)1/p′

)q·(1+δ(1/q−1/r)) |Q|α(−q+δq/r−δ)+α(q+δ(1−q/r))

· σ(Q)−q/p′·(1+δ(1/q−1/r))+q−δ/p′(q/r−1)

≤ [ω, σ]q(1+δ(1/q−1/r))σ(Q)q/p.

For the last estimate we need δ(

1q −

1r

)≥ −1. For q > r this is equivalent to

δ ≤ 11r −

1q

= r · 1

1− rq

.

The second factor on the right is bigger than 1 and therefore we find asuitable δ satisfying our new restriction together with (4.2). Putting alltogether and using δ

(1 − r

q

)+ r(1 + δ

(1q −

1r

))= r for the power of [ω, σ],

34 STEPHAN FACKLER AND TUOMAS P. HYTONEN

we obtain by Lemma 4.2

∥∥∥∥∥ ∑Q∈S:Q⊂R

|Q|−αr σ(Q)r1Q

∥∥∥∥∥Lq/rω

. [ω, σ]r

( ∑Q∈S:Q⊂R

σ(Q)q/p

)r/q

≤ [ω, σ]r

(σ(R)q/p−1

∑Q∈S:Q⊂R

σ(Q)

)r/q

. [ω, σ]r(σ(R)q/p−1[σ]A∞σ(R)

)r/q≤ [ω, σ]r[σ]

r/qA∞

σ(R)r/p.

This finishes the estimate for T in the case q > r. We let Tr denote thevalue of T for a particular choice of r. For the case q > p or α = 1 nowsuppose that q ≤ r. We choose some 0 < s < q ≤ r. We then have

T 1/rr = sup

R∈Sσ(R)−1/p

∥∥∥∥∥( ∑Q∈S:Q⊂R

|Q|−αr σ(Q)r1Q

)1/r∥∥∥∥∥Lqω

(4.3)

≤ supR∈S

σ(R)−1/p

∥∥∥∥∥( ∑Q∈S:Q⊂R

|Q|−αs σ(Q)s1Q

)1/s∥∥∥∥∥Lqω

≤ T 1/ss . [ω, σ][σ]

1/qA∞

.

Taking both sides of the inequality to the power r gives the desired estimate.Let us come to the very last case, namely q ≤ r, p = q and α < 1. Here westart with the special choice r = q = p. Then by (4.1) for δ = r∥∥∥∥∥ ∑Q∈S:Q⊂R

|Q|−αr σ(Q)1Q

∥∥∥∥∥Lq/rω

=∑

Q∈S:Q⊂R|Q|−αr σ(Q)rω(Q)

≤ [ω, σ]r∑

Q∈S:Q⊂Rσ(Q) . [ω, σ]r[σ]A∞σ(R).

Hence, Tq . [ω, σ]r[σ]A∞ . Now, if q < r, then we choose s = p = q, and thesame reasoning as in (4.3) gives

T 1/rr ≤ T 1/q

q . [ω, σ]Aαpp [σ]1/pA∞

.

Part II. We now come to the estimate for the testing constant T ∗. Recallthat here we are only interested in the case p > r. We write s = (p/r)′. We

TWO-WEIGHT ESTIMATES FOR SPARSE OPERATORS 35

use Lemma 4.1 again to rewrite the involved norm as

∥∥∥∥∥ ∑Q∈S:Q⊂R

|Q|−αr σ(Q)r−1ω(Q)1Q

∥∥∥∥∥L(p/r)′σ

'

( ∑Q∈S:Q⊂R

σ(Q) |Q|−αr σ(Q)r−1ω(Q)

·

[σ(Q)−1

∑Q′⊂Q

∣∣Q′∣∣−αr σ(Q′)r−1ω(Q′)σ(Q′)

]s−1)1/s

=

( ∑Q∈S:Q⊂R

|Q|−αr σ(Q)rω(Q)

[σ(Q)−1

∑Q′⊂Q

∣∣Q′∣∣−αr σ(Q′)rω(Q′)

]s−1)1/s

.

