Exotic Calderón–Zygmund Operators

We study singular integral operators with kernels that are more singular than standard Calderón–Zygmund kernels, but less singular than bi-parameter product Calderón–Zygmund kernels. These kernels arise as restrictions to two dimensions of certain three-dimensional kernels adapted to so-called Zygmund dilations, which is part of our motivation for studying these objects. We make the case that such kernels can, in many ways, be seen as part of the extended realm of standard kernels by proving that they satisfy both a T1 theorem and commutator estimates in a form reminiscent of the corresponding results for standard Calderón–Zygmund kernels. However, we show that one-parameter weighted estimates, in general, fail.


Introduction
Working on the Euclidean product space R 2 = R × R, we define for x = (x 1 , x 2 ) and y = (y 1 , y 2 ) the decay factor whenever x 1 = y 1 and x 2 = y 2 . Notice that this decay factor becomes larger and larger as θ shrinks. The point is that when θ = 1 it is at its smallest, and then That is, in this case, the bi-parameter size estimate multiplied with this decay factor yields the usual one-parameter size estimate. When θ < 1, the decay factor is larger and the corresponding product is something between the bi-parameter and one-parameter size estimate.
We say that kernels that decay like for some θ and satisfy some similar continuity estimates are C Z X kernels-one can pronounce the "X " in "C Z X" as "exotic". Such kernels are more singular than the standard Calderón-Zygmund kernels, but less singular than the product Calderón-Zygmund(-Journé) kernels [5,13,18]. Even with θ = 1, they are different from the standard Calderón-Zygmund kernels-in this case, the difference is only in the Hölder estimates (see Sect. 2). The C Z X kernels can, for example, be motivated by looking at Zygmund dilations [4,[19][20][21]. Zygmund dilations are a group of dilations lying in between the standard product theory and the one-parameter setting-in R 3 = R × R 2 they are the dilations (x 1 , x 2 , x 3 ) → (δ 1 x 1 , δ 2 x 2 , δ 1 δ 2 x 3 ). Recently, in [8] and subsequently in [3,9] general convolution form singular integrals invariant under Zygmund dilations were studied. In these papers the decay factor controls the additional, compared to the product setting, decay with respect to the Zygmund ratio See also our recent paper [12] which attacks the Zygmund setting from the point of view of new multiresolution methods. Essentially, in the current paper, we isolate the conditions on the lower-dimensional kernels obtained by fixing the variables x 1 , y 1 in the Zygmund setting [8,12] and ignoring the dependence on these variables. A class of C Z X operators is also induced by the Fefferman-Pipher multipliers [4]importantly, they satisfy θ = 1 but with an additional logarithmic growth factor. This subtle detail has a key relevance for the weighted estimates as we explain below.
There is a useful operator-valued viewpoint to multi-parameter analysis-Journé [13] views, e.g. bi-parameter operators as "operator-valued one-parameter operators". For recent work using this viewpoint see e.g. [11]. Developing such an approach to Zygmund SIOs is interesting. The operator-valued viewpoint is useful for example when proving the necessity of T 1 type assumptions in the product setting, see e.g. [6], and the full product BMO type T 1 theory of Zygmund SIOs is still to be developed. The operator-valued approach will necessarily be complicated in the Zygmund setting, since the parameters are tied and it is not as simple as fixing a single variable. Our new exotic operators are pertinent to the operator-valued viewpoint, where Zygmund SIOs could partly be seen as operator-valued one-parameter operators the values being exotic operators.

Theorem Let B( f , g) be a bilinear form defined on finite linear combinations of indicators of cubes of R 2 , and such that
. Then the following are equivalent: • the weak boundedness property |B(1 I , 1 I )| |I | for all cubes I ⊂ R 2 , and • the T (1) conditions Moreover, under these conditions, In fact, our proof also gives a representation of B( f , g), Theorem 4.9, which includes both one-parameter [10] and bi-parameter [18] elements. The following commutator bounds follow from the representation; however, the argument is not entirely standard due to the hybrid nature of the model operators.

Theorem
Let T ∈ L(L 2 (R 2 )) be an operator associated with a CZX kernel K . Then Thus, the commutator estimate holds with the one-parameter BMO space. This is another purely one-parameter feature of these exotic operators. As the weighted estimates do not, in general, hold, the commutator estimate cannot be derived from the well-known Cauchy integral trick. Over the past several years, a standard approach to weighted norm inequalities has been via the methods of sparse domination pioneered by Lerner. For θ = 1, we can derive our weighted estimates directly from our representation theorem. However, we also provide some additional sparse estimates that give a solid quantitative dependence on the A p constant and yield two-weight commutator estimates for free.

