ANNALES DE
L’INSTITUT FOURIER
LesAnnales de l’institut Fouriersont membres du
Francesco Di Plinio, Tuomas P. Hytönen & Kangwei Li Sparse bounds for maximal rough singular integrals via the Fourier transform
Tome 70, no5 (2020), p. 1871-1902.
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SPARSE BOUNDS FOR MAXIMAL ROUGH SINGULAR INTEGRALS VIA THE FOURIER TRANSFORM
by Francesco DI PLINIO,
Tuomas P. HYTÖNEN & Kangwei LI (*)
Abstract. —We prove a quantified sparse bound for the maximal truncations of convolution-type singular integrals with suitable Fourier decay of the kernel. Our result extends the sparse domination principle by Conde-Alonso, Culiuc, Ou and the first author to the maximally truncated case, and covers the rough homogeneous singular integralsTΩonRdwith bounded angular part Ω having vanishing integral on the sphere. Among several consequences, we obtain new quantitative weighted norm inequalities for the maximal truncation ofTΩ, extending a result by Roncal, Tapiola and the second author.
A convex-body valued version of the sparse bound is also deduced and employed towards novel matrix-weighted norm inequalities for the maximal truncations of TΩ. Our result is quantitative, but even the qualitative statement is new, and the present approach via sparse domination is the only one currently known for the matrix weighted bounds of this class of operators.
Résumé. —Nous démontrons un contrôle épars qualitatif pour la troncature maximale des noyaux de convolution dont la transformée de Fourier satisfait des conditions de décroissance appropriées. Notre résultat étend le principe de contrôle épars de Conde-Alonso, Culiuc, Ou et du premier auteur au cas de la troncature maximale, et inclut le cas des intégrales singulières homogènes TΩ sur Rd dont la composante angulaire Ω est bornée et a une moyenne nulle. Parmi les diverses conséquences, nous obtenons de nouvelles inégalités quantitatives pondérées pour la troncature maximale de TΩ, l’extension d’un résultat de Roncal, Tapiola et du second auteur. De plus, une extension appropriée aux valeurs vectorielles du contrôle épars implique de nouvelles estimations à poids matriciels pour les tron- catures maximales deTΩ. Notre résultat est quantitatif, mais il est nouveau même d’un point de vue qualitatif. L’approche actuelle basée sur le contrôle épars est la seule actuellement connue pour les estimations à poids matriciels de cette classe d’opérateurs.
Keywords:Sparse domination, rough singular integrals, weighted norm inequalities.
2020Mathematics Subject Classification:42B20, 42B25.
(*) F. Di Plinio was partially supported by the National Science Foundation under the grants NSF-DMS-1650810 and NSF-DMS-1800628. T. Hytönen was supported by the Academy of Finland via project No. 314829 and via the Centre of Excellence in Analysis and Dynamics Research (project No. 307333). K. Li was supported by Juan de la Cierva-Formación 2015 FJCI-2015-24547. Di Plinio and Li are also supported by the Severo Ochoa Program SEV-2013-0323 and by Basque Government BERC Program 2014-2017.
1. Introduction and main results
Let η ∈ (0,1). A countable collection S of cubes of Rd is said to be η-sparse if there exist measurable sets{EI :I∈ S}such that
EI ⊂I,|EI|>η|I|, I, J∈ S, I 6=J =⇒ EI∩EJ=∅. Let T be a sublinear operator mapping the space L∞0 (Rd) of complex- valued, bounded and compactly supported functions on Rd into locally integrable functions. We say thatT has the sparse (p1, p2) bound [10] if there exists a constant C > 0 such that for all f1, f2 ∈ L∞0 (Rd) we may find a 12-sparse collection S=S(f1, f2) such that
|hT f1, f2i |6C X
Q∈S
|Q|
2
Y
j=1
hfjipj,Q
in which case we denote bykTk(p1,p2),sparse the least such constantC. As customary,
hfip,Q= kf1Qkp
|Q|1p , p∈(0,∞].
Estimating the sparse norm(s) of a sublinear or multisublinear operator entails a sharp control over the behavior of such operator in weightedLp- spaces; this theme has been recently pursued by several authors, see for instance [1, 8, 19, 20, 21, 33]. This sharp control is exemplified in the fol- lowing proposition, which is a collection of known facts from the indicated references.
Proposition 1.1. — Let T be a sublinear operator on Rd mapping L∞0 (Rd)toL1loc(Rd). Then the following hold.
