Endpoint estimates for compact Calderón-Zygmund operators
EENDPOINT ESTIMATES FOR COMPACTCALDER ´ON-ZYGMUND OPERATORS
JAN-FREDRIK OLSEN AND PACO VILLARROYA
Abstract.
We prove necessary and sufficient conditions for aCalder´on-Zygmund operator to be compact at the endpoint from L ( R d ) into L , ∞ ( R d ). Introduction
The paper [9] introduced a new T (1) Theory to study compactnessof singular integral operators. Its main result provided necessary andsufficient conditions for operators associated with classical Calder´on-Zygmund kernels to be compact on L p ( R ) for all 1 < p < ∞ . Thischaracterization was expressed in terms of three conditions: the de-cay of the derivative of the kernel along the direction of the diagonal,an appropriate ’weak compactness condition’, and the membership ofproperly constructed T (1) and T ∗ (1) functions to the space CMO( R ).Here, the latter space is defined as the closure in BMO( R ) of the spaceof continuous functions vanishing at infinity. Later, in [6], the endpointcase of compactness from L ∞ ( R ) into CMO( R ) was obtained.We note that, although the results in the two above-mentioned pa-pers were proven in the context of functions defined on R , the resultsand techniques developed also hold in the multi-dimensional setting.See, for instance, the preprint [10] which contains the proof of a global T ( b ) theorem for compactness of singular integrals in R d .A natural question is whether one can obtain the two remainingendpoint results, namely, compactness from H ( R d ) into L ( R d ) andfrom L ( R d ) into L , ∞ ( R d ). A little bit of thought shows that the for-mer case is an immediate consequence of [6] and Schauder’s Theorem,which states that an operator between two Banach spaces, T : X → Y ,is compact if and only if the same holds true for T ∗ : Y ∗ → X ∗ (see Date : September 10, 2018.
Key words and phrases.
Compact singular integral operators, T (1) Theory.The second author has been partially supported by Spanish project MTM2011-23164. a r X i v : . [ m a t h . C A ] D ec JAN-FREDRIK OLSEN AND PACO VILLARROYA e.g. [8]). The point of this paper is to prove that the latter endpointresult also holds.Although both results are the natural extensions of the classical end-point theorems for boundedness, the method used to prove compactnessfrom L ( R d ) to L , ∞ ( R d ) is very different from the standard one. Itis true that the demonstration follows the same general scheme andshares identical initial steps as in the proof of boundedness. However,the standard reasoning comes to a halt when applied to the orthogonalprojection operator, which is an element completely absent in the clas-sical proof. This difficulty forces one to perform the operator analysisin a different way, more in accordance with the ideas carried out toshow compactness at the non-endpoint case [9].Since the current project is the continuation of [9], we often cite thispaper for detailed references about the notation and the definitions weuse, and also for proofs of those results that we merely state. And yet,we intend to present a paper as self-contained as possible.2. Definitions
Notation.
We say that I = (cid:81) di =1 [ a i , b i ) is a cube in R d if thequantity | b i − a i | remains constant for all indices i ∈ { , . . . , d } . Forevery cube I ⊂ R d we denote its centre by c ( I ) = (2 − ( a i + b i )) di =1 ,its side length by (cid:96) ( I ) = | b i − a i | , and its volume by | I | = (cid:96) ( I ) d . Forany λ >
0, we denote by λI the cube such that c ( λI ) = c ( I ) and | λI | = λ d | I | . Accordingly, we also write B = B d = ( − / , / d and B λ = λ B = ( − λ/ , λ/ d .We denote by | · | p , with 0 < p ≤ ∞ , the (cid:96) p -norm in R d and by | · | the modulus of a complex number. Hopefully, this notation will notcause any confusion with the one we use for the volume of a cube.Given two cubes I, J ⊂ R d , we denote by (cid:104) I, J (cid:105) any cube with mini-mal side length containing I ∪ J and write its side length by diam( I ∪ J ).If there is more than one cube satisfying these conditions, we will sim-ply select one and refer to it as (cid:104) I, J (cid:105) regardless of the choice.We note that if I = (cid:81) di =1 I i , J = (cid:81) di =1 J i , with I i , J i intervals in R ,we have diam( I ∪ J ) = max i diam( I i ∪ J i ), where diam( I i ∪ J i ) is thelength of (cid:104) I i , J i (cid:105) , the smallest interval containing I i and J i . Therefore,we have the following equivalencesdiam( I ∪ J ) ≈ (cid:96) ( I ) + (cid:96) ( J )2 + | c ( I ) − c ( J ) | ∞ ≈ max( (cid:96) ( I ) , (cid:96) ( J )) + | c ( I ) − c ( J ) | ∞ . NDPOINT ESTIMATES FOR COMPACT CALDER ´ON-ZYGMUND OPERATORS We also define the relative distance between I and J byrdist( I, J ) = diam( I ∪ J )max( (cid:96) ( I ) , (cid:96) ( J )) , which is comparable to max(1 , n ) where n is the smallest number oftimes the larger cube needs to be shifted a distance equal to its sidelength so that it contains the smaller one. Note that from the above,we have12 (cid:16) | c ( I ) − c ( J ) | ∞ max( (cid:96) ( I ) , (cid:96) ( J )) (cid:17) ≤ rdist( I, J ) ≤ | c ( I ) − c ( J ) | ∞ max( (cid:96) ( I ) , (cid:96) ( J )) . We also define the eccentricity of I and J to beecc( I, J ) = min( | I | , | J | )max( | I | , | J | ) . Finally, we say that a cube I is dyadic if I = 2 j (cid:81) di =1 [ k i , k i + 1) forsome j, k , . . . , k d ∈ Z , and denote by C and D the families of all cubesand all dyadic cubes in R d , respectively. Definition 2.1.
For every M ∈ N , we define C M to be the family ofall cubes in R d such that − M ≤ (cid:96) ( I ) ≤ M and rdist( I, B M ) ≤ M .We also define D M to be the intersection of C M with D .For every fixed M , we will call the cubes in C M and D M lagom cubesand dyadic lagom cubes respectively. Remark 2.2.
Note that I ∈ C M implies that − M (2 M + | c ( I ) | ∞ ) ≤ M ,and so | c ( I ) | ∞ ≤ ( M − M . Therefore, in this case, I ⊂ B M M with − M ≤ (cid:96) ( I ) .On the other hand, I / ∈ C M implies either (cid:96) ( I ) > M or (cid:96) ( I ) < − M ,or − M ≤ (cid:96) ( I ) ≤ M with | c ( I ) | ∞ > ( M − M . Compact Calder´on-Zygmund kernels and associated op-erators.
We define the type of kernels that can be associated withcompact operators.
Definition 2.3.
Three bounded functions
L, S, D : [0 , ∞ ) → [0 , ∞ ) constitute a set of admissible functions if the following limits hold (1) lim x →∞ L ( x ) = lim x → S ( x ) = lim x →∞ D ( x ) = 0 . Remark 2.4.
Since any fixed dilation of an admissible function isagain admissible, we will often omit all universal constants appearingin the argument of these functions. ‘Lagom’ is a Swedish word with the following meanings: adequate, moderate,in balance, just right. JAN-FREDRIK OLSEN AND PACO VILLARROYA
Definition 2.5.
A function K : ( R d × R d ) \{ ( t, x ) ∈ R d × R d : t = x } → C is called a compact Calder´on-Zygmund kernel if it is bounded in itsdomain and there exist < δ < , C > , and admissible functions L, S, D such that (2) | K ( t, x ) − K ( t (cid:48) , x (cid:48) ) | ≤ C ( | t − t (cid:48) | ∞ + | x − x (cid:48) | ∞ ) δ | t − x | d + δ ∞ F ( t, x ) , whenever | t − t (cid:48) | ∞ + | x − x (cid:48) | ∞ ) < | t − x | ∞ , where F ( t, x ) = L ( | t − x | ∞ ) S ( | t − x | ∞ ) D ( | t + x | ∞ ) . We use the standard definition of multi-indices: α = ( α , . . . , α d ) ∈ N d , | α | = (cid:80) di =1 α i and ∂ α = ∂ | α | ∂ α x ··· ∂ αdxd . Definition 2.6.
For every N ∈ N , N ≥ , we define S N ( R d ) to be theset of all functions f ∈ C N ( R d ) such that (cid:107) f (cid:107) m,n = sup x ∈ R d | x | β | ∂ α f ( x ) | < ∞ for all α, β ∈ N d with | α | , | β | ≤ N . Clearly, S N ( R d ) equipped with thefamily of seminorms (cid:107) · (cid:107) α,β is a Fr´echet space. Then, we can alsodefine its dual space S N ( R d ) (cid:48) equipped with the dual topology whichturns out to be a subspace of the space of multidimensional tempereddistributions. We write S ( R d ) for the classical Schwartz space. Definition 2.7.
