Analysis of hyper-singular, fractional, and order-zero singular integral operators
AANALYSIS OF HYPER-SINGULAR, FRACTIONAL, AND ORDER-ZERO SINGULARINTEGRAL OPERATORS
LUCAS CHAFFEE, JAROD HART, AND LUCAS OLIVEIRAA
BSTRACT . In this article, we conduct a study of integral operators defined in terms of non-convolution type kernels with singularities of various degrees. The operators that fall within ourscope of research include fractional integrals, fractional derivatives, pseudodifferential operators,Calder´on-Zygmund operators, and many others. The main results of this article are built around thenotion of an operator calculus that connects operators with different kernel singularities via vanish-ing moment conditions and composition with fractional derivative operators. We also provide sev-eral boundedness results on weighted and unweighted distribution spaces, including homogeneousSobolev, Besov, and Triebel-Lizorkin spaces, that are necessary and sufficient for the operator’svanishing moment properties, as well as certain behaviors for the operator under composition withfractional derivative and integral operators. As applications, we prove T S , , fractional order and hyper-singular paraproduct boundedness,a smooth-oscillating decomposition for singular integrals, sparse domination estimates that quantifyregularity and oscillation, and several operator calculus results. It is of particular interest that manyof these results do not require L -boundedness of the operator, and furthermore, we apply our resultsto some operators that are known not to be L -bounded.
1. I
NTRODUCTION
Our primary goal in this article is to develop a theory that connects integral operators of dif-ferent singularities through composition with fractional derivatives, and to understand how theseoperators are related through vanishing moment conditions and distribution space boundednessproperties. The operators we consider are, formally speaking, ν -order singular integral operatorsof the form T f ( x ) = (cid:90) R n K ( x , y ) f ( y ) dy , where K is a kernel that satisfies | K ( x , y ) | ≤ | x − y | − ( n + ν ) , which will be defined precisely in Sec-tion 2 as members of the ν -order Singular Integral Operator class SIO ν . When ν > T is ahyper-singular operator and resembles differentiation in some sense. When ν < T is a frac-tional order operator and resembles a fractional integral or anti-differentiation operator. The pro-totypical example for such operators are the ν -order fractional derivatives | ∇ | ν ∈ SIO ν for ν (cid:54) = | ξ | ν . Such operators are typically viewed as ν -order Date : September 17, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Operator Calculus, Operator Algebra, Calder´on-Zygmund Operator, Fractional Integral,Fractional Derivative, Hyper-Singular Operator, Forbidden Pseudodifferential operator, Exotic Pseudodifferential Op-erator, T1 Theorem, Non-Convolution, Vanishing Moment, Weighted Estimate. a r X i v : . [ m a t h . F A ] J a n LUCAS CHAFFEE, JAROD HART, AND LUCAS OLIVEIRA derivatives when ν > ν -order fractional integrals when ν <
0, which agrees with our roughinterpretation of
SIO ν . We will also work with the zero-order, or critical index, class SIO , whichinclude several mainstays in harmonic analysis, like the Hilbert transform, Riesz transforms, otherCalder´on-Zygmund operators, zero-order pseudodifferential operators, and others. Zero-order op-erators have been studied extensively. Some of the work most closely related to the current articleinclude [11, 14, 18, 16, 44, 20, 22, 23]. There appears to be much less theory developed for theclasses SIO ν for ν (cid:54) =
0. The most relevant sources are [44] for ν (cid:54) = ν < SIO ν in several different capacities, which we will eventually showare all equivalent in some sense. For T ∈ SIO ν , we consider the following questions: Under whatconditions on T and s , t ∈ R does | ∇ | − s T | ∇ | t belong to SIO ν + t − s ? Under what condition can T ∈ SIO ν be extended to a bounded operator on various distribution spaces, including weightedand unweighted homogeneous Sobolev, Besov, and Triebel-Lizorkin spaces? We will show that theappropriate conditions for T are a ν -order Weak Boundedness Property, which we refer to as W BP ν ,and vanishing moment conditions of the form T ∗ ( x α ) =
0. Furthermore, we show that, in manysituations, both the operator calculus properties and distribution space boundedness properties ofan operator T ∈ SIO ν are sufficient conditions for T ∗ ( x α ) =
0. Thus, we develop many equivalentconditions between vanishing moment properties, distribution space boundedness, and operatorcomposition properties.The original motivation for this work came from the notion of an operator calculus for differentclasses of integral operators, some examples of which include classes of linear and bilinear pseu-dodifferential operators, Calder´on-Zygmund operators, and fractional integral operator. It appearsto be a consensus that the origins of a pseudodifferential symbolic calculus lie in the theory ofsingular integral operators, but it is not clear exactly where or when such a calculus first appeared.Some have credited early development of the topic to Bokobza, Calder´on, Mihlin, H ¨ormander,Kohn, Nirenberg, Seeley, Unterberger, and Zygmund; see [39, 36, 40] for more information on theearly history of this. An operator calculus for a forbidden class, sometimes also referred to as anexotic class, of pseudodifferential operators was formulated by Bourdaud [7], and the notion of asymbolic calculus for bilinear pseudodifferential operators was introduced by B´enyi, Maldonado,Naibo, and Torres [3]. Several algebras of Calder ´on-Zygmund operators have been formulated, forexample, by Coifman and Meyer in [12]. These subclasses of operators can be defined in termsof almost diagonal operators, which is a notion depending on wavelet decompositions, but theyultimately bare out to be equivalent to vanishing moment conditions for the operator. Finally, thecurrent authors developed a restricted calculus for linear and bilinear fractional integral operatorsin [10] that in some senses resembles the one we present here. One of the main results of thisarticle is a restricted calculus for
SIO ν (see Theorems 4.2 and 4.3), and as an application we alsointroduce some new operator algebras associated to SIO ν in Section 7.6.Our first objective is to develop the restricted operator calculus for SIO ν being acted on by | ∇ | s for s ∈ R . A little more precisely, in Theorem 4.2 we show that if T ∈ SIO ν satisfies a ν -orderweak boundedness property and T ∗ ( x α ) = α ∈ N n , then | ∇ | − s T | ∇ | t agrees modulo polynomials with an operator in SIO ν + t − s for certain ranges of s , t ∈ R . In theprocess of formulating this restricted calculus, we prove some estimates for functions of the form ψ k ∗ T f ( x ) for T ∈ SIO ν , which are of interest and useful on their own right; see Theorem 3.1 andCorollary 3.2. NALYSIS OF HYPER-SINGULAR, FRACTIONAL, AND ORDER-ZERO SINGULAR INTEGRAL OPERATORS 3
Our second goal is to show that the same conditions on T mentioned in the previous para-graph are also sufficient for the boundedness of | ∇ | − s T | ∇ | t , and hence of T , on certain distributionspaces. We show that | ∇ | − s T | ∇ | t can be extended, modulo polynomials, to a bounded linear oper-ator from ˙ W ν + t − s , p into L p , which implies T is can be extended to a bounded linear operator from˙ W ν − t into ˙ W − t , p for appropriate t > ν ; see Theorem 4.3 for more on this. We extend the bound-edness properties of both | ∇ | − s T | ∇ | t and T to other functions spaces as well, including weightedBesov and Triebel-Lizorkin spaces, in Theorem 5.1, Theorem 5.2, Corollary 5.3, and Corollary5.4. A significant feature of all the results mentioned to this point, including the ones in the pre-ceding paragraph, is that no a priori boundedness of T is required. Indeed, if T ∈ SIO , one neednot require even L -boundedness for T to apply these results. Furthermore, it is even possible toapply these results to operators that are not L -bounded, and we provide some example of suchoperators. This notion will be explored in more detail in the applications provided in Section 7,specifically in Sections 7.2-7.5.Our third objective is to establish several sufficient conditions for vanishing moment propertiesof the form T ∗ ( x α ) =
0. We show that under slightly stronger initial assumptions on T , the resultspertaining to | ∇ | − s T | ∇ | t and boundedness of T discussed in the last two paragraphs are not onlynecessary for T ∗ ( x α ) = T T ∗ ( x α ) = T are necessary and sufficientto well-define | ∇ | − s T | ∇ | t as a singular integral operator. The results pertaining to sufficiencyconditions for T ∗ ( x α ) = T T ∗ ( x α ) = T is a Calder´on-Zygmund operator that can be extended to a bounded operator on the weightedHardy spaces H p for p ≤
1, then T ∗ ( x α ) = α ∈ N n depending on the size of p .We refer to p here as a Lebesgue parameter or Lebesgue index since H p is defined in terms ofan L p norm. Thus, if T is bounded on distribution spaces with small enough Lebesgue index p ,then T ∗ must vanish on polynomials up to some degree. This is one way to formulate sufficientconditions for T ∗ ( x α ) =
0. We show that if T is bounded on spaces ˙ W − t , p for certain ranges of t > < p < ∞ , then T ∗ ( x α ) =
0; the same holds for boundedness for Triebel-Lizorkin andBesov spaces. This shows that boundedness on negative smoothness index spaces provide anotherway to formulate sufficient conditions for T ∗ ( x α ) =
0. Working formally, it then also follow byduality that if T is bounded on positive index smoothness spaces, then T ( x α ) =
0. These twoways to formulate sufficient conditions for vanishing moments are well-understood, for examplesome results along these lines can be found in [1, 12, 22]. We provide two other types of sufficientconditions for T ∗ ( x α ) =
0. One is to require T to be bounded on weighted distribution spaces wherethe weights are outside the natural weight class for the Lebesgue index of the space. For instance,if T is bounded on H w for all Muckenhoupt weights w ∈ A ∞ , then T ∗ ( x α ) = α ∈ N n .Similar results hold for other Triebel-Lizorkin spaces, for Lebesgue indices other than 2, and for A q in place of A ∞ . The final way we formulate sufficient conditions for T ∗ ( x α ) = | ∇ | − s T | ∇ | t to agree modulo polynomials with operators in SIO ν + t − s , which of courseare closely related to the boundedness of T on distribution spaces, but nonetheless provide another LUCAS CHAFFEE, JAROD HART, AND LUCAS OLIVEIRA sufficient condition for T ∗ ( x α ) =
0. All of these approaches to formulating sufficient conditionsfor T ∗ ( x α ) = T ν ∈ R . In Corollary 7.1, we imposea little more on a given operator a priori, that T belong to CZO ν rather than just SIO ν , and in doingso we obtain necessary and sufficient conditions using some of the results from Sections 4-6.In Section 7.2, we verify that pseudodifferential operators with symbols in the forbidden class S , can be treated with our results. We show that such operators that satisfy T ( x α ) = L -bounded, aswell as their transposes.In Section 7.3, we construct some paraproduct operators that belong to SIO ν , for any given ν ∈ R , to which we can apply our operator calculus and boundedness results. Furthermore, weconstruct paraproducts in SIO that are not L -bounded, but are bounded on homogeneous Sobolevspaces ˙ W − t , for all t >
0, as well as other negative smoothness indexed spaces. Hence we areoutside of the class of Calder ´on-Zygmund operators, but still obtain several boundedness results.The paraproducts we construct are of interest in their own right as well. In form and function,they resemble the Bony paraproduct, however we construct them for any class
SIO ν with ν ∈ R ,and we construct them to reproduce higher order moments as apposed to just the typical condition Π b = b . See (7.1) for the definition of these paraproducts, as well as Corollary 7.4 and Lemma7.7 for more information on the relevant properties they satisfy.In Section 7.4, we provide a decomposition of operators in SIO ν into two terms, an oscillation-preserving term and a regularity-preserving term. Roughly speaking, we show in Theorem 7.8 thatunder some mild moment conditions on T ∈ SIO ν , we can write T = S + O where S , O ∈ SIO ν , S isbounded on ˙ W t , p for a range of t > < p < ∞ , and O is bounded on ˙ W − t , p for all t > < p < ∞ . We actually show that our decomposition satisfies the conditions S ( x α ) = α ∈ N n and O ∗ ( x α ) = α ∈ N n . Then by our results in Section 5, we obtainboundedness results for both S and O on different classes of spaces. In some senses, Theorem7.8 describes how any operator T ∈ SIO ν can be decomposed T = S + O , where S behaves likea convolution operator with respect to smoothness properties and O behaves like a convolutionoperator with respect to oscillatory properties. See Section 7.4 for more information on this. Itshould be noted that this decomposition is valid for operators T ∈ SIO ν that, once again, are notbounded from ˙ W ν , into L .In Section 7.5, we prove smooth and oscillatory sparse domination principles for operators in SIO ν . This application is included to demonstrate the following notion. We expend a lot of effortto provide conditions for an operator T ∈ SIO ν that imply | ∇ | − ( ν + t ) T | ∇ | t is a Calder ´on-Zygmundoperator (modulo polynomials). Hence we can obtain new results for any such T by applyingexisting results from Calder´on-Zygmund theory to | ∇ | − ( ν + t ) T | ∇ | t . In a way, our restricted operatorcalculus allows us to translate CZO theory to SIO ν theory for ν (cid:54) =
0. We demonstrate this principlethrough the sparse domination principle for
SIO ν in Corollary 7.10.In Section 7.6, we develop some new operator calculus results. It appears that this is the firstoperator algebra that includes operators of non-convolution type containing hypersingular and frac-tional integral operators. Furthermore, in Theorem 7.12 we describe several operator algebras that NALYSIS OF HYPER-SINGULAR, FRACTIONAL, AND ORDER-ZERO SINGULAR INTEGRAL OPERATORS 5 include operators of different singularities. Some are made up of differential operators, fractionalintegral operators, and/or order-zero operators in various combinations.This article is organized as follows. In Section 2, we provide several definitions, notation, andpreliminary results. Section 3 contains the bulk of the work of truncating and approximatingsingular integral operators, and provides crucial support for the results in the sections that follow.In Section 4, we formulate our restricted operator calculus by verifying | ∇ | − s T | ∇ | t has a ( ν + t − s ) -order standard function kernel and satisfies certain boundedness properties for appropriate T ∈ SIO ν . Section 5 is dedicated to proving several boundedness results for operators T ∈ SIO ν thatsatisfy T ∗ ( x α ) = T ∗ ( x α ) =
0. Finally, we present several applications in Section 7.2. P
RELIMINARIES , D
EFINITIONS , AND N OTATION
Let S be the Schwartz class, S P be the subspace of S made up of functions with vanishingmoments of all order up to P , and S ∞ be the intersection of S P for all P ∈ N . We give S thestandard Schwartz semi-norm topology defined via ρ α , β ( f ) = sup x ∈ R n | x α · D β f ( x ) | . It is easy to verify that S P for P ∈ N and S ∞ are closed subspaces of S . Hence we can give S P and S ∞ the Frech´et topology endowed by the Schwartz semi-norms for S . Let S (cid:48) , S (cid:48) P , and S (cid:48) ∞ be the dual spaces of S , S P , and S ∞ respectively, which we refer to as tempered distribu-tions, tempered distributions modulo polynomials of degree P , and tempered distributions modulopolynomials, respectively. Let D = C ∞ be the space of smooth compactly supported functions,and define D P to be the subspace of D made up of all function with vanishing moments up to or-der P . We endow D with the topology characterized by the following sequential convergence: for f j , f ∈ D , we say f j → f in D if there exists a compact set K ⊂ R n such that supp ( f ) , supp ( f j ) ⊂ K for all j and D α f j → D α f uniformly as j → ∞ for all α ∈ N n . It follows that D P is a closed sub-space of D for any P ∈ N , and hence we endow D P with the topology inherited from D . Let D (cid:48) and D (cid:48) P be the dual spaces of D and D P , respectively, which we refer to as distributions anddistributions modulo polynomials of degree at most M .For 1 < p < ∞ , a non-negative locally integrable function w belongs to the Muckenhoupt weightclass A p if [ w ] A p = sup Q (cid:18) | Q | (cid:90) Q w ( x ) dx (cid:19) (cid:18) | Q | (cid:90) Q w ( x ) − p (cid:48) / p dx (cid:19) p / p (cid:48) < ∞ , where the supremum is taken over all cubes Q ⊂ R n , and w belongs to A if there exists a constant C > M w ( x ) ≤ Cw ( x ) , where M is the Hardy-Littlewood maximal operator. Also define A ∞ to be the union of all A p for 1 ≤ p < ∞ .Let ψ ∈ S ∞ such that (cid:98) ψ is supported in the annulus 1 / < | ξ | < (cid:98) ψ ( ξ ) ≥ c > c > / < | ξ | < /
3. For w ∈ A ∞ , define ˙ F s , qp , w to be the collection of f ∈ S (cid:48) ∞ such that || f || ˙ F s , qp , w = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:32) ∑ k ∈ Z ( sk | ψ k ∗ f | ) q (cid:33) / q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L pw < ∞ LUCAS CHAFFEE, JAROD HART, AND LUCAS OLIVEIRA for 0 < p , q < ∞ and s ∈ R , and define ˙ B s , qp , w to be the collection of all f ∈ S (cid:48) ∞ such that || f || ˙ B s , qp , w = (cid:32) ∑ k ∈ Z ( sk || ψ k ∗ f || L pw ) q (cid:33) / q < ∞ for 0 < p ≤ ∞ , 0 < q < ∞ , and s ∈ R . Also define ˙ F s , q ∞ , w to be the collection of all f ∈ S (cid:48) ∞ such that (cid:107) f (cid:107) ˙ F s , q ∞ , w = sup Q w ( Q ) (cid:90) Q ∑ k ∈ Z :2 − k ≤ (cid:96) ( Q ) ( sk | ψ k ∗ f ( x ) | ) q / q < ∞ , where the supremum is taking over all cubes Q ⊂ R n with sides parallel to the axes and (cid:96) ( Q ) denotes the side length of Q . Finally define ˙ B s , ∞∞ , w = ˙ B s , ∞∞ for s ∈ R to be the collection of f ∈ S (cid:48) ∞ such that (cid:107) f (cid:107) ˙ B s , ∞∞ = sup k ∈ Z sk (cid:107) ψ k ∗ f (cid:107) L ∞ < ∞ . Taking these spaces modulo polynomials makes them Banach spaces for 1 ≤ p , q < ∞ and s ∈ R and quasi-Banach spaces when 0 < p , q < ∞ and s ∈ R . Note that by the work in [17] for theunweighted setting and [8] for the weighted setting, it follows that S ∞ is dense in ˙ F s , qp for all0 < p , q < ∞ and s ∈ R . It also follows that H pw = L pw = ˙ F , p , w for all 1 < p < ∞ and w ∈ A p , H pw = ˙ F , p , w for all 0 < p < ∞ and w ∈ A ∞ , and ˙ W s , pw = ˙ F s , p , w for all 1 < p < ∞ and w ∈ A p . Here L pw denote weighted Lebesgue spaces, H pw denote weighted Hardy spaces, and ˙ W s , pw denote weightedhomogeneous Sobolev spaces. Even more, ˙ B s , ∞∞ = ˙ Λ s is the (homogeneous) space of Lipschitzfunctions when s > s ∈ N , B s , ∞∞ is the Zygmund class of smooth functions,which strictly contain ˙ Λ s ; see [46] for more on Zygmund’s smooth functions. For s =
0, the space˙ B , ∞∞ is sometimes referred to as the Bloch space, and it closely related to certain Bergman spaces;see for example [12] for more on this.Finally, we note also that ˙ F s , ∞ , w = ˙ F s , ∞ = I s ( BMO ) for all s > w ∈ A ∞ , which was proved in[23]. Here I s ( BMO ) are Sobolev- BMO spaces for s >
0, and we take the convention I ( BMO ) =
BMO . See [37, 41, 42, 23] for more information on these spaces.Let X be a closed subspace of S ( R n ) . We say that a linear operator T mapping X into S (cid:48) ( R n ) is continuous if there exists a distribution kernel K ∈ S (cid:48) ( R n ) such that (cid:104) T f , g (cid:105) = (cid:104) K , g ⊗ f (cid:105) = (cid:90) R n K ( x , y ) g ( x ) f ( y ) dy dx for all f ∈ X and g ∈ S ( R n ) . Here and throughout this article, any integral that has K ( x , y ) in theintegrand should be interpreted as a dual pairing between S (cid:48) ( R n ) and S ( R n ) . We will use thisnotion of continuity when X is S and S P for P ∈ N at various points throughout the article. Itis obvious that continuity from S into S (cid:48) implies continuity from S P into S (cid:48) for P ∈ N , whichimplies continuity from S P + into S (cid:48) and from S ∞ into S (cid:48) .We consider operators that are continuous from S M into S (cid:48) since it make it easier in somesituations to initially define and work with operators. For example, consider the negative indexderivative operator | ∇ | − ν f defined via the Fourier multiplier | ξ | − ν for ν >
0. For 0 < ν < n , it iseasy to define | ∇ | − ν f for f ∈ S since | ξ | − ν is locally integrable for such ν , but it is a little moretedious to define | ∇ | − ν when ν ≥ n . Since we allow for our operators to be defined a priori only NALYSIS OF HYPER-SINGULAR, FRACTIONAL, AND ORDER-ZERO SINGULAR INTEGRAL OPERATORS 7 on S P for some P ∈ N , it is much easier to work with such operators. For any ν >
0, we choose P ≥ ν , and it follows that (cid:92) | ∇ | − ν f ( ξ ) is uniformly bounded for f ∈ S P . So | ∇ | − ν trivially definesa continuous operator from S P into S (cid:48) as long as P ≥ ν . This type of issue is less severe forthe fractional derivative operator | ∇ | ν , defined in the same way via the Fourier multiplier | ξ | ν , butusing S P rather than S may still be of value. This is because | ξ | ν is not smooth at the origin (forcertain ν > f ∈ S P for some P ∈ N smooths this non-regularity, at least to somedegree.Assuming that T is continuous from S P into S (cid:48) makes defining the transpose of T a little tricky.In this paper we will impose on a given operator T that both T and T ∗ are continuous from S P into S (cid:48) for some P ∈ N . By this we mean that T is continuous from S P into S (cid:48) , and there existsanother operator S that is also continuous from S P into S (cid:48) such that (cid:104) T f , g (cid:105) = (cid:104) Sg , f (cid:105) for all f , g ∈ S P . We call this operator S = T ∗ the transpose of T . It should be noted that we assume that T ∗ is continuous from S P into S (cid:48) . This does not necessarily follow from the continuity of T from S P into S (cid:48) . Definition 2.1.