(1) [7, Appendix B]Let16p1, p2<∞. There is an absolute constant Cp2 >0such that
kT :Lp1(Rd)→Lp1,∞(Rd)k6Cp2kTk(p1,p2),sparse (2) [11, Proposition 4.1]If
(1.1) Ψ(t) :=kTk(1+1t,1+1t),sparse<∞ ∀t >1, then there is an absolute constantC >0such that
kTkL2(w,Rd)6C[w]A2Ψ (C[w]A2). In particular,
sup
t>1
Ψ(t)<∞ =⇒ kTkL2(w,Rd)6C[w]A2.
In this article, we are concerned with the sparse norms (1.1) of a class of convolution-type singular integrals whose systematic study dates back to the celebrated works by Christ [4], Christ–Rubio de Francia [5], and Duoandikoetxea–Rubio de Francia [12], admitting a decomposition with good decay properties of the Fourier transform. To wit, let{Ks : Rd → C, s∈Z} be a sequence of (smooth) functions with the properties that
suppKs⊂As:={x∈Rd: 2s−4<|x|∞<2s−2}, sup
s∈Z
2sdkKsk∞61, sup
s∈Z
sup
ξ∈Rd
max
|2sξ|α,|2sξ|−α |Kcs(ξ)|61, (1.2)
for someα >0. We consider truncated singular integrals of the type T f(x, t1, t2) = X
t1<s6t2
Ks∗f(x), t1, t2∈Z, and their maximal version
(1.3) T?f(x) := sup
t16t2
|T f(x, t1, t2)|.
Theorem 1.2. — Let T? be the maximal truncated singular integral defined in(1.3), under assumptions (1.2)on{Ks:s∈Z}. Then
sup
0<ε<1
εkT?k(1+ε,1+ε),sparse.1.
The implicit constant depends on dimension d only and is in particular uniform over families{Ks}satisfying (1.2).
Theorem 1.2 entails immediately a variety of novel corollaries involving weighted norm inequalities for the maximally truncated operatorsT?. In addition to, for instance, those obtained by suitably applying the points of Proposition 1.1, we also detail the quantitative estimates below, whose proof will be given in Section 7.
Theorem 1.3. — Let T be a sublinear operator satisfying the sparse bound(1.1)withΨ(t)6Ct.
(1) Let1< p <∞,w∈Ap be a Muckenhoupt weight andσ=w−p−11 be itsAp0 Muckenhoupt dual. We have the estimate
kTkLp(w).[w]
1 p
Ap([w]
1 p0
Ap+ [σ]
1 p
A∞) max{[σ]A∞,[w]A∞}
with implicit constant possibly depending onpand dimensiond; in particular,
(1.4) kTkLp(w).[w]2 max{1,
1 p−1}
Ap .
(2) The Fefferman–Stein type inequality
kT fkLp(w).p2(p0)1p(r0)1+p10kfkLp(Mrw), 1< r < p <∞ holds with implicit constant possibly depending ondonly.
(3) TheAq-A∞ estimate (1.5) kT fkLp(w).[w]
1 p
Aq[w]1+
1 p0
A∞ kfkLp(w),
holds for 1 6 q < p < ∞ and w ∈ Aq, with implicit constant possibly depending onp, qanddonly.
(4) The following Coifman–Fefferman type inequality kT fkLp(w).[w]2A∞
ε kM1+εfkLp(w), 0< p <∞
holds for allε >0 with implicit constant possibly depending onp anddonly.
Remark 1.4. — Take Ω :Sd−1→CwithkΩk∞61 and having vanishing integral onSd−1, and consider the associated truncated integrals and their maximal function
TΩ,δf(x) :=
Z
δ<|t|<1δ
f(x−t)Ω(t/|t|)
|t|d dt, TΩ,?f(x) := sup
δ>0
|TΩ,δf(x)|, x∈Rd. (1.6)
It is well known (see, e.g., the recent contribution [16, Section 3]) that TΩ,?f(x).Mf(x) +T?f(x), x∈Rd
withT? being defined as in (1.3) for a suitable choice of {Ks:s∈Z} sat- isfying (1.2) withα=1d. AskMk(1,1),sparse.1, a corollary of Theorem 1.2 is that
(1.7) kTΩ,?k(1+ε,1+ε),sparse. 1 ε as well. The main result of [7] is the stronger control
(1.8) sup
δ>0
kTΩ,δk(1,1+ε),sparse. 1 ε.
The above estimate, in particular, is stronger than the uniform weak type (1,1) for the operatorsTΩ,δ, a result originally due to Seeger [31]. As the weak type (1,1) ofTΩ,?under no additional smoothness assumption on Ω is a difficult open question, estimating the (1,1 +ε) sparse norm ofTΩ,?as in (1.8) seems out of reach.