Let T : S N ( R d ) → S N ( R d ) (cid:48) be a linear operator whichis continuous with respect to the topology of S N ( R d ) and the dual topol-ogy of S N ( R d ) (cid:48) .We say that T is associated with a compact Calder´on-Zygmund kernel K if for all f, g ∈ S N ( R d ) with disjoint compact supports, the action of T f as a distribution satisfies the following integral representation (cid:104)
T f, g (cid:105) = (cid:90) R d (cid:90) R d f ( t ) g ( x ) K ( t, x ) dt dx. The weak compactness condition.Definition 2.8.
For < p ≤ ∞ , we say that a function φ ∈ S N ( R d ) isan L p ( R d ) -normalized bump function adapted to I with constant C > and order N ∈ N if, for all multi-indices ≤ | α | ≤ N , it holds that | ∂ α φ ( x ) | ≤ C | I | /p (cid:96) ( I ) | α | (cid:18) | x − c ( I ) | ∞ (cid:96) ( I ) (cid:19) − N . Observe that, for
N p > d , the bump functions in Definition 2.8are normalized to be uniformly bounded in L p ( R d ). The order of thebump functions will always be denoted by N , even though its value NDPOINT ESTIMATES FOR COMPACT CALDER ´ON-ZYGMUND OPERATORS might change from line to line. We will often use the greek letters φ , ϕ for general bump functions while we reserve the use of ψ to denotebump functions with mean zero. If not otherwise stated, we will usuallyassume that the bump functions are L ( R d )-normalized.We now state the weak compactness condition. Definition 2.9.
A linear operator T : S N ( R d ) → S N ( R d ) (cid:48) satisfies theweak compactness condition if there exist admissible functions L, S, D such that: for every (cid:15) > there exists M ∈ N so that for any cube I and every pair φ I , ϕ I of L -normalized bump functions adapted to I with constant C > and order N , we have |(cid:104) T φ I , ϕ I ) (cid:105)| (cid:46) C (cid:18) L (cid:16) (cid:96) ( I )2 M (cid:17) · S (cid:16) M (cid:96) ( I ) (cid:17) · D (cid:16) rdist (cid:0) I, B M ) M (cid:17) + (cid:15) (cid:19) , where the implicit constant only depends on the operator T . There are other alternative and less technical formulations of thisconcept. For example, we can say that T satisfies the weak compactnesscondition if and only if, for every pair φ I , ϕ I of L -normalized bumpfunctions adapted to I , we havelim M →∞ sup I / ∈D M |(cid:104) T φ I , ϕ I ) (cid:105)| = 0 , where the lagom dyadic cubes D M appear in Definition 2.1. However,we prefer the formulation used in Definition 2.9 because it is particu-larly well-suited for the calculations performed in [9] and thus, the onescarried out in the current paper.We introduce the following notation to simplify otherwise cumber-some formulas, which appear both in the statement of Proposition 2.22and in the proof of Theorem 4.1, below. Namely, we write F ( I ; M ) = L K (cid:0) (cid:96) ( I ) (cid:1) · S K (cid:0) (cid:96) ( I ) (cid:1) · D K (cid:0) rdist( I, B ) (cid:1) + F W (cid:16) (cid:96) ( I )2 M (cid:17) · S W (cid:16) M (cid:96) ( I ) (cid:17) · D W (cid:16) rdist( I, B M ) M (cid:17) , where L K , S K , D K are the functions appearing in the definition of acompact Calder´on-Zygmund kernel, while L W , S W , D W and the con-stant M are as in the definition of the weak compactness condition.We also set F ( I , . . . ,I n ; M ) = n (cid:88) i =1 L K (cid:0) (cid:96) ( I i ) (cid:1) · n (cid:88) i =1 S K (cid:0) (cid:96) ( I i ) (cid:1) · n (cid:88) i =1 D K (cid:0) rdist( I i , B ) (cid:1) + n (cid:88) i =1 L W (cid:16) (cid:96) ( I i )2 M (cid:17) · n (cid:88) i =1 S W (cid:16) M (cid:96) ( I i ) (cid:17) · n (cid:88) i =1 D W (cid:16) rdist( I i , B ) M (cid:17) . JAN-FREDRIK OLSEN AND PACO VILLARROYA
The following lemma is proven at the beginning of the proof of The-orem 2.21, below, as it is given in [9].
Lemma 2.10.
Given (cid:15) > , then there exists exists M so that for all M > M we have F ( I , . . . , I ; M T,(cid:15) ) (cid:46) (cid:15) whenever all I i ∈ D cM . We end this subsection with two results that we will use to provethe reverse implication in our main result. Their proofs can be foundin [4, Theorem 10.1] and [1, Theorem 3.1], respectively.
Theorem 2.11.
Let T be an operator with a standard Calder´on-Zygmundkernel and bounded from L ( R d ) into L , ∞ ( R d ) . Then, T is bounded on L p ( R d ) for any < p < ∞ with (cid:107) T (cid:107) L p ( R d ) → L p ( R d ) (cid:46) (cid:107) T (cid:107) L ( R d ) → L , ∞ ( R d ) and the implicit constant only depends on p and the dimension d . Theorem 2.12.
Let A = ( A , A ) and B = ( B , B ) be quasi-Banachcouples and let T : A → B such that T : A → B compactly. Then,for any < θ < and < q ≤ ∞ , T : ( A , A ) θ,q → ( B , B ) θ,q iscompact. Characterization of compactness and the lagom projec-tion operator.
The following characterization of compact operatorsin a Banach space with a Schauder basis (see for example [2]) was usedin [9] to study compact Calderon-Zygmund operators.
Theorem 2.13.
Suppose that { e n } n ∈ N is a Schauder basis of a Banachspace E . For each positive integer k , let P k be the canonical projection P k (cid:16) (cid:88) n ∈ N α n e n (cid:17) = (cid:88) n ≤ k α n e n . Then, a bounded linear operator T : E → E is compact if and only if P k ◦ T converges to T in operator norm. Let E be one of the following Banach spaces: the Lebesgue space L p ( R d ), 1 < p < ∞ , the Hardy space H ( R d ), or the space CMO( R d ),defined in Subsection 2.5 below. In each of these cases, E is equippedwith a wavelet basis which is also a Schauder basis (see [3] and Lemma2.19). Moreover, in these cases, we can assume that the wavelets belongto S N ( R d ) and, if needed, that they are compactly supported. How-ever, we intentionally decide to use more general wavelets to explicitlyshow that our results hold in settings where, for example, compactlysupported wavelets are not available. Definition 2.14.
Let E be one of the previously mentioned Banachspaces. Let ( ψ iI ) I ∈D ,i =1 ,..., d − be a normalized wavelet basis of E and NDPOINT ESTIMATES FOR COMPACT CALDER ´ON-ZYGMUND OPERATORS ( ˜ ψ iI ) ∈D ,i =1 ,..., d − its dual wavelet basis. Then, for every M ∈ N , wedefine the lagom projection operator P M f = (cid:88) I ∈D M d − (cid:88) i =1 (cid:104) f, ˜ ψ iI (cid:105) ψ iI , where (cid:104) f, ˜ ψ iI (cid:105) = (cid:82) R d f ( x ) ˜ ψ iI ( x ) dx . We also define P ⊥ M f = f − P M f , and we remark that the equality(3) P ⊥ M f = (cid:88) I ∈D cM d − (cid:88) i =1 (cid:104) f, ˜ ψ iI (cid:105) ψ iI is to be interpreted in the sense of Schauder bases, i.e.,lim M (cid:48) →∞ (cid:13)(cid:13)(cid:13) P ⊥ M f − (cid:88) I ∈D M (cid:48) \D M d − (cid:88) i =1 (cid:104) f, ˜ ψ iI (cid:105) ψ iI (cid:13)(cid:13)(cid:13) E = 0 . In the language of the lagom projection, we can give yet anotheralternative formulation of weak compactness (Definition 2.9). Namely,an operator T is weakly compact if and only if, for every pair φ I , ϕ I of L -normalized bump functions adapted to I , we havelim M →∞ |(cid:104) ( P ⊥ M ◦ T )( φ I ) , ϕ I ) (cid:105)| = 0 , ∀ I ∈ D . Strictly speaking, the characterization given in Theorem 2.13 is notsufficient for our purposes since, in Section 4, we also consider compactoperators into the space L , ∞ ( R d ), which is a quasi-Banach space. Thisis addressed in Definition 2.15, where we define compact operators from L ( R d ) into L , ∞ ( R d ) in the topological sense. Definition 2.15.
An operator T : L ( R d ) → L , ∞ ( R d ) is compact if,for every bounded set A ⊂ L ( R d ) , the set T ( A ) is relatively compact in L , ∞ ( R d ) . Equivalently, T is compact if, for every sequence ( f n ) n ∈ N ⊂ L ( R d ) with (cid:107) f n (cid:107) L ( R d ) (cid:46) , there exist a subsequence ( f n k ) k ∈ N and g ∈ L , ∞ ( R d ) such that λm ( { x ∈ R d : | T f n k ( x ) − g ( x ) | > λ } ) tends tozero when k tends to infinity uniformly for all λ > . Remark 2.16.
Observe that finite rank operators are compact in thissense, and that the limit of finite rank operators is a compact operator.