Let ν ∈ R , M ≥ < γ ≤
1. A linear operator T is in the classof ν -order Singular Integral Operators, denoted T ∈ SIO ν ( M + γ ) , if T and T ∗ are continuous from S P into S (cid:48) from some P ∈ N , there is a kernel function K ( x , y ) such that (cid:104) T f , g (cid:105) = (cid:90) R n K ( x , y ) f ( y ) g ( x ) dy dx for any pair ( f , g ) ∈ D P × C ∞ or ( f , g ) ∈ C ∞ × D P with disjoint support, | D α x D β y K ( x , y ) | (cid:46) | x − y | n + ν + | α | + | β | for all x (cid:54) = y and α , β ∈ N n with | α | + | β | ≤ M , and | D α x D β y K ( x + h , y ) − D α x D β y K ( x , y ) | + | D α x D β y K ( x , y ) − D α x D β y K ( x , y + h ) | (cid:46) | h | γ | x − y | n + ν + M + γ for all x , y , h ∈ R n satisfying | h | < | x − y | / α , β ∈ N n satisfying | α | + | β | = M . We will referto SIO ν as the union of all SIO ν ( M + γ ) for M ∈ N n and 0 < γ ≤ SIO ν ( ∞ ) as the intersectionof SIO ν ( M + γ ) for M ∈ N n and 0 < γ ≤ T ∈ SIO ν ( M + γ ) implies that T f can be realizedas a function for f ∈ D P and x / ∈ supp ( f ) . Indeed, for such f , T f ∈ S (cid:48) by assumption. Thenby taking appropriate kernel functions ϕ k ∈ C ∞ that generate an approximation to the identity, thekernel representation of T and dominated convergence imply that ϕ k ∗ T f ( x ) = (cid:104) T f , ϕ x (cid:105) convergesas k → ∞ as long as x / ∈ supp ( f ) . Hence we can realize the distribution T f ∈ S (cid:48) pointwise as T f ( x ) = lim k → ∞ ϕ k ∗ T f ( x ) = (cid:90) R n K ( x , y ) f ( y ) dy when f ∈ D P and x / ∈ supp ( f ) . Definition 2.2.
For ν ∈ R , M ∈ N , and 0 < γ ≤
1, an operator T ∈ SIO ν ( M + γ ) is a ν -orderCalder´on-Zygmund Operator, denoted T ∈ CZO ν ( M + γ ) , if T can be continuously extended toan operator from ˙ W ν , p into L p for all 1 < p < ∞ . We will also refer to CZO ν as the union of all LUCAS CHAFFEE, JAROD HART, AND LUCAS OLIVEIRA
CZO ν ( M + γ ) for M ∈ N and 0 < γ ≤ CZO ν ( ∞ ) as the intersection of all CZO ν ( M + γ ) for M ∈ N and 0 < γ ≤ Definition 2.3.
An operator T ∈ SIO ν satisfies the ν -order Weak Boundedness Property (WBP ν )if there are integers M , N ≥ C > |(cid:104) T ψ , ϕ (cid:105)| + |(cid:104) T ∗ ψ , ϕ (cid:105)| ≤ C | B | − ν / n (2.1)for any ball B ⊂ R n , ψ ∈ D M and ϕ ∈ C ∞ with supp ( ψ ) ∪ supp ( ϕ ) ⊂ B and || D α ψ || L ∞ , || D α ϕ || L ∞ ≤| B | −| α | / n for | α | ≤ N . Remark . Note that the definition of
SIO ν and W BP ν are both symmetric under T and T ∗ , butthis is not always the case for CZO ν . When ν (cid:54) = T ∈ CZO ν and T ∗ ∈ CZO ν are not equivalentconditions. On the other hand when ν = CZO is closed under transposes and actually collapsesto the traditional definition of a Calder´on-Zygmund operator.For a function F on R n , x ∈ R n , and an integer L ≥
0, we define the Taylor polynomial (some-times also called the jet) centered at x by J Lx [ F ]( x ) = ∑ | α |≤ L D α F ( x ) α ! ( x − x ) α . Definition 2.5.
Let ν ∈ R , M ≥ < γ ≤ T ∈ SIO ν ( M + γ ) . Let P ∈ N be the integer specified for the kernel representation in the definition of T ∈ SIO ν ( M + γ ) , andwithout loss of generality assume P ≥ M + | ν | . Let η ∈ D P such that η = B ( , ) , and define η R ( x ) = η ( x / R ) for R > x ∈ R n . For α ∈ N n with | α | < M + ν + γ , define T ∗ ( x α ) ∈ D (cid:48) P by (cid:104) T ( x α ) , ψ (cid:105) = lim R → ∞ (cid:104) T ( x α · η R ) , ψ (cid:105) = lim R → ∞ (cid:90) R n K ( x , y ) y α η R ( y ) ψ ( x ) dy dx for ψ ∈ D P . Here K ∈ S (cid:48) ( R n ) denotes the distributional kernel of T , and the integrals aboveshould be interpreted as the dual pairing between S ( R n ) and S (cid:48) ( R n ) . Note that if M + ν + γ < T ( x α ) for any α ∈ N . Furthermore, we will simply state that T ( x α ) = P ≥ T ( x α ) = D (cid:48) P .It is a somewhat standard argument by now to show that T ( x α ) is well-defined. We provide abrief sketch of this since our definition is slightly different than others that have appeared; compare,for example, with the corresponding definitions in [18, 16, 44, 21, 22, 23]. Proof.
Let η R ∈ D P be as in Definition 2.5. Let ψ ∈ D P with supp ( ψ ) ⊂ B ( , R / ) . For | α | < M + ν + γ and R > (cid:104) T ( x α · η R ) , ψ (cid:105) = (cid:104) T ( x α · η R ) , ψ (cid:105) + lim R → ∞ (cid:90) R n (cid:0) K ( x , y ) − J M [ K ( · , y )]( x ) (cid:1) y α ( η R ( y ) − η R ( y )) ψ ( x ) dy dx = (cid:104) T ( x α · η R ) , ψ (cid:105) + (cid:90) R n (cid:0) K ( x , y ) − J M [ K ( · , y )]( x ) (cid:1) y α ( − η R ( y )) ψ ( x ) dy dx . The first term is well-defined since T is maps S P into S (cid:48) and η ∈ D P implies x α · η R ∈ D P for | α | < M + ν + γ . The second term is also well-defined by the support properties of η R ( x ) − η R ( x ) , NALYSIS OF HYPER-SINGULAR, FRACTIONAL, AND ORDER-ZERO SINGULAR INTEGRAL OPERATORS 9 the kernel representation of T , and the vanishing moment properties of ψ . In fact, dominatedconvergence can be applied to the second term since (cid:12)(cid:12)(cid:0) K ( x , y ) − J M [ K ( · , y )]( x ) (cid:1) y α ( η R ( y ) − η R ( y )) (cid:12)(cid:12) (cid:46) R M + γ ( R + | y | ) n + ν + M + γ −| α | for x ∈ supp ( ψ ) . Therefore T ( x α ) is also well defined. Furthermore, it is not hard to see thatthe definition of T ( x α ) does not depend on the particular function η R ∈ D P chosen in Definition2.5. (cid:3) Throughout, we will use the notation Φ Nk ( x ) = kn ( + k | x | ) − N for N > k ∈ Z , and x ∈ R n .It is well known that Φ Nk ∗ | f | ( x ) (cid:46) M f ( x ) for any locally integrable function f and N > n , wherethe constant may depend on N and M is the Hardy-Littlewood maximal operator. It is also wellknown that Φ Nk ∗ Φ Mj ( x ) (cid:46) Φ min ( M , N ) min ( j , k ) ( x ) for any j , k ∈ Z , M , N > M , N > n , and theconstant depends on M , N , but not on j , k , x .The following lemmas are also well-known. More information can be found for example in [17]. Lemma 2.6.
Let P ≥ be an integer. There exist functions ψ ∈ D P and (cid:101) ψ ∈ S ∞ such thatf ( x ) = ∑ k ∈ Z ψ k ∗ (cid:101) ψ k ∗ f ( x ) in S ∞ for any f ∈ S ∞ . Furthermore, ψ and (cid:101) ψ can be chosen to be radial. Lemma 2.7.
Let P ≥ be an integer. There exist functions φ ∈ D P and (cid:101) φ ∈ S ∞ and an integerN ∈ Z such that f ( x ) = ∑ k ∈ Z ∑ Q : (cid:96) ( Q )= − ( k + N ) (cid:101) φ k ∗ f ( c Q ) φ k ( x − c Q ) in S ∞ for any f ∈ S ∞ . The sum in Q here is over all dyadic cubes of side length − ( k + N ) and c Q denotes the lower-left hand corner of Q. Furthermore, (cid:101) φ and φ can be chosen to be radial. Throughout this article, we will choose convolution kernel functions ψ , (cid:101) ψ , φ , and (cid:101) φ to be radialso that they are self-transpose; that is, so that (cid:10) ψ j ∗ f , g (cid:11) = (cid:10) ψ j ∗ g , f (cid:11) . We will make the sameconvention when working with approximation to the identity operators and functions ψ used todefine Besov and Triebel-Lizorkin spaces. Of course, this simplification is not necessary, but iteases the need for complicated notation.Define M rj f ( x ) = M ∑ Q : (cid:96) ( Q )= − ( j + N ) f ( c Q ) · χ Q r ( x ) / r . Lemma 2.8.
Let f : R n → C be a non-negative continuous function, µ > , and nn + µ < r ≤ . Then ∑ Q : (cid:96) ( Q )= − ( j + N ) | Q | Φ n + µ min ( j , k ) ( x − c Q ) f ( c Q ) (cid:46) max ( , j − k ) µ M rj f ( x ) for all x ∈ R n . The properties of M rj in Lemma 2.8 can, at least in part, be attributed to Frazier and Jawerth,but can be found in several other places as well. A proof of it, as stated here, can also be found in[22, Proposition 2.2]. Lemma 2.9.
For any < p , q < ∞ , < r < min ( p , q ) , t ∈ R , w ∈ A p / r and f ∈ ˙ F t , qp , w , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:32) ∑ j ∈ Z (cid:104) t j M rj ( (cid:101) φ j ∗ f ) (cid:105) q (cid:33) / q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L pw (cid:46) || f || ˙ F t , qp , w , where (cid:101) φ is chosen as in Lemma 2.7. Lemma 2.9 is implicit in the work of Bui in [8]. Indeed Lemma 2.9 can be proved with a standardargument using the weighted version of the Fefferman-Stein vector-valued maximal inequalityproved in [2] and the discrete Littlewood-Paley characterization of ˙ F t , qp , w proved in [8]. See also[32], where they prove this result in the setting of weighted Hardy spaces (i.e. for t = q = MOOTHLY T RUNCATED K ERNEL E STIMATES
In this section, we start to work with operators T ∈ SIO ν ( M + γ ) . A priori, we are very limitedin what we can do with such operators. We will impose a little more structure on T through the W BP ν and T ∗ ( x α ) = T that make it easier to work with. Our first result provides an integral representation andestimates for ψ k ∗ T f , which will be useful for working with boundedness properties of T onBesov and Triebel-Lizorkin spaces. These estimates are similar to ones proved in [22]. Thoughit should be noted that the corresponding result in [22] is only for order-zero operators and itassumes that the operator is L -bounded, whereas we address ν order operators and only assumeWeak Boundedness Properties here. Theorem 3.1.