The study of sharp weighted norm inequalities forTΩ,δ (the uniformity in δ is of course relevant here) was initiated in the recent article [16] by
Hytönen, Roncal and Tapiola. Improved quantifications have been obtained in [7] as a consequence of the domination result (1.8), and further weighted estimates (including a Coifman–Fefferman type inequality, that is a norm control ofTΩ,δ by M on allLp(w), 0< p <∞when w∈A∞) have been later derived from (1.8) in the recent preprint by the third named author, Pérez, Roncal and Rivera-Rios [28].
Although (1.7) is a bit weaker than (1.8), we see from comparison of (1.4) from Theorem 1.3 with the results of [7, 28] that the quantification of the L2(w)-norm dependence on [w]A2entailed by the two estimates is the same (quadratic); on the contrary, for p 6= 2, (1.8) yields the better estimate kTΩ,δkLp(w) . [w]pA0
p. We also observe that the proof of the mixed esti- mate (1.5) actually yields the following estimate for the non-maximally truncated operators, improving the previous estimate given in [28]
kTΩ,δfkLp(w).[w]
1 p
Aq[w]
1 p0
A∞kfkLp(w).
Finally, we emphasize that (1.7) also yields a precise dependence onp of the unweightedLp operator norms. Namely, from the sparse domination, we get
kTΩ,?kLp(Rd)→Lp,∞(Rd).max{p, p0}, kTΩ,?kLp(Rd).pp0max{p, p0} (1.9)
with absolute dimensional implicit constant, which improves on the implicit constants in [12]. Moreover, we note that the main result of [29] implies that if (1.9) is sharp, then our quantitative weighted estimate (1.4) is also sharp.
Remark 1.5. — We note that Theorem 1.3 holds with T being the com- mutator of a Calderón–Zygmund operator with a BMO symbol. This fol- lows by comparing Theorem 1.2 with the sparse domination formula for commutators from [24], with the help of the John–Nirenberg inequality.
1.1. Matrix weighted estimates for vector valued rough singular integrals
Let (e`)n`=1,h ·,· iFnand| · |Fn be the canonical basis, scalar product and norm onFnoverF, whereF∈ {R,C}. A recent trend in Harmonic Analysis (see, among others, [2, 3, 9, 15, 30]) is the study of quantitative matrix weighted norm inequalities for the canonical extension of the (integral) linear operatorT
hT f(x), e`iFn:=hT⊗IdFnf(x), e`iFn=T(hf, e`iFn)(x), x∈Rd
toFn-valued functionsf. In Section 6 of this paper, we introduce anLp, p > 1, version of the convex body averages first brought into the sparse domination context by Nazarov, Petermichl, Treil and Volberg [30], and use them to produce a vector valued version of Theorem 1.2. As a corollary, we obtain quantitative matrix weighted estimates for the maximal truncated vector valued extension of the rough singular integralsTΩ,δ from (1.6). In fact, the next corollary is a special case of the more precise Theorem 6.4 from Section 6.
Corollary 1.6. — LetW be a positive semidefinite and locally inte- grableL(Fn)-valued function onRd andTΩ,δ be as in(1.6). Then
(1.10)
sup
δ>0
W12TΩ,δf Fn
L2(Rd)
.[W]A52
2
W12f Fn
L2(
Rd)
with implicit constant depending ond, nonly, where thematrixA2constant is given by
[W]A2:= sup
Qcube ofRd
1
|Q|
Z
Q
W(x) dx 12
1
|Q|
Z
Q
W−1(x) dx 12
2
L(Fn)
. As the left hand side of (1.10) dominates the matrix weighted norm of the vector valued maximal operator first studied by Christ and Goldberg in [6], the finiteness of [W]A2 is actually necessary for the estimate to hold.
To the best of the authors’ knowledge, Theorem 6.4 has no predecessors, in the sense that no matrix weighted norm inequalities for vector rough singular integrals were known before, even in qualitative form. At this time we are unable to assess whether the power 52appearing in (1.10) is optimal.
For comparison, if the angular part Ω is Hölder continuous, the currently best known result [30] is that (1.10) holds with power 32; see also [9].
