Singular integrals with variable kernels in dyadic settings
aa r X i v : . [ m a t h . C A ] J a n Singular integrals with variable kernels in dyadic settings
Hugo Aimar, Raquel Crescimbeni and Luis Nowak ∗ Abstract
In this paper we explore conditions on variable symbols with respect to Haar systems,defining Calder´on-Zygmund type operators with respect to the dyadic metrics associatedto the Haar bases.We show that Petermichl’s dyadic kernel can be seen as a variable kernelsingular integral and we extend it to dyadic systems built on spaces of homogeneous type.
The seminal work of A.P. Calder´on and A. Zygmund during the fifties of the last cen-tury, regarding singular integrals and their relation to partial differential equations, can beconsidered the corner stone of modern Harmonic Analysis, see E. Stein in [16] for historicaldevelopment of the ideas and their impact in the actual and future research in the area. Letus point out two aspects of their contributions that will help us at introducing the problemsthat we consider in this paper. These aspects are contained in the two papers [7] and [8]. In[7] the authors consider convolution type singular integral operators and in [8] they introducenon-convolution type kernels, also called variable kernels.In the Calder´on-Zygmund singular integral theory in metric and quasi-metric spaces (see[9], [13],[14], [1] and [10]), the distinction between convolution and non-convolution kernelsdoes not a priori make sense because convolution is not generally defined in this setting.Nevertheless, there is still another way to consider a convolution operator. The idea goes backto the works of Mikhlin, Giraud and Tricomi (see [11], [12] and the references therein) which,aside from the depth of the analytic tools, it becomes relevant at generating convolution typefilters in machine learning when the analysis is considered on non euclidean data.This way isprovided by the spectral analysis of the operators, when it is available. Let us briefly sketchthe basic idea in a general framework. Assume that { ϕ k } is an orthonormal basis for the space L ( X, µ ), where X is a measure space and µ is a Borel measure. In analogy with the Fouriercase we consider convolution type operators, bounded in L ( X, µ ), as a multiplier operators ofthe form T η f ( x ) = X k η k < f, ϕ k > ϕ k ( x ) , ∗ This research is partially supported by Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas, Univer-sidad Nacional del Litoral and Universidad Nacional del Comahue, Argentina.Keywords and phrases: Singular integrals, Spaces of homogeneous type, Petermichl’s operator, Haar basis. η = { η k } a bounded scalar sequence. Here < f, g > denotes the usual scalar product in L ( X, µ ). On the other hand, if instead of a sequence { η k } we consider in the definition of T a sequence of bounded functions of x , { η k ( x ) } , i.e. T f ( x ) = X k η k ( x ) < f, ϕ k > ϕ k ( x ) , we say that T is an operator with variable kernel given at least formally by K ( x, y ) = X k η k ( x ) ϕ k ( x ) ϕ k ( y ) . In the analysis of unconditionality wavelet bases in functional Banach spaces, as L p ( R n ),the operator defined by T η f ( x ) = Z R n K η ( x, y ) f ( y ) dy with a kernel given by K η ( x, y ) = P h ∈H η ( h ) h ( x ) h ( y ) where H is the classical Haar system in R n and η is some bounded se-quence defined on H , is a singular integral operator when we give to R n a suitable metricstructure (see [4]). Since K η is not translation invariant, the operator T η is not a convolutiontype operator in the classical euclidean sense. Nevertheless, the spectral form of K η ( x, y ) givenby its symbol η : H −→ R , with respect to the Haar basis H which is independent of the points x and y , is a good reason to consider K η as a standard convolution type kernel.On the other hand, a kernel whose spectral Haar analysis takes the form K η ( x, y ) = X h ∈H η ( h, x ) h ( x ) h ( y )for some η : H × R n −→ R , can be considered a variable kernel. A special case of variablekernel K η is considered by S. Petermichl in [15] as we shall see in Section 2.In this work we aim to explore conditions on the variable symbol η ( h, x ) in order to getkernels defining Calder´on-Zygmund type operators with respect to a suitable dyadic metric.The construction of dyadic cubes due to M. Christ (see [6]) in spaces of homogeneous typebecomes a basic tool in order to consider the problem in these general settings.The paper is organized as follows. In Section 2 we consider the variable kernel structureof Petermichl’s operator in R . In Section 3 we introduce the basic properties of spaces ofhomogeneous type and we define the dyadic