Theorem
Let T ∈ L(L 2 (R 2 )) be an operator with a CZX kernel with θ = 1. Then for every p ∈ (1, ∞) and every w ∈ A p (R 2 ) the operator T extends boundedly to L p (w) with norm where the supremum is over cubes I ⊂ R 2 , then The quantitative bound (in particular quadratic in [w] A 2 when p = 2) is worse than the linear A 2 theorem valid for classical Calderón-Zygmund operators [10]. We conclude the introduction with an outline of how the paper is organized. In Sect. 2, we define the C Z X kernels and prove part of Theorem 1.3 in Proposition 2.4. Section 3 begins with the definition of C Z X forms. Lemma 3.3 proves estimates for C Z X forms acting on Haar functions, which will be used in the representation theorem, Theorem 4.9. In Proposition 4.4, we prove certain weighted maximal function estimates, which are at the heart of proving that C Z X forms with decay parameter θ 2 = 1 satisfy weighted estimates. The dyadic operators used to represent C Z X forms are defined in Definition 4.5, and estimates for them are proved in Lemma 4.6. The representation identity and the T 1 theorem, and the weighted estimates when θ 2 = 1, for C Z X forms are recorded in Theorem 4.9. Theorem 1.4 is proved in Sect. 5. In the beginning of Sect. 6, we construct the counterexamples required to prove (1) of Theorem 1.2. The sparse domination of C Z X operators with θ 2 = 1 is recorded in Corollary 6.7. Theorem 1.5 is proved in Corollary 6.8 and in the discussion after Proposition 6.10.

CZX Kernels
We work in R 2 = R × R. Let θ 1 , θ 2 ∈ (0, 1]. For x 1 = y 1 and x 2 = y 2 define We assume that the kernel K : R 2 \{x 1 = y 1 or x 2 = y 2 } → C satisfies the size estimate and the mixed Hölder and size estimate whenever |x 1 − w 1 | ≤ |x 1 − y 1 |/2, together with the other three symmetric mixed Hölder and size estimates. If this is the case, we say that K ∈ C Z X(R 2 ). Again, such kernels are more singular than standard Calderón-Zygmund kernels, but less singular than the product Calderón-Zygmund(-Journé) kernels. See Remark 4.10 for some additional logarithmic factors when θ 2 = 1 and why they are relevant from the point of view of Fefferman-Pipher multipliers [4].

Lemma
Also, for L > 0 there holds that which is a useful estimate if L |x 2 − y 2 |.

Proof By elementary calculuŝ
and the logic for the second estimate is also clear from this.
The first sharper estimate in the next lemma is only needed to derive the weighted estimates in the case θ 2 = 1.

Lemma
Proof There holds that Suppose for instance that dist(y 1 , J 1 ) ≥ dist(y 2 , J 2 ). Then the sum simplifies to where further with θ := 1 2 min(θ 1 , θ 2 ). A combination of the previous two lemmas shows that C Z X-kernels satisfy the Hörmander integral condition: where the first two components on the right-hand side are symmetric. For these, we simply estimatê where the first was an application of Lemma 2.1. The estimate for K (c J , y) is of course a special case of this with x = c J . For the remaining component of the integration domain, there holds that where the first was an application of Lemma 2.2.
At this point, we can already provide a proof of part (1.3) of Theorem 1.3, which we restate as

Proposition Let T ∈ L(L 2 (R 2 )) be an operator associated with a CZX kernel K . Then T extends boundedly from L
Proof By Lemma 2.3, the kernel K satisfies the Hörmander integral condition; the symmetry of the assumption on K ensures that it also satisfies the version with the roles of the first and second variable interchanged. It is well known that any L 2 (R 2 )-bounded operator with a Hörmander kernel satisfies the mapping properties stated in the proposition. (See e.g. [22, §I.5] for the boundedness from L 1 (R 2 ) into L 1,∞ (R 2 ), and from L p (R 2 ) into itself for p ∈ (1, 2), and [22,§IV.4.1] for the boundedness from L ∞ (R 2 ) into BMO(R 2 ). The latter is formulated for convolution kernels K (x, y) = K (x − y), but an inspection of the proof shows that it extends to the general case with trivial modifications. The case of p ∈ (2, ∞) can be inferred either by duality (observing that the adjoint T * satisfies the same assumption) or by interpolation between the L 2 (R 2 ) and the L ∞ (R 2 )-to-BMO(R 2 ) estimates.)