1.2. Strategy of proof of the main results
We will obtain Theorem 1.2 by an application of an abstract sparse dom- ination principle, Theorem 3.3 from Section 3, which is a modification of [7, Theorem C]. At the core of our approach lies a special configuration of stop- ping cubes, the so-calledstopping collectionsQ, and their related atomic spaces. The necessary definitions, together with a useful interpolation prin- ciple for the atomic spaces, appear in Section 2. In essence, Theorem 3.3 can be summarized by the inequality
kT?k(p1,p2),sparse.kT?kL(L2(Rd))+ sup
Q,t1,t2
kQtt21kX˙p1×Yp2+kQtt21kY∞×X˙p2
where the supremum is taken over all stopping collectionsQand all measur- able linearizations of the truncation parameterst1, t2, andQtt21are suitably adapted localizations of (the adjoint form to the linearized versions of)T?. In Section 4, we prove the required uniform estimates for the localizations Qtt2
1 coming from Dini-smooth kernels. The proof of Theorem 1.2 is given in Section 5, relying upon the estimates of Section 4 and the Littlewood–Paley decomposition of the convolution kernels (1.2) whose first appearance dates back to [12].
Remark 1.7. — We remark that while this article was being finalized, an alternative proof of (1.8) was given by Lerner [23]. It is of interest whether the strategy of [23], relying on bumped bilinear grand local maximal func- tions, can be applied towards estimate (1.7) as well.
Notation
With q0 = q−1q we indicate the Lebesgue dual exponent to q ∈ (1,∞), with the usual extension 10 = ∞, ∞0 = 1. The center and the (dyadic) scale of a cubeQ∈Rd will be denoted bycQ andsQ respectively, so that
|Q|= 2sQd. We use the notation Mp(f)(x) = sup
Q⊂Rd
hfip,Q1Q(x)
for thep-Hardy Littlewood maximal function and write M in place of M1. Unless otherwise specified, the almost inequality signs . imply absolute dimensional constants which may be different at each occurrence.
Acknowledgments
This work was carried out during F. Di Plinio’s stay at the Basque Cen- ter for Applied Mathematics (BCAM), Bilbao as a visiting fellow. The author gratefully acknowledges the kind hospitality of the staff and re- searchers at BCAM and in particular of Carlos Pérez. Some of the results were discovered at the Mathematical Sciences Research Institute (MSRI) in Berkeley, during the workshop “Recent trends in harmonic analysis”, in which Di Plinio and Hytönen participated with the support of the MSRI and the Clay Mathematics Institute. The authors also want to thank José Conde-Alonso, Amalia Culiuc, Yumeng Ou and Ioannis Parissis for several inspiring discussions on sparse domination principles.
2. Stopping collections and interpolation in localized spaces
The notion of stopping collection Q with the top (dyadic) cube Q has been introduced in [7, Section 2]: we proceed by recalling the relevant def- initions. A stopping collection Q with top Q is a collection of pairwise disjoint dyadic cubes contained in 3Qand satisfying suitable Whitney type properties. More precisely,
[
L∈cQ
9L⊂shQ:= [
L∈Q
L⊂3Q, cQ:={L∈ Q: 3L∩2Q6=∅};
(2.1)
L, L0 ∈ Q, L∩L06=∅ =⇒ L=L0; (2.2)
L∈ Q, L0∈N(L) =⇒ |sL−sL0|68, N(L) :={L0∈ Q: 3L∩3L06=∅}.
(2.3)
A consequence of (2.3) is that the cardinality ofN(L) is bounded by an absolute constant.
We proceed with the definition of the localized spaces Yp(Q), Xp(Q), X˙p(Q), whose first appearance is in [7, Section 2]. The spaceYp(Q) is the subspace ofLp(Rd) of functions satisfying
suppf ⊂3Q,kfkYp(Q)<∞, kfkYp(Q):=
(max
kf1Rd\shQk∞,supL∈Qinf
x∈bL
Mpf(x) p <∞
kfk∞ p=∞
(2.4)
whereLb stands for the (non-dyadic) 25-fold dilate ofL, and thatXp(Q) is the subspace ofYp(Q) of functions satisfying
(2.5) b= X
L∈Q
bL, suppbL⊂L.
Finally, we write b ∈ X˙p(Q) if b ∈ Xp(Q) and each bL has mean zero.
We will omit (Q) from the subscript of the norms whenever the stopping collectionQis clear from context.
There is a natural interpolation procedure involving the Yp-spaces. We do not strive for the most general result but restrict ourselves to proving a significant example, which is also of use to us in the proof of Theorem 1.2.
Proposition 2.1. — LetBbe a bisublinear form andA1, A2be positive constants such that the estimates
|B(b, f)|6A1kb1kX˙1(Q)kfkY1(Q), |B(g1, g2)|6A2kg1kY2(Q)kg2kY2(Q) hold true. Then for all0< ε <1
|B(f1, f2)|.(A1)1−ε(A2)εkf1kX˙p(Q)kf2kYp(Q), p= 1 +ε.