Let ν ∈ R , M , L ≥ be integers satisfying ν ≤ L ≤ M + ν , < γ ≤ , and T ∈ SIO ν ( M + γ ) . Assume that T satisfies the W BP ν and that T ∗ ( x α ) = for | α | ≤ L. Then for any ψ ∈ D P , with P sufficiently large, there is a kernel function θ k ( x , y ) such that ψ k ∗ T f ( x ) = (cid:90) R n θ k ( x , y ) f ( y ) dyfor f ∈ S P , where θ k ( x , y ) satisfies | D α y θ k ( x , y ) | (cid:46) ( ν + | α | ) Φ n + ν + M + γ k ( x − y ) for | α | ≤ (cid:98) L − ν (cid:99) (3.1) | D α y θ k ( x , y + h ) − D α y θ k ( x , y ) | (cid:46) ( ν + | α | ) k ( k | h | ) γ (cid:48) Φ n + ν + M + γ k ( x − y ) for | α | = (cid:98) L − ν (cid:99) (3.2) for all x , y , h ∈ R n with | h | < ( − k + | x − y | ) / and any < γ (cid:48) < γ .Proof. Fix P ∈ N large enough so that P ≥ M , T satisfies W BP ν with parameters M = N = P ,the kernel representation for T is valid for f , g ∈ D P with disjoint support, and T ∗ ( x α ) = D (cid:48) P for all | α | ≤ L . Let ψ ∈ D P , and assume without loss of generality that supp ( ψ ) ⊂ B ( , ) . NALYSIS OF HYPER-SINGULAR, FRACTIONAL, AND ORDER-ZERO SINGULAR INTEGRAL OPERATORS 11
Since T is continuous from S P into S (cid:48) , we have T ∗ ψ xk ∈ S (cid:48) for all x ∈ R n and k ∈ Z , where ψ xk ( y ) = ψ k ( x − y ) . This is the distribution kernel of ψ k ∗ T f in the sense that ψ k ∗ T f ( x ) = (cid:104) T f , ψ xk (cid:105) = (cid:104) T ∗ ψ xk , f (cid:105) for f ∈ S P . So we would like to define θ k ( x , y ) = T ∗ ψ xk ( y ) , but this expression may not be well-defined pointwise since T ∗ ψ xk is a priori only a distribution (not necessarily a function in y ). How-ever, we will show that this is merely a technicality, and that T ∗ ψ xk , as a distribution, agrees withintegration against a function T ∗ ψ xk ( y ) .Let ϕ ∈ C ∞ with integral 1 such that (cid:101) ψ ( x ) = n ϕ ( x ) − ϕ ( x ) and (cid:101) ψ ∈ D P . Define P k f = ϕ k ∗ f and (cid:101) Q (cid:96) f = (cid:101) ψ (cid:96) ∗ f . Note that P N f → f as N → ∞ in S for f ∈ S , and so P N T ∗ ψ xk → T ∗ ψ xk as N → ∞ in S (cid:48) . Furthermore, P N T ∗ ψ xk ( y ) = (cid:10) T ∗ ψ xk , ϕ yN (cid:11) is a function in y for all x ∈ R n and k , N ∈ Z by the definition of distributional convolution (in fact, it is a C ∞ function of tempered growth since T ∗ ψ xk ∈ S (cid:48) ). Define θ k ( x , y ) = lim N → ∞ P N T ∗ ψ xk ( y ) . (3.3)It should be noted that from what we have shown so far, it is not clear yet that the limit in (3.3)exits for all x , y ∈ R n and k ∈ Z . We will show that this is indeed the case and that the kernelfunction defined in (3.3) satisfies estimates (3.1) and (3.2). Assuming for the moment that (3.3)holds pointwise and that θ k satisfies estimates (3.1) and (3.2) (which we will prove), by the S (cid:48) convergence P N T ∗ ψ xk → T ∗ ψ xk and by dominated convergence, we have ψ k ∗ T f ( x ) = (cid:104) T ∗ ψ xk , f (cid:105) = lim N → ∞ (cid:90) R n P N T ∗ ψ xk ( y ) f ( y ) dy = (cid:90) R n θ k ( x , y ) f ( y ) dy for all f ∈ S P . So to complete the proof, we must show that the limit in (3.3) exists for each x , y , k and that θ k satisfies (3.1) and (3.2).For | x − y | > − k , it follows from the kernel representation of T that T ∗ ψ xk is a continu-ous function on a sufficiently small neighborhood of y for any fixed x ∈ R n and k ∈ Z , andthat P N T ∗ ψ xk ( y ) → T ∗ ψ xk ( y ) pointwise as N → ∞ . In particular, the limit in (3.3) exists when | x − y | > − k . Furthermore, still assuming that | x − y | > − k , by the kernel representation for T and the support properties of ψ k , it follows that T ∗ ψ xk is M -times differentiable on a neighborhoodof y . For | α | ≤ (cid:98) L − ν (cid:99) , we have | D α y θ k ( x , y ) | = | D α y T ∗ ψ xk ( y ) | = (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) R n (cid:16) D α K ( u , y ) − J M −| α | x [ D α K ( · , y )]( u ) (cid:17) ψ k ( u − x ) du (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:90) R n | x − u | M −| α | + γ | x − y | n + ν + M + γ | ψ k ( u − x ) | du (cid:46) ( ν + | α | ) k Φ n + ν + M + γ k ( x − y ) . If | h | ≥ − k , then it trivially follows that | D α y θ k ( x , y + h ) − D α y θ k ( x , y ) | (cid:46) ( ν + | α | ) k (cid:104) Φ n + ν + M + γ k ( x − y ) + Φ n + ν + M + γ k ( x − y − h ) (cid:105) (cid:46) ( ν + | α | ) k ( k | h | ) γ (cid:104) Φ n + ν + M + γ k ( x − y ) + Φ n + ν + M + γ k ( x − y − h ) (cid:105) . Otherwise we assume that | h | < − k , and it follows that | x − y − h | ≥ | x − y | − | h | > | x − y | / ≥ − k .In this situation, we consider two cases: if | α | = M and if | α | < M . When | α | = M , we have | D α y θ k ( x , y + h ) − D α y θ k ( x , y ) | = (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) R n (cid:0) D α y K ( u , y + h ) − D α y K ( u , y ) (cid:1) ψ k ( u − x ) du (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:90) R n | h | γ | u − y | n + ν + | α | + γ | ψ k ( u − x ) | du (cid:46) ( ν + | α | ) k ( k | h | ) γ Φ n + ν + M + γ k ( x − y ) . Now assume that | α | < M . Then | D α y θ k ( x , y + h ) − D α y θ k ( x , y ) | = (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) R n (cid:0) D α y K ( u , y + h ) − D α y K ( u , y ) (cid:1) ψ k ( u − x ) du (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) R n (cid:16)(cid:0) D α y K ( u , y + h ) − D α y K ( u , y ) (cid:1) − J M − −| α | x [ D α y K ( · , y + h ) − D α y K ( · , y )]( u ) (cid:17) ψ k ( u − x ) du (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) R n ∑ | β | = M −| α | | D β D α K ( ξ , y + h ) − D β D α K ( ξ , y ) | | u − x | M −| α | | ψ k ( u − x ) | du for some ξ = cx + ( − c ) u with 0 < c < (cid:46) (cid:90) R n | h | γ | x − u | M −| α | | ξ − y | n + ν + M + γ | ψ k ( u − x ) | du (cid:46) ( ν + | α | ) k ( k | h | ) γ Φ n + ν + M + γ k ( x − y ) . So θ k ( x , y ) is well-defined and satisfies (3.1) and (3.2) when x and y are far apart.When | x − y | ≤ − k , we decompose P N T ∗ ψ xk further, P N T ∗ ψ xk ( y ) = N − ∑ (cid:96) = k (cid:101) Q (cid:96) T ∗ ψ xk ( y ) + P k T ∗ ψ xk ( y ) . (3.4)Recall that ϕ ∈ C ∞ was chosen so that (cid:101) ψ (cid:96) = ϕ (cid:96) + − ϕ (cid:96) ∈ D P . For the remainder of the proof, wedrop the tilde on top of (cid:101) ψ for the sake of simplifying notation. This causes an overlap in notationbetween (cid:101) ψ (cid:96) and ψ k , which is ultimately harmless, but the distinction can still be recovered at anypoint in the remainder of the proof by identifying the subscripts, (cid:96) versus k .Let α ∈ N n with | α | ≤ (cid:98) L − ν (cid:99) . Using the hypothesis T ∗ ( x µ ) = | µ | ≤ L we write (cid:12)(cid:12) D α y (cid:10) T ψ y (cid:96) , ψ xk (cid:11)(cid:12)(cid:12) ≤ | A (cid:96), k ( x , y ) | + | B (cid:96), k ( x , y ) | , where A (cid:96), k ( x , y ) = (cid:96) | α | (cid:90) R n T (( D α ψ ) y (cid:96) )( u ) (cid:0) ψ xk ( u ) − J Ly [ ψ xk ]( u ) (cid:1) η − (cid:96) ( u − y ) du , B (cid:96), k ( x , y ) = lim R → ∞ (cid:96) | α | (cid:90) R n T (( D α ψ ) y (cid:96) )( u ) (cid:0) ψ xk ( u ) − J Ly [ ψ xk ]( u ) (cid:1) ( η R ( u ) − η − (cid:96) ( u − y )) du , where η R ∈ D P with η R ( x ) = η ( x / R ) , supp ( η ) ⊂ B ( , ) , and η = B ( , ) . We apply the W BP ν property to estimate A (cid:96), k as follows, | A (cid:96), k ( x , y ) | = (cid:96) | α | ( (cid:96) + k ) n ( L + γ )( k − (cid:96) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:42) T (cid:18) ( D α ψ ) y (cid:96) (cid:96) n (cid:19) , ψ xk − J Ly [ ψ xk ] kn ( L + γ )( k − (cid:96) ) η − (cid:96) ( · − y ) (cid:43)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:96) | α | (cid:96) ν ( L + γ )( k − (cid:96) ) kn = k ( ν + | α | ) ( L + γ − ν −| α | )( k − (cid:96) ) kn . NALYSIS OF HYPER-SINGULAR, FRACTIONAL, AND ORDER-ZERO SINGULAR INTEGRAL OPERATORS 13
Note that supp (cid:0) − (cid:96) n ( D α ψ ) y (cid:96) (cid:1) ⊂ B ( y , − (cid:96) ) and || D µ (cid:0) − (cid:96) n ( D α ψ ) y (cid:96) (cid:1) || L ∞ (cid:46) | µ | (cid:96) for all µ ∈ N n , wherethe associated constants are independent of x , y , (cid:96), k . Similarly, we have supp ( η − (cid:96) ( · − y )) ⊂ B ( y , − (cid:96) ) and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D µ (cid:32) ψ xk ( u ) − J Ly [ ψ xk ]( u ) kn ( L + γ )( k − (cid:96) ) η − (cid:96) ( u − y ) (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:96) | µ | for all µ ∈ N n as long as k ≤ (cid:96) , where the associated constant does not depend on u , x , y , k , (cid:96) . Henceit is an appropriate function to apply W BP ν here.Let γ (cid:48) , γ (cid:48)(cid:48) > γ (cid:48) < γ (cid:48)(cid:48) < γ . The B (cid:96), k term is bounded using the kernel representation of T in the following way | B (cid:96), k ( x , y ) | ≤ (cid:96) | α | lim sup R → ∞ (cid:90) | u − y | > − (cid:96) (cid:90) R n (cid:12)(cid:12) K ( u , v ) − J My [ K ( u , · )]( v ) (cid:12)(cid:12) | ( D α ψ ) y (cid:96) ( v ) | dv × (cid:12)(cid:12) ψ xk ( u ) − J Ly [ ψ xk ]( u ) (cid:12)(cid:12) | η R ( u ) | du (cid:46) (cid:96) | α | ∞ ∑ m = (cid:90) m − (cid:96) < | u − y |≤ m + − (cid:96) (cid:32) (cid:90) R n − ( M + γ ) (cid:96) ( n + ν + M + γ )( m − (cid:96) ) | ( D α ψ ) y (cid:96) ( v ) | dv (cid:33) kn ( k m − (cid:96) ) L + γ (cid:48)(cid:48) du (cid:46) k ( n + ν + | α | ) ( L + γ (cid:48)(cid:48) − ν −| α | )( k − (cid:96) ) ∞ ∑ m = − ( ν + M + γ − L − γ (cid:48)(cid:48) ) m (cid:46) k ( n + ν + | α | ) ( L + γ (cid:48)(cid:48) − ν −| α | )( k − (cid:96) ) . Here we used that ν + M + γ > L + γ (cid:48)(cid:48) since L ≤ M + ν and γ (cid:48)(cid:48) < γ . It is not crucial here that wetook γ (cid:48) < γ (cid:48)(cid:48) < γ , but this estimate will be used again later where our choice of γ (cid:48) < γ (cid:48)(cid:48) will beimportant. It follows that ∑ (cid:96) ≥ k (cid:96) | α | (cid:12)(cid:12)(cid:10) T (( D α ψ ) y (cid:96) ) , ψ xk (cid:11)(cid:12)(cid:12) (cid:46) k ( n + ν + | α | ) (cid:46) k ( ν + | α | ) Φ n + ν + M + γ k ( x − y ) . Here we use that α ∈ N n must satisfy | α | ≤ L − ν , which implies that ν + | α | < L + γ (cid:48)(cid:48) . This verifiesthat the limit of (3.4) as N → ∞ exists for | x − y | ≤ − k as well, and that the first term on the righthand side of (3.4) satisfies (3.1). Hence θ k is well-defined by (3.3). Since T satisfies the W BP ν , italso follows that | D α y P k T ∗ ψ xk ( y ) | = | α | k kn (cid:12)(cid:12)(cid:12)(cid:12)(cid:28) T ∗ (cid:18) ψ xk ( u ) kn (cid:19) , ( D α ϕ ) k ( · − y ) kn (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ( n + ν + | α | ) k . Then second term on the right hand side of (3.4) also satisfies (3.1). Therefore θ k satisfies (3.1).We also verify the γ (cid:48) -H¨older regularity estimate (3.2): let α ∈ N n with | α | = L . It triviallyfollows from the estimates already proved that ∑ (cid:96) ≥ k : 2 − (cid:96) < | h | (cid:12)(cid:12)(cid:12)(cid:68) D α y T ( ψ y + h (cid:96) − ψ y (cid:96) ) , ψ xk (cid:69)(cid:12)(cid:12)(cid:12) (cid:46) k ( ν + | α | ) ( k | h | ) γ (cid:48) (cid:16) Φ n + ν + M + γ k ( x − y ) + Φ n + ν + M + γ k ( x − y − h ) (cid:17) ∑ (cid:96) ≥ k : 2 − (cid:96) < | h | ( L + γ (cid:48)(cid:48) − γ (cid:48) − ν −| α | )( k − (cid:96) ) (cid:46) k ( ν + | α | ) ( k | h | ) γ (cid:48) (cid:16) Φ n + ν + M + γ k ( x − y ) + Φ n + ν + M + γ k ( x − y − h ) (cid:17) . Note that ν + | α | < L + γ (cid:48)(cid:48) − γ (cid:48) since we chose γ (cid:48) < γ (cid:48)(cid:48) . On the other hand, for the situation where | h | ≤ − (cid:96) , we consider ∑ (cid:96) ≥ k : 2 − (cid:96) ≥| h | (cid:12)(cid:12)(cid:12)(cid:68) D α T ( ψ y + h (cid:96) − ψ y (cid:96) ) , ψ xk (cid:69)(cid:12)(cid:12)(cid:12) ≤ | A (cid:96), k ( x , y , h ) | + | B (cid:96), k ( x , y , h ) | , where A (cid:96), k ( x , y , h ) = (cid:96) | α | (cid:90) R n T (( D α ψ ) y + h (cid:96) − ( D α ψ ) y (cid:96) )( u ) (cid:0) ψ xk ( u ) − J Ly [ ψ xk ]( u ) (cid:1) η − (cid:96) ( u − y ) du and B (cid:96), k ( x , y , h ) = lim R → ∞ (cid:96) | α | (cid:90) R n T (( D α ψ ) y + h (cid:96) − ( D α ψ ) y (cid:96) )( u ) (cid:0) ψ xk ( u ) − J Ly [ ψ xk ]( u ) (cid:1) × ( η R ( u ) − η − (cid:96) ( u − y )) du . Note that when | h | ≤ − (cid:96) , the function 2 − (cid:96) n ( (cid:96) | h | ) − γ (cid:48) (cid:104) ( D α ψ ) y + h (cid:96) − ( D α ψ ) y (cid:96) (cid:105) ∈ D P is supported in B ( y , − (cid:96) ) ∪ B ( y + h , − (cid:96) ) ⊂ B ( y , | h | + − (cid:96) ) ⊂ B ( y , − (cid:96) ) with (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:96) n ( (cid:96) | h | ) − γ (cid:48) D µ (cid:104) ( D α ψ ) y + h (cid:96) − ( D α ψ ) y (cid:96) (cid:105)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ∞ (cid:46) (cid:96) | µ | for | µ | ≤ N , where the constant does not depend on y , h , or (cid:96) ; here N is the integer specified in the W BP ν condition for T . Then using the W BP ν for T , we have have | A (cid:96), k ( x , y , h ) | ≤ (cid:96) | α | ( (cid:96) | h | ) γ (cid:48) kn ( k − (cid:96) )( L + γ ) (cid:96) n × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:42) T (cid:32) ( D α ψ ) y + h (cid:96) − ( D α ψ ) y (cid:96) (cid:96) n ( (cid:96) | h | ) γ (cid:48) (cid:33) , ψ xk − J Ly [ ψ xk ] kn ( k − (cid:96) )( L + γ ) η − (cid:96) ( · − y ) (cid:43)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:96) ( ν + | α | ) ( (cid:96) | h | ) γ (cid:48) kn ( L + γ )( k − (cid:96) ) = k ( n + ν + | α | ) ( γ − γ (cid:48) − ν )( k − (cid:96) ) ( k | h | ) γ (cid:48) . Recall the selection of γ (cid:48)(cid:48) such that 0 < γ (cid:48) < γ (cid:48)(cid:48) < γ . The B (cid:96), k term is bounded using the kernelrepresentation of T | B (cid:96), k ( x , y , h ) | ≤ (cid:96) | α | (cid:90) | u − y | > − (cid:96) (cid:90) R n | K ( u , v ) − J My [ K ( u , · )]( v ) |× | ( D α ψ ) y + h (cid:96) ( v ) − ( D α ψ ) y (cid:96) ( v ) | | ψ xk ( u ) − J Ly [ ψ xk ]( u ) | du dv (cid:46) (cid:96) | α | ∞ ∑ m = (cid:90) m − (cid:96) < | u − y |≤ m + − (cid:96) (cid:90) R n − ( M + γ ) (cid:96) ( n + ν + M + γ )( m − (cid:96) ) ( (cid:96) | h | ) γ (cid:48) × (cid:0) Φ n + (cid:96) ( y − v ) + Φ n + (cid:96) ( y + h − v ) (cid:1) dv kn ( k | u − y | ) L + γ (cid:48)(cid:48) du (cid:46) k ( n + ν + | α | ) ( k | h | ) γ (cid:48) ( L − ν −| α | + γ (cid:48)(cid:48) − γ (cid:48) )( k − (cid:96) ) ∞ ∑ m = − ( ν + M + γ − L − γ (cid:48)(cid:48) ) m (cid:46) k ( n + ν + | α | ) ( k | h | ) γ (cid:48) ( L − ν −| α | + γ (cid:48)(cid:48) − γ (cid:48) )( k − (cid:96) ) NALYSIS OF HYPER-SINGULAR, FRACTIONAL, AND ORDER-ZERO SINGULAR INTEGRAL OPERATORS 15
Again we use that ν + M + γ > L + γ (cid:48)(cid:48) . Then it follows that ∞ ∑ (cid:96) = k | A (cid:96), k ( x , y , y (cid:48) ) | + | B (cid:96), k ( x , y , y (cid:48) ) | (cid:46) k ( n + ν + | α | ) ( k | h | ) γ (cid:48) since | α | + ν + γ (cid:48) < L + γ (cid:48)(cid:48) . This completes the estimate in (3.2) for the first term on the right handside of (3.4). Now we check that P k T ∗ ψ xk ( y ) , the second term from the right hand side of (3.4), alsosatisfies the γ (cid:48) -H¨older estimate. The estimate is trivial when | h | ≥ − k . Since ( k | h | ) − γ (cid:48) ( ψ x + hk − ψ xk ) ∈ D P with the appropriate derivative estimates. When | h | ≤ − k , it follows from the WBP ν for T that | D α y P k T ∗ ( ψ x + hk − ψ xk )( y ) | = ( k | h | ) γ (cid:48) | α | k kn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:42) T ∗ (cid:32) ( ψ x + hk − ψ xk ) kn ( k | h | ) γ (cid:48) (cid:33) , ( D α ϕ yk ) kn (cid:43)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ( k | h | ) γ (cid:48) ( n + ν + | α | ) k . Hence both terms on the right hand side of (3.4) satisfy the appropriate estimates, and hence sodoes θ k ( x , y ) . This completes the proof. (cid:3) Corollary 3.2.
Let ν ∈ R , M , L ≥ be integers satisfying ν ≤ L ≤ M + ν , < γ ≤ , and T ∈ SIO ν ( M + γ ) . Assume that T ∗ ( x α ) = for | α | ≤ L and that T ∈ W BP ν . Fix ψ , ˜ ψ ∈ D P for Psufficiently large, and define λ j , k ( x , y ) = (cid:68) T ∗ ψ xj , ˜ ψ yk (cid:69) for j , k ∈ Z and x , y ∈ R n . Then | D α x D β y λ j , k ( x , y ) | (cid:46) ( ν + | α | ) j + | β | k ( (cid:101) L + γ (cid:48) ) min ( , j − k ) Φ n + ν + M + γ min ( j , k ) ( x − y ) | D α x D β y λ j , k ( x + h , y ) − D α x D β y λ j , k ( x , y ) | (cid:46) j ( ν + | α | )+ k | β | ( j | h | ) γ (cid:48) − δ min ( , j − k )( (cid:101) L + δ ) Φ n + ν + M + γ min ( j , k ) ( x − y ) for all α , β ∈ N n , ≤ δ ≤ γ (cid:48) < γ and x , y , h ∈ R n satisfying | h | < ( − min ( j , k ) + | x − y | ) / , where (cid:101) L = (cid:98) L − ν (cid:99) .Proof. This follows immediately by applying Theorem 3.1 with ( D α ψ ) j ∗ T f in place of ψ k ∗ T f .When k ≥ j , we have | D α x D β y λ j , k ( x , y ) | ≤ j | α | + k | β | (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) R n (cid:16) θ j ( x , u ) − J (cid:101) Ly [ θ j ( x , · )]( u ) (cid:17) ( D β ˜ ψ ) yk ( u ) du (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) j | α | + k | β | (cid:90) R n ν j ( j | u − y | ) (cid:101) L + γ (cid:48) (cid:16) Φ n + ν + M + γ j ( x − u ) + Φ n + ν + M + γ j ( x − y ) (cid:17) | ( D β ˜ ψ ) yk ( u ) | du (cid:46) ν j j | α | + k | β | ( (cid:101) L + γ (cid:48) )( j − k ) Φ n + ν + M + γ min ( j , k ) ( x − y ) . When k < j , we have | D α x D β y λ j , k ( x , y ) | ≤ j | α | + k | β | (cid:90) R n | T ∗ ( D α ψ ) xj ( u )( D β ˜ ψ ) yk ( u ) | du (cid:46) ν j j | α | + k | β | Φ n + ν + M + γ min ( j , k ) ( x − y ) . If | h | ≥ − j , then the second estimate trivially follows from the first. When | h | ≤ − j , we have | D α x D β y λ j , k ( x + h , y ) − D α x D β y λ j , k ( x , y ) | (cid:46) j | α | + k | β | (cid:90) R n ν j ( j | h | ) γ (cid:48) ( j | u − y | ) (cid:101) L + γ | ( D β ˜ ψ ) yk ( u ) | × (cid:16) Φ n + ν + M + γ j ( x − u ) + Φ n + ν + M + γ j ( x − y ) (cid:17) du (cid:46) j ( ν + | α | )+ k | β | ( j | h | ) γ (cid:48) ( (cid:101) L + γ ) min ( , j − k ) Φ n + ν + M + γ min ( j , k ) ( x − y ) . Combining this with the first estimate yields the second one. (cid:3)
4. A R
ESTRICTED O PERATOR C ALCULUS
In this section, we prove two operator calculus type results, in Theorems 4.2 and 4.3. The firstprovides conditions on T ∈ SIO ν of the form T ∗ ( x α ) = | ∇ | − s T | ∇ | t ∈ SIO ν + t − s (tech-nically, this hold modulo polynomials), and the second provides conditions so that | ∇ | − s T | ∇ | t ∈ CZO ν + t − s . It is of particular interest to note that in Theorem 4.3, we can actually conclude that | ∇ | − s T | ∇ | t ∈ CZO ν + t − s while T only belongs to SIO ν . We provide some example later that showthere are operators T in SIO ν but not CZO ν such that | ∇ | − s T | ∇ | t ∈ CZO ν + t − s . In fact, we constructtwo classes of such example, pseudodifferential operators with symbols in the forbidden class S , and a variant of the Bony paraproduct. Lemma 4.1.