Observe that the inner sum∑

Q′⊂Q |Q′|−αr σ(Q′)rω(Q′) is exactly the same

sum as in the first part of the proof. Hence, exactly the same considerationsand estimates apply here. Again ignoring the additional constants appearingin the case α < 1 and p = q, we get

∥∥∥∥∥ ∑Q∈S:Q⊂R

|Q|−αr σ(Q)r−1ω(Q)1Q

∥∥∥∥∥L(p/r)′σ

. [ω, σ]δ/s′

( ∑Q∈S:Q⊂R

|Q|−αr σ(Q)rω(Q)

·(σ(Q)−1σ(Q)r−δ/p

′ |Q|α(δ−r) ω(Q)1−δ/q)s−1

) 1s

= [ω, σ]δrp

( ∑Q∈S:Q⊂R

|Q|α(−r+(δ−r)(s−1)) σ(Q)r+(r−1−δ/p′)(s−1)

· ω(Q)1+(1−δ/q)(s−1)

) 1s

= [ω, σ]δrp

∑Q∈S:Q⊂R

|Q|α(−rs+δ(s−1)) σ(Q)rs−(1+δ/p′)(s−1)ω(Q)s−δ/q(s−1)

1s

.

36 STEPHAN FACKLER AND TUOMAS P. HYTONEN

In the following paragraph we will verify the following estimate step by step:

|Q|α(−rs+δ(s−1)) σ(Q)rs−(1+δ/p′)(s−1)ω(Q)s−δ/q(s−1)(4.4)

=(|Q|−α σ(Q)1/p′ω(Q)1/q

)p′·(rs−(1+δ/p′)(s−1))

· |Q|α(−rs+δ(s−1))+αp′(rs−(1+δ/p′)(s−1))

· ω(Q)s−δ/q(s−1)−p′/q(rs−(1+δ/p′)(s−1))

. [ω, σ]p′(rs−(1+δ/p′)(s−1))ω(Q)s/q(q−r),

which holds provided we can find δ > 0 for which the power of [ω, σ] isnonnegative. For this consider the following:

rs−(

1 +δ

p′

)(s− 1) ≥ 0 ⇔ r −

(1 +

δ

p′

)r

p≥ 0

⇔ 1 +δ

p′≤ p

⇔ δ ≤ p′(p− 1) = p.

Note that we can find δ > 0 that additionally satisfies the condition δ ≤ pbecause of the assumption p > r. Let us verify the identities for the powersof |Q| and ω(Q) used in (4.4). For the power of |Q| we have

− rs+ δ(s− 1) + p′(rs−

(1 +

δ

p′

)(s− 1)

)= −rs+ p′rs− p′(s− 1)

= s(p′(r − 1)− r) + p′

=p(p(r − 1)− r(p− 1)) + p(p− r)

(p− 1)(p− r)= 0,

whereas for the power of ω(Q) the following calculation shows the claimedidentity

s− p′

q(rs− (s− 1)) = s

(1− p′

q

(r −

(1− 1

s

)))= s

(1− p′

q

(r − 1

s′))

= s

(1− rp′

q

(1− 1

p

))= s

(1− r

q

)= s

(q − rq

).

TWO-WEIGHT ESTIMATES FOR SPARSE OPERATORS 37

Hence, we have as desired∥∥∥∥∥ ∑Q∈S:Q⊂R

|Q|−αr σ(Q)r−1ω(Q)1Q

∥∥∥∥∥L(p/r)′σ

. [ω, σ]δr/p+p′/s·(rs−(1+δ/p′)(s−1))

( ∑Q∈S:Q⊂R

ω(Q)s/(q/r)′

)1/s

= [ω, σ]r

(ω(R)s/(q/r)

′−1∑

Q∈S:Q⊂Rω(Q)

)1/s

. [ω, σ]r(ω(R)s/(q/r)

′−1[ω]A∞ω(R)

)1/s

= [ω, σ]r[ω]1/sA∞

ω(R)1/(q/r)′ .

Let us again verify the power of [ω, σ] explicitly by a small calculation:

δr

p+p′

s

(rs−

(1 +

δ

p′

)(s− 1)

)= δ

r

p− δ(

1− 1

s

)+ p′

(r −

(1− 1

s

))= p′

(r − r

p

)= r.