Proof. — We may assumeA2< A1, otherwise there is nothing to prove.
We are allowed to normalize A1 = 1. Fixing now 0 < ε < 1, so that 1< p <2, it will suffice to prove the estimate
(2.6) |B(f1, f2)|.(A2)ε
for each pair f1 ∈ X˙p(Q), f2 ∈ Yp(Q) with kf1kX˙p = kf2kYp = 1 with implied constant depending on dimension only. Let λ > 1 to be chosen later. Using the notationf>λ:=f1|f|>λ, we introduce the decompositions
f1=g1+b1, b1:= X
Q∈Q
(f1)>λ− 1
|Q|
Z
Q
(f1)>λ
1Q, f2=g2+b2, b2:= (f2)>λ
which verify the properties
g1∈X˙2(Q), kg1kX˙p .1, kg1kX˙2 .λ1−p2, b1∈X˙1(Q), kb1kX˙1 .λ1−p kg2kX˙2 .λ1−p2, kb2kX1 .λ1−p.
We have used thatb1 is supported on the union of the cubes Q∈ Q and has mean zero on each Q, and thereforeg1 has the same property, given thatf1∈X˙p(Q). Therefore
|B(f1, f2)|
6|B(b1, b2)|+|B(b1, g2)|+|B(g1, b2)|+|B(g1, g2)|
6kb1kX˙1kb2kY1+kb1kX˙1kg2kY1+kg1kX˙1kb2kY1+A2kg1kY2kg2kY2
.λ2−2p+ 2λ1−p+A2λ2−p.λ1−p(1 +A2λ)
which yields (2.6) with the choiceλ=A−12 .
3. A sparse domination principle for maximal truncations
We consider families of functions [K] ={Ks:s∈Z}satisfying suppKs⊂
(x, y)∈Rd×Rd:|x−y|<2s , k[K]k:= sup
s∈Z
2sd sup
x∈Rd
(kKs(x,·)k∞+kKs(·, x)k∞)<∞ (3.1)
and associate to them the linear operators (3.2) T[K]f(x, t1, t2) := X
t1<s6t2
Z
Rd
Ks(x, y)f(y) dy, x∈Rd, t1, t2∈Z
and their sublinear maximal versions T?t2
t1[K]f(x) := sup
t16τ16τ26t2
|T[K]f(x, τ1, τ2)|, T?[K]f(x) = sup
t16t2
|T[K]f(x, t1, t2)|.
We assume that there exists 1< r <∞such that (3.3) k[K]kr,?:=kT?[K]kLr(Rd)<∞.
For pairs of bounded measurable functionst1, t2:Rd→Z, we also consider the linear operators
(3.4) T[K]tt2
1f(x) :=T[K]f(x, t1(x), t2(x)), x∈Rd. Remark 3.1. — From the definition (3.2), it follows that
t1, t2∈Z, t1>t2 =⇒ T[K]f(x, t1, t2) = 0.
In consequence, for the linearized versions defined in (3.4) we have suppT[K]tt2
1f ⊂ {x∈Rd:t2(x)−t1(x)>0}.
A related word on notation: we will be using linearizations of the type T[K]stQ
1 and similar, where sQ is the (dyadic) scale of a (dyadic) cubeQ.
With this we mean we are using the constant function equal tosQ as our upper truncation function. Finally, we will be using the notationst2∧sQfor the linearizing functionx7→min{t2(x), sQ}andt1∨sL for the linearizing functionx7→max{t1(x), sL}.
Given two bounded measurable functionst1, t2and a stopping collection Qwith topQ, we define the localized truncated bilinear forms
(3.5) Q[K]tt21(f1, f2)
:= 1
|Q|
D
T[K]tt2∧sQ
1 (f11Q), f2E
− X
L∈Q L⊂Q
T[K]tt2∧sL
1 (f11L), f2
.
Remark 3.2. — Note that we have normalized by the measure ofQ, un- like the definitions in [7, Section 2]. Observe that as a consequence of the support assumptions in (3.1) and of the largest allowed scale beingsQ, we have
Q[K]tt21(f1, f2) =Q[K]tt21(f11Q, f213Q).
Similarly we remark that T[K]tt21∧sL(f1L) is supported on the set 3L∩ {x∈Rd:sL−t1(x)>0}; see Remark 3.1.
Within the above framework, we have the following abstract theorem.