For any M > − n such that N > n + M and x (cid:54) = ∑ j ∈ Z jM Φ Nj ( x ) (cid:46) | x | n + M . Proof.
This estimate is straightforward to prove. For M and N as above and x (cid:54) = ∑ j ∈ Z jM Φ Nj ( x ) ≤ ∑ j ∈ Z :2 j ≤| x | − j ( n + M ) + | x | N ∑ j ∈ Z :2 j > | x | − − j ( N − n − M ) (cid:46) | x | n + M . (cid:3) Theorem 4.2.
Let ν ∈ R , M , L ≥ be integers satisfying ν ≤ L ≤ M + ν , < γ ≤ , and T ∈ SIO ν ( M + γ ) . Assume that T ∗ ( x α ) = for | α | ≤ L and that T ∈ W BP ν . Also fix s , t ∈ R satisfyingt < (cid:98) L − ν (cid:99) + γ , s > ν , and t − s < n + M + γ . Then there exists T s , t ∈ SIO ν + t − s ( γ (cid:48) ) for < γ (cid:48) < γ such that (cid:104)| ∇ | − s T | ∇ | t f , g (cid:105) = (cid:104) T s , t f , g (cid:105) for all f , g ∈ S ∞ . The kernel K s , t ( x , y ) of T s , t satisfies | D α D β K s , t ( x , y ) | (cid:46) | x − y | n + ν + t − s + | α | + | β | for all x (cid:54) = y and α , β ∈ N n satisfying | α | < s − ν , | β | < (cid:98) L − ν (cid:99) + γ − t, and | α | + | β | < n + M + γ + s − t, and | D α D β K s , t ( x + h , y ) − D α D β K s , t ( x , y ) | (cid:46) | h | γ (cid:48) | x − y | n + ν + t − s + | α | + | β | + γ (cid:48) | D α D β K s , t ( x , y + h ) − D α D β K s , t ( x , y ) | (cid:46) | h | γ (cid:48) | x − y | n + ν + t − s + | α | + | β | + γ (cid:48) for all x , y , h ∈ R n with | h | < | x − y | / , < γ (cid:48) < γ , and α , β ∈ N n satisfying | α | < s − ( ν + γ (cid:48) ) , | β | ≤ (cid:98) L − ν (cid:99) − t + γ − γ (cid:48) , and | α | + | β | < n + M + γ + s − ( t + γ (cid:48) ) . Furthermore, T s , t and T ∗ s , t arecontinuous from S P into S (cid:48) for P sufficiently larger, and can be defined (cid:104) T s , t f , g (cid:105) = ∑ j , k ∈ Z tk − s j (cid:90) R n (cid:10) T ∗ ψ xj , ψ yk (cid:11) ( | ∇ | t (cid:101) ψ ) k ∗ f ( y )( | ∇ | − s (cid:101) ψ ) j ∗ g ( x ) dx dy , NALYSIS OF HYPER-SINGULAR, FRACTIONAL, AND ORDER-ZERO SINGULAR INTEGRAL OPERATORS 17 where ψ and (cid:101) ψ as chosen as in Lemma 2.6.Proof. Let ψ ∈ D P and (cid:101) ψ ∈ S ∞ be as in Lemma 2.6. Define Λ j , k = Q k T Q j , whose kernel is givenby λ j , k ( x , y ) = (cid:68) T ∗ ψ xj , ψ yk (cid:69) for j , k ∈ Z and x , y ∈ R n . For any f , g ∈ S ∞ , it follows that (cid:10) | ∇ | − s T | ∇ | t f , g (cid:11) = ∑ j , k ∈ Z (cid:68) | ∇ | − s (cid:101) Q j Q j T Q k (cid:101) Q k | ∇ | t f , g (cid:69) = ∑ j , k ∈ Z tk − s j (cid:90) R n Q j T Q k (cid:101) Q tk f ( u ) (cid:101) Q − sj g ( u ) du = ∑ j , k ∈ Z tk − s j (cid:90) R n (cid:18) (cid:90) R n λ j , k ( u , v ) (cid:101) ψ tk ( v − y ) (cid:101) ψ − sj ( u − x ) dv du (cid:19) g ( x ) f ( y ) dy dx . Here we denote (cid:101) Q tk f = ( | ∇ | t ψ ) k ∗ f and likewise for (cid:101) Q − sj . For x (cid:54) = y , define K s , t ( x , y ) = ∑ j , k ∈ Z tk − s j (cid:90) R n λ j , k ( u , v ) (cid:101) ψ tk ( v − y ) (cid:101) ψ − sj ( u − x ) dv du . Let 0 < γ (cid:48) < γ and α , β ∈ N n satisfying | α | < s − ν , | β | < (cid:101) L + γ − t , and | α | + | β | < n + M + γ + s − t ,where (cid:101) L = (cid:98) L − ν (cid:99) . Then for all x (cid:54) = y , using Corollary 3.2, we have ∑ j , k ∈ Z tk − s j (cid:12)(cid:12)(cid:12)(cid:12) D α x D β y (cid:90) R n λ j , k ( u , v ) (cid:101) ψ tk ( v − y ) (cid:101) ψ − sj ( u − x ) dv du (cid:12)(cid:12)(cid:12)(cid:12) ≤ ∑ j , k ∈ Z ( | α |− s ) j +( t + | β | ) k (cid:90) R n | λ j , k ( u , v )( D β (cid:101) ψ ) tk ( v − y )( D α (cid:101) ψ ) − sj ( u − x ) | dv du ≤ ∑ j , k ∈ Z ( ν + | α |− s ) j +( t + | β | ) k ( (cid:101) L + γ (cid:48) ) min ( , j − k ) × (cid:90) R n Φ n + ν + M + γ min ( j , k ) ( u − v ) Φ n + ν + M + γ k ( v − y ) Φ n + ν + M + γ j ( x − u ) dv du ≤ ∑ j , k ∈ Z ( ν + | α |− s ) j +( t + | β | ) k ( (cid:101) L + γ (cid:48) ) min ( , j − k ) Φ n + ν + M + γ min ( j , k ) ( x − y ) (cid:46) ∑ j , k ∈ Z : j ≤ k ( (cid:101) L + γ (cid:48) + ν + | α |− s ) j ( t + | β |− (cid:101) L − γ (cid:48) ) k Φ n + ν + M + γ j ( x − y )+ ∑ j , k ∈ Z : j > k ( ν + | α |− s ) j ( t + | β | ) k Φ n + ν + M + γ k ( x − y ) (cid:46) ∑ j ∈ Z ( ν + | α | + | β | + t − s ) j Φ n + ν + M + γ j ( x − y ) + ∑ k ∈ Z ( ν + | α | + | β | + t − s ) k Φ n + ν + M + γ k ( x − y ) (cid:46) | x − y | n + ν + | α | + | β | + t − s . Here we use that t + | β | < (cid:101) L + γ and | α | < s − ν to assure that the summations above converge andLemma 4.1 to justify the last inequality. It follows that for x (cid:54) = y , we have | D α x D β y K s , t ( x , y ) | ≤ ∑ j , k ∈ Z tk − s j (cid:12)(cid:12)(cid:12)(cid:12) D α x D β y (cid:90) R n λ j , k ( u , v ) (cid:101) ψ tk ( v − y ) (cid:101) ψ − sj ( u − x ) dv du (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) | x − y | n + ν + | α | + | β | + t − s . Now suppose α , β ∈ N n with | α | < s − ( ν + γ (cid:48) ) , | β | < (cid:101) L + γ − γ (cid:48) − t , and | α | + | β | < n + M + γ + s − ( t + γ (cid:48) ) . For any x , y , h ∈ R n satisfying | h | < | x − y | /
2, we have ∑ j , k ∈ Z tk − s j (cid:12)(cid:12)(cid:12)(cid:12) D α x D β y (cid:90) R n λ j , k ( u , v ) (cid:101) ψ tk ( v − y )( (cid:101) ψ − sj ( u − x ) − (cid:101) ψ − sj ( u − x − h )) dv du (cid:12)(cid:12)(cid:12)(cid:12) ≤ | h | γ (cid:48) ∑ j , k ∈ Z ( ν + | α |− s + γ (cid:48) ) j +( t + | β | ) k ( (cid:101) L + γ (cid:48) ) min ( , j − k ) × (cid:90) R n Φ n + ν + M + γ min ( j , k ) ( u − v ) Φ n + ν + M + γ k ( v − y ) Φ n + ν + M + γ j ( x − u ) dv du (cid:46) | h | γ (cid:48) ∑ j ∈ Z ( ν + | α | + | β | + γ (cid:48) + t − s ) j Φ n + ν + M + γ j ( x − y )+ | h | γ (cid:48) ∑ k ∈ Z ( ν + | α | + | β | + γ (cid:48) + t − s ) k Φ n + ν + M + γ k ( x − y ) (cid:46) | h | γ (cid:48) | x − y | n + ν + | α | + | β | + γ (cid:48) + t − s . Here we use that | β | < (cid:101) L + γ − γ (cid:48) − t and | α | < s − ( ν + γ (cid:48) ) so that the summations above converge,and Lemma 4.1 for the last line. Fix γ (cid:48) < γ (cid:48)(cid:48) < γ such that | β | < (cid:101) L + γ (cid:48)(cid:48) − γ (cid:48) − t . By a similarargument, but applying Corollary 3.2 with γ (cid:48)(cid:48) in place of γ (cid:48) , it follows that ∑ j , k ∈ Z tk − s j (cid:12)(cid:12)(cid:12)(cid:12) D α x D β y (cid:90) R n λ j , k ( u , v )( (cid:101) ψ tk ( v − y ) − (cid:101) ψ tk ( v − y − h )) (cid:101) ψ − sj ( u − x ) dv du (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) | h | γ (cid:48) ∑ j , k ∈ Z : j ≤ k ( (cid:101) L + γ (cid:48)(cid:48) + ν + | α |− s ) j ( t + | β | + γ (cid:48) − (cid:101) L − γ (cid:48)(cid:48) ) k Φ n + ν + M + γ j ( x − y )+ | h | γ (cid:48) ∑ j , k ∈ Z : j > k ( ν + | α |− s ) j +( t + | β | + γ (cid:48) ) k Φ n + ν + M + γ k ( x − y ) (cid:46) | h | γ (cid:48) | x − y | n + ν + | α | + | β | + γ (cid:48) + t − s , for which we use that | β | < (cid:101) L + γ (cid:48)(cid:48) − γ (cid:48) − t and | α | < s − ν . (cid:3) Theorem 4.3.
Let ν ∈ R , L be an integer with L ≥ | ν | , ( L − ν ) ∗ < γ ≤ , and M ≥ max ( L , L − ν ) .If T ∈ SIO ν ( M + γ ) satisfies W BP ν and T ∗ ( x α ) = for all | α | ≤ L, then for each ν < s < ν + (cid:98) L − ν (cid:99) + γ and < t < (cid:98) L − ν (cid:99) + γ , there exists T s , t ∈ CZO ν + t − s such that | ∇ | − s T | ∇ | t f − T s , t f is apolynomial for all f ∈ S ∞ . NALYSIS OF HYPER-SINGULAR, FRACTIONAL, AND ORDER-ZERO SINGULAR INTEGRAL OPERATORS 19
Proof.
Assume that T ∈ SIO ν ( M + γ ) satisfies W BP ν and that there is an integer P ≥ M such that T ∗ ( x α ) = D (cid:48) P for all | α | ≤ L . Fix ψ ∈ D P and (cid:101) ψ ∈ S ∞ as in Lemma 2.6. Define Λ j , k f = Q j T Q k ,whose kernel is given by λ j , k ( x ) = (cid:68) T ∗ ψ xj , ψ yk (cid:69) . Fix s , t ∈ R so that 0 < s − ν , t < (cid:101) L + γ , and definethe operator T s , t , which is continuous from S into S (cid:48) , by T s , t f ( x ) = ∑ j , k ∈ Z tk − js (cid:101) Q − sj Λ j , k (cid:101) Q tk f ( x ) . By Theorem 4.2, it follows that T s , t ∈ SIO ν + t − s ( γ (cid:48) ) for all 0 < γ (cid:48) < γ and that | ∇ | − s T | ∇ | t f − T s , t f is a polynomial for all f ∈ S ∞ . By Corollary 3.2, it follows that | Q (cid:96) (cid:101) Q − sj Λ j , k f ( x ) | (cid:46) ν j − K | (cid:96) − j | ( (cid:101) L + γ (cid:48) ) min ( , j − k ) M f ( x ) , where we choose γ (cid:48) so that max ( , s − ν − (cid:101) L , t − (cid:101) L ) < γ (cid:48) < γ and K >
0. Then for all f ∈ S P and1 < p , q < ∞ , it follows that || T s , t f || ˙ F , qp ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:32) ∑ (cid:96) ∈ Z (cid:34) ∑ j , k ∈ Z tk − js | Q (cid:96) (cid:101) Q − sj Λ j , k (cid:101) Q tk f | (cid:35) q (cid:33) / q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p (cid:46) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:32) ∑ (cid:96) ∈ Z (cid:34) ∑ j , k ∈ Z tk − js ν j − K | (cid:96) − j | ( (cid:101) L + γ (cid:48) ) min ( , j − k ) M ( (cid:101) Q tk f ) (cid:35) q (cid:33) / q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p (cid:46) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:32) ∑ (cid:96), j , k ∈ Z q ( ν + t − s ) k ( s − ν )( k − j ) − K | (cid:96) − j | ( (cid:101) L + γ (cid:48) ) min ( , j − k ) (cid:104) M ( (cid:101) Q tk f ) (cid:105) q (cid:33) / q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p (cid:46) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:32) ∑ k ∈ Z q ( ν + t − s ) k (cid:104) M ( (cid:101) Q tk f ) (cid:105) q (cid:33) / q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p (cid:46) || f || ˙ F ν + t − s , qp . Therefore T s , t can be extended to a bounded linear operator from ˙ F ν + t − s , qp into ˙ F , qp by density. Inparticular taking q = T s , t can be extended to a bounded linear operator from ˙ W ν + t − s , p into L p for all 1 < p < ∞ . Then we have verified that T s , t ∈ CZO ν + t − s ( γ (cid:48) ) for all 0 < γ (cid:48) < γ . (cid:3)
5. B
OUNDEDNESS OF S INGULAR I NTEGRAL O PERATORS
In this section, we show that T ∗ ( x α ) = T ∈ SIO ν are sufficient for severalboundedness properties. The first two results, in Theorems 5.1 and 5.2, are on negative smooth-ness Besov and Triebel-Lizorkin spaces, and they are proved directly with the help of Corollary 3.2and Theorem 4.2. The last two boundedness results, in Corollaries 5.3 and 5.4, are consequencesof Theorems 5.1 and 5.2 by duality. To aid in the discussion of the boundedness of these spaces, wedescribe some geometric properties of the parameter space related to the Triebel-Lizorkin spaceestimates for T . Another interesting feature of the results in this section is that cancellation proper-ties T ∗ ( x α ) = T allow for us to conclude boundedness on certain weighted Triebel-Lizorkinspaces ˙ F s , qp , w , where the weight w does not belong to A p . This type of behavior for operators wasobserved in [32, 23] in relation to Hardy spaces. We should also note that the boundedness results proved in this section are also sufficient for T ∗ ( x α ) = Theorem 5.1.
Let ν ∈ R , L be an integer with L ≥ | ν | , M ≥ max ( L , L − ν ) , ( L − ν ) ∗ < γ ≤ , andT ∈ SIO ν ( M + γ ) satisfy W BP ν . If T ∗ ( x α ) = for all | α | ≤ L, then T can be extended to a boundedoperator from ˙ B ν − t , qp , w into ˙ B − t , qp , w for all < p < ∞ , < q < ∞ , w ∈ A p , and ν < t < ν + (cid:98) L − ν (cid:99) + γ .Proof. We denote (cid:101) L = (cid:98) L − ν (cid:99) . Fix ν < t < ν + (cid:101) L + γ , 1 < p < ∞ , and 0 < γ (cid:48) < γ such that t < ν + (cid:101) L + γ (cid:48) . Let T t be as in Theorem 4.2 such that | ∇ | − t T | ∇ | t f − T t f is a polynomial for all f ∈ S ∞ . It follows that (cid:101) Q (cid:96) T t f ( x ) = ∑ j , k ∈ Z t ( k − j ) (cid:101) Q (cid:96) (cid:101) Q − tj Λ j , k (cid:101) Q tk f ( x ) . For any K > ε > | (cid:101) Q (cid:96) (cid:101) Q − sj Λ j , k f ( x ) | (cid:46) ν j − K | (cid:96) − j | M f ( x ) | (cid:101) Q (cid:96) (cid:101) Q − sj Λ j , k f ( x ) | (cid:46) ν j ( (cid:101) L + γ (cid:48) + ε ) min ( , j − k ) M f ( x ) . The first estimate here is a standard almost orthogonality estimate, and the second is obtained byapplying Corollary 3.2. Then it follows that | (cid:101) Q (cid:96) (cid:101) Q − sj Λ j , k f ( x ) | (cid:46) ν ( j − k ) − ˜ K | (cid:96) − j | ( (cid:101) L + γ (cid:48) ) min ( , j − k ) ν k M f ( x ) for some ˜ K >
0. Using that ν < t < ν + (cid:101) L + γ (cid:48) , for w ∈ A p it follows that || T t f || ˙ B , qp , w (cid:46) (cid:32) ∑ (cid:96) ∈ Z (cid:34) ∑ j , k ∈ Z : j ≤ k − ˜ K | (cid:96) − j | ( (cid:101) L + γ (cid:48) + ν − t )( j − k ) ν k || M ( (cid:101) Q tk f ) || L pw (cid:35) q (cid:33) / q + (cid:32) ∑ (cid:96) ∈ Z (cid:34) ∑ j , k ∈ Z : j > k − ˜ K | (cid:96) − j | ( t − ν )( k − j ) ν k || M ( (cid:101) Q tk f ) || L pw (cid:35) q (cid:33) / q (cid:46) (cid:32) ∑ (cid:96), j , k ∈ Z : j ≤ k − ˜ K | (cid:96) − j | ( (cid:101) L + γ (cid:48) + ν − t )( j − k ) (cid:104) ν k || M ( (cid:101) Q tk f ) || L pw (cid:105) q (cid:33) / q + (cid:32) ∑ (cid:96), j , k ∈ Z : j > k − ˜ K | (cid:96) − j | ( t − ν )( k − j ) (cid:104) ν k || M ( (cid:101) Q tk f ) || L pw (cid:105) q (cid:33) / q (cid:46) (cid:32) ∑ k ∈ Z (cid:104) ν k || M ( (cid:101) Q tk f ) || L pw (cid:105) q (cid:33) / q (cid:46) || f || ˙ B ν , qp , w . Therefore T t is bounded from ˙ B ν , qp , w into ˙ B , qp , w for all 1 < p , q < ∞ , w ∈ A p , and ν < t < ν + (cid:101) L + γ .Then for all f ∈ S ∞ , it follows that || T f || ˙ B − t , qp , w = || | ∇ | − t T | ∇ | t ( | ∇ | − t f ) || ˙ B , qp , w = || T t ( | ∇ | − t f ) || ˙ B , qp , w (cid:46) || | ∇ | − t f || ˙ B , qp , w = || f || ˙ B − t , qp , w . Therefore T can be extended to a bounded linear operator on ˙ B − t , qp , w for all 1 < p , q < ∞ , w ∈ A p ,and ν < t < ν + (cid:101) L + γ . (cid:3) NALYSIS OF HYPER-SINGULAR, FRACTIONAL, AND ORDER-ZERO SINGULAR INTEGRAL OPERATORS 21
A similar argument to the one above can be made to show that T is bounded on from ˙ F ν − t , qp , w into ˙ F − t , qp , w under the same assumptions on T and for the same ranges of parameters (in particularimposing that w ∈ A p ). However, we do not pursue this argument since we can prove somethingstronger for Triebel-Lizorkin spaces, where the weight is allowed to range outside of the A p classcorresponding to the Lebesgue space parameter p . This stronger result is the following. Theorem 5.2.