5. The fractional square function

We now specialize our findings to the case of classical fractional squarefunctions, i.e. α ∈ (0, 1) and r = 2 for the condition α = 1

p′ + 1q . In the

one-weighted theory one here considers estimates for Lpωp → Lqωq . For aweight ω : Rn → R≥0, p, q ∈ (1,∞) and α ∈ (0, 1] one is interested in sharpestimates in the one-weight characteristic

[ω]Aαpq := supQ

(1

|Q|

∫Qωq)(

1

|Q|

∫Qω−p

′)q/p′

.

Its relation to the two-weight characteristic is [ωq, ω−p′]Apq = [ω]

1/qApq

. Hence,

Theorem 1.1 for σ = ω−1/(p−1) gives the following mixed Apq−A∞ estimate.

Corollary 5.1. Let α ∈ (0, 1) and 1 < p ≤ q <∞ with 1q + 1

p′ = α. Then

‖A2,αS ‖Lpωp→Lqωq . [ω]

1q

Apq

([ω−p

′]1q

A∞+ [ωq]

( 12− 1p

)+

A∞

).

One has [ωq]A1+q/p′ = [ω]Apq and [ω−p′]A1+p′/q = [ω]

p′/qApq

. In particular,

ω ∈ Apq implies the finiteness of the above A∞-characteristics. Using thisrelation to the Apq-characteristic we a fortiori obtain the following pureApq-estimate.

38 STEPHAN FACKLER AND TUOMAS P. HYTONEN

Corollary 5.2. Let α ∈ (0, 1) and 1 ≤ p ≤ q <∞ with 1q + 1

p′ = α. Then

‖A2,αS ‖Lpωp→Lqωq . [ω]

1q

Apq

([ω]

p′

q2

Apq+ [ω]

( 12− 1p

)+

Apq

). [ω]

max( p′qα,α− 1

2)

Apq.

This estimate is optimal in the following sense.

Proposition 5.3. Let α ∈ (0, 1) and 1 < p ≤ q < ∞ with 1q + 1

p′ = α. If

Φ: [1,∞)→ R>0 is a monotone function such that for all ω ∈ Apq,

‖A2,αS ‖Lpωp→Lqωq . Φ([ω]Apq)

with an implicit S-dependent constant, then Φ(t) & tmax( p′qα,α− 1

2).

Proof. For k ∈ N0 choose Ik = [0, 2−k]. Then the family S = (Ik)k∈N0

is 12 -sparse. We consider its associated operator A = A2,α

S . Following[LacMPT10, Section 7], for ε ∈ (0, 1) we let

ωε(x) = |x|(1−ε)/p′

and f(x) = |x|ε−11[0,1].

Then [ωε]Apq ' ε−q/p′

and ‖ωεf‖Lp ' ε−1/p. Now, let x ∈ [2−(k+1), 2−k] fork ∈ N0. Then

Af(x) ≥ |Ik|−α∫Ik

|y|ε−1 dy1Ik(x) = 2αk∫ 2−k

0|y|ε−1 dy ' |x|−α · ε−1 |x|ε .

Consequently,∫RAf qωqε ≥

∞∑k=0

∫ 2−k

2−(k+1)

Af qωqε ≥ ε−q∫ 1

0|x|q(ε−α) |x|

qp′ (1−ε) dx

= ε−q∫ 1

0|x|

εp−1dx ' ε−1−q.

This shows that ‖Af‖Lqωqε

& ε−1−1/q, and hence

Φ(ε−q/p′) & ε−1−1/q+1/p = ε−(1/p′+1/q) = ε−α.

This finishes the first part of the estimate.The second upper bound follows from a duality argument: If A = A2,α

Swith S as above is reinterpreted as a vector-valued operator in a natural way,i.e., we have a bounded linear operator A : Lpωp(R) → Lqωq(R; `2), then its

adjoint with respect to the unweighted duality maps Lq′

ω−q′(R; `2) boundedly

into Lp′

ω−p′(R). Applying this adjoint to (aIω

q)I∈S , for a sequence (aI)I∈Sof measurable functions, one has the estimate∥∥∥∥∥∑

I∈S|I|−α

∫IaIω

q1I

∥∥∥∥∥Lp′

ω−p′

. Φ([ω]Apq)

∥∥∥∥∥∥(∑I∈I|aI |2

)1/2∥∥∥∥∥∥Lq′ωq

.