family D , the Haar system H and the dyadicmetric δ in this general setting. Section 4 is devoted to introduce and prove the main result ofthis work providing sufficient conditions in the multiplier sequence in order to obtain obtaina Calder´on-Zygmund operator. Finally, in Section 5 we build Petermichl type operators onspaces of homogeneous type.Throughout this work, we denote by C a constant that may change from one occurrenceto other. In [15], S. Petermichl introduce a dyadic kernel given in terms of the Haar functions by P ( x, y ) = X I ∈D h I ( y )[ h I − ( x ) − h I + ( x )]for x, y ∈ R + , D the dyadic intervals in R + , h I the Haar wavelets with support in the dyadicinterval I and h I − , h I + the Haar wavelets in the left and right halves of the dyadic interval I .The corresponding operator is given by P f ( x ) = X I ∈D < f, h I > ( h I − ( x ) − h I + ( x )) . This operator is used in [15] to provide an outstanding formula for the Hilbert transform.In [5] the authors proved that the kernel P ( x, y ) has a standard Calder´on-Zygmund struc-ture when we consider the theory of singular integrals extended to metric measure spaces or,more precisely, to spaces of homogeneous type (see definition in Section 3). In other words,they show that P ( x, y ) = Ω( x, y ) δ ( x, y )with δ ( x, y ) = | I ( x, y ) | where I ( x, y ) is the smallest dyadic interval in R + containing x and y .They also prove that Ω is bounded and smooth with respect to the ultrametric δ . Before movingto the abstract setting in order to extend P and P , in this section we prove two elementaryproperties of the Petermichl’s kernel that we shall explore later in the general frame work. Set H and D to denote the Haar system and dyadic family respectively in R + . For h ∈ H wedenote with I ( h ) the interval support of h , and we consider as I −− h the left quarter of I ( h ), I − + h as the second quarter, I + − h as the third quarter and I ++ h as the last quarter of I ( h ). Proposition 2.1. (a) The operator P can be written as a variable kernel singular integral operator, in fact P f ( x ) = 1 √ X h ∈H η ( x, h ) < f, h > h ( x ) with η ( x, h ) = 1 if x ∈ I −− h ∪ I + − h and m ( x, h ) = − if x ∈ I − + h ∪ I ++ h .(b) If P ∗ denotes the adjoint of P , then PP ∗ = P ∗ P = 2 I , twice the identity in L ( R + ) .Proof. Let us start by proving (a) . If we denote with h − and h + the Haar wavelets in the leftand right halves of the support of h , respectively, we have that the supports of h ( y ) h ( x ) and h ( y )[ h − ( x ) − h + ( x )] coincide as subsets of ( R + ) . Then in the support of h ( x ) h ( y ) we havethat h ( y )[ h − ( x ) − h + ( x )] = h ( y ) [ h − ( x ) − h + ( x )] h ( y ) h ( x ) h ( y ) h ( x )= 1 √ η ( x, h ) h ( y ) h ( x ) , as desired.In order to prove (b) observe that P ∗ f ( y ) = X I ∈D ( h f, h I − i − h f, h I + i ) h I ( y ) . On the other hand, from the orthonormality of the system H , for each I ∈ D we have that * X J ∈D < f, h J > ( h J − − h J + ) , h I − + = < f, h I >< h I − , h I − > and * X J ∈D < f, h J > ( h J − − h J + ) , h I + + = < f, h I >< h I + , h I + > . Therefore P ∗ ( P f )( y ) = X I ∈D ( < P f, h I − > − < P f, h I + > ) h I ( y )= X I ∈D h f, h I i h h I − , h I − i h I ( y ) + X I ∈D h f, h I i h h I + , h I + i h I ( y )= 2 f, as desired. Let us first briefly recall the basic properties of the general theory of spaces of homogeneoustype. Assume that X is a set, a nonnegative symmetric function d on X × X is called a quasi-distance if there exists a constant K such that d ( x, y ) ≤ K [ d ( x, z ) + d ( z, y )] , for every x, y, z ∈ X , and d ( x, y ) = 0 if and only if x = y .We shall say that ( X, d, µ ) is a space of homogeneous type if d is a quasi-distance on X , µ is a positive Borel measure defined on a σ -algebra of subsets of X which contains the balls,and there exists a constant A such that0 < µ ( B ( x, r )) ≤ A µ ( B ( x, r )) < ∞ (3.1)holds for every x ∈ X and every r >
0. This property is usually named as the doublingcondition.The construction of dyadic type families of subsets in metric or quasi-metric spaces with someinner and outer metric control of the sizes of the dyadic sets is given in [6]. These familiessatisfy all the relevant properties of the usual dyadic cubes in R n and are the basic tool to buildwavelets on a metric space of homogeneous type (see [1] or [2]). Actually Christ’s constructionin [6] shows the existence of dyadic families in spaces of homogeneous type. Nevertheless, inorder to define Haar wavelets all we need is a dyadic family satisfying the following propertiesthat we state as a definition and we borrow from [2]. Definition 3.1.