Let ν ∈ R , L be an integer with L ≥ | ν | , M ≥ max ( L , L − ν ) , ( L − ν ) ∗ < γ ≤ , andT ∈ SIO ν ( M + γ ) . Further assume that T satisfies W BP ν . If T ∗ ( x α ) = for all | α | ≤ L, then Tcan be extended to a bounded operator from ˙ F ν − t , qp , w into ˙ F − t , qp , w for all ν < t < ν + (cid:98) L − ν (cid:99) + γ , / λ < p < ∞ , min ( , p ) ≤ q < ∞ , and w ∈ A λ p where λ = n + ν + (cid:98) L − ν (cid:99) + γ − tn . Furthermore, there isan increasing function N : [ , ∞ ) → ( , ∞ ) (possibly depending on ν , p, t, and q) such that, for thesame range of indices, we have || T f || ˙ F − t , qp , w ≤ N ([ w ] A p ) || f || ˙ F ν − t , qp , w . Proof.
Fix ν < t < ν + (cid:101) L + γ , 1 / λ < p < ∞ , and w ∈ A λ p . Here we denote (cid:101) L = (cid:98) L − ν (cid:99) and λ = n + ν + (cid:101) L + γ − tn . Then there exists 1 / λ < r < min ( , p ) such that w ∈ A λ r . Also let 0 < γ (cid:48) < γ and0 < µ < ν + (cid:101) L + γ (cid:48) − t so that t < ν + (cid:101) L + γ (cid:48) and 1 / λ < nn + µ < r < min ( , p ) . Let T t be defined asin Theorem 4.2. Now we fix φ , (cid:101) φ ∈ S as in Lemma 2.7. Then it follows that (cid:101) Q (cid:96) T t f ( x ) = ∑ j , k ∈ Z t ( k − j ) Q (cid:96) (cid:101) Q − tj Λ j , k (cid:101) Q tk f ( x )= ∑ j , k , m ∈ Z ∑ Q : (cid:96) ( Q )= − m t ( k − j ) (cid:101) Q (cid:96) (cid:101) Q − tj Λ j , k (cid:101) Q tk φ c Q m ( x ) ˜ φ m ∗ f ( c Q ) . For any any K > ε > | (cid:101) Q (cid:96) (cid:101) Q − tj Λ j , k (cid:101) Q tk φ c Q m ( x ) | (cid:46) ν j − K | (cid:96) − j | Φ n + M + γ min ( (cid:96), j , k , m ) ( x − c Q ) , | (cid:101) Q (cid:96) (cid:101) Q − tj Λ j , k (cid:101) Q tk φ c Q m ( x ) | (cid:46) ν j − K | k − m | Φ n + M + γ min ( (cid:96), j , k , m ) ( x − c Q ) , | (cid:101) Q (cid:96) (cid:101) Q − tj Λ j , k (cid:101) Q tk φ c Q m ( x ) | (cid:46) ν j ( (cid:101) L + γ (cid:48) + ε ) min ( , j − k ) Φ n + M + γ min ( (cid:96), j , k , m ) ( x − c Q ) . As before, the first two lines here follow from standard almost orthogonality estimates, and thethird follows from Corollary 3.2. Combining these estimates, it also follows that | Q (cid:96) (cid:101) Q − tj Λ j , k (cid:101) Q tk φ c Q m ( x ) | (cid:46) ν j − ˜ K | (cid:96) − j | − ˜ K | k − m | ( L + γ (cid:48) ) min ( , j − k ) Φ n + ν + M + γ min ( (cid:96), j , k , m ) ( x − c Q ) for some ˜ K > µ + | t | , as long as K is selected sufficiently large, depending on L , γ , and ν . Thenusing Lemma 2.8, it follows that ∑ Q : (cid:96) ( Q )= − m | Q (cid:96) (cid:101) Q − tj Λ j , k (cid:101) Q tk φ c Q m ( x ) ˜ φ m ∗ f ( c Q ) |≤ ν j − ˜ K | (cid:96) − j | − ˜ K | k − m | ( (cid:101) L + γ (cid:48) ) min ( , j − k ) µ max ( , m − j , m − k , m − (cid:96) ) M rm ( ˜ φ m ∗ f ) ≤ ν j − ( ˜ K − µ ) | (cid:96) − j | − ( ˜ K − µ ) | k − m | ( (cid:101) L + γ (cid:48) − µ ) min ( , j − m ) M rm ( ˜ φ m ∗ f ) . Using that ˜ K > µ + | t | and that ν < t < ν + (cid:101) L + γ (cid:48) − µ , we also have ∑ (cid:96) ∈ Z | Q (cid:96) T t f | q (cid:46) ∑ (cid:96) ∈ Z (cid:34) ∑ j , k , m ∈ Z : j ≤ m − ( ˜ K − µ ) | (cid:96) − j | − ( ˜ K − µ −| t | ) | k − m | ( ν + (cid:101) L + γ (cid:48) − t − µ )( j − m ) ν m M rm ( ˜ φ m ∗ f ) (cid:35) q + ∑ (cid:96) ∈ Z (cid:34) ∑ j , k , m ∈ Z : j > m ( t − ν )( m − j ) − ( ˜ K − µ ) | (cid:96) − j | − ( ˜ K − µ −| t | ) | k − m | ν m M rm ( ˜ φ m ∗ f ) (cid:35) q (cid:46) ∑ (cid:96), j , k , m ∈ Z : j ≤ m − ( ˜ K − µ ) | (cid:96) − j | − ( ˜ K − µ −| t | ) | k − m | ( ν + (cid:101) L + γ (cid:48) − t − µ )( j − m ) (cid:2) ν m M rm ( ˜ φ m ∗ f ) (cid:3) q + ∑ (cid:96), j , k , m ∈ Z : j > m ( t − ν )( m − j ) − ( ˜ K − µ ) | (cid:96) − j | − ( ˜ K − µ −| t | ) | k − m | (cid:2) ν m M rm ( ˜ φ m ∗ f ) (cid:3) q (cid:46) ∑ m ∈ Z (cid:2) ν m M rm ( ˜ φ m ∗ f ) (cid:3) q , and hence that || T t f || ˙ F , qp , w = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:32) ∑ (cid:96) ∈ Z | Q (cid:96) T t f | q (cid:33) / q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L pw (cid:46) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:32) ∑ m ∈ Z (cid:2) ν m M rm ( ˜ φ m ∗ f ) (cid:3) q (cid:33) / q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L pw (cid:46) || f || ˙ F ν , qp , w , where we use Lemma 2.9 in the last inequality. Therefore T t is bounded from ˙ F ν , qp , w into ˙ F , qp , w forall ν < t < ν + (cid:101) L + γ , 1 / λ < p < ∞ , min ( , p ) ≤ q < ∞ , and w ∈ A λ p . Then for the same range ofparameters and for all f ∈ S ∞ , it follows that || T f || ˙ F − t , qp , w = || | ∇ | − t T | ∇ | t ( | ∇ | − t f ) || ˙ F , qp , w = || T t ( | ∇ | − t f ) || ˙ F , qp , w (cid:46) || | ∇ | − t f || ˙ F ν , qp , w = || f || ˙ F ν − t , qp , w . Therefore T can be extended to a bounded linear operator form ˙ F ν − t , qp , w into ˙ F − t , qp , w for all ν < t < ν + (cid:101) L + γ , 1 / λ < p < ∞ , min ( , p ) ≤ q < ∞ , and w ∈ A λ p . It is not hard to note that the dependenceon w ∈ A p from the argument about yields an estimate depending on a positive power of [ w ] A p , andhence N can be taken as such (in particular, such a function exists). (cid:3) Below we represent ˙ F t , p in parameter space ( t , p ) for t ∈ R and 0 < p ≤ ∞ . Most of the dis-cussion here will apply for ˙ F t , qp when q (cid:54) =
2, but for simplicity we only discuss restrict to q = < q ≤ F t , qp does not behave the same as the q = ( t , p ) = ( , p ) representsthe Hardy spaces ˙ F , p = H p for 0 < p < ∞ and Lebesgue spaces ˙ F , p = L p for 1 < p < ∞ . Thehorizontal lines given by ( t , p ) when 1 < p < ∞ describe the homogeneous Sobolev spaces ˙ F t , p = ˙ W t , p . We will always identify the origin ( t , p ) = ( , ) as ˙ F , ∞∞ = BMO , but for ( t , p ) = ( t , ) ournotation is somewhat inconsistent. Sometimes these locations will represent Sobolev- BMO via˙ F t , ∞ = I t ( BMO ) for t >
0, and other times they should be interpreted as the Besov-Lipschitz spaces˙ B t , ∞∞ when t >
0. Recall that ˙ B t , ∞∞ is a Lipschitz space only when t > NALYSIS OF HYPER-SINGULAR, FRACTIONAL, AND ORDER-ZERO SINGULAR INTEGRAL OPERATORS 23
Let us first consider boundedness results for T ∈ SIO ν ( M + γ ) when ν =
0. The graph on the leftin Figure 1 is a depiction of the boundedness properties of T provided in Theorem 5.2 (restrictedto the unweighted version). That is, if T ∗ ( x α ) = | α | ≤ L , then T is bounded on ˙ F t , p when ( t , p ) lies in the blue shaded region (excluding the boundary) in the left picture. If we were toassume in addition that T is L -bounded, then it also follows that T is bounded on ˙ F , p = H p when ( t , p ) = ( , p ) satisfies nN + L + γ < p < ∞ . This does not follow from what we’ve proved here, butthese boundedness properties are classical in the Lebesgue space setting when 1 < p < ∞ and wereproved in [22] for 0 < p ≤ t t t p = 2 p = 2 p = 1 L + ( L + ) || ( L + ) | L + ( L + ) || ( , ) (1 , ) n ( p
1) = L + + t n ( p
1) = L + + 1 n ( p
1) = L + + t p p p F IGURE
1. Parameter space ( t , p ) for boundedness properties of T and T ∗ on ˙ F t , p ,pictured with n = L =
1, and γ = T to where 1 < p < ∞ , wherethe spaces ˙ F t , p are reflexive. In this situation, we have ( ˙ F t , p ) ∗ = ˙ F − t , p (cid:48) for t ∈ R and 1 < p < ∞ .Geometrically, the dual of ˙ F t , p can be found by reflecting over the vertical line t = p =
2, as pictured. Then by duality, T ∗ ( x α ) = | α | ≤ L implies that T ∗ isbounded on ˙ F t , p for all 1 < p < ∞ and 0 < t < L + γ . This boundedness result for T ∗ is shown inthe green shaded region in Figure 1.In the picture on the right in Figure 1, we describe the dual boundedness implications for T ∗ when 0 < p ≤
1, which are more delicate that the ones already discussed. We consider two sit-uations: where p = < p <
1. It was proved by Frazier and Jawerth [17] that ( ˙ F t , ) ∗ = ˙ F − t , ∞ . Then the boundedness of T on ˙ F t , for indices on the horizontal line segmentgiven by ( t , p ) = ( t , ) with − ( L + γ ) < t < T ∗ is bounded on the Sobolev-BMOspaces ˙ F t , ∞ = I t ( BMO ) for 0 < t < L + γ . Geometrically, this summarizes boundedness for T ∗ onthe horizontal line ( t , p ) = ( t , ) with 0 ≤ t < L + γ , where we make the Triebel-Lizorkin spaceidentification ˙ F t , ∞ . This duality still obeys the geometric rule of reflecting over the lines t = p = < p <
1. The remainingportion of the red region lying to the left of the axis is where − ( L + γ ) < t < nn + L + γ − t < p < ( ˙ F t , p ) ∗ = ˙ B − t + n ( / p − ) , ∞∞ for0 < p < t ∈ R . Note that the duals of ˙ F t , p coincide for several values of t and p here. Inparticular, for any t ∈ R and 0 < p < − t + n ( / p − ) = s satisfies ( ˙ F t , p ) ∗ = ˙ B s , ∞∞ . This is Plots appearing in this article were generated using the Desmos.com online graphing tool and Matlab. depicted above by the highlighted red line, and the associated red x on the horizontal axis locatedat ( t , p ) = ( , ) . Then by duality T ∗ ( x α ) = | α | ≤ L implies that T ∗ is bounded on ˙ B t , ∞∞ for0 < t < L + γ . This conclusion can be made by duality from the boundedness of T at any point alongthe appropriate line. Geometrically, this deviates slightly from the previous cases. In particular,when 0 < p < t ∈ R , in order to obtain the appropriate indices for the dual of ˙ F t , p , first project ( t , p ) along a line with slope 1 / n onto the line p =
1, then reflect over t = p =
2. Makingthese geometric manipulations lands the dual indices on the horizontal axis, overlapping with theprevious case where p =
1. We emphasize that when 0 < p <
1, the appropriate interpretation ofthe pictures above is that ( ˙ F t , p ) ∗ = ˙ B − t + n ( / p − ) , ∞∞ , with this Besov space in place of ˙ F − t , ∞ . Hencethe distinction 0 < p < p = F t , ∞ versus ˙ B t , ∞∞ .The estimates for T on ˙ F t , p indicated in the blue shaded region in the left picture of Figure1 describes only the the unweighted estimates proved in Theorem 5.2, but we can extend thisrepresentation to weighted estimates as well. In order to do so, consider the parameter space madeup of ordered triples of the form ( t , p , q ) for which T is bounded on ˙ F t , p , w when w ∈ A q . Theorem5.2 says that the triple ( t , p , q ) represents where T is bounded on the weighted spaces ˙ F t , p , w for all w ∈ A q when nn + L + γ − t p ≤ q ≤
1. This defines a solid in R lying under the blue shaded region onthe left picture in Figure 1, which is shown in Figure 2. p t t q p p t q | n + L + n ( L + ) | | n + L + n ( L + ) | | | | n + L + n ( L + ) | F IGURE
2. Parameter space ( t , p , q ) for boundedness properties of T on ˙ F t , p , w with w ∈ A q , pictured with n = L =
1, and γ =
1. The plot on the left depicts thesame region as the plot on the left of Figure 1, which coincides with the q = ν ∈ R . Similar geometric depictions of the boundedness results above for ν (cid:54) = ( t , p ) of the boundedness properties of T from ˙ F t , p into ˙ F t − ν , p for ν < − t < ν + (cid:101) L + γ and nn + ν + (cid:101) L + γ + t < p < ∞ . Figure 3 briefly demonstrateshow the boundedness of T and T ∗ is expressed in parameter space when ν (cid:54) = Corollary 5.3.
Let ν ∈ R , L be an integer with L ≥ | ν | , M ≥ max ( L , L − ν ) , ( L − ν ) ∗ < γ ≤ ,and T ∈ SIO ν ( M + γ ) . Further assume that T satisfies W BP ν . If T ∗ ( x α ) = for all | α | ≤ L, then
NALYSIS OF HYPER-SINGULAR, FRACTIONAL, AND ORDER-ZERO SINGULAR INTEGRAL OPERATORS 25 F IGURE
3. Parameter space for boundedness properties of T from ˙ F ν + t , p into ˙ F t , p and for T ∗ from ˙ F t , p into ˙ F t − ν , p , pictured with n = L =
3, and γ = . Theleft, middle, and right plots correspond to T belonging to SIO ν for ν = − , , respectively, which yields (cid:101) L = , , respectively. T ∗ can be extended to a bounded operator from ˙ F t , qp into ˙ F t − ν , qp and from ˙ B t , qp into ˙ B t − ν , qp for all < p , q < ∞ and ν < t < ν + (cid:98) L − ν (cid:99) + γ . This corollary is immediate given Theorems 5.1 and 5.2 applied only in the unweighted and1 < p , q < ∞ situation. Corollary 5.4.
Let ν ∈ R , L be an integer with L ≥ | ν | , M ≥ max ( L , L − ν ) , ( L − ν ) ∗ < γ ≤ , andT ∈ SIO ν ( M + γ ) . Further assume that T satisfies W BP ν . If T ∗ ( x α ) = for all | α | ≤ L, then T ∗ isbounded from ˙ F t , q ∞ into ˙ F t − ν , q ∞ and from ˙ B t , ∞∞ into ˙ B t − ν , ∞∞ for all ν < t < ν + (cid:101) L + γ and < q < ∞ .Proof. For 0 < t < ν + (cid:101) L + γ and 1 < q < ∞ , Theorem 5.2 implies that T is bounded from ˙ F ν − t , q into ˙ F − t , q . Then by duality (see for example the Frazier and Jawerth article [17, Theorem 5.13]),it follows that T ∗ is bounded from ˙ F t , q ∞ into ˙ F t − ν , q ∞ for ν < t < ν + (cid:101) L + γ and 1 < q < ∞ . For ν ≤ t < ν + (cid:101) L + γ , choose ν < s < L + γ and nn + ν + (cid:101) L + γ − s < p < s + n ( / p − ) = t . Thenit follows that T is bounded from ˙ F ν − s , p into ˙ F − s , p . So by duality, it follows that T ∗ is boundedfrom ˙ B s + n ( / p − ) , ∞∞ into ˙ B s − ν + n ( / p − ) , ∞∞ . That is, T ∗ is bounded from ˙ B t , ∞∞ into ˙ B t − ν , ∞∞ . Here weuse the duality result of Jawerth [26, Theorem 4.2]. (cid:3)
6. N
ECESSITY OF V ANISHING M OMENT C ONDITIONS
In this section, we establish the necessity of the T ∗ ( x α ) = T T (or T ∗ ) to be bounded based on Weak Boundedness Properties and cancellationconditions. It is interesting to note that T ∗ ( x α ) = T ( x α ) . Some of these implicationscome from Proposition 6.1 and Lemma 6.3, both of which are interesting in their own right. Proposition 6.1.