TWO-WEIGHT ESTIMATES FOR SPARSE OPERATORS 39

Since the right hand side is independent of the sign of aI , averaging and theKhintchine inequality give∥∥∥∥∥∥

(∑I∈S

(|I|−α

∫IaIω

q)21I

)1/2∥∥∥∥∥∥Lp′

ω−p′

(5.1)

. Φ([ω]Apq)

∥∥∥∥∥∥(∑I∈S|aI |2

)1/2∥∥∥∥∥∥Lq′ωq

.

Now, for ε ∈ (0, 1) choose ωε(x) = |x|(ε−1)/q. Then

[ω]Apq = [ωq]1+q/p′ ' ε−1.

Further, imitating the argument in [LacSc12, Section 3], we choose

ak(x) = ε1/2 |Ik|−ε |x|ε 1Ik(x).

With this, for x ∈ (2−(l+1), 2−l] and l ∈ N0

∞∑k=0

a2k(x) = ε |x|2ε

∞∑k=0

|Ik|−2ε1[0,2−k](x) = ε |x|2ε

l∑k=0

(22ε)k

= ε |x|2ε 22(l+1)ε − 1

22ε − 1. |x|2ε 22lε . 1.

This directly gives for the right hand side of (5.1)∥∥∥∥∥∥( ∞∑k=0

|ak|2)1/2

∥∥∥∥∥∥Lq′ωq

.

(∫ 1

0|x|ε−1 dx

)1/q′

= ε−1/q′ .

Let us now come to the left hand side of (5.1). First,

|Ik|−α∫Ik

akωq = ε1/22k(α+ε)

∫ 2−k

0|x|2ε−1 =

1

2ε−1/22k(α−ε).

Consequently, for x ∈ [2−(l+1), 2−l), l ∈ N0 and ε ∈ (0, α0] with α0 < α

∞∑k=0

(|Ik|−α

∫Ik

akωq

)2

1Ik(x) =1

4ε−1

l∑k=0

22k(α−ε)

& ε−1(2l)2(α−ε) ' ε−1 |x|−2(α−ε) .

40 STEPHAN FACKLER AND TUOMAS P. HYTONEN

Thus, ∥∥∥∥∥∥(∑I∈S

(|I|−α

∫IaIω

q

)2

1I

)1/2∥∥∥∥∥∥Lp′

ω−p′

& ε−12

(∫ 1

0|x|−p

′(α−ε) |x|−(ε−1) p′q dx

) 1p′

= ε−12

(1− p′

(α− ε+

1

q(ε− 1)

)) 1p′

=

(p′

q′

) 1p′

ε− 1

2− 1p′ .

Hence, one has for sufficiently small ε the estimate

Φ(ε−1) & ε−12− 1p′+

1q′ = ε

12−α,

which gives the desired bound by the monotonicity of Φ.

References

[BFP16] Bernicot, Frederic; Frey, Dorothee; Petermichl, Stefanie.Sharp weighted norm estimates beyond Calderon-Zygmund theory.Anal. PDE 9 (2016), 1079–1113. MR3531367, Zbl 1344.42009,arXiv:1510.00973, doi: 10.2140/apde.2016.9.1079.

[CaOV04] Cascante, Carme; Ortega, Joaquin M.; Verbitsky, Igor E. Non-linear potentials and two weight trace inequalities for general dyadic andradial kernels. Indiana Univ. Math. J. 53 (2004), 845–882. MR2086703(2005g:42053), Zbl 1077.31007, arXiv:math/0309286, http://www.iumj.indiana.edu/oai/2004/53/2443/2443.xml.

[CoACDPO17] Conde-Alonso, Jose M.; Culiuc, Amalia; Di Plinio, Francesco;Ou, Yumeng. A sparse domination principle for rough singular in-tegrals. Anal. PDE 10 (2017), 1255–1284. MR3668591, Zbl 06740746,arXiv:1612.09201, doi: 10.2140/apde.2017.10.1255.

[CrU17] Cruz-Uribe, David. Two weight inequalities for fractional integral op-erators and commutators. Advanced courses of mathematical analysisVI, 25–85. World Sci. Publ., Hackensack, NJ, 2017. MR3642364, Zbl06751414.

[CrUM13a] Cruz-Uribe, David; Moen, Kabe. A fractional Muckenhoupt–Wheedentheorem and its consequences. Integral Equations Operator Theory76 (2013) 421–446. MR3065302, Zbl 1275.42029, arXiv:1303.3424,doi: 10.1007/s00020-013-2059-z.