We also note that, in light of Theorem 2.13, it would be naturalto assume that the above definition is equivalent to asking that P ⊥ M T converges to zero in the operator norm (cid:107) · (cid:107) L ( R d ) → L , ∞ ( R d ) . However,this is not the case as we see from the following example. JAN-FREDRIK OLSEN AND PACO VILLARROYA
Example 2.17.
Let ( ψ I ) I ∈D be the Haar wavelet of L ( R ) and P M theassociated lagom projection operator. Then, the operator defined by T f = (cid:104) f, ψ [0 , (cid:105) χ [0 , is compact from L ( R ) to L , ∞ ( R ) (since it is bounded and of finiterank), but P ⊥ M T does not converge to zero in L , ∞ ( R ) . Indeed, it fol-lows from the computation below that P ⊥ M T ψ [0 , = 2 − M χ [0 , M ] , whence (cid:107) P ⊥ M T (cid:107) L ( R ) → L , ∞ ( R ) ≥ for all M ∈ N .First, we observe that P ⊥ M T ψ [0 , = P ⊥ M χ [0 , = χ [0 , − (cid:88) I ∈D M (cid:104) χ [0 , , ψ I (cid:105) ψ I . Now, (cid:104) χ [0 , , ψ I (cid:105) (cid:54) = 0 if and only if I = (0 , k ) with ≤ k ≤ M and, inthat case, (cid:104) χ [0 , , ψ I (cid:105) = | I | − / . With this, we obtain P ⊥ M T ψ [0 , = χ (0 , − (cid:88) ≤ k ≤ M − k − k ( χ (0 , k − ) − χ (2 k − , k ) )= 2 − M χ (0 , + (cid:88) ≤ j ≤ M − M χ (2 j − , j ) = 2 − M χ (0 , M ) as claimed. The space
CM O ( R d ) and the construction of T (1) .Definition 2.18. We define
CMO( R d ) as the closure in BMO( R d ) ofthe space of continuous functions vanishing at infinity. The next lemma gives two characterizations of CMO( R d ): the firstone in terms of the average deviation from the mean, and the secondone in terms of a wavelet decomposition. See [7] and [5] for the proofs.We will only use the latter formulation. Lemma 2.19.
The following statements are equivalent: (i) f ∈ CMO( R d ) , (ii) f ∈ BMO( R d ) and lim M →∞ sup I / ∈I M | I | (cid:90) I (cid:12)(cid:12)(cid:12) f ( x ) − | I | (cid:90) I f ( y ) dy (cid:12)(cid:12)(cid:12) dx = 0 , (iii) f ∈ BMO( R d ) and lim M →∞ sup Ω ⊂ R d (cid:16) | Ω | (cid:88) I / ∈ D M I ⊂ Ω 2 d − (cid:88) i =1 |(cid:104) f, ψ iI (cid:105)| (cid:17) / = 0 , where the supremum is taken over all measurable sets Ω ⊂ R d . NDPOINT ESTIMATES FOR COMPACT CALDER ´ON-ZYGMUND OPERATORS Next, we state a technical lemma needed to give meaning to T (1)and T ∗ (1). To this end, we introduce some notation. For a ∈ R and λ > , we define the translation operator as T a f ( x ) = f ( x − a ) and thedilation operator as D λ f ( x ) = f ( x/λ ). Let Φ ∈ S ( R d ) be such thatΦ( x ) = 1 for | x | ∞ ≤
1, 0 < Φ( x ) < < | x | ∞ < x ) = 0for | x | ∞ > Lemma 2.20.
Let T be a linear operator associated with a compactCalder´on-Zygmund kernel K with parameter < δ < .Let I ⊂ R d be a cube and let f ∈ S N ( R d ) have compact support in I and mean zero. Then, the limit L ( f ) = lim k →∞ (cid:104) T ( T a D k (cid:96) ( I ) Φ) , f (cid:105) exists and is independent of the translation parameter a ∈ R and thecut-off function Φ . The previous lemma allows one to define T (1) as an element on thedual of the space of functions in S N ( R d ) with compact support andmean zero. Namely, define (cid:104) T (1) , f (cid:105) = L ( f ) for all f ∈ S N ( R d ).2.6. Compactness on L p ( R d ) . We now state the main result in [6],whose proof, although proven only for the one-dimensional case, alsoholds in the setting of several variables.
Theorem 2.21.
Let T be a linear operator associated with a standardCalder´on-Zygmund kernel.Then, T extends to a compact operator on L p ( R d ) , for any < p < ∞ , if and only if T is associated with a compact Calder´on-Zygmundkernel and it satisfies both the weak compactness condition and thecancellation conditions T (1) , T ∗ (1) ∈ CMO( R d ) .Under the same hypotheses, T is also compact as a map from L ∞ ( R d ) into CMO( R d ) . Moreover, with the extra assumption T (1) = T ∗ (1) =0 , T is compact from BMO( R d ) into CMO( R d ) . We end this section stating the main auxiliary result in the proofof Theorem 2.21, which is also the starting point of the proof of theendpoint result in this paper. To this end, we provide the followingdefinitions: given two cubes I and J , we denote K min = J and K max = I if (cid:96) ( J ) ≤ (cid:96) ( I ), and K min = I and K max = J otherwise. We denoteby ˜ K max the translate of K max with the same centre as K min . Proposition 2.22.
Let T be a linear operator associated with a com-pact Calder´on-Zygmund kernel with parameter δ . We assume T satis-fies the weak compactness condition and the special cancellation condi-tion T (1) = 0 and T ∗ (1) = 0 . Then, for any θ ∈ (0 , small enough, there exist < δ (cid:48) < δ , N ≥ and C δ (cid:48) > such that for every (cid:15) > , all cubes I, J and all meanzero bump functions ψ I , ψ J , L -adapted to I and J respectively withconstant C > and order N , we have |(cid:104) T ψ I , ψ J (cid:105)| ≤ C δ (cid:48) C ecc( I, J ) + δ (cid:48) d rdist( I, J ) d + δ (cid:48) (cid:16) F ( I , . . . , I ; M T,(cid:15) ) + (cid:15) (cid:17) , where I = I , I = J , I = (cid:104) I, J (cid:105) , I = λ ˜ K max , I = λ ˜ K max and I = λ K min with λ = (cid:96) ( K max ) − diam( I ∪ J ) , λ = (cid:96) ( K min ) − θ diam( I ∪ J ) θ . Localization properties of bump functions
In this section, we prove two technical results. Lemma 3.1 concernsthe localization of multi-variable bump functions while Lemma 3.2 es-timates the interaction of bump functions with atoms. The proofs ofboth results in the one-dimensional case can be found in [11]. See also[9] for a more detailed proof of the latter result.
Lemma 3.1.
Let φ I and ψ J be bump functions L -adapted to I and J respectively with order N ≥ d and constant C . For (cid:96) ( J ) ≤ (cid:96) ( I ) , then (4) |(cid:104) φ I , ψ J (cid:105)| (cid:46) C (cid:18) | J || I | (cid:19) (cid:18) | c ( I ) − c ( J ) | ∞ (cid:96) ( I ) (cid:19) − N . If, in addition, φ I and ψ J have order N > d and ψ J has vanishingmean, i.e., (cid:82) ψ J ( x ) dx = 0 , then (5) |(cid:104) φ I , ψ J (cid:105)| (cid:46) C (cid:18) | J || I | (cid:19) + d (cid:18) | c ( I ) − c ( J ) | ∞ (cid:96) ( I ) (cid:19) − N + d . Proof.