Haar Coefficients of CZX Forms
We recall the weak boundedness property and the T 1 assumptions, which are just the same as in the classical theory for usual Calderón-Zygmund forms.

Definition
Let B( f , g) be a bilinear form defined on finite linear combinations of indicators of cubes of R 2 , and such that

Definition A C Z X(R 2 )-form satisfies the weak boundedness property if
where the supremum is over all cubes I ⊂ R 2 and b I = 1 |I |´I b. For an interval I ⊂ R, we denote by I l and I r the left and right halves of the interval I , respectively. We define h 0 . Let now I = I 1 × I 2 be a cube, and define the Haar function h

Lemma Let B be a C Z X (R 2 )-form satisfying the weak boundedness property. There holds that
whenever I , J are dyadic cubes with equal side lengths (I ) = (J ) and at least β = 0 or γ = 0.
Proof We consider several cases. Adjacent cubes: By this, we mean that dist(I , J ) = 0, but I = J . Here, we simply put absolute values inside. We are thus led to estimatê By symmetry, we may assume for instance that I 2 = J 2 . Lemma 2.1 gives that The assumption I 2 = J 2 implies that The dependence on x 1 has already disappeared, and integration with respect to x 1 ∈ I 1 results in another (I ). Then we are only left with observing that (I ) 2 /|I | = 1.

Equal cubes: Now
For J = I , we can estimate the term as in the case of adjacent I = J , recalling that only the size and no cancellation of the Haar functions was used there.

Cubes separated in one direction:
By this, we mean that, say, dist(I 1 , J 1 ) = 0 < dist(I 2 , J 2 ), or the same with 1 and 2 interchanged. We still apply only the non-cancellative estimate (3.4) (in contrast to what one would do with standard Calderón-Zygmund operators). From Lemma 2.1, we deduce that There is no more dependence on the remaining variables x 1 , x 2 , y 2 , so integrating over these gives the factor (I ) 3 . After dividing by |I | = (I ) 2 in (3.4), we arrive at the bound Cubes separated in both directions: By this, we mean that dist(I i , J i ) > 0 for both i = 1, 2. It is only here that we make use of the assumed cancellation of at least one of the Haar functions, say h β I . Thus, which readily simplifies to the claimed bound after |I | 2 /|I | = (I ) 2 .

Dyadic Representation and T1 Theorem
Let D 0 be the standard dyadic grid in R. For ω ∈ {0, 1} Z , ω = (ω i ) i∈Z , we define the shifted lattice Let P ω be the product probability measure on {0, 1} Z . We recall the notion of k-good cubes from [7]. We say that We will need an estimate for the maximal operator Before bounding it, we recall the following interpolation result due to Stein and Weiss, see [23, Theorem 2.11].

Proposition
Suppose that 1 ≤ p 0 , p 1 ≤ ∞ and let w 0 and w 1 be positive weights. Suppose that T is a sublinear operator that satisfies the estimates Let t ∈ (0, 1) and define 1/ p = (1 − t)/ p 0 + t/ p 1 and w = w  We denote (by slightly abusing notation) a general cancellative Haar function h η I , η = (0, 0), simply by h I . For k = (k 1 , k 2 ), k i ≥ 0, we define that the operator Q k,σ has either the form

Definition
or the symmetric form, and here I (k) = I (k 1 ) × I (k 2 ) and the constants a I J K satisfy Proof We consider σ fixed here and drop it from the notation. Suppose, e.g. k 1 ≥ k 2 . We write Notice now that for w ∈ A 2 , there holds that where we used the standard weighted square function estimate twice in the end.
To bound Q k f we need to estimate while the second piece is bounded by which is estimated similarly except that the bound for M D 2 k 1 −k 2 is replaced by the standard result for M D . This proves the claimed bounds in L 2 (w), and the results for L p (w) follow by Rubio de Francia's extrapolation theorem (the correct 1 − η dependence is maintained by the extrapolation, see Remark 4.7).
For the unweighted estimate in L p (with better complexity dependence), simply run the above argument using the Fefferman-Stein L p ( 2 ) estimate for the strong maximal function instead of the L 2 (w) estimate of M D 2 k 1 −k 2 , and use the analogous L p ( 2 ) estimate for the square function involving γ K ,k 1 that follows via Rubio de Francia extrapolation from the proved L 2 (w) estimate of the same square function.