Let T be an operator, s , t ∈ R , < p , q < ∞ , and λ ≥ / p . Assume that T isbounded from ˙ F s , qp , w into ˙ F t , qp , w , and there is an increasing function N : R → [ , ∞ ) such that || T f || ˙ F t , qp , w ≤ N ([ w ] A λ p ) || f || ˙ F s , qp , w for all w ∈ A λ p . Then T is bounded from ˙ F s , qp , w into ˙ F t , qp , w for all / λ < p < ∞ and w ∈ A λ p . Proof.
Define ( F , G ) = (cid:32) ∑ k ∈ Z ( sk | Q k f | ) q (cid:33) q λ , (cid:32) ∑ k ∈ Z ( tk | Q k T f | ) q (cid:33) q λ For all w ∈ A λ p || F || L λ p w = || T f || / λ ˙ F t , qp , w ≤ N ([ w ] A λ p ) / λ || f || / λ ˙ F s , qp , w = N ([ w ] A λ p ) / λ || G || L λ p w We apply extrapolation to the pairs of functions ( F , G ) indexed by f ∈ S ∞ . Note that 1 ≤ λ p < ∞ .Then, by extrapolation, it follows that || G || L rw ≤ K ( w ) / λ || F || L rw for all 1 < r < ∞ and w ∈ A r , where K ( w ) is specified in [15]. Therefore || T f || ˙ F t , qr / λ , w = || G || λ L rw ≤ K ( w ) || F || λ L rw = K ( w ) || f || ˙ F s , qr / λ , w for all 1 < r < ∞ and w ∈ A r . Now we simply shift notation to p = r / λ , and it follows that || T f || ˙ F t , qp , w ≤ K ( w ) || f || ˙ F t , qp , w for all 1 / λ < p < ∞ , w ∈ A λ p , and f ∈ S ∞ . By density, T is bounded from ˙ F t , qp , w into ˙ F s , qp , w for thesame range of indices. (cid:3) Remark . Though Proposition 6.1 is a relatively trivial application of Rubio de Francia’s ex-trapolation, there are some interesting subtleties that can be observed in this result. It demonstratesa way to “trade” the seemingly unnatural weighted estimates on L pw for w ∈ A r when r > p for theability to move the index p below 1. In particular, this provides a way to avoid a typical difficultythat arrises in Hardy space theory for indices smaller than 1. Suppose you’d like to prove that agiven operator T is bounded on H p for some 1 / < p <
1. For such p , the duality theory of H p can be cumbersome. However, by Proposition 6.1 it is sufficient to prove that T is bounded on H w = ˙ F , , w for all w ∈ A p (note that H w (cid:54) = L w for all such w since p > / H w may beeasier to work with since, for example, it is a Banach space rather than only a quasi-Banach spacelike H p . Being able to “bump up” the index from p to 2 may also make it possible to use dualityarguments that may not be viable for quasi-Banach spaces. Lemma 6.3.
Let L ≥ be an integer and γ > . If f ∈ ˙ B − t , ∞ p ∩ L ( + | x | L + γ ) for all < t < L + γ and < p < ∞ , then (cid:90) R n f ( x ) x α dx = for all | α | ≤ L.Consequently, for any < q < ∞ , the same conclusion holds if f ∈ ˙ F − t , qp ∩ L ( + | x | L + γ ) orf ∈ ˙ B − t , qp ∩ L ( + | x | L + γ ) for all < t < L + γ and < p < ∞ . NALYSIS OF HYPER-SINGULAR, FRACTIONAL, AND ORDER-ZERO SINGULAR INTEGRAL OPERATORS 27
Proof.
We proceed by induction. First assume that L =
0. Let 0 < t < γ and 1 < p < ∞ be smallenough so that n / p (cid:48) < t < n / p (cid:48) + γ . Assume f ∈ ˙ B − t , ∞ p ∩ L ( + | x | γ ) . Then for any ψ ∈ S ∞ , wehave || f || ˙ B − t , ∞ p ≥ sup k < ( n / p (cid:48) − t ) k || ψ || L p (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) R n f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) − sup k < − tk (cid:20) (cid:90) R n (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) R n ( ψ k ( x − y ) − ψ k ( x )) f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) p dx (cid:21) / p . The second term above is bounded since we havesup k < ( γ − t ) k (cid:20) (cid:90) R n (cid:18) (cid:90) R n | y | γ (cid:0) Φ Nk ( x − y ) + Φ Nk ( x ) (cid:1) | f ( y ) | dy (cid:19) p dx (cid:21) / p (cid:46) sup k < ( γ − t ) k (cid:20) (cid:90) R n Φ Nk ∗ ( | y | γ | f | )( x ) p dx (cid:21) / p + || f || L ( | y | γ ) sup k < ( γ − t ) k (cid:20) (cid:90) R n Φ Nk ( x ) p dx (cid:21) / p (cid:46) || f || L ( | x | γ ) . Note that we chose t so that n / p (cid:48) < t < n / p (cid:48) + γ , which implies 2 ( γ − t ) k || Φ Nk || L p is bounded uni-formly in k (as long as N > n / p ) and that 2 ( n / p (cid:48) − t ) k is unbounded for k <
0. Then it follows that f must have integral zero. Now assume that Lemma 6.3 holds for all M ≤ L −
1. Let 0 < t < L + γ and1 < p < ∞ be small enough so that n / p (cid:48) + L < t < n / p (cid:48) + L + γ . Assume f ∈ ˙ B − t , ∞ p ∩ L ( + | x | L + γ ) .Then for any ψ ∈ S ∞ || f || ˙ B − t , ∞ p ≥ sup k < − tk (cid:20) (cid:90) R n (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) R n J Lx [ ψ k ]( y ) f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) p dx (cid:21) / p − sup k < − tk (cid:20) (cid:90) R n (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) R n ( ψ k ( x − y ) − J Lx [ ψ k ]( y )) f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) p dx (cid:21) / p . Again the second term is bounded above sincesup k < − tk (cid:20) (cid:90) R n (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) R n ( ψ k ( x − y ) − J Lx [ ψ k ]( y )) f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) p dx (cid:21) / p (cid:46) sup k < − tk (cid:20) (cid:90) R n (cid:18) (cid:90) R n ( k | y | ) L + γ (cid:0) Φ Nk ( x − y ) + Φ Nk ( x ) (cid:1) | f ( y ) | dy (cid:19) p dx (cid:21) / p (cid:46) sup k < ( L + γ + n / p (cid:48) − t ) k || f || L ( | x | L + γ ) ≤ || f || L ( | x | L + γ ) . We also have, by the inductive hypothesis, that2 − tk (cid:20) (cid:90) R n (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) R n J Lx [ ψ k ]( y ) f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) p dx (cid:21) p = ( L + n / p (cid:48) − t ) k (cid:34) (cid:90) R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∑ | α | = L D α ψ ( x ) α ! (cid:90) R n f ( y ) y α dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dx (cid:35) p . Since 2 ( L + n / p (cid:48) − t ) k is unbounded for k <
0, it follows that (cid:90) R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∑ | α | = L D α ψ ( x ) α ! (cid:90) R n f ( y ) y α dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dx = ψ ∈ S ∞ , and hence that (cid:90) R n f ( y ) y α dy = | α | = L . By induction, this completes the proof when f ∈ ˙ B − t , ∞ p . Note that the remainingproperties trivially follow since ˙ F − t , qp ⊂ ˙ B − t , ∞ p and ˙ B − t , qp ⊂ ˙ B − t , ∞ p for all 0 < q < ∞ , 1 < p < ∞ , and t ∈ R . (cid:3) Remark . It is known that H p quantifies vanishing moment properties for its members (see e.g.[19] by Grafakos and He on weak Hardy spaces), and by Lemma 6.3 we have vanishing momentproperties for negative smoothness index Triebel-Lizorkin and Besov spaces. In particular, Lemma6.3 should be interpreted as follows. The spaces ˙ F − t , qp for L ≤ t + n ( / p − ) < L + | α | = L for its members, as described in Lemma 6.3. Theorem 6.5.
Let ν ∈ R , L be an integer with L ≥ | ν | , M ≥ max ( L , L − ν ) , ( L − ν ) ∗ < γ ≤ , andT ∈ CZO ν ( M + γ ) . If any one of the conditions hold, then T ∗ ( x α ) = for all | α | ≤ L. (1) For every < p < ∞ and ν < t < ν + (cid:101) L + γ , there exists < q ≤ ∞ such that T is boundedfrom ˙ F ν − t , qp into ˙ F − t , qp (2) For every < p < ∞ and ν < t < ν + (cid:101) L + γ , there exists < q < ∞ such that T ∗ is boundedfrom ˙ F t , qp into ˙ F t − ν , qp (3) For every < p < ∞ and ν < t < ν + (cid:101) L + γ , there exists < q ≤ ∞ such that T is boundedfrom ˙ B ν − t , qp into ˙ B − t , qp (4) For every < p < ∞ and ν < t < ν + (cid:101) L + γ , there exists < q < ∞ such that T ∗ is boundedfrom ˙ B t , qp into ˙ B t − ν , qp (5) For each ν < t < ν + (cid:101) L + γ , there exist < q ≤ ∞ and / λ < p < ∞ such that T is boundedfrom ˙ F ν − t , qp , w into ˙ F − t , qp , w and there is an increasing function N : [ , ∞ ) → ( , ∞ ) that does notdepend on w such that || T f || ˙ F − t , qp , w ≤ N ([ w ] A λ p ) || f || ˙ F ν − t , qp , w for all w ∈ A λ p , where λ = n + ν + (cid:101) L + γ − tn (6) For every ν < t < ν + (cid:101) L + γ , T ∗ is bounded from ˙ B t , ∞∞ into ˙ B t − ν , ∞∞ . (7) For every ν < t < ν + (cid:101) L + γ , there exists a < q < ∞ such that T ∗ is bounded from ˙ F t , q ∞ into ˙ F t − ν , q ∞ (8) For every ν < t < ν + (cid:101) L + γ , there exists a < q < ∞ such that T is bounded from ˙ F ν − t , q into ˙ F − t , q (9) For each ν < s < ν + (cid:98) L − ν (cid:99) + γ and < t < (cid:98) L − ν (cid:99) + γ there exists T s , t ∈ CZO ν + t − s suchthat | ∇ | − s T | ∇ | t f − T s , t f is a polynomial for all f ∈ S ∞ . NALYSIS OF HYPER-SINGULAR, FRACTIONAL, AND ORDER-ZERO SINGULAR INTEGRAL OPERATORS 29
Proof.
Assume that (1) holds, and let ψ ∈ D P , for P ∈ N sufficiently large, with supp ( ψ ) ⊂ B ( , R / ) for some R >
1. Note that T ψ is locally integrable by the T ∈ CZO ν assumption. Alsoif x / ∈ B ( , R ) , then it follows that | T ψ ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) R n (cid:0) K ( x , y ) − J M [ K ( x , · )] ( y ) (cid:1) ψ ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:90) R n | y | M + γ | x | n + ν + M + γ | ψ ( y ) | dy (cid:46) R L + γ − ν || ψ || L | x | n + L + γ . Then it follows that T ψ ∈ L ( + | x | L + γ (cid:48) ) for any 0 < γ (cid:48) < γ since (cid:90) R n | T ψ ( x ) | ( + | x | L + γ (cid:48) ) dx (cid:46) ( + R L + γ ) (cid:90) | x |≤ R | T ψ ( x ) | dx + R γ (cid:48) − ν . This, in addition to (1), says that T ψ ∈ ˙ B − t , ∞∞ ∩ L ( + | x | L + γ (cid:48) ) for all 0 < t < L + γ (cid:48) and 1 < p < ∞ .So by Lemma 6.3, it follows that T ψ has vanishing moments up to order L . But this means exactlythat T ∗ ( x α ) = α ’s since (cid:104) T ∗ x α , ψ (cid:105) = lim R → ∞ (cid:90) R n T ψ ( x ) η R ( x ) x α dx = (cid:90) R n T ψ ( x ) x α dx = , where we use dominated convergence and that T ψ ∈ L ( + | x | L + γ (cid:48) ) to handle the limit in R .Therefore T ∗ ( x α ) = | α | ≤ L .By exactly the same argument, it follows that condition (3) also implies T ∗ ( x α ) = | α | ≤ L .Furthermore, by duality (2) implies (1) and (4) implies (3). Hence we have shown that any one ofthe conditions (1)–(4) implies T ∗ ( x α ) = | α | ≤ L .Assume that (5) holds. Then by Proposition 6.1 it follows that T is bounded from ˙ F ν − t , qp , w into˙ F − t , qp , w for all 0 < t < ν + (cid:101) L + γ , 1 / λ < p < ∞ , and w ∈ A λ p , where λ = n + ν + (cid:101) L + γ − tn . In particular, (5)implies (1) which in turn implies that T ∗ ( x α ) = | α | ≤ L .Assume that (6) holds. Let α ∈ N n with | α | ≤ L . Note that ( L − ν ) ∗ < γ implies that L < ν + (cid:101) L + γ . Then there is a t / ∈ Z such that max ( ν , | α | ) < t < ν + (cid:101) L + γ . Also let 0 < p < n ( / p − ) = t − ν . Then for ψ ∈ D P , with P sufficiently large, we have | (cid:104) T ∗ ( x α ) , ψ (cid:105) | (cid:46) lim sup R → ∞ || η R x α || ˙ B t , ∞∞ || ψ || H p (cid:46) lim sup R → ∞ R | α |− t || ψ || H p = . Here, we simply note that when t > B t , ∞∞ is the class of t -Lipschitz functions,and it easily follows that || φ ( · / R ) || ˙ B t , ∞∞ (cid:46) R − t ∗ for any φ ∈ C ∞ , where again t ∗ is the decimal part of t . Therefore T ∗ ( x α ) = | α | ≤ L .Assume that (7) holds. Note that ˙ F t − ν , q ∞ ⊂ ˙ B t − ν , ∞∞ and || f || ˙ B t − ν , ∞∞ ≤ || f || ˙ F t − ν , q ∞ for any ν < t < ν + (cid:101) L + γ and 1 < q < ∞ . Then we argue as we did in the previous case. For | α | ≤ L , let max ( ν , | α | ) < t < ν + (cid:101) L + γ and ψ ∈ D P . Then | (cid:104) T ∗ ( x α ) , ψ (cid:105) | (cid:46) lim sup R → ∞ || η R x α || ˙ F t − ν , q ∞ || ψ || ˙ F ν − t , q (cid:48) ≤ lim sup R → ∞ || η R x α || ˙ B t − ν , ∞∞ || ψ || ˙ F ν − t , q (cid:48) = T ∗ ( x α ) = | α | ≤ L .By duality (8) implies (7) and by density (9) implies (1). Hence both (8) and (9) also imply T ∗ ( x α ) = | α | ≤ L . (cid:3)
7. A
PPLICATIONS
Necessary and Sufficient Conditions for Classes of Singular Integral Operators.
In thissection we collect and summarize the results in the preceding three sections to form several equiva-lent conditions for cancellation and boundedness of operators T ∈ CZO ν . This result is essentiallya combination of a few of the operator calculus results from Section 4, the boundedness resultsfrom Section 5, and the sufficiency for vanishing moments from Section 6. As a result we obtainthe following T SIO ν . Corollary 7.1.
Let ν ∈ R , L ≥ | ν | be an integer, ( L − ν ) ∗ < γ ≤ , (cid:101) L = (cid:98) L − ν (cid:99) , and T ∈ CZO ν ( L + γ ) . Then the following are equivalent. (1) T ∗ ( x α ) = for all | α | ≤ L (2) For every ν < s < ν + (cid:101) L + γ and < t < (cid:101) L + γ , there exists T s , t ∈ CZO ν + t − s ( γ (cid:48) ) for < γ < γ such that | ∇ | − s T | ∇ | t f − T s , t f is a polynomial for all f ∈ S ∞ (3) For all ν < t < ν + (cid:101) L + γ , / λ < p < ∞ , and min ( , p ) ≤ q < ∞ , T is bounded from ˙ F ν − t , qp , w into ˙ F − t , qp , w and there is an increasing function N : [ , ∞ ) → ( , ∞ ) that does not depend onw such that || T f || ˙ F − t , qp , w ≤ N ([ w ] A λ p ) || f || ˙ F ν − t , qp , w for all w ∈ A λ p , where λ = n + ν + (cid:101) L + γ − tn . This corollary follows immediately from Theorems 4.3, 5.2, and 6.5. We should also note thatone could obtain many other equivalent conditions to put on this list by turning to Theorems 5.2and 6.5, as well as Corollaries 5.3 and 5.4.There is a long history of results along the lines Corollary 7.1. Several boundedness results for ν = A λ p for λ > p . The only articles we are aware ofwhere such estimates are proved are [32, 23], where the results are limited to ν = ν = CZO ν for ν (cid:54) = ν = SIO ν to be bounded on homogeneous Besov andTriebel-Lizorkin spaces. However, conditions of the form T ∗ ( x α ) = T = T =
0, showthat such boundedness properties are also sufficient for T ∗ ( x α ) = ν < NALYSIS OF HYPER-SINGULAR, FRACTIONAL, AND ORDER-ZERO SINGULAR INTEGRAL OPERATORS 31
Pseudodifferential Operators.
In this application we consider the forbidden class of pseu-dodifferential operators
OpS , . They are defined as follows. We say σ ∈ S , if | D αξ D β x σ ( x , ξ ) | (cid:46) ( + | ξ | ) | β |−| α | . for all α , β ∈ N n , and T σ ∈ OpS , is the associated operator defined T σ f ( x ) = (cid:90) R n σ ( x , ξ ) (cid:98) f ( ξ ) e ix ξ d ξ for f ∈ S . The reason OpS , is referred to as a forbidden class, or sometimes an exotic class,of operators is because it is not closed under transpose, and they are not necessarily L -bounded.However, any T σ ∈ OpS , has a standard kernel and is bounded on several smooth function spaces.For instance, such T σ is bounded on several classes of inhomogeneous Lipschitz, Sobolev, Besov,and Triebel-Lizorkin spaces; see for example [33, 38, 7, 24, 25, 43, 40, 12]. All of these inhomoge-neous space estimates are obtained in the absence of vanishing moment assumptions. On the otherhand, Meyer showed that under vanishing moment conditions T σ ( x α ) = σ ∈ S , , T σ is alsobounded on homogeneous Lipschitz and Sobolev spaces; see [34]. Our next result provides moreestimates along the lines of Meyer’s that require vanishing moments for the operator. We also notethat Bourdaud proved a noteworthy result in [7] about the largest sub-algebra of OpS , . In partic-ular, he showed that the subclass of OpS , made up of operators T σ ∈ OpS , so that T ∗ σ ∈ OpS , is an algebra and that all such operators are L -bounded. Corollary 7.2.