Theorem 3.3. — Let [K] = {Ks : s ∈ Z} be a family of functions satisfying (3.1)and (3.3)above. Assume that there exist 1 6p1, p2 <∞ such that
(3.6) CL[K](p1, p2)
:= sup
kbkX˙p1(Q)=1 kfkYp2(Q)=1
Q[K]tt21(b, f)
+ sup
kfkY∞(Q)=1 kbkX˙p2(Q)=1
Q[K]tt21(f, b) <∞.
hold uniformly over all bounded measurable functionst1, t2, and all stop- ping collectionsQ. Then
(3.7) kT?[K]k(p1,p2),sparse.k[K]kr,?+CL[K](p1, p2).
Proof. — The proof follows essentially the same scheme of [7, Theo- rem C]; for this reason, we limit ourselves to providing an outline of the main steps.
Step 1. Auxiliary estimate. — First of all, an immediate consequence of the assumptions of the Theorem is that the estimate
(3.8)
Q[K]tt21(f1, f2)
6CΘ[K],p1,p2kf1kYp
1(Q)kf2kYp
2(Q)
where Θ[K],p1,p2 := k[K]kr,?+CL[K](p1, p2), holds with C > 0 uniform over bounded measurable functionst1, t2. See [7, Lemma 2.7]. Therefore, (3.9)
D
T[K]tt21∧sQ(f11Q), f2
E
6CΘ[K],p1,p2|Q|kf1kYp
1(Q)kf2kYp
2(Q)
+ X
L∈Q L⊂Q
T[K]tt2∧sL
1 (f11L), f2
Step 2. Initialization. — The argument begins as follows. Fixing fj ∈ Lpj(Rd), j = 1,2 with compact support, we may find measurable func- tionst1, t2 which are bounded above and below and a large enough dyadic cubeQ0 from one of the canonical 3d dyadic systems such that suppf1⊂ Q0,suppf2⊂3Q0 and
hT?[K]f1,|f2|i62 D
T[K]tt21∧sQ0(f11Q0),|f2|E and we clearly can replacef2by|f2| in what follows.
Step 3. Iterative process. — Then, the argument proceeds via itera- tion overkof the following construction, which follows from (3.9) and the Calderón–Zygmund decomposition and is initialized by takingSk ={Q0} for k = 0. Given a disjoint collection of dyadic cubes Q ∈ Sk with the further Whitney property that (2.3) holds forSk in place ofQ, there exists a further collection of disjoint dyadic cubesL∈ Sk+1such that
• (2.3) forSk in place ofQcontinues to hold,
• each subcollectionSk+1(Q) ={L ∈ Sk+1 :L ⊂3Q} is a stopping collection with topQ,
and for which for allQ∈ Sk there holds (3.10)
D
T[K]tt21∧sQ(f11Q), f2
E
6C|Q|Θ[K],p1,p2hf1ip1,3Qhf2ip2,3Q
+ X
L∈Sk+1(Q) L⊂Q
T[K]tt2∧sL
1 (f11L), f2 .
More precisely,Sk+1is composed by the maximal dyadic cubesLsuch that (3.11) 9L⊂ [
Q∈Sk
EQ, EQ:=
x∈3Q: max
j=1,2
Mpj(fj13Q)(x) hfjipj,3Q
> C
for a suitably chosen absolute large dimensional constantC. This construc- tion, as well as the Whitney property (2.3) results into
(3.12)
Q∩ [
L∈Sk+1
L
=
Q∩ [
Q0∈Sk:Q0∈N(Q)
EQ0
61 2|Q|
∀Q∈ Sk, k= 0,1, . . . guaranteeing thatTk:=∪kκ=0Sκis a sparse collection for allk. Whenk= ¯k is such that inf{sQ:Q∈ Sk¯}<inft1, the iteration stops and the estimate
hT?[K]f1, f2i.Θ[K],p1,p2 X
Q∈Tk¯
|Q|hf1ip1,3Qhf2ip2,3Q
is reached. This completes the proof of Theorem 3.3.
4. Preliminary localized estimates for the truncated forms (3.5)
We begin by introducing our notation for the Dini constant of a family of kernels [K] as in (3.1). We write
(4.1) k[K]kDini:=k[K]k+
∞
X
j=0
$j([K])
where
$j([K]) := sup
s∈Z
2sd sup
x∈Rd
sup
h∈Rd
|h|<2s−j−3
kKs(x, x+·)−Ks(x+h, x+·)k∞
+kKs(x+·, x)−Ks(x+·, x+h)k∞
!
The estimates contained within the lemmata that follow are meant to be uniform over all measurable functionst1, t2and all stopping collectionsQ.
The first one is an immediate consequence of the definitions: for a full proof, see [7, Lemma 2.3].
Lemma 4.1. — Let1< r <∞. Then Q[K]tt21(f1, f2)
.k[K]kr,?kf1kYr(Q)kf2kYr0(Q)
The second one is a variant of [7, Lemma 3.2]; we provide a full proof.