Remark
It is clear that when p = 2, η depends only on the A 2 constant of w. In fact, in the proof of Proposition 4.4 we get η ∼ 1/[w] A 2 . Thus, where K is an increasing function. Hence N is also an increasing function. Then standard extrapolation (see e.g. [2, Theorem 3.1]) gives that the L p (w) bound of Q k,σ is

Definition
We say that π b is a (one-parameter) paraproduct if it has the form or the symmetric form.
It is well known (and follows readily from H 1 -BMO duality) that paraproducts are L p bounded for p ∈ (1, ∞) (and L p (w) bounded for w ∈ A p ) precisely when b ∈ BMO.
Finally, notice that Lemma 3.3 implies that The unweighted boundedness follows immediately from the L p bounds of the paraproducts and the bound Q k,σ f L p (1 + k max ) 1/2 f L p , since the exponentially decaying factor 2 −θ 2 (k max −k min ) 2 −θ 1 k min clearly make the series summable for any θ 1 , θ 2 > 0.
Let us finally consider the weighted case with θ 2 = 1. Then for some η = η( p, w) > 0, there holds that and again we have exponential decay that makes the series over k 1 , k 2 summable.

Remark
If θ 2 = 1, we may redefine D 1 (x, y) to be the slightly larger quantity and still prove the weighted estimates essentially like above. This is pertinent in the sense that if we take a Fefferman-Pipher multiplier [4]-a singular integral of Zygmund type-and use it to induce a CZX operator, a logarithmic term appears. In this threshold a weighted estimate still holds. See also [12].

Commutator Estimates
We show that our exotic Calderón-Zygmund operators also satisfy the usual oneparameter commutator estimates. Since weighted estimates with one-parameter weights do not in general hold (see Sect. 6), this does not follow from the well-known Cauchy integral trick.

Theorem Let T ∈ L(L 2 (R 2 )) be an operator associated with a CZX kernel K . Then
Proof By Theorem 4.9 and the well-known commutator estimates for the paraproducts π , we only need to prove that where ϕ is some polynomial. We consider σ fixed and drop it from the notation. Recall the usual paraproduct decomposition of b f : Invoking the above decomposition, the well-known boundedness of paraproducts and Lemma 4.6, it suffices to control We may assume k 1 ≥ k 2 . There holds that On the other hand, since we only need to consider those Q such that Q f , H I ,J = 0, i.e. either I ⊂ Q or J ⊂ Q, there holds that However, this is clear because

Counterexample to Weighted Estimates and Sparse Bounds
We begin by showing that bounded C Z X(R 2 ) operators need not be bounded with respect to the one-parameter weights if θ 2 < 1.

Lemma
For scalars θ 2 ∈ (0, 1], t 1 , t 2 > 0 and a bump function ϕ we define Then, uniformly on t 1 , t 2 , K ∈ C Z X(R 2 ) with θ 1 = 1 and θ 2 appearing in the very definition of K , and Proof Suppose by symmetry that t 1 ≤ t 2 . Then, using the rapid decay of all the derivatives of φ, we conclude for all N that In the last two lines of (6.2), if N is large enough, each factor in front of the last one can be bounded by one. Thus, we get that From α ∈ {(0, 0), (0, 1), (1, 0)}, we get the desired kernel estimates. For the boundedness, notice that We fix t 1 , t 2 , θ 2 momentarily and denote K = K t 1 ,t 2 ,θ 2 . For any rectangle R of sidelengths t 1 , t 2 , it is clear that is the eccentricity of R. Suppose now that for p ∈ (1, ∞) there holds that K * f L p (w) ≤ C([w] A p )N f L p (w) for all w ∈ A p and f ∈ L p (w). Then If all L 2 bounded C Z X operators would satisfy the L p (w) boundedness for all w ∈ A p with a bound C([w] A p )N , where N depends only on the kernel constants and boundedness constants of the operator, then for all rectangles R ⊂ R 2 , the estimate holds. This is because for the kernels K = K t 1 ,t 2 ,θ 2 the constant N is uniformly bounded on t 1 , t 2 .

Definition
We say that an operator T satisfies pointwise L p -sparse domination with constants C and , if for every compactly supported f ∈ L p (R d ), there exists an -sparse collection S of cubes such that where f S, p := | f | p 1/ p S , and a collection S of cubes is called -sparse, if there are pairwise disjoint sets E(S) ⊂ S for every S ∈ S with |E(S)| ≥ |S|.
There are by now several approaches to proving sparse domination. We will use one by Lerner and Ombrosi [14], which depends on bounds on the following maximal function related to the operator T under investigation where the supremum, once again, is over all cubes J .