Let T σ ∈ OpS , and L ∈ N . If T ∗ σ ( x α ) = for all | α | ≤ L, then Theorem 4.3,Theorem 5.1, Theorem 5.2, Corollary 5.3, and Corollary 5.4 can all be applied to T σ . If T σ ( x α ) = for all | α | ≤ L, then the same results can be applied to T ∗ σ . Note that Corollary 7.2 does not require, nor imply, that T σ is bounded on L . In fact, there arestandard constructions of operators to which we can apply Corollary 7.2 that are not L -bounded,as is shown in the next example.It should also be noted here that even though OpS , is not closed under transposes, Corollary7.2 applies to both T σ and its transpose for any σ ∈ S , . This is because S , ⊂ SIO ( ∞ ) and SIO ( ∞ ) is closed under transposes. Hence for any T σ ∈ OpS , , both T σ , T ∗ σ ∈ SIO ( ∞ ) , and soCorollary 7.2 is even capable of concluding operator estimates for operators that do not belong to OpS , . of Corollary 7.2. It is well known that σ ∈ S , implies T σ ∈ SIO ( ∞ ) . That is, it is known that such T σ are continuous from S into S (cid:48) , and have a standard functional kernel K ( x , y ) . It is also easy toshow that | (cid:104) T σ f , g (cid:105) | (cid:46) (cid:107) (cid:98) f (cid:107) L (cid:107) g (cid:107) L for all f , g ∈ S , and so T σ trivially satisfies W BP . Recall that SIO ( ∞ ) and W BP are closed under transposition, and the corollary easily follows. (cid:3) Example 7.3.
Let ψ ∈ S ∞ be such that (cid:98) ψ is supported in an annulus, and define σ ( x , ξ ) = ∑ k ∈ Z e − i k x (cid:98) ψ ( − k ξ ) , as well as the associated pseudodifferential operator T σ . It is known that σ ∈ S , and hence T σ ∈ OpS , ; see for example [40] for more details. It is easy to verify that ( T ∗ σ ) ∗ ( x α ) = T σ ( x α ) = α ∈ N n . Then Corollary 7.2 can be applied to T ∗ σ . Furthermore, since T σ ( x α ) = α ∈ N n ,the restrictions involving L can be removed entirely and one can allow t > T σ is bounded, for example, on ˙ F t , qp , w for all 1 < p , q < ∞ , t >
0, and w ∈ A ∞ . Also, for every s , t > T s , t ∈ CZO t − s such that | ∇ | − s T ∗ σ | ∇ | t f − T s , t f is a polynomial for all f ∈ S ∞ .In particular, | ∇ | − t T ∗ σ | ∇ | t − P f and | ∇ | t T σ | ∇ | − t − ˜ P f are Calder ´on-Zygmund operators in CZO forall t >
0, where P f and ˜ P f are polynomials depending on f and t .7.3. Paraproducts.
In this section, we consider a generalization of the Bony paraproduct, con-structed originally in [6]. The crucial properties of this operator are, for a given b ∈ BMO , theBony paaproduct Π b is a Calder´on-Zygmund operator (in particualr L -bounded), Π b = b , and Π ∗ b =
0. This operator played a crucial role in the proof of the T Π α b ∈ SIO ν for b ∈ ˙ B | α |− ν∞ , α ∈ N n , and ν ∈ R . Theysatisfy prescribed polynomial moment conditions, including ( Π α b ) ∗ ( x α ) = CZO ν or satisfy any continuous mapping properties into Lebesguespaces. See the Corollary 7.4, Lemma 7.7, and the remarks at the end of Sections 7.3 and 7.4 formore on this.Let ν ∈ R , α ∈ N n , and b ∈ ˙ B | α |− ν , ∞∞ . Also let (cid:101) ψ and ψ be as in Lemma 2.6, and ϕ ∈ S withintegral 1. Define the paraproduct operator Π α b f ( x ) = ( − ) | α | α ! ∑ k ∈ Z (cid:101) Q k ( Q k b · P k D α f )( x ) . (7.1)We can apply our results to these paraproducts as well, as is shown in the next corollary. Corollary 7.4.
Let ν ∈ R , α ∈ N n , and b ∈ ˙ B | α |− ν , ∞∞ . Then Π α b ∈ SIO ν ( ∞ ) satisfies W BP ν and ( Π α b ) ∗ ( x β ) = for all β ∈ N n , where Π α b is as in (7.1) . Hence Theorem 4.3, Theorem 5.1, Theorem5.2, Corollary 5.3, and Corollary 5.4 can all be applied to Π α b with any number of vanishingmoments.Proof. It is trivial to see that Π b is continuous from S P into S (cid:48) for an appropriately chosen P ∈ N .Indeed, taking M ∈ N to be an even integer larger than | ν | , and g ∈ S , we have | (cid:104) Π b f , g (cid:105) | ≤ (cid:107) b (cid:107) ˙ B | α |− ν , ∞∞ ∑ k ∈ Z (cid:90) R n ( M + ν ) k | ( | ∇ | M D α ϕ ) k ∗ ( | ∇ | − M f )( x ) (cid:101) Q k g ( x ) | dx (cid:46) (cid:107) b (cid:107) ˙ B | α |− ν , ∞∞ (cid:107) | ∇ | − M f (cid:107) L ∑ k ∈ Z ( M + ν ) k (cid:107) (cid:101) Q k g (cid:107) L (cid:46) (cid:107) b (cid:107) ˙ B | α |− ν , ∞∞ (cid:107) f (cid:107) ˙ W − M , (cid:107) g (cid:107) ˙ B M + ν , as long as M > | ν | (which assures that S ⊂ ˙ B M + ν , since M + ν > f ∈ S and g ∈ S P | (cid:104) Π b f , g (cid:105) | ≤ (cid:107) b (cid:107) ˙ B | α |− ν , ∞∞ ∑ k ∈ Z (cid:90) R n ν k | ( D α ϕ ) k ∗ f ( x ) (cid:101) Q k g ( x ) | dx (cid:46) (cid:107) b (cid:107) ˙ B | α |− ν , ∞∞ (cid:107) f (cid:107) L ∑ k ∈ Z ν k (cid:107) (cid:101) Q k g (cid:107) L (cid:46) (cid:107) b (cid:107) ˙ B | α |− ν , ∞∞ (cid:107) f (cid:107) L (cid:107) g (cid:107) ˙ B ν , . NALYSIS OF HYPER-SINGULAR, FRACTIONAL, AND ORDER-ZERO SINGULAR INTEGRAL OPERATORS 33
Here we choose P ∈ N large enough so that D P ⊂ ˙ W − M , ∩ ˙ B ν , . Note that this is also sufficient toshow that Π α b and ( Π α b ) ∗ both satisfy W BP ν . The kernel of Π b is π α b ( x , y ) = ( − ) | α | α ! ∑ k ∈ Z k | α | (cid:90) R n (cid:101) ψ k ( x − u ) Q k b ( u )( D α ϕ ) k ( u − y ) du . For β , µ ∈ N n and x (cid:54) = y , it follows that | D β x D µy π α b ( x , y ) | (cid:46) (cid:107) Q k b (cid:107) ˙ B | α |− ν∞ ∑ k ∈ Z ( ν + | β | + | µ | ) k Φ n + ν + | β | + | µ | + k ( x − y ) (cid:46) (cid:107) Q k b (cid:107) ˙ B | α |− ν∞ | x − y | n + ν + | β | + | µ | . Therefore Π b ∈ SIO ν ( ∞ ) . It is easy to see that ( Π α b ) ∗ ( x α ) = (cid:101) ψ k ∈ S ∞ . (cid:3) Remark . It is worth noting that Π α b may not belong to CZO ν , but we still conclude manyboundedness results for it. In particular, in when ν = b ∈ ˙ B , ∞∞ \ BMO , the T Π b is not L -bounded. However, we still conclude boundedness results for Π b onnegative smoothness spaces, just not on any Lebesgue spaces. It is likely, by analogy, that b ∈ ˙ B | α |− ν , ∞∞ \ I | α |− ν ( BMO ) implies that Π α b is not bounded from ˙ W ν , into L (and hence Π α b would notbelong to CZO ν ), but we don’t pursue this property here.7.4. Smooth and Oscillating Operator Decompositions.
In this application, we decompose asingular integral operator T into a sum of two terms S + O , one that preserves smoothness andone that preserves oscillatory properties of the input function. We achieve this by constructingseveral paraproducts which satisfy Π ∗ ( x α ) = α . A sum of such operators define O , andhence O enjoys all of the oscillatory preserving properties associated to the cancellation conditionsof the form O ∗ ( x α ) =
0. Furthermore, S will be constructed so that S ( x α ) = α , which is sufficient for S to be bounded on many smooth function spaces.To motivate this, let’s consider for a moment an operator T ∈ SIO ν ( ∞ ) of convolution type. Thatis, assume there is a distribution kernel k ∈ S (cid:48) ( R n ) such that T f ( x ) = (cid:104) k , f ( x − · ) (cid:105) for f ∈ S P for some P ∈ N n . Such an operator preserves both regularity and oscillation since convolutionoperators commute. For instance, suppose that T is bounded from X into Y , where X , Y ⊂ S (cid:48) / P are Banach spaces. It follows that T is bounded from I s ( X ) into I s ( Y ) for all s ∈ R , where I s ( X ) = {| ∇ | s f : f ∈ X } with the natural norm (cid:107) f (cid:107) I s ( X ) = (cid:107) | ∇ | s f (cid:107) X . This is because (cid:107) T f (cid:107) I s ( X ) = (cid:107) | ∇ | s ( T f ) (cid:107) X = (cid:107) T ( | ∇ | s f ) (cid:107) X (cid:46) (cid:107) | ∇ | s f (cid:107) Y = (cid:107) f (cid:107) I s ( Y ) . When s >
0, this says that if f has s -order derivatives in X , then T f has s -order derivatives in Y .For s <
0, it says that f has s -order anti-derivatives in X , then T f has s -order anti-derivatives in Y ,which in many situations quantify oscillatory properties of f and T f . Hence convolution opera-tors simultaneously preserve both regularity and oscillatory properties of its input functions. Thiscannot be expected for non-convolution operators, but the main result of this section formulates adecomposition that extends this principle to non-convolution operators in some sense. We show,roughly, that for any operator T ∈ SIO ν , we can decompose T = S + O , where S preserves smooth-ness and O preserves oscillation. This is achieved by constructing S and O so that S | ∇ | s ≈ | ∇ | s S and O | ∇ | − s ≈ | ∇ | − s O for s >
0, in the appropriate sense, so that S and O each behave like a con-volution operator on one side. Based on our operator calculus from Section 4, to construct S and O in this way, it is sufficient to make sure that S ( x α ) = O ∗ ( x α ) = α . This ismade precise below.We will have to use the non-convolutional moment for singular integral operators, which weredefined for SIO in [22, 23]. For T ∈ SIO ν ( M + γ ) , α ∈ N n with | α | ≤ ν + M , define [[ T ]] α ∈ D P by (cid:104) [[ T ]] α , ψ (cid:105) = lim R → ∞ (cid:90) R n K ( x , y )( x − y ) α η R ( y ) ψ ( x ) dy dx . Following the same argument used to justify the definition of T ( x α ) , we can define [[ T ]] α ∈ D (cid:48) P for the same ranges of indices. See also [22, 23] for more information on this construction. Lemma 7.6.
Let T ∈ SIO ν ( M + γ ) and L ≤ ν + M. Then T ( x α ) = for all | α | ≤ L if and only if [[ T ]] α = for all | α | ≤ L.Proof.
This proof follows immediately from the following formula, which is just expanding thepolynomial ( x − y ) α . Let | α | ≤ L , η R ∈ D P and ψ ∈ D P , for P ∈ N sufficiently larger, be as inthe definition of [[ T ]] α , and we have (cid:104) [[ T ]] α , ψ (cid:105) = lim R → ∞ ∑ β + µ = α c β , µ (cid:90) R n K ( x , y ) y µ η R ( y ) x β ψ ( x ) dy dx = ∑ β + µ = α c β , µ (cid:68) T ( x µ ) , x β ψ (cid:69) . It immediately follows that T ( x α ) vanishes for all | α | ≤ L if and only if [[ T ]] α does. (cid:3) Lemma 7.7.
Let ν ∈ R , α ∈ N n , b ∈ ˙ B | α |− ν , ∞∞ , and Π α b be defined as in (7.1) . Then [[ Π α b ]] β ∈ ˙ B | β |− ν , ∞∞ for all β ∈ N n , Π α b ( x β ) = for all | β | ≤ | α | with β (cid:54) = α , and [[ Π α b ]] α = b.Proof. For any β ∈ N n and ψ ∈ D P with P sufficiently large, we have | ψ j ∗ [[ Π α b ]] β ( x ) | = lim R → ∞ (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) R n π α b ( u , y )( u − y ) β η R ( y ) ψ xj ( u ) dy du (cid:12)(cid:12)(cid:12)(cid:12) ≤ lim sup R → ∞ ∑ ρ + µ = β c ρ , µ ∑ k ∈ Z k | α | (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) R n ( D α ϕ ) k ( v − y )( v − y ) µ η R ( y ) × Q k b ( v ) (cid:101) ψ k ( u − v )( u − v ) ρ ψ xj ( u ) dv du dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ lim sup R → ∞ ∑ ρ + µ = β c ρ , µ ∑ k ∈ Z k ( | α |−| β | ) (cid:90) R n (cid:18) (cid:90) R n | ( D α ϕ ) k ( y )( k y ) µ η R ( y ) | dy (cid:19) × | Q k b ( v ) | | (cid:101) ψ ρ k ∗ ψ xj ( v ) | dv (cid:46) (cid:107) b (cid:107) ˙ B | α |− ν , ∞∞ ∑ k ∈ Z − K | j − k | k ( ν −| β | ) (cid:46) ( ν −| β | ) j (cid:107) b (cid:107) ˙ B | α |− ν , ∞∞ , NALYSIS OF HYPER-SINGULAR, FRACTIONAL, AND ORDER-ZERO SINGULAR INTEGRAL OPERATORS 35 where (cid:101) ψ ρ k ( x ) = (cid:101) ψ k ( x ) x ρ . Then [[ Π α b ]] β ∈ ˙ B | β |− ν , ∞∞ for all β ∈ N n . For ψ ∈ D P , with P ∈ N n fixedsufficiently large, we have (cid:104) [[ Π α b ]] α , ψ (cid:105) = lim R → ∞ (cid:90) R n π α b ( x , y )( x − y ) α η R ( y ) ψ ( x ) dy dx = lim R → ∞ ( − ) | α | α ! ∑ k ∈ Z k | α | (cid:90) R n (cid:18) (cid:90) R n ( D α ϕ ) k ( u − y )( x − y ) α η R ( y ) dy (cid:19) × (cid:101) ψ k ( x − u ) Q k b ( u ) ψ ( x ) du dx = ∑ k ∈ Z (cid:90) R n (cid:101) ψ k ( x − u ) Q k b ( u ) ψ ( x ) du dx = (cid:104) b , ψ (cid:105) . Similarly, for ψ ∈ D P with P sufficiently large and | β | ≤ | α | such that α (cid:54) = β (cid:68) Π α b ( x β ) , ψ (cid:69) = lim R → ∞ α ! ∑ k ∈ Z (cid:90) R n Q k b ( x ) (cid:18) (cid:90) R n ϕ k ( x − y ) D α (cid:16) η R ( y ) y β (cid:17) dy (cid:19) (cid:101) Q k ψ ( x ) dx = α ! ∑ k ∈ Z (cid:90) R n Q k b ( x ) (cid:18) (cid:90) R n ϕ k ( x − y ) D α ( y β ) dy (cid:19) (cid:101) Q k ψ ( x ) dx = . Note that | β | ≤ | α | and β (cid:54) = α implies that D α ( y β ) = (cid:3) Theorem 7.8.
Let T ∈ SIO ν ( M + γ ) for some M ∈ N and < γ ≤ . If [[ T ]] α ∈ ˙ B | α |− ν , ∞∞ for all | α | ≤ M. Then there exist operators S ∈ SIO ν ( M + γ ) and O ∈ SIO ν ( ∞ ) such that T = S + O,S ( x α ) = for | α | ≤ M, and O ∗ ( x α ) = for α ∈ N n . Furthermore, if T satisfies W BP ν , then bothS and O satisfy W BP ν , in which case Theorem 4.3, Theorem 5.1, Theorem 5.2, Corollary 5.3, andCorollary 5.4 can be applied to S ∗ and O. Note that the results from Sections 4 and 5 can be applied to O regardless of whether T satisfies W BP ν since O is a sum of paraproducts of the form (7.1) and by Corollary 7.4 the results can beapplied to each of these paraproducts. of Theorem 7.8. Let T be as above. Define b = [[ T ]] = T ( ) ∈ ˙ B − ν , ∞∞ . For 1 ≤ | α | ≤ M , define b α = [[ T ]] α − ∑ | β | < | α | [[ Π β b β ]] α ∈ ˙ B | α |− ν , ∞∞ . Also define S = T − ∑ | α |≤ M Π α b α and O = ∑ | α |≤ M Π α b α . It immediately follows that S ∈ SIO ν ( M + γ ) and O ∈ SIO ν ( ∞ ) . Using Lemmas 7.6 and 7.7, wealso have [[ S ]] = [[ T ]] − ∑ | α |≤ M [[ Π α b α ]] = [[ T ]] − [[ Π b ]] = , and for 0 < | α | ≤ M we have [[ S ]] α = [[ T ]] α − ∑ | β |≤ M [[ Π β b β ]] α = [[ T ]] α − b α − ∑ | β | < | α | [[ Π β b β ]] α = . By Lemma 7.6, it follows that S ( x α ) = | α | ≤ M . It also follows from Corollary 7.4 that O ∗ ( x α ) = α ∈ N n . Note also that O always satisfies the W BP ν , and if T satisfies W BP ν ,then so does S . (cid:3) Remark . It may seem a little strange that we use the non-convolution moments [[ T ]] α , ratherthan T ( x α ) , in Lemma 7.7 and Theorem 7.8. By Lemma 7.7, when using vanishing momentconditions the two are equivalent. However, there is a crucial difference when the moments arenot required to vanish, but instead some other conditions as we do in Theorem 7.8. This differencemanifests in our setting when computing [[ Π α b ]] β versus Π α b ( x β ) for | β | > | α | . In Lemma 7.7, weshowed that [[ Π α b ]] β ∈ ˙ B | β |− ν , ∞∞ , which is the natural condition to expect in this setting. However, itmay not be the case that Π α b ( x β ) ∈ ˙ B | β |− ν , ∞∞ . To demonstrate this, let the dimension n = ν = α = b ∈ ˙ B , ∞∞ , and β =
1. Then for x ∈ R | ψ j ∗ Π b ( x β )( x ) | = lim R → ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∑ k ∈ Z (cid:90) R ϕ k ( u − y ) y η R ( y ) Q k b ( u ) (cid:101) ψ k ∗ ψ xj ( u ) du dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ | x | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∑ k ∈ Z (cid:90) R Q k b ( u ) (cid:101) ψ k ∗ ψ xj ( u ) du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − | ψ j ∗ [[ Π b ]] ( x ) |≥ | x | | ψ j ∗ b ( x ) | − | ψ j ∗ [[ Π b ]] ( x ) | . Taking j =
0, we have (cid:107) Π b ( x ) (cid:107) ˙ B , ∞∞ ≥ | x | | ψ ∗ b ( x ) | − (cid:107) [[ Π b ]] (cid:107) ˙ B , ∞∞ . By Lemma 7.7, (cid:107) [[ Π b ]] (cid:107) ˙ B , ∞∞ < ∞ , and it is not hard to construct b ∈ ˙ B , ∞∞ such that | x | | ψ ∗ b ( x ) | is unbounded (for example, b ( x ) = sin ( x ) would do). Hence Π b ( x ) / ∈ ˙ B , ∞∞ for such b . Similarconstructions can be done in any dimension and for b ∈ ˙ B | α |− ν , ∞∞ to produce the property Π α b ( x β ) / ∈ ˙ B | β |− ν∞ Sparse Domination.