[CrUM13b] Cruz-Uribe, David; Moen, Kabe. One and two weight norm inequali-ties for Riesz potentials. Illinois J. Math. 57 (2013), 295–323. MR3224572,Zbl 1297.42022, arXiv:1207.5551, https://projecteuclid.org/euclid.ijm/1403534497.

[CrUMP12] Cruz-Uribe, David; Martell, Jose Marıa; Perez, Carlos. Sharpweighted estimates for classical operators. Adv. Math. 229 (2012),408–441. MR2854179 (2012k:42020), Zbl 1236.42010, arXiv:1001.4254,doi: 10.1016/j.aim.2011.08.013.

TWO-WEIGHT ESTIMATES FOR SPARSE OPERATORS 41

[H12] Hytonen, Tuomas P. The sharp weighted bound for gen-eral Calderon–Zygmund operators. Ann. of Math. (2) 175(2012), 1473–1506. MR2912709, Zbl 1250.42036, arXiv:1007.4330,doi: 10.4007/annals.2012.175.3.9.

[H14] Hytonen, Tuomas P. The A2 theorem: remarks and complements.Harmonic analysis and partial differential equations, 91–106. Contemp.Math., 612. Amer. Math. Soc., Providence, RI, 2014. MR3204859, Zbl1354.42022, arXiv:1212.3840.

[HLa12] Hytonen, Tuomas P.; Lacey, Michael T. The Ap-A∞ inequality forgeneral Calderon-Zygmund operators. Indiana Univ. Math. J. 61 (2012),2041–2092. MR3129101, Zbl 1290.42037, arXiv:1106.4797, http://www.

iumj.indiana.edu/oai/2012/61/4777/4777.xml.[HLi15] Hytonen, Tuomas P.; Li, Kangwei. Weak and strong Ap-A∞ estimates

for square functions and related operators. Proc. Amer. Math. Soc., toappear. arXiv:1509.00273.

[HP13] Hytonen, Tuomas P.; Perez, Carlos. Sharp weighted bounds in-volving A∞. Anal. PDE 6 (2013), 777–818. MR3092729, Zbl 1283.42032,arXiv:1103.5562, doi: 10.2140/apde.2013.6.777.

[HRT17] Hytonen, Tuomas P.; Roncal, Luz; Tapiola, Olli. Quanti-tative weighted estimates for rough homogeneous singular integrals.Israel J. Math. 218 (2017), 133–164. MR3625128, Zbl 1368.42015,arXiv:1510.05789, doi: 10.1007/s11856-017-1462-6.

[Lac14] Lacey, Michael T. Two-weight inequality for the Hilbert transform: areal variable characterization, II. Duke Math. J. 163 (2014), 2821–2840.MR3285858, Zbl 1312.42010, arXiv:1301.4663, https://projecteuclid.org/euclid.dmj/1417442573.

[Lac17] Lacey, Michael T. An elementary proof of the A2 bound. Israel J. Math.217 (2017), 181–195. MR3625108, Zbl 1368.42016, arXiv:1501.05818,doi: 10.1007/s11856-017-1442-x.

[LacL16] Lacey, Michael T.; Li, Kangwei. On Ap–A∞ type estimates for squarefunctions. Math. Z. 284 (2016), 1211–1222. MR3563275, Zbl 1370.42015,arXiv:1505.00195, doi: 10.1007/s00209-016-1696-8.

[LacMPT10] Lacey, Michael T.; Moen, Kabe; Perez, Carlos; Torres,Rodolfo H. Sharp weighted bounds for fractional integral operators.J. Funct. Anal. 259 (2010), 1073–1097. MR2652182 (2011h:42023), Zbl1196.42014, arXiv:0905.3839, doi: 10.1016/j.jfa.2010.02.004.

[LacSaSUT14] Lacey, Michael T.; Sawyer, Eric T.; Shen, Chun-Yen; Uriarte-Tuero, Ignacio. Two-weight inequality for the Hilbert transform: areal variable characterization, I. Duke Math. J. 163 (2014), 2795–2820.MR3285857, Zbl 1312.42011, arXiv:1201.4319, https://projecteuclid.org/euclid.dmj/1417442572.