Lemma Let T be an operator with a CZX kernel satisfying
where the right-hand side is the strong maximal function, with supremum over all (axes-parallel) rectangles R ⊂ R 2 containing x.
Proof Let us fix a cube J ⊂ R 2 with centre c J , and some x, y, z ∈ J . Note that and As usual, we split For the integral over the last component, Lemma 2.2 implies that where c and A need to be chosen so that each of the sets have measure at most d |Q| for some small dimensional d . However, On the other hand, using among other things that q ≤ s and Hölder's inequality, there holds that so to make this at most d , it is enough to take since q ≥ 1. Similarly, with M # T ,3 in place of T and r in place of q in order that So an admissible choice is c = c d and and hence An immediate consequence of the previous results is the following:

Corollary
Let T ∈ L(L 2 (R 2 )) be an operator with a CZX kernel satisfying θ 2 = 1. Then for every p > 1, the operator T satisfies pointwise L p -sparse domination with constants C p and an absolute > 0.
Proof We know from Theorem 1.3 that T is bounded from L 1 (R 2 ) to L 1,∞ (R 2 ). From Lemma 6.5 we know that M # T , 3 f M * f . Since the strong maximal operator is bounded from L p (R 2 ) to itself, and hence to L p,∞ (R 2 ), the assumptions of Theorem 6.6 are satisfied. Since d = 2, it is immediate that = 2 provided by that theorem is absolute. In order to obtain the claim C p , we observe for completeness the following (probably well known) estimate for M * f ≤ M 2 M 1 f , where M i is the one-dimensional maximal operator with respect to the ith variable: It follows that using the fact that the L p norm of the usual maximal operator is O( p ), while its L p -to-L p,∞ norm can be estimated independently of p. In fact, 6.8 Corollary Let T ∈ L(L 2 (R 2 )) be an operator with a CZX kernel satisfying θ 2 = 1. Then for every p ∈ (1, ∞) and every w ∈ A p (R 2 ), the operator T extends boundedly to L p (w) with norm Proof This follows by the same reasoning as [17,Theorem 1.6]. The said theorem is stated for different operators, but its proof only uses a certain sparse domination estimate for these operators, which is a slightly weaker variant (so-called form domination) of what we proved (pointwise domination) for operators with CZX kernel in Corollary 6.7, and hence the same reasoning applies to the case at hand. A curious feature of the above proof of the weighted estimates is that it passes through estimates involving the strong maximal operator, which in principle should be forbidden in the theory of standard A p weights; indeed, the strong maximal operator is bounded in L p (w) for strong A p weights only. The resolution of this paradox is that we only use the strong maximal operator as an intermediate step, in a part of the argument with no weights yet present, to establish some sparse bounds, which in turn imply the weighted estimates.
To conclude this section, we discuss commutator estimates. In fact, combining the ideas from [15] and [14] we can establish the following sparse domination principle. r ). Then there exists an -sparse family S such that the commutator

Proof
The proof is actually similar to [14, Theorem 1.1] but one should adapt ideas used in [15]. Let c, A, d be the same as those in the proof of Theorem 6.6. Apart from the sets for the same reason, we can also let each of the sets have measure at most d |Q|. Denote the union of the above six sets by E and set = E ∩ Q. Manipulating in the same way as in [14, Theorem 1.1], we get a family of pairwise disjoint cubes {P j } ⊂ Q such that j |P j | ≤ 1 2 |Q| and | \ ∪ j P j | = 0. The latter implies that for a.e. x ∈ Q \ ∪ j P j there holds that On the other hand, similarly as in [14, Theorem 1.1] we also have for a.e. x ∈ P j that Note that the linearity of T is used in the second step. With the recursive inequality at hand, the rest is standard (see e.g. [14, Lemma 2.1]). And since the constants c and A are the same as those in Theorem 6.6, we get the desired constant in the sparse domination.
Analogous to Corollary 6.7, we have the following result 6.10 Proposition Let T ∈ L(L 2 (R 2 )) be an operator with a CZX kernel satisfying θ 2 = 1. Then for every p > 1, the commutator [b, T ] satisfies where S is an -sparse family with > 0 absolute and C is an absolute constant depending only on the operator T .
In [16, Theorem 5.2] a two-weight commutator estimate for rough homogeneous singular integrals was formulated. As our sparse forms are the same as there, the two-weight commutator estimate of Theorem 1.5 follows as a direct consequence of Proposition 6.10.
Structures "FiRST". K. Li was supported by the National Natural Science Foundation of China through Project Numbers 12222114 and 12001400.
Funding Open Access funding provided by University of Helsinki including Helsinki University Central Hospital.
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