There has been a lot of interest lately in sparse domination results forCalder´on-Zygmund operators. The first such result is due to Lerner [30], but there have been manyextensions and improvements; see for example [35, 9, 5, 13, 27, 28, 29, 45]. However, there donot appear to be any results that apply to regularity estimates for operators or to hyper-singularoperators. There are some sparse estimates for fractional integral operators, for example in [35].We will apply the sparse domination result from [9], which can be summarized as follows. Fora collection of dyadic cubes S , define the dyadic operator A S f ( x ) = ∑ Q ∈ S (cid:104) f (cid:105) Q χ Q ( x ) , where (cid:104) f (cid:105) Q = | Q | (cid:82) Q f ( x ) dx . The collection of dyadic cubes belonging to the same dyadic grid S is sparse if for every Q ∈ S there exists a measurable subset E ( Q ) ⊂ Q with | E ( Q ) | > | Q | / E ( Q ) ∩ E ( Q (cid:48) ) = /0 for ever Q (cid:48) ∈ S with Q (cid:48) (cid:40) Q . They prove that if T ∈ CZO and f is an integrablefunction supported in a cube Q , then there exist sparse dyadic collections S , ..., S n (possibly NALYSIS OF HYPER-SINGULAR, FRACTIONAL, AND ORDER-ZERO SINGULAR INTEGRAL OPERATORS 37 associated to different dyadic grids) such that | T f ( x ) | (cid:46) n ∑ i = A S i ( | f | )( x ) (7.2)almost everywhere on Q . We use this result to prove our next result. Corollary 7.10.
Let ν ∈ R , L be an integer with L ≥ | ν | , M ≥ max ( L , L − ν ) , ( L − ν ) ∗ < γ ≤ , andT ∈ SIO ν ( M + γ ) satisfies W BP ν . • Assume that T ∗ ( x α ) = for all | α | ≤ L. Then for any < p < ∞ , < t < (cid:101) L + γ , cubeQ ⊂ R n , and f ∈ ˙ W − t , p with supp ( | ∇ | − t f ) ⊂ Q , there is a polynomial P f and sparsecollections of dyadic cubes S , ..., S n such that | | ∇ | − ( ν + t ) T f ( x ) − P f ( x ) | (cid:46) n ∑ i = A S i ( | | ∇ | − t f | )( x ) a.e. x ∈ Q . (7.3) • Assume that T ( x α ) = for all | α | ≤ L. Then for any < p < ∞ , < t < (cid:101) L + γ , cubeQ ⊂ R n , and f ∈ ˙ W ν + t , p with supp ( | ∇ | ν + t f ) ⊂ Q , there is a polynomial P f and sparsecollections of dyadic cubes S , ..., S n such that | | ∇ | t T f ( x ) − P f ( x ) | (cid:46) n ∑ i = A S i ( | | ∇ | ν + t f | )( x ) a.e. x ∈ Q . (7.4) Proof.
Let ν ∈ R , L be an integer with L ≥ | ν | , M ≥ max ( L , L − ν ) , ( L − ν ) ∗ < γ ≤
1, and T ∈ SIO ν ( M + γ ) satisfy W BP ν . Assume that T ∗ ( x α ) = | α | ≤ L . By Theorem 4.3, for any0 < t < L + γ there exists T t ∈ CZO such that P f = | ∇ | − ( ν + t ) T | ∇ | t T f − T t f is a polynomial forall f ∈ S ∞ . Note that T t ∈ CZO implies that T t is already well-defined on L p for 1 < p < ∞ .Hence T t ( | ∇ | − t f ) is well-defined and belongs to L p for any f ∈ ˙ W − t , p . It also follows that | ∇ | − ( ν + t ) T f = T t ( | ∇ | − t f ) for all f ∈ S ∞ , and (cid:107) | ∇ | − ( ν + t ) T f − P f (cid:107) L p = (cid:107) T t ( | ∇ | − t f ) (cid:107) L p (cid:46) (cid:107) f (cid:107) ˙ W − t , p .Then | ∇ | − ( ν + t ) T f can be defined pointwise for every f ∈ ˙ W − t , p with 1 < p < ∞ via the equation | ∇ | − ( ν + t ) T f = T t ( | ∇ | − t f ) + P f . Now it is just a matter of applying the pointwise sparse operatorestimate from [9]. Let f ∈ ˙ W − t , p with supp ( | ∇ | − t f ) ⊂ Q , and since T t ∈ CZO it follows that | | ∇ | − ( ν + t ) T f ( x ) − P f ( x ) | = | T t ( | ∇ | − t f )( x ) | (cid:46) n ∑ i = A S i ( | | ∇ | − t f | )( x ) . The estimate in (7.4) can be proved in the same way. (cid:3)
Operator Calculus.
Throughout this article, we have been working with a restricted operatorcalculus, where we only consider compositions of the form | ∇ | − s T | ∇ | t . In this application, weconstruct a true operator calculus (or operator algebra) made up of singular integrals with differentsingularities. In order to make our notation and computations a little simpler here, we will onlywork with operators in SIO ν ( ∞ ) that satisfy T ( x α ) = T ∗ ( x α ) = α ∈ N n . These assumptionare necessary for some of the algebras we construct, but not for all. Before we continue, wewill need some additional information about the operators we defined in Corollary 3.2, which weprovide in the next lemma. Lemma 7.11.
Let ν ∈ R and T ∈ SIO ν ( ∞ ) . Assume that T ( x α ) = T ∗ ( x α ) = for all α ∈ N n andT ∈ W BP ν . Fix L ∈ N , ψ , ˜ ψ ∈ D P for P sufficiently large, and define λ j , k ( x , y ) = (cid:68) T ∗ ψ xj , ˜ ψ yk (cid:69) forj , k ∈ Z and x , y ∈ R n . Then (cid:90) R n λ j , k ( x , y ) x α dx = (cid:90) R n λ j , k ( x , y ) y α dx = for all | α | ≤ L.Proof.
Let L ∈ N , | α | ≤ L , and η ∈ D P with η = B ( , ) and η R ( x ) = η ( x / R ) , where P ischosen sufficiently large. By Theorem 3.1, we have (cid:90) R n λ j , k ( x , y ) y α dy = (cid:90) R n (cid:10) T ψ xj , ψ yk (cid:11) y α dy = lim R → ∞ (cid:10) T ψ xj , F α , R , k (cid:11) , where F α , R , k ( u ) = (cid:90) | y | < R ψ k ( y − u ) y α dy . It follows that supp ( F α , R , k ) ⊂ B ( , R + k + ) \ B ( , R − k + ) and F α , R , k ∈ D P . Then as long as R > | x | + k + + j + , it follows that | (cid:10) T ψ xj , F α , R , k (cid:11) | = (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) R n ( K ( u , v ) − J Mx [ K ( · , v )]( u )) ψ xj ( u ) F α , R , k ( v ) du dv (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:90) R n | x − u | M + γ | u − v | n + ν + M + γ | ψ xj ( u ) F α , R , k ( v ) | du dv (cid:46) ( M + γ ) j R n + ν + M + γ (cid:90) | y | < R (cid:18) (cid:90) R n | ψ xj ( u ) ψ k ( v ) | du dv (cid:19) | y α | dy (cid:46) ( M + γ ) j R n + ν + M + γ R n + | α | = ( M + γ ) j R − ( ν + M + γ −| α | ) , which tends to zero as R → ∞ . Here we take M > L + | ν | + γ and P > M ; note that P then dependson L , and so we cannot completely remove the restriction | α | ≤ L in the statement of Lemma 7.11.Therefore the first integral condition in (7.5) holds for | α | ≤ L , and by symmetry the same holdsfor the second one. (cid:3) Theorem 7.12.
Let V ⊂ R be a set that is closed under addition. Then the collection of operators A V = { T ∈ SIO ν ( ∞ ) : ν ∈ V , T ( x α ) = T ∗ ( x α ) = , T ∈ W BP ν } is an operator algebra in the sense that it is closed under composition and transpose.Proof. Fix two real numbers ν , ν ∈ V . In order to show S , T ∈ A V implies S ◦ T ∈ SIO ν + ν ( ∞ ) where T ∈ SIO ν and S ∈ SIO ν , we must first show that S ◦ T and ( S ◦ T ) ∗ are defined as (or can beextended to) operators from S P into S (cid:48) for some P ∈ N sufficiently large. We first note that thevanishing moment and weak boundedness properties of operators in A V imply that all membersof A V ∩ SIO ν are bounded from ˙ W ν + s , p into ˙ W s , p for all s ∈ R such that s < − ν and for s > f ∈ S P and g ∈ S (with P ≥ | ν | + | ν | ), wenote that for any 1 < p < ∞ | (cid:104) S ◦ T f , g (cid:105) | = | (cid:104) T f , S ∗ g (cid:105) | ≤ (cid:107) T f (cid:107) ˙ W − µ , p (cid:107) S ∗ g (cid:107) ˙ W µ , p (cid:48) NALYSIS OF HYPER-SINGULAR, FRACTIONAL, AND ORDER-ZERO SINGULAR INTEGRAL OPERATORS 39 ≤ (cid:107) T (cid:107) ˙ W ν − µ , p → ˙ W − µ , p (cid:107) S ∗ (cid:107) ˙ W µ , p (cid:48) → ˙ W ν + µ , p (cid:48) (cid:107) f (cid:107) ˙ W ν − µ , p (cid:107) g (cid:107) ˙ W ν + µ , p (cid:48) , where µ > max ( ν , − ν ) . Note that T is bounded from ˙ W ν − µ , p into ˙ W − µ , p since − µ < − ν , and S ∗ is bounded from ˙ W µ , p (cid:48) into ˙ W ν + µ , p (cid:48) since ν + µ >
0. Since S P and S embed continuouslyinto ˙ W − µ , p and ˙ W ν + µ , p (cid:48) , respectively, it follows that S ◦ T is continuous from S P into S (cid:48) . Bysymmetry, it follows that ( S ◦ T ) ∗ is also continuous from S P into S (cid:48) . Furthermore, this inequality,and a similar one for ( S ◦ T ) ∗ , imply that S ◦ T satisfies the W BP ν + ν Next we show that S ◦ T has a standard kernel. Let ψ and (cid:101) ψ be as in Lemma 2.6, Q k f = ψ k ∗ f ,and (cid:101) Q k f = (cid:101) ψ k ∗ f . For f , g ∈ D P , we have (cid:104) S ◦ T f , g (cid:105) = ∑ j , k ,(cid:96), m ∈ Z (cid:68) (cid:101) Q (cid:96) Q (cid:96) T Q m (cid:101) Q m f , (cid:101) Q k Q k S ∗ Q j (cid:101) Q j g (cid:69) = ∑ j , k ,(cid:96), m ∈ Z (cid:90) R n ω j , k ,(cid:96), m ( x , y ) f ( y ) g ( x ) dy dx , where ω j , k ,(cid:96), m ( x , y ) = (cid:90) R n (cid:101) ψ (cid:96) ( u − w ) (cid:101) ω j , k ,(cid:96), m ( w , ξ ) (cid:101) ψ j ( ξ − x ) dz d ξ dv dw du and (cid:101) ω j , k ,(cid:96), m ( w , ξ ) = (cid:90) R n λ T (cid:96), m ( w , z ) (cid:101) ψ m ( z − y ) (cid:101) ψ k ( u − v ) λ S ∗ k , j ( v , ξ ) dz dv du . For any fixed K , N ≥
0, it follows from Corollary 3.2 and Lemma 7.11, as well as similar argumentsto those in the proof of Corollary 3.2, that (cid:12)(cid:12)(cid:101) ω j , k ,(cid:96), m ( w , ξ ) (cid:12)(cid:12) (cid:46) ν min ( (cid:96), m )+ ν min ( j , k ) − K max ( | j − k | , | k − (cid:96) | , | (cid:96) − m | ) Φ N min ( j , k ,(cid:96), m ) ( w − ξ ) . From this bound, it easily follows that | ω j , k ,(cid:96), m ( x , y ) | (cid:46) ν min ( (cid:96), m )+ ν min ( j , k ) − K max ( | j − k | , | k − (cid:96) | , | (cid:96) − m | ) Φ N min ( j , k ,(cid:96), m ) ( x − y ) . Then for any α , β ∈ N n and x (cid:54) = y , we have ∑ j , k ,(cid:96), m ∈ Z | D α y D β x ω j , k ,(cid:96), m ( x , y ) | (cid:46) ∑ j , k ,(cid:96), m ∈ Z ( ν + | α | ) m +( ν + | β | ) j − K max ( | j − k | , | k − (cid:96) | , | (cid:96) − m | ) Φ N min ( j , k ,(cid:96), m ) ( x − y ) (cid:46) ∑ j , k ,(cid:96), m ∈ Z ( ν + ν + | α | + | β | ) min ( j , k ,(cid:96), m ) − ˜ K max ( | j − k | , | k − (cid:96) | , | (cid:96) − m | ) Φ N min ( j , k ,(cid:96), m ) ( x − y ) (cid:46) ∑ j ∈ Z ( ν + ν + | α | + | β | ) j Φ Nj ( x − y ) (cid:46) | x − y | n + ν + ν + | α | + | β | . These sums converges as long as K > ( | ν | + | ν | + | α | + | β | ) and N > n + ν + ν + | α | + | β | . Itfollows that K S ◦ T ( x , y ) = ∑ j , k ,(cid:96), m ∈ Z ω j , k ,(cid:96), m ( x , y ) is the kernel of S ◦ T and is a ( ν + ν ) -order standard kernel that is C ∞ off of the diagonal. There-fore S ◦ T ∈ SIO ν + ν ( ∞ ) . Using the estimate for ω j , k ,(cid:96), m above, for f , g ∈ S ∞ and t ∈ R we have | (cid:104) S ◦ T f , g (cid:105) | ≤ ∑ j , k ,(cid:96), m ∈ Z t ( m − j ) (cid:12)(cid:12)(cid:12)(cid:68) (cid:101) Q (cid:96) Q (cid:96) T Q m (cid:101) Q tm ( | ∇ | − t f ) , (cid:101) Q k Q k S ∗ Q j (cid:101) Q − tj ( | ∇ | t g ) (cid:69)(cid:12)(cid:12)(cid:12) (cid:46) ∑ j , k ,(cid:96), m ∈ Z t ( m − j ) (cid:90) R n (cid:12)(cid:12)(cid:12)(cid:101) ω j , k ,(cid:96), m ( x , y ) (cid:101) Q tm ( | ∇ | − t f )( y ) (cid:101) Q − tj ( | ∇ | t g )( x ) (cid:12)(cid:12)(cid:12) dy dx (cid:46) (cid:90) R n ∑ j , k ,(cid:96), m ∈ Z ( ν + ν ) m − ˜ K max ( | j − k | , | k − (cid:96) | , | (cid:96) − m | ) M ( (cid:101) Q tm ( | ∇ | − t f ))( x ) | (cid:101) Q − tj ( | ∇ | t g )( x ) | dx (cid:46) (cid:107) f (cid:107) ˙ W ν + ν + t , p (cid:107) g (cid:107) ˙ W − t , p (cid:48) . In this estimate, we fix a P ∈ N large enough depending on t . Therefore S ◦ T can be extended to abounded linear operator from ˙ W ν + ν + t , p into ˙ W t , p for all t ∈ R and 1 < p < ∞ . Taking t =
0, we seethat S ◦ T ∈ CZO ν + ν ( ∞ ) , and we can apply Theorem 6.5, which implies that ( S ◦ T ) ∗ ( x α ) = α ∈ N n . By symmetry S ◦ T ( x α ) = α ∈ N n as well. Also ν + ν ∈ V , and so S ◦ T ∈ A V which verifies that A V closed under composition as long as V is closed under addition. Since SIO ν ( ∞ ) , W BP ν , and the condition T ( x α ) = T ∗ ( x α ) = A V is closed under transposes too. Therefore A V is an algebra that is closed undercomposition and transposes. (cid:3) Remark . Theorem 7.12 defines many operator algebras for different classes of singular inte-gral operators. If one takes V = { } , then A { } is a set of Calder ´on-Zygmund operators and one ofthe operator algebras discussed by Coifman and Meyer in [12]. Some other interesting examplesof algebras A V can be constructed by taking V to be, for example, V = R or V = { µ ν : ν ∈ Z } for some fixed µ ∈ R . One can also modify any of these examples by replacing V with V ∩ ( , ∞ ) , V ∩ [ , ∞ ) , V ∩ ( − ∞ , ) , or V ∩ ( − ∞ , ] , which amounts to restricting an algebra to differentialor fractional integral operators (strictly or including order-zero operators). Furthermore, one cancombine different algebras by defining A V = A V + A V , where V ⊂ R is the smallest set thatis closed under addition and contains V ∪ V . Using this convention, we can consider the set V = { µ − µ π : µ , µ ∈ Q , µ ≥ , µ ≥ , µ · µ (cid:54) = } ⊂ R , which is closed under addition. Since π is transcendental, it follows that 0 / ∈ V . This generates a somewhat peculiar example of A V sincewith this selection of V , it follows that A V is an operator that has both derivative and fractionalintegral operators, but does not contain any Calder´on-Zygmund operators (order zero operators).Of course many other examples can be generated by selecting V in different ways, and even furtherunderstand A V through the algebraic properties of V .Note that the properties imposed on the operators in A V imply that if A V ∩ SIO ν ⊂ CZO ν . Alsonote that any algebra A V contains all convolutions operators in CZO ν ( ∞ ) as long as ν ∈ V . That is (cid:91) ν ∈ V { T ∈ CZO ν ( ∞ ) : T is a convolution } ⊂ A V ⊂ (cid:91) ν ∈ V CZO ν ( ∞ ) . Furthermore, it is not hard to see that both inclusions here are strict for non-empty sets V ⊂ R . NALYSIS OF HYPER-SINGULAR, FRACTIONAL, AND ORDER-ZERO SINGULAR INTEGRAL OPERATORS 41 ACKNOWLEDGEMENTS
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12 (1945) 47–76.L. C
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