[LacSaU09] Lacey, Michael T.; Sawyer, Eric T.; Uriarte-Tuero, Ignacio.Two weight inequalities for discrete positive operators. arXiv:0911.3437.

[LacSc12] Lacey, Michael T.; Scurry, James. Weighted weak type estimates forsquare functions. arXiv:1211.4219.

[LacSp15] Lacey, Michael T.; Spencer, Scott. On entropy bumps for Calderon–Zygmund operators. Concr. Oper. 2 (2015), 47–52. MR3357767, Zbl1333.42021, arXiv:1504.02888, doi: 10.1515/conop-2015-0003.

[Ler11] Lerner, Andrei K. Sharp weighted norm inequalities for Littlewood–Paley operators and singular integrals. Adv. Math. 226 (2011), 3912–3926. MR2770437 (2012c:42048), Zbl 1226.42010, arXiv:1005.1422,doi: 10.1016/j.aim.2010.11.009.

42 STEPHAN FACKLER AND TUOMAS P. HYTONEN

[Ler13] Lerner, Andrei K. On an estimate of Calderon–Zygmund opera-tors by dyadic positive operators. J. Anal. Math. 121 (2013), 141–161.MR3127380, Zbl 1285.42015, arXiv:1202.1860, doi: 10.1007/s11854-013-0030-1.

[Ler16] Lerner, Andrei K. On pointwise estimates involving sparse operators.New York J. Math. 22 (2016) 341–349. MR3484688, Zbl 1347.42030,arXiv:1512.07247, http://nyjm.albany.edu/j/2016/22-15v.pdf.

[LiPRR17] Li, Kangwei; Perez, Carlos; Rivera-Rıos, Israel P.; Roncal,Luz. Weighted norm inequalities for rough singular integral operators.arXiv:1701.05170.

[Neu83] Neugebauer, C. J. Inserting Ap-weights. Proc. Amer. Math.Soc. 87 (1983), 644–648. MR0687633 (84d:42026), Zbl 0521.42019,doi: 10.2307/2043351.

[Per94a] Perez, Carlos. Weighted norm inequalities for singular integral op-erators. J. London Math. Soc. (2) 49 (1994), 296–308. MR1260114(94m:42037), Zbl 0797.42010, doi: 10.1112/jlms/49.2.296.

[Per94b] Perez, Carlos. Two weighted inequalities for potential and fractionaltype maximal operators. Indiana Univ. Math. J. 43 (1994), 663–683.MR1291534 (95m:42028), Zbl 0809.42007, http://www.iumj.indiana.

edu/oai/1994/43/43028/43028.xml.[RS16] Rahm, Robert; Spencer, Scott. Entropy bump conditions for frac-

tional maximal and integral operators. Concr. Oper. 3 (2016), 112–121.MR3543678, Zbl 1346.42020, doi: 10.1515/conop-2016-0013.

[Saw82] Sawyer, Eric T. A characterization of a two-weight norm inequal-ity for maximal operators. Studia Math. 75 (1982), 1–11. MR0676801(84i:42032), Zbl 0508.42023, doi: 10.4064/sm-75-1-1-11.

[Saw88] Sawyer, Eric T. A characterization of two weight norm inequalities forfractional and Poisson integrals. Trans. Amer. Math. Soc. 308 (1988),533–545. MR0930072 (89d:26009), Zbl 0665.42023, doi: 10.1090/S0002-9947-1988-0930072-6.

[TV16] Treil, Sergei; Volberg, Alexander. Entropy conditions intwo weight inequalities for singular integral operators. Adv. Math.301 (2016), 499–548. MR3539383, Zbl 06620627, arXiv:1408.0385,doi: 10.1016/j.aim.2016.06.021.

[ZK16] Zorin-Kranich, Pavel. Ap–A∞ estimates for multilinear maximal andsparse operators. arXiv:1609.06923.

(Stephan Fackler) Institute of Applied Analysis, Ulm University, Helmholtzstr.18, 89069 Ulm, Germanystephan.fackler@alumni.uni-ulm.de

(Tuomas P. Hytonen) Department of Mathematics and Statistics, P.O.B. 68 (Gus-taf Hallstromin katu 2b), FI-00014 University of Helsinki, Finlandtuomas.hytonen@helsinki.fi

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