We start by proving inequality (4). Let c be the midpoint be-tween c ( I ) and c ( J ), let L be the line going through c ( I ) and c ( J ) andlet H ⊂ R d be the hyperplane perpendicular to L passing through c .Let also H I and H J be the two half-spaces defined by the connectedcomponents of R d \ H so that c ( I ) ∈ H I and c ( J ) ∈ H J . We split (cid:104) φ I , ψ J (cid:105) = (cid:90) H I φ I ( x ) ψ J ( x ) dx + (cid:90) H J φ I ( x ) ψ J ( x ) dx. Applying H¨older’s inequality and Definition 2.8, we get(6) |(cid:104) φ I , ψ J (cid:105)| ≤ (cid:107) φ I (cid:107) L ( R d ) (cid:107) ψ J (cid:107) L ∞ ( H I ) + (cid:107) φ I (cid:107) L ∞ ( H J ) (cid:107) ψ J (cid:107) L ( R d ) ≤ C (cid:18) | I || J | (cid:19) (cid:18) | c − c ( J ) | ∞ (cid:96) ( J ) (cid:19) − N + C (cid:18) | J || I | (cid:19) (cid:18) | c − c ( I ) | ∞ (cid:96) ( I ) (cid:19) − N . NDPOINT ESTIMATES FOR COMPACT CALDER ´ON-ZYGMUND OPERATORS Since | c − c ( I ) | ∞ = | c − c ( J ) | ∞ = | c ( I ) − c ( J ) | ∞ / (cid:96) ( J ) ≤ (cid:96) ( I ) and N ≥ d , we have that the first term is smaller than the second one,which is of the desired form.To prove (5), we assume without loss of generality that | c ( I ) − c ( J ) | ∞ = | c ( I ) − c ( J ) | . Then, for all x ∈ R d we write x = ( x , x (cid:48) )with x (cid:48) ∈ R d − . We define the operators D − ( ψ J )( x ) = (cid:90) x −∞ ψ J ( s, x (cid:48) ) ds and, for t ∈ R , D − ( ψ J )( t ) = (cid:90) R d − (cid:90) t −∞ ψ J ( x , x (cid:48) ) dx dx (cid:48) = (cid:90) te + H − ψ J ( x ) dx, where H − = { x ∈ R d : x ≤ } . Note that, due to the vanishing meanof ψ J , we have(7) D − ( ψ J )( t ) = − (cid:90) R d − (cid:90) ∞ t ψ J ( x , x (cid:48) ) dsdx (cid:48) = − (cid:90) te + H +1 ψ J ( x ) dx, where H +1 = { x ∈ R d : 0 ≤ x } . Then, it is readily checked that (cid:104) φ I , ψ J (cid:105) = − (cid:90) R d ∂ φ I ( x ) · D − ψ J ( x ) dx. Now, the function D − ψ J can be expressed as the sum of four positivefunctions D − ψ J = f − f + i ( f − f ) = (cid:80) k =0 i k f k . Hence, applyingthe Mean Value Theorem for integrals to each positive function f k withrespect to the variable x (cid:48) ∈ R d − , we obtain (cid:104) φ I , ψ J (cid:105) = − (cid:88) k =0 i k (cid:90) R ∂ φ I (cid:0) x , g k ( x ) (cid:1)(cid:18) (cid:90) R d − f k ( x ) dx (cid:48) (cid:19) dx , where the functions g k denote the dependence of all coordinates from x . Hence, |(cid:104) φ I , ψ J (cid:105)| ≤ (cid:90) R sup k | ∂ φ I ( x , g k ( x )) | (cid:18) (cid:88) k =0 (cid:90) R d − f k ( x ) dx (cid:48) (cid:19) dx . Since D − ψ J ( t ) = (cid:90) R d − D − ψ J ( t, x (cid:48) ) dx (cid:48) = (cid:88) k =0 i k (cid:90) R d − f k ( t, x (cid:48) ) dx (cid:48) = (cid:88) k =0 i k F k ( t ) , we have that (cid:88) k =0 (cid:90) R d − f k ( t, x (cid:48) ) dx (cid:48) ≤ / (cid:18)(cid:16) (cid:88) k even F k ( t ) (cid:17) + (cid:16) (cid:88) k odd F k ( t ) (cid:17) (cid:19) / = 2 / | D − ψ J ( t ) | . Therefore, we can write(8) |(cid:104) φ I , ψ J (cid:105) L ( R d ) | (cid:46) (cid:104) sup k | ∂ φ I ( t, g k ( t )) | , | D − ( ψ J )( t ) |(cid:105) L ( R ) . Now, on the one hand, we have by Definition 2.8, | ∂ φ I ( t, g k ( t )) | ≤ C | I | (cid:96) ( I ) (cid:18) | ( t, g k ( t )) − c ( I ) | ∞ (cid:96) ( I ) (cid:19) − N ≤ C | I | (cid:96) ( I ) (cid:96) ( I ) (cid:18) | t − c ( I ) | (cid:96) ( I ) (cid:19) − N . This is the decay estimate in Definition 2.8 of a function being adaptedto the interval I with constant C | I | − / (cid:96) ( I ) − / .On the other hand, to control the second factor in (8), we make thefollowing computation: since | · | ≤ d | · | ∞ , | D − ( ψ J )( t ) | ≤ C | J | (cid:90) te + H +1 (cid:18) | x − c ( J ) | ∞ (cid:96) ( J ) (cid:19) − N d x ≤ C | J | (cid:90) te − c ( J )+ H +1 (cid:18) | x | (cid:96) ( J ) d (cid:19) − N d x = C | J | (cid:90) R d − (cid:90) ∞ t − c ( J ) (cid:18) | x | + | x (cid:48) | (cid:96) ( J ) d (cid:19) − N d x d x (cid:48) (cid:46) C | J | (cid:96) ( J ) d d d (cid:18) | t − c ( J ) | (cid:96) ( J ) d (cid:19) − N + d ≤ Cd N | J | (cid:18) d + | t − c ( J ) | (cid:96) ( J ) (cid:19) − N + d (cid:46) C | J | (cid:96) ( J ) (cid:96) ( J ) (cid:18) | t − c ( J ) | (cid:96) ( J ) (cid:19) − N + d . Here, we tacitly assumed that t − c ( J ) >
0. If the opposite is true, weuse (7) in the first line of the argument to make the same calculationwork. We note that this is the decay estimate in Definition 2.8 of afunction being adapted to the interval I with constant C | J | / (cid:96) ( J ) / .Now, we combine the above two estimates. Repeating the proof of(4) in the one-dimensional case for the expression in (8), starting with NDPOINT ESTIMATES FOR COMPACT CALDER ´ON-ZYGMUND OPERATORS the splitting in (6), we obtain, for N > d , the bound |(cid:104) φ I , ψ J (cid:105)| (cid:46) C (cid:18) | J | (cid:96) ( J ) | I | (cid:96) ( I ) (cid:19) (cid:16) (cid:96) ( J ) (cid:96) ( I ) (cid:17) (cid:18) | c ( I ) − c ( J ) | (cid:96) ( I ) (cid:19) − N + d = C (cid:16) | J || I | (cid:17) + d (cid:18) | c ( I ) − c ( J ) | ∞ (cid:96) ( I ) (cid:19) − N + d . (cid:3) Lemma 3.2.
Let I be a cube and f be an integrable function supportedon I with mean zero. For each dyadic cube J , let φ J be a bump functionadapted to J with constant C > and order N .Then, for all dyadic cubes J such that (cid:96) ( J ) ≤ (cid:96) ( I ) , we have (9) |(cid:104) f, φ J (cid:105)|| J | (cid:46) C (cid:107) f (cid:107) L ( R d ) (cid:16) | c ( I ) − c ( J ) | ∞ (cid:96) ( I ) (cid:17) − N , while for (cid:96) ( I ) ≤ (cid:96) ( J ) , we get (10) |(cid:104) f, φ J (cid:105)|| J | (cid:46) C (cid:107) f (cid:107) L ( R d ) (cid:96) ( I ) (cid:96) ( J ) (cid:16) | c ( I ) − c ( J ) | ∞ (cid:96) ( J ) (cid:17) − N . Proof.