Lemma 4.2. — There holds Q[K]tt21(b, f)
.k[K]kDinikbkX˙1(Q)kfkY1(Q).
Proof. — We consider the familyKfixed and use the simplified notation Qtt21 in place ofQ[K]tt21, and similarly for the truncated operatorsT[K]. By horizontal rescaling we can assume |Q| = 1. Let b ∈ X˙1. Recalling the definition (2.5) and using bilinearity of Qtt21 it suffices for each stopping cubeR∈ Q to prove that
Qtt21(bR, f)
.k[K]kDinikbRk1kfkY1 (4.2)
askbRk1.|R|kbkX˙1, and conclude by summing up over the disjointR∈ Q, whose union is contained in 3Q. We may further assumeR⊂Q; otherwise Qtt21(bR, f) = 0. In addition we can assume f is positive, by repeating the same argument below with the real and imaginary, and positive and negative parts off. Using the definition of the truncated forms (3.5) and the disjointness ofL∈ Q,
Qtt21(bR, f) =
Ttt12∧sQ(bR)−Ttt12∧sR(bR), f
=
Ttt12∨s∧sRQ(bR), f 6
T?sQ
sRbR, f .
Thus, ifRsdenotes the cube concentric toRand whose sidelength is 210+s, using the support and Dini conditions
Qtt21(bR, f) 6
T?ssQ
RbR, f 6
sQ
X
s=sR+1
Z
Rs
Z
R
Ks(x, y)bR(y) dy
f(x) dx
=
sQ
X
s=sR+1
Z
Rs
Z
R
[Ks(x, y)−Ks(x, cR)]bR(y) dy
f(x) dx
6kbRk1 sQ
X
s=sR+1
ωs−sR([K])2−sd Z
Rs
f(x) dx .k[K]kDinikbRk1sup
j
hfi1,Rj
which is bounded by the right hand side of (4.2).
The third localized estimate is new. However, its roots lie in the well- known principle that the maximal truncations of a Dini-continuous kernel to scales larger than s do not oscillate too much on a ball of radius 2s, see (4.7). This was recently employed, for instance, in [13, 16, 18].
Lemma 4.3. — There holds Q[K]tt21(f, b)
.(k[K]kDini∨ kKkr,?)kfkY∞(Q)kbkX1(Q).
Proof. — We use similar notation as in the previous proof and again we rescale to|Q|= 1, and work with positiveb∈ X1. We can of course assume that suppf ⊂Q. We begin by removing an error term; namely, referring to notation (2.1), if
bo= X
R6∈cQ
bR
then (4.3)
Q[K]tt2
1(f, bo)
6h|T f(·, sQ−1, sQ)|, boi.k[K]kkbkX1kfkY∞ The first inequality holds because dist(suppf,suppbo)>2sQ−1, so at most thesQ scale may contribute, and in particular no contribution comes from cubes L (Q. The second inequality is a trivial estimate, see [7, Appen- dix A] for more details. Thus we may assume bR = 0 whenever R 6∈cQ.
We begin the main argument by fixingR∈cQ. Then by support consid- erations
Ttt12∧sL(f1L), bR
6= 0 =⇒ L∈N(R).
Similarly, Ttt12∧sRf, bR
=
Ttt12∧sR(f1shQ), bR
= X
L∈N(R) L⊂Q
Ttt12∧sR(f1L), bR
.
In fact, using (2.1) we learn that dist(shQ, R) > 2sR, whence the first equality. Therefore, subtracting and adding the last display to obtain the second equality,
Qtt21(f, bR) =D
Ttt12∧sQf, bR
E− X
L∈N(R) L⊂Q
Ttt12∧sL(f1L), bR
=D
Ttt12∨s∧sRQf, bR
E− X
L∈N(R) L⊂Q
sign(sL−sR)D
Ttt2∧(sL∨sR)
1∨(sL∧sR)(f1L), bR
E .