The proofs of both inequalities follow the pattern from the pre-vious lemma with the required modifications to take advantage of thecompact support of f .In order to prove (9), we divide the argument into two cases. When | c ( I ) − c ( J ) | ∞ ≤ (cid:96) ( I ), the inequality follows from H¨older: |(cid:104) f, ϕ J (cid:105)| ≤ (cid:107) f (cid:107) L ( R d ) (cid:107) ϕ J (cid:107) L ∞ ( R d ) ≤ C (cid:107) f (cid:107) L ( R d ) | J | / . When | c ( I ) − c ( J ) | ∞ > (cid:96) ( I ), we denote by c the midpoint between c ( I ) and c ( J ), and by H ⊂ R d the hyperplane passing through c andperpendicular to the line containing c ( I ) and c ( J ). Let also H J bethe half-space defined by the connected component of R d \ H so that c ( J ) ∈ H J . It can be readily checked that I ∩ H J = ∅ , whence supp f ⊂ I ⊂ H cJ , and thus, |(cid:104) f, φ J (cid:105)| ≤ (cid:90) H cJ | f ( x ) || φ J ( x ) | dx ≤ (cid:107) f (cid:107) L ( R d ) (cid:107) φ J (cid:107) L ∞ ( H cJ ) ≤ (cid:107) f (cid:107) L ( R d ) C | J | / (cid:18) | c ( I ) − c ( J ) | ∞ (cid:96) ( J ) (cid:19) − N , which is smaller than the bound in (9) since (cid:96) ( J ) ≤ (cid:96) ( I ).To prove (10), we divide in two similar cases. We first assume that | c ( I ) − c ( J ) | ∞ ≥ (cid:96) ( J ). Let c = ( c , c (cid:48) ) ∈ R × R d − be the midpoint between c ( I ) and c ( J ) and let H and H J be as before. Since now (cid:96) ( I ) ≤ (cid:96) ( J ), we have again that c ( J ) ∈ H J and supp f ⊂ H cJ .As in the previous lemma, we assume without loss of generality that | c ( I ) − c ( J ) | ∞ = | c ( I ) − c ( J ) | . Then, we consider again the operator D − ( f )( t ) = (cid:90) te + H − f ( x ) dx = − (cid:90) te + H +1 f ( x ) dx, where H − = { x ∈ R d : x ≤ } and H +1 = { x ∈ R d : 0 ≤ x } , due tothe vanishing mean of f . Moreover, the support of D − f is includedin I , which is, in turn, included in ( −∞ , c ).From the computations developed in the proof of (5), we have that (cid:104) f, φ I (cid:105) = − (cid:88) k =0 i k (cid:90) R (cid:18) (cid:90) R d − f k ( x , x (cid:48) ) dx (cid:48) (cid:19) ∂ φ J (cid:0) x , g k ( x ) (cid:1) dx , where D − f = (cid:80) k =0 i k f k , with f k positive functions and the functions g k denote the dependence of all coordinates from x . Now, as in theproof of Lemma 3.1, we have the inequalities | ∂ φ J ( t, g k ( t )) | ≤ C | J | / (cid:96) ( J ) (cid:18) | t − c ( J ) | (cid:96) ( J ) (cid:19) − N , and (cid:88) k =0 (cid:90) R d − f k ( t, x (cid:48) ) dx (cid:48) ≤ / | D − f ( t ) | . Let ϕ J ( t ) = sup k | ∂ φ J ( t, g k ( t )) | . Then, |(cid:104) f, φ J (cid:105)| (cid:46) (cid:90) c −∞ | D − f ( t ) | ϕ J ( t ) dt ≤ (cid:107) D − f (cid:107) L ( −∞ ,c ) (cid:107) ϕ J (cid:107) L ∞ ( −∞ ,c ) ≤ (cid:107) D − f (cid:107) L ( R ) C | J | / (cid:96) ( J ) (cid:18) | c ( I ) − c ( J ) | (cid:96) ( J ) (cid:19) − N . Now, from the bound (cid:107) D − f (cid:107) L ( R ) ≤ (cid:96) ( I ) |(cid:107) f (cid:107) L ( R d ) and the assump-tion about the first coordinate, we obtain the bound stated in (10).Finally, when | c ( I ) − c ( J ) | ≤ (cid:96) ( J ), we use the easier estimate |(cid:104) f, φ J (cid:105)| (cid:46) (cid:90) c −∞ | D − f ( t ) | ϕ J ( t ) dt ≤ (cid:107) D − f (cid:107) L ( R ) (cid:107) ϕ J (cid:107) L ∞ ( R ) ≤ (cid:96) ( I ) (cid:107) f (cid:107) L ( R d ) C | J | − / (cid:96) ( J ) − . This ends the proof. (cid:3)
NDPOINT ESTIMATES FOR COMPACT CALDER ´ON-ZYGMUND OPERATORS Compactness from L ( R d ) into L , ∞ ( R d )In this section, we state and prove our main result. Theorem 4.1.
Let T be a linear operator associated with a standardCalder´on-Zygmund kernel. Then, T can be extended to a compact op-erator from L ( R d ) into L , ∞ ( R d ) if and only if it is associated witha compact Calder´on-Zygmund kernel satisfying the weak compactnesscondition and the cancellation conditions T (1) , T ∗ (1) ∈ CMO( R d ) . Remark 4.2. If T is a linear operator with a standard Calder´on-Zygmund kernel which can be extended compactly on L p ( R d ) for any < p < ∞ then, we know by Theorem 2.21 that T satisfies the samethree hypotheses for compactness of Theorem 4.1 and so, it can also beextended as a compact operator from L ( R d ) into L , ∞ ( R d ) . We first justify the converse, which is essentially a consequence ofTheorem 2.11 and compact real interpolation. We assume that T iscompact from L ( R d ) into L , ∞ ( R d ). Since T is bounded between thesame spaces, by Theorem 2.11, we have that T is also bounded on,say, L ( R d ). Then, by the interpolation Theorem 2.12 with θ = − / − / ,we obtain that T is compact on L ( R d ). Now, the reverse implicationof Theorem 2.21 implies the required hypotheses: T has a compactCalder´on-Zygmund kernel and it satisfies both the weak compactnesscondition and the cancellation conditions T (1) , T ∗ (1) ∈ CMO( R d ).The remainder of the paper is devoted to show sufficiency. As in thestudy of boundedness, the proof is split into two cases: first a specialcase, when extra cancellation properties are assumed (Proposition 4.3);and second, the general case, which is dealt with by proving compact-ness of paraproducts (Proposition 4.4). Proposition 4.3.
Let T be a linear operator associated with a compactCalder´on-Zygmund kernel satisfying the weak compactness conditionand the cancellation conditions T (1) = 0 and T ∗ (1) = 0 . Then, T iscompact from L ( R d ) into L , ∞ ( R d ) . Proposition 4.4.
Given a function b ∈ CMO( R d ) , there exists a linearoperator T b associated with a compact Calder´on-Zygmund kernel suchthat T b and T ∗ b are compact from L ( R d ) into L , ∞ ( R d ) and satisfy (cid:104) T b (1) , g (cid:105) = (cid:104) b, g (cid:105) and (cid:104) T b ( f ) , (cid:105) = 0 , for all f, g ∈ S ( R d ) . We now remind the reader how to deduce Theorem 4.1 from thesepropositions. The argument follows the well-known scheme provided inthe proof of the classical T (1) theorem. Namely, when b = T (1), b = T ∗ (1) are functions in CMO( R d ), we use Proposition 4.4 to construct the paraproduct operators T b i . As proved in [9], they have compactCalder´on-Zymund kernels, are compact operators on L ( R d ) (and thus,they satisfy the weak compactness condition), and satisfy T b (1) = b , T b (1) = b and T ∗ b (1) = T ∗ b (1) = 0. It now follows from Proposition4.3 that the operator T − T b − T ∗ b is compact from L ( R d ) into L , ∞ ( R d ). Finally, after proving that T b and T ∗ b are compact from L ( R d ) into L , ∞ ( R d ), we deduce that thesame holds for the initial operator T .4.1. Proof of Proposition 4.3.
Let ( ψ iI ) I ∈D ,i =1 ,..., d − be an orthogo-nal wavelet basis of L ( R d ) such that every function ψ iI is adapted to adyadic cube I with constant C > N . We denote by P M thelagom projection of Definition 2.14 associated with ( ψ iI ) I ∈D ,i =1 ,..., d − .Since the index i ∈ { , . . . , d − } and the dual wavelet play no signif-icant role in the proof, in order to simplify notation, we will write thewavelet decomposition in L ( R d ) simply as f = (cid:80) I ∈D (cid:104) f, ψ I (cid:105) ψ I .By the classical theory, we know that T extends to a bounded oper-ator on L p ( R d ), and from L ( R d ) into L , ∞ ( R d ). Therefore, for every f ∈ S ( R d ), T f and also P ⊥ M T f are meaningful as functions in theintersection of L p ( R d ) and L , ∞ ( R d ).Fix 1 < p < ∞ . By Theorem 2.21, we already know that T extendsto a compact operator on L p ( R d ). Hence, for every (cid:15) >
0, there existsan M ∈ N such that, for all M > M and f ∈ S ( R d ), we have(11) (cid:107) P ⊥ M T f (cid:107) L p ( R d ) (cid:46) (cid:15) (cid:107) f (cid:107) L p ( R d ) , where the implicit constant depends only on T and p .According to Remark 2.16, it suffices to prove that for any given (cid:15) > M ∈ N , we have for all M > M with M − Mδ + M − δ < (cid:15) ,(12) m ( { x ∈ R d : | P ⊥ M T f ( x ) | > λ } ) (cid:46) (cid:15)λ (cid:107) f (cid:107) L ( R d ) for all f ∈ S ( R d ) and all λ >
0. The implicit constant is allowed todepend on δ >
0, the parameter of the compact Calder´on-Zygmundkernel, and the constant given by the wavelet basis, but is to be inde-pendent of (cid:15) , f and λ .To prove (12), we perform a classical Calder´on-Zygmund decompo-sition of f at level (cid:15) − λ >