Now, the summation in the above display is then bounded in absolute value by
X
L∈N(R)
T?sL∧sR
sL∨sR(f1L), bR
.k[K]k X
L∈N(R)
kbRk1hfi1,L
.|[K]kDinikbRk1kfkY∞,
using that|sL−sR|68 wheneverL∈N(R). Therefore whenR∈cQ
(4.4)
Qtt21(f, bR) 6
Ttt12∨s∧sRQf, bR
+Ck[K]kDinikbRk1kfkY∞ with absolute constantC. Now, define the function
F(x) =
(supsR6τ16τ26sQ|T f(x, τ1, τ2)| x∈R∈cQ
0 x6∈S
R∈cQR
and notice that|Ttt12∨s∧sRQf|6F onR∈cQ. Since bis positive, using (4.3), summing (4.4) over R ∈ cQ and using that this is a pairwise disjoint collection, we obtain that
(4.5)
|Qtt21(f, b)|6|Qtt21(f, bo)|+ X
R∈cQ
|Qtt21(f, bR)|
6Ck[K]kDinikbkY1kfkY∞+ X
R∈cQ
Ttt12∨s∧sRQf, bR
6Ck[K]kDinikbkX1kfkY∞+ X
R∈cQ
hF, bRi
=Ck[K]kDinikbkX1kfkY∞+hF, bi.
Therefore, we are left with boundinghF, bi. This is actually done using both theLr estimate and the Dini cancellation condition. In fact, decompose
b=g+z,
g= X
R∈cQ
gR:= X
R∈cQ
hbi1,R1R, z= X
R∈cQ
zR:= X
R∈cQ
(b− hbi1,R)1R
so that
kgkY∞ 6kbkX1, kzkX˙162kbkX1
Then
hF, gi6hT?f, gi6k[K]kr,?kfkrkgkr0 6k[K]kr,?kgkY∞kfkY∞
6k[K]kr,?kfkY∞kbkX1 (4.6)
and we are left to control |hF, zi|. We recall from [16, Lemma 2.3] the inequality
(4.7) |T f(x, τ1, τ2)−T f(ξ, τ1, τ2)|.k[K]kDini sup
s>sR
hfi1,Rs,
x, ξ∈R, τ2>τ1>sR
where Rs is the cube concentric with R and sidelength 2s, whence for suitable absolute constantC
F(x)6F(ξ) +Ck[K]kDinikfkY1, x, ξ∈R and taking averages there holds
sup
x∈R
|F(x)− hFi1,R|.k[K]kDinikfkY1.
Finally, using the above display and the fact that eachzRhas zero average and is supported onR,
|hF, zi|6 X
R∈Q
|hF, zRi|= X
R∈cQ
|hF− hFi1,R1R, zRi|
.k[K]kDinikfkY1
X
R∈cQ
kzRk16k[K]kDinikfkY∞kbkX1
(4.8)
and collecting (4.5), (4.6) and (4.8) completes the proof of the Lemma.
By Proposition 2.1 applied to the formsQtt21[K], we may interpolate the bound of Lemma 4.2 with the one of Lemma 4.1 withr= 2. A similar but easier procedure allows to interpolate Lemma 4.3 with Lemma 4.1 with r = 2. We summarize the result of such interpolations in the following lemma.
Lemma 4.4. — For06ε61 andp= 1 +εthere holds CL[K](p, p).(|[K]kDini∨ k[K]k2,?)1−ε(k[K]k2,?)ε whereCL[K](p1, p2)is defined in (3.6).
Remark 4.5 (Calderón–Zygmund theory). — LetT be anL2(Rd)-boun- ded singular integral operator with Dini-continuous kernel K. Then its maximal truncations obey the estimate
T?f(x) := sup
δ>0
Z
δ<|h|<1δ
K(x, x+h)f(x+h) dh .Mf(x) +T?[K]f(x),
(4.9)
with the family [K] :={Ks:s∈Z} defined by
Ks(x, x+h) :=K(x, x+h)ψ(2−sh), x, h∈Rd where the smooth radial functionψsatisfies
suppψ⊂ {h∈Rd: 2−2<|h|<1}, X
s∈Z
ψ(2−sh) = 1, h6= 0.
We know from classical theory [32, Ch. I.7] that k[K]k2,?.kTkL2(Rd)+kKkDini.
Therefore, in consequence of (4.9) and of the boundkMk(1,1),sparse.1, an application of Theorem 3.3 in conjunction with Lemmata 4.2 and 4.3 yields that
kT?k(1,1),sparse.kTkL2(Rd)+kKkDini.
This is a well-known result. The dual pointwise version was first obtained in this form in [16] quantifying the initial result of Lacey [18]; see also [22].
An extension to multilinear operators with less regular kernels was recently obtained in [27].
5. Proof of Theorem 1.2
In this section, we will prove Theorem 1.2 by appealing to Theorem 3.3 for the family [K] = {(x, y) 7→ Ks(x−y) : s ∈ Z} of (1.2). First of all, we notice that the assumption (3.1) is a direct consequence of (1.2). It is known from e.g. [12] (and our work below actually reproves this) that, with reference to (1.3)
kT?[K]kL2(Rd).1,