0. For this, we consider the collection I ofmaximal dyadic cubes I with respect to set inclusion such that1 | I | (cid:90) I | f ( x ) | dx > λ(cid:15) . NDPOINT ESTIMATES FOR COMPACT CALDER ´ON-ZYGMUND OPERATORS Let E be the disjoint union of all I ∈ I , which satisfies m ( E ) ≤ (cid:15)λ − (cid:107) f (cid:107) L ( R d ) . With this, we define the usual Calder´on-Zygmund de-composition f = ˜ g + ˜ b , where˜ g = (cid:88) I ∈I m I ( f ) χ I + f χ E c , ˜ b = (cid:88) I ∈I f I = (cid:88) I ∈I (cid:0) f − m I ( f ) (cid:1) χ I , with m I ( f ) = | I | − (cid:82) I f ( x ) dx .By standard arguments, it follows that (cid:107) ˜ g (cid:107) L ∞ ( R d ) ≤ d λ/(cid:15) , and more-over, that (cid:107) ˜ g (cid:107) L ( R d ) ≤ (cid:107) f (cid:107) L ( R d ) . From this, the inequality (11), andthe fact that M > M > M , we get (cid:107) P ⊥ M T ˜ g (cid:107) pL p ( R d ) (cid:46) (cid:15) p (cid:107) ˜ g (cid:107) pL p ( R d ) (cid:46) (cid:15) p (cid:90) R d | ˜ g ( x ) | λ p − (cid:15) p − dx ≤ (cid:15)λ p − (cid:107) f (cid:107) L ( R d ) . Whence, m (cid:0) { x ∈ R d : | P ⊥ M T ˜ g ( x ) | > λ/ } (cid:1) (cid:46) λ p (cid:107) P ⊥ M T ˜ g (cid:107) pL p ( R d ) (cid:46) (cid:15)λ (cid:107) f (cid:107) L ( R d ) . Now we need to prove the same estimate for ˜ b . To do so, we define˜ E as the union of all cubes 10 I with I ∈ I . Writing R d = ˜ E ∪ ˜ E C ,yields m ( { x ∈ R d : | P ⊥ M T ˜ b ( x ) | > λ/ } ) (cid:46) m ( ˜ E ) + 1 λ (cid:107) P ⊥ M T ˜ b (cid:107) L ( ˜ E C ) . Since m ( ˜ E ) (cid:46) (cid:15)λ − (cid:107) f (cid:107) L ( R d ) , it remains to show that (cid:107) P ⊥ M T ˜ b (cid:107) L ( ˜ E C ) (cid:46) (cid:15) (cid:107) f (cid:107) L ( R d ) . To prove this, it suffices to show that for each I ∈ I , we have(13) (cid:107) P ⊥ M T f I (cid:107) L ( ˜ E C ) (cid:46) (cid:15) (cid:107) f I (cid:107) L ( R d ) . Indeed, by sub-linearity, this would imply (cid:107) P ⊥ M T ˜ b (cid:107) L ( ˜ E C ) (cid:46) (cid:15) (cid:88) I ∈I (cid:107) f I (cid:107) L ( R d ) ≤ (cid:15) (cid:107) f (cid:107) L ( R d ) , whence, m ( { x ∈ R d : | P ⊥ M T ˜ b ( x ) | > λ/ } ) (cid:46) (cid:15)λ (cid:107) f (cid:107) L ( R d ) , which would complete the proof.The remainder of the proof therefore deals with obtaining (13). Tothis end, since f I = ( f − m I ( f )) χ I and apply Fatou’s Lemma to obtain (cid:107) P ⊥ M T f I (cid:107) L ( ˜ E C ) ≤ lim inf R →∞ (cid:107) P ⊥ M T f I (cid:107) L ( ˜ E C ∩ [ − R,R ] d ) . To estimate the last quantity, it suffices, by duality, to check that forall g ∈ S ( R d ) in the unit ball of L ∞ ( R d ) with compact support in K R = ˜ E C ∩ [ − R, R ] d , we have(14) |(cid:104) P ⊥ M T f I , g (cid:105)| (cid:46) (cid:15) (cid:107) f I (cid:107) L ( R d ) . We now justify this claim. Observe that since f I ∈ L ( R d ), it fol-lows by the continuity of T and P ⊥ M on L ( R d ) that we also have P ⊥ M T f I ∈ L ( R d ). Hence, the function h = sign( P ⊥ M T f I ) χ K R can beapproximated in the norm of L ( K R ) by a function g ∈ S ( R d ) withcompact support in K R such that (cid:107) g (cid:107) L ∞ ( R d ) ≤ (cid:107) h (cid:107) L ∞ ( R d ) = 1. Withthis, we have (cid:107) P ⊥ M T f I (cid:107) L ( K R ) ≤ |(cid:104) P ⊥ M T f I , g (cid:105)| + |(cid:104) P ⊥ M T f I , h − g (cid:105)| , where the last term can be bounded by a constant times (cid:107) P ⊥ M T f I (cid:107) L ( R d ) (cid:107) h − g (cid:107) L ( K R ) (cid:46) (cid:107) f I (cid:107) L ( R d ) (cid:15) (cid:107) f I (cid:107) L ( R d ) (cid:107) f I (cid:107) L ( R d ) . This ends the desired justification.We work now to obtain (14). We start by justifying the equality(15) (cid:104) P ⊥ M T f I , g (cid:105) = (cid:88) J ∈D (cid:88) K ∈D c M (cid:104) f I , ψ J (cid:105)(cid:104) g, ψ K (cid:105)(cid:104) T ψ J , ψ K (cid:105) , for functions g as described above. Since g ∈ S ( R d ), we have that P ⊥ M g = g − (cid:80) K ∈D M (cid:104) g, ψ K (cid:105) ψ K is a well defined bounded smoothfunction. Therefore, we can give sense to (cid:104) P ⊥ M T f I , g (cid:105) = (cid:104) T f I , P ⊥ M g (cid:105) . Moreover, since f I ∈ L ( R d ), we can write f I = (cid:80) J ∈D (cid:104) f I , ψ J (cid:105) ψ J withconvergence in L ( R d ). Also g ∈ L ( R d ) and so, according to Defini-tion 2.14, we have P ⊥ M g = (cid:80) K ∈D c M (cid:104) g, ψ K (cid:105) ψ K with convergence alsoin L ( R d ). We now write for all M (cid:48) , M (cid:48)(cid:48) > M , (cid:12)(cid:12) (cid:104) T f I , P ⊥ M g (cid:105) − (cid:88) J ∈D M (cid:48) (cid:88) K ∈D M (cid:48)(cid:48) \D M (cid:104) f I , ψ J (cid:105)(cid:104) g, ψ K (cid:105)(cid:104) T ψ J , ψ K (cid:105) (cid:12)(cid:12) ≤ (cid:107) T f I (cid:107) L ( R d ) (cid:13)(cid:13)(cid:13) P ⊥ M g − (cid:88) K ∈D M (cid:48)(cid:48) \D M (cid:104) g, ψ K (cid:105) ψ K (cid:13)(cid:13)(cid:13) L ( R d ) + (cid:107) g (cid:107) L ( R d ) (cid:13)(cid:13)(cid:13) T (cid:0) f I − (cid:88) J ∈D M (cid:48) (cid:104) f I , ψ J (cid:105) ψ J (cid:1)(cid:13)(cid:13)(cid:13) L ( R d ) By all the stated relationships and the continuity of T on L ( R d ), bothterms in previous inequality tend to zero when M (cid:48) , M (cid:48)(cid:48) tend to infinity.This justifies the equality (15). NDPOINT ESTIMATES FOR COMPACT CALDER ´ON-ZYGMUND OPERATORS Then, it follows from the triangle inequality that |(cid:104) P ⊥ M T f I , g (cid:105)| = |(cid:104) T f I , P ⊥ M g (cid:105)|≤ (cid:88) J ∈D (cid:88) K ∈D c M |(cid:104) f I , ψ J (cid:105)||(cid:104) g, ψ K (cid:105)||(cid:104) T ψ J , ψ K (cid:105)| . Now, for any given (cid:15) > M T,(cid:15) ∈ N , we have by Proposition 2.22,(16) |(cid:104) T ψ J , ψ K (cid:105)| (cid:46) ecc( J, K ) + δd rdist( J, K ) d + δ (cid:16) F ( J , . . . , J ; M T,(cid:15) ) + (cid:15) (cid:17) , where we wrote the parameter δ (cid:48) simply as δ , J = J , J = K , J = (cid:104) J, K (cid:105) , J = λ ˜ K max , J = λ ˜ K max and J = λ K min , with parameters λ , λ ≥ F ( J i ) + (cid:15) .Applying (16), we get |(cid:104) P ⊥ M T f I , g (cid:105)| (cid:46) (cid:88) J ∈D (cid:88) K ∈D c M |(cid:104) f I , ψ J (cid:105)||(cid:104) g, ψ K (cid:105)| ecc( J, K ) + δd rdist( J, K ) d + δ (cid:0) F ( J i ) + (cid:15) (cid:1) . Now, we parametrise both sums according to the eccentricities andrelative distances: first of J with respect to the fixed cube I and, later,of K with respect each cube J . To this end, for every k ∈ Z and m ∈ N , m ≥
1, we define the family I k,m = { J ∈ D : (cid:96) ( I ) = 2 k (cid:96) ( J ) , m ≤ rdist( I, J ) < m + 1 } . We note that the cardinality of I k,m is 2 max( k, d d (2 m ) d − .In the same way, for every e ∈ Z and n ∈ N , n ≥
1, and every givencube J ∈ I k,m , we define the family J e,n = { K ∈ D : (cid:96) ( J ) = 2 e (cid:96) ( K ) , n ≤ rdist( J, K ) < n + 1 } whose the cardinality is 2 max( e, d d (2 n ) d − . With all this, we have |(cid:104) P ⊥ M T f I , g (cid:105)| (cid:46) (cid:88) k ∈ Z m ∈ N (cid:88) e ∈ Z n ∈ N (cid:88) J ∈ I k,m (cid:88) K ∈ J e,n ∩D c M |(cid:104) f I , ψ J (cid:105)||(cid:104) g, ψ K (cid:105)|· −| e | d ( + δd ) n − ( d + δ ) ( F ( J i ) + (cid:15) ) (cid:46) (cid:88) e ∈ Z n ∈ N −| e | ( d + δ ) n − ( d + δ ) max( e, d n d − · (cid:88) k ∈ Z m ∈ N (cid:88) J ∈ I k,m |(cid:104) f I , ψ J (cid:105)| sup K ∈ J e,n ∩D c M |(cid:104) g, ψ K (cid:105)| ( F ( J i ) + (cid:15) ) . A crude estimate yields |(cid:104) g, ψ K (cid:105)| ≤ (cid:107) g (cid:107) L ∞ ( R d ) (cid:90) ˜ E C | K | − (cid:16) | x − c ( K ) | ∞ (cid:96) ( K ) (cid:17) − N dx (cid:46) | K | / (cid:16) E C , c ( K )) (cid:96) ( K ) (cid:17) − N = | J | / − ed/ w ( ˜ E C , K ) − N , (17)where the expression w ( ˜ E C , K ) is defined by the last equality. Usingthis and 2 − e/ max( e, = 2 | e | / , the above inequality becomes:(18) |(cid:104) P ⊥ M T f I , g (cid:105)| (cid:46) (cid:88) e ∈ Z (cid:88) n ∈ N −| e | δ n − (1+ δ ) · (cid:88) k ∈ Z (cid:88) m ∈ N (cid:88) J ∈ I k,m |(cid:104) f I , ψ J (cid:105)|| J | / sup K ∈ J e,n ∩D c M w ( ˜ E C , K ) − N ( F ( J i ) + (cid:15) ) . To keep the notation simple, we take the supremum over the emptyset to be zero (recall that the support of g is contained in ˜ E C ). Also,observe that, even though it is hidden by our choice of notation, F ( J i )depends on both J and K and thus, depends on k , m , e and n .In order to estimate (18), we need to control the terms of the doubleinner sum. We split the argument into two cases, depending on thesize of F ( J i ):( I ) J i / ∈ D M for all i = 1 , . . . , II ) J i ∈ D M for some i = 1 , . . . , I ) we know, according to Lemma 2.10,that F ( J i ) < (cid:15) and thus it suffices to merely bound the sum by someconstant. On the other hand, in case ( II ) we only know that F ( J i ) isbounded and so, we need to use the size and location of the cubes J and K to deduce an estimate that depends on (cid:15) . Proof of (I).
As already noted, in this first case we have F ( J i ) < (cid:15) .We divide the study in two cases: (cid:96) ( I ) ≤ (cid:96) ( J ) and (cid:96) ( I ) > (cid:96) ( J ), whichcorrespond to the cases k ≤ k > k ≤ f I is supported on I with zero mean, by (10) we have |(cid:104) f I , ψ J (cid:105)|| J | (cid:46) (cid:107) f I (cid:107) L ( R d ) (cid:96) ( I ) (cid:96) ( J ) (cid:18) | c ( I ) − c ( J ) | (cid:96) ( J ) (cid:19) − N (cid:46) (cid:107) f I (cid:107) L ( R d ) k m − N . (19) NDPOINT ESTIMATES FOR COMPACT CALDER ´ON-ZYGMUND OPERATORS Moreover, the cardinality of J ∈ I k,m is comparable to m d − and so, inlight of (19) and the inequality F ( J i ) < (cid:15) , the inequality (18) becomes |(cid:104) P ⊥ M T f I , g (cid:105)| (cid:46) (cid:15) (cid:88) e ∈ Z (cid:88) n ∈ N −| e | δ n − (1+ δ ) (cid:88) k ≤ (cid:88) m ∈ N k (cid:107) f I (cid:107) L ( R d ) m d − − N (cid:46) (cid:15) (cid:107) f I (cid:107) L ( R d ) , for N > d , with the implicit constant depending exponentially on d .In the second case, however, we need to be more careful. Now wehave (cid:96) ( J ) < (cid:96) ( I ), or equivalently k >
0, and we further divide in twomore cases: (cid:96) ( I ) < (cid:96) ( K ) and (cid:96) ( K ) ≤ (cid:96) ( I ). In the first case, we use (9)in Lemma 3.2, and so, |(cid:104) f I , ψ J (cid:105)|| J | (cid:46) (cid:107) f I (cid:107) L ( R d ) (cid:18) | c ( I ) − c ( J ) | (cid:96) ( I ) (cid:19) − N (cid:46) (cid:107) f I (cid:107) L ( R d ) m − N . Moreover, we have that (cid:96) ( I ) < (cid:96) ( K ) = 2 − ( e + k ) (cid:96) ( I ) which implies 0
32 ) (cid:96) ( I ) ≥ m −
12 2 k (cid:96) ( J )and | x − c ( J ) | ∞ ≤ (2 m + 32 ) (cid:96) ( I ) ≤ m + 1)2 k (cid:96) ( J ) . Moreover, for every fixed integer ( m − k − ≤ r ≤ k +1 ( m + 1), thereare at most d d ( r + 1) d − cubes J ∈ I k,m with r ≤ (cid:96) ( J ) − | x − c ( J ) | ∞
1, and so, withthe estimate (20) and the inequality F ( J i ) < (cid:15) , the bound for thecorresponding terms of (18) becomes (cid:15) (cid:88) e ∈ Z (cid:88) n ∈ N −| e | δ n − (1+ δ ) (cid:88) k ≥ (cid:88) m ≥ (cid:88) J ∈ I k,m |(cid:104) f I , ψ J (cid:105)|| J | / (cid:46) (cid:15) (cid:107) f I (cid:107) L ( R d ) (cid:88) e ∈ Z (cid:88) n ∈ N −| e | δ n − (1+ δ ) (cid:88) k ≥ (cid:88) m ≥ m d − N k ( d − N ) (cid:46) (cid:15) (cid:107) f I (cid:107) L ( R d ) as long as N > d + 1.In the case J ∩ I (cid:54) = ∅ , we have m = 1 and | c ( I ) − c ( J ) | ∞ ≤ (cid:96) ( I ) / K ∩ I = ∅ and K ∩ I (cid:54) = ∅ .In the former case, we have that | c ( K ) − c ( I ) | ∞ ≥ (cid:96) ( I ) / n + 1 > rdist( J, K ) ≥ (cid:18) | c ( J ) − c ( K ) | ∞ max (cid:0) (cid:96) ( J ) , (cid:96) ( K ) (cid:1) (cid:19) ≥ (cid:18) | c ( K ) − c ( I ) | ∞ − | c ( I ) − c ( J ) | ∞ max (cid:0) (cid:96) ( J ) , (cid:96) ( K ) (cid:1) (cid:19) ≥ (cid:18) (cid:96) ( I )max (cid:0) (cid:96) ( J ) , (cid:96) ( K ) (cid:1) (cid:19) . Since (cid:96) ( J ) = 2 − k (cid:96) ( I ) and (cid:96) ( K ) = 2 − e (cid:96) ( J ) = 2 − ( e + k ) (cid:96) ( I ), this yields n + 1 > (cid:0) k min(1 , e ) (cid:1) , NDPOINT ESTIMATES FOR COMPACT CALDER ´ON-ZYGMUND OPERATORS whence 1 ≤ k ≤ n + 1min(1 , e ) ≤ n (1 + 2 − e ) . With the estimate (20) and the inequality F ( J i ) < (cid:15) , the bound forthe corresponding terms of (18) becomes (cid:15) (cid:88) e ∈ Z (cid:88) n ∈ N −| e | δ n − (1+ δ )log(3 n (1+2 − e )) (cid:88) k =1 (cid:88) J ∈ I k, |(cid:104) f I , ψ J (cid:105)|| J | / (cid:46) (cid:15) (cid:107) f I (cid:107) L ( R d ) (cid:88) e ∈ Z (cid:88) n ∈ N −| e | δ n − (1+ δ ) log (cid:0) n (1 + 2 − e ) (cid:1) (cid:46) (cid:15) (cid:107) f I (cid:107) L ( R d ) as long as N > d .On the other hand, when K ∩ I (cid:54) = ∅ and (cid:96) ( K ) ≤ (cid:96) ( I ), we havedist( ˜ E C , c ( K )) ≥ (cid:96) ( I ). Moreover, (cid:96) ( I ) = 2 k + e (cid:96) ( K ) with k + e ≥ w ( ˜ E C , K ) = 1 + dist( ˜ E C , c ( K )) (cid:96) ( K ) ≥ k + e . With this, the estimate (20), and the inequality F ( J i ) < (cid:15) , the boundfor the corresponding terms of (18) becomes (cid:15) (cid:88) e ∈ Z (cid:88) n ∈ N −| e | δ n − (1+ δ ) (cid:88) k ≥ k + e ≥ (cid:88) J ∈ I k, |(cid:104) f I , ψ J (cid:105)|| J | / w ( ˜ E C , K ) − N (cid:46) (cid:15) (cid:107) f I (cid:107) L ( R d ) (cid:88) e ∈ Z (cid:88) n ∈ N −| e | δ n − (1+ δ ) (cid:88) k ≥ k + e ≥ − ( k + e ) N (cid:46) (cid:15) (cid:107) f I (cid:107) L ( R d ) . Combining all the obtained estimates, we get the desired bound for(18) under the assumption of case ( I ). Proof of (II).
As previously stated, in this case we use the size andlocation of the cubes J and K to deduce an estimate that depends on (cid:15) . This leads to the following sub-cases:( II ) J = J ∈ D M ( II ) J = K ∈ D M ( II ) J = (cid:104) J ∪ K (cid:105) ∈ D M ( II ) J / ∈ D M but J = λ ˜ K max ∈ D M ( II ) J / ∈ D M but J = λ ˜ K max ∈ D M ( II ) J / ∈ D M but J = λ K min ∈ D M We can use the fact that K ∈ J e,n ∩ D c M to immediately rule out thecase ( II ). We note that the property K / ∈ D M plays a crucial role inthe remaining cases. We prove only the case ( II ) since, as explained in more detail in [9], all other cases can be dealt with by a similarreasoning.(II ): We recall that the cubes J and K in the sum (18) satisfy (cid:96) ( J ) = 2 e (cid:96) ( K ) and n ≤ rdist( J, K ) < n + 1.By assumption, we have J ∈ D M . That is, 2 − M ≤ (cid:96) ( J ) ≤ M andrdist( J, B M ) ≤ M . Also, since F is bounded, we have F ( J i ) + (cid:15) (cid:46) K ∈ D c M , we separate the study into three cases:( II . ) (cid:96) ( K ) > M ,( II . ) (cid:96) ( K ) < − M ( II . ) 2 − M ≤ (cid:96) ( K ) ≤ M with rdist( K, B M ) > M .( II . ) : The inequalities (cid:96) ( K ) > M and 2 e (cid:96) ( K ) = (cid:96) ( J ) ≤ M imply2 e ≤ M (cid:96) ( K ) − ≤ − M and so, e ≤ − M .Using this and repeating the arguments from ( I ), the inequality (18)becomes |(cid:104) P ⊥ M T f I , g (cid:105)| (cid:46) (cid:107) f I (cid:107) L ( R d ) (cid:88) e ≤− M (cid:88) n ∈ N −| e | δ n − (1+ δ ) ( | e | + log n ) (cid:46) M − Mδ (cid:107) f I (cid:107) L ( R d ) < (cid:15) (cid:107) f I (cid:107) L ( R d ) , where the last inequality holds by the choice of M .( II . ) : The case (cid:96) ( K ) < − M is totally symmetrical with respectto the previous one, and amounts to changing e ≤ − M by e ≥ M .( II . ) : When 2 − M ≤ (cid:96) ( K ) ≤ M and rdist( K, B M ) ≥ M , wehave by Remark 2.2 that | c ( K ) | ∞ ≥ (2 M − M . Moreover, since J ∈ D M , by the same remark we have | c ( J ) | ∞ ≤ ( M − M . Then, | c ( J ) − c ( K ) | ∞ ≥ | c ( K ) | ∞ − | c ( J ) | ∞ ≥ M M . Furthermore, max( (cid:96) ( J ) , (cid:96) ( K )) ≤ M and so, n + 1 > rdist( J, K ) ≥ | c ( J ) − c ( K ) | ∞ max( (cid:96) ( J ) , (cid:96) ( K )) ≥ M. Using this in combination with the arguments from ( I ), the inequality(18) now becomes |(cid:104) P ⊥ M T f I , g (cid:105)| (cid:46) (cid:107) f I (cid:107) L ( R d ) (cid:88) e ∈ Z (cid:88) n ≥ M − −| e | δ n − (1+ δ ) ( | e | + log n ) (cid:46) M − δ (cid:107) f I (cid:107) L ( R d ) < (cid:15) (cid:107) f I (cid:107) L ( R d ) , again by the choice of M . (cid:3) NDPOINT ESTIMATES FOR COMPACT CALDER ´ON-ZYGMUND OPERATORS Proof of Proposition 4.4.
Let ( ψ iI ) I ∈D ,i =1 ,..., d − be an orthog-onal wavelet basis of L ( R d ) such that every function ψ iI is adaptedto a dyadic cube I with constant C > N . As in theproof of Proposition 4.3, we suppress the dependence on the index i ∈ { , . . . , d − } .We denote by ϕ ∈ S ( R d ) a positive bump function adapted to[ − / , / d with order N and constant C > (cid:82) R d ϕ ( x ) dx =1. In particular, we have that 0 ≤ ϕ ( x ) ≤ C (1+ | x | ∞ ) − N and | ∂ i ϕ ( x ) | ≤ C (1 + | x | ∞ ) − N for all i = 1 , . . . , d . Let ( ϕ I ) I ∈D be the family of bumpfunctions defined by ϕ I ( x ) = | I | ϕ (cid:16) x − c ( I ) (cid:96) ( I ) (cid:17) .Given a function b ∈ CMO( R d ), we define the linear operator T b by (cid:104) T b f, g (cid:105) = (cid:88) J ∈D (cid:104) b, ψ J (cid:105)(cid:104) f, ϕ J (cid:105)(cid:104) g, ψ J (cid:105) , for all f, g ∈ S ( R d ). It was shown in [9] that T b and T ∗ b are associatedwith a compact Calder´on-Zygmund kernel, are compact on L p ( R d ) forevery 1 < p < ∞ , and they satisfy (cid:104) T b (1) , g (cid:105) = (cid:104) b, g (cid:105) and (cid:104) T b ( f ) , (cid:105) = 0.Now, we prove that T b , T ∗ b are compact from L ( R d ) into L , ∞ ( R d ).To prove compactness of the former operator, we show first the equality P ⊥ M T b = T P ⊥ M b . Let f, g ∈ D ( R d ). Since P ⊥ M g = (cid:80) J ∈D cM (cid:104) g, ψ J (cid:105) ψ J , (cid:104) P ⊥ M T b f, g (cid:105) = (cid:104) T b f I , P ⊥ M g (cid:105) = (cid:88) J ∈D cM (cid:104) b, ψ J (cid:105)(cid:104) f, ϕ J (cid:105)(cid:104) g, ψ J (cid:105) = (cid:88) J ∈D (cid:104) P ⊥ M b, ψ J (cid:105)(cid:104) f, ϕ J (cid:105)(cid:104) g, ψ J (cid:105) = (cid:104) T P ⊥ M b ( f ) , g (cid:105) , where the second last equality holds because b ∈ CMO( R d ) and so, wealso have P ⊥ M b = (cid:80) J ∈D cM (cid:104) b, ψ J (cid:105) ψ J . Moreover, b ∈ CMO( R d ) implies that for any given (cid:15) >
0, thereexists M ∈ N such that (cid:107) P ⊥ M b (cid:107) BMO( R d ) < (cid:15) for all M > M . Also,since T P ⊥ M b is a Calder´on-Zygmund operator, we know by the classicaltheory that it is bounded from L ( R d ) into L , ∞ ( R d ) with constantbounded by (cid:107) P ⊥ M b (cid:107) BMO( R d ) . With all this we can write m ( { x ∈ R d : | P ⊥ M T b f ( x ) | > λ } ) = m ( { x ∈ R d : | T P ⊥ M b f ( x ) | > λ } ) (cid:46) λ (cid:107) P ⊥ M b (cid:107) BMO( R d ) (cid:107) f (cid:107) L ( R d ) (cid:46) (cid:15)λ (cid:107) f (cid:107) L ( R d ) , which is the result we seek. Finally, we turn to the operator T ∗ b f ( x ) = (cid:88) J ∈D (cid:104) b, ψ J (cid:105) (cid:104) f, ψ J (cid:105) ϕ J ( x ) . Our previous reasoning does not apply because, in general, P ⊥ M T ∗ b doesnot converge to zero. Namely, for d = 1, b = ψ [0 , and ϕ = χ [0 , , wehave that T ∗ b f = (cid:10) f, ψ [0 , (cid:11) χ [0 , , which is the operator we studied inexample 2.17. As we saw, T ∗ b is compact at the endpoint but P M T ∗ b does not converge to T ∗ b in L , ∞ ( R ).However, by linearly, we still have T ∗ b = T ∗ P M b + T ∗ P ⊥ M b . Now, T ∗ P M b isof finite rank, and therefore compact. Moreover, a similar argument asbefore shows that m ( { x ∈ R d : | T ∗ P ⊥ M b f ( x ) | > λ } ) can be made smallerthan (cid:15)/λ by choosing M large. This proves compactness of T ∗ b . (cid:3) Acknowledgements
We extend our most sincere gratitude to ˚Agot Holand Olsen forarranging an enjoyable and productive stay in Harstad, Norway, wherethe seeds of this project first sprouted. We also acknowledge FernandoCobos, Michael Lacey and Xavier Tolsa for helpful comments whichimproved the final version of our work. Last, the second author wouldalso like to express his appreciation to and George Janbing Leefor their invaluable help.We also thank the support received from the Center for Mathemat-ical Sciences at Lund University, Sweden, and by in Sunnyvale,USA, the places where this research was successfully conducted.
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Centre for Mathematical Sciences, University of Lund, Lund, Swe-den
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