Let (
X, d, µ ) be a metric space of homogeneous type. We say that D = S j ∈ Z D j is a dyadic family on X with parameter λ ∈ (0 ,
1) if each D j is a family of Borelsubsets Q of X , such that (d.1) for every j ∈ Z the cubes in D j are pairwise disjoint;(d.2) for every j ∈ Z the family D j covers X in the sense that X = S Q ∈D j Q ; (d.3) if Q ∈ D j and i < j , then there exists a unique ˜ Q ∈ D i such that Q ⊆ ˜ Q ;(d.4) if Q ∈ D j and ˜ Q ∈ D i with i ≤ j , then either Q ⊆ ˜ Q or Q ∩ ˜ Q = ∅ ;(d.5) there exist two constants a and a such that for each Q ∈ D j there exists a point x ∈ Q that satisfies B ( x, a λ j ) ⊆ Q ⊆ B ( x, a λ j ) . The following properties can be deduced from (d.1) to (d.5) , see [3]. Lemma 3.2.
Let D be a dyadic family, then(d.6) there exists a positive integer M depending on a i , i = 1 , in ( d. and on the doublingconstant A in (3.1) such that for every j ∈ Z and all Q ∈ D j the inequalities ≤ L ( Q )) ≤ M hold, where L ( Q ) = { Q ′ ∈ D j +1 : Q ′ ⊆ Q } and B ) denote the cardinal of B ;(d.7) there exists a positive constant C such that µ ( Q ) ≤ Cµ ( Q ′ ) for all Q ∈ ˜ D and every Q ′ ∈ L ( Q ) . It is easy to give examples of dyadic systems D such that a dyadic cube Q belong to differentlevels j ∈ Z . Since we are interested in the identification of those scales and places of partitionwhich shall give rise to the Haar functions, we consider the subfamily ˜ D of D given by˜ D = [ j ∈ Z ˜ D j , with ˜ D j = { Q ∈ D j : { Q ′ ∈ D j +1 : Q ′ ⊆ Q } ) > } . Properties (d.1) to (d.6) allow us to obtain the following aditional properties for ˜ D . (d.8) The families ˜ D j , j ∈ Z are pairwise disjoints. (d.9) The function J : ˜ D −→ Z given by Q
7→ J ( Q ) if Q ∈ ˜ D J ( Q ) is well defined.Let D be a dyadic family. We define, for each dyadic cube Q in D , the quadrant of X thatcontain the cube Q , C ( Q ), by C ( Q ) = [ { Q ′ ∈D : Q ⊆ Q ′ } Q ′ . Following the lines in [2] for the case of Christ’s dyadic cube, from (d.6) and since all thedyadic cubes Q in D are spaces of homogeneous type with uniform doubling constant, we getthat if ( X, d, µ ) is a space of homogeneous type and if D is a dyadic family, then there existsa positive integer N (that depend of the geometric constants of ( X, d, µ )) and disjoint dyadiccubes Q α , α = 1 , ..., N such that X = [ α =1 ,...,N C α , where C α = C ( Q α ). That is, there exists a finite number of quadrants these are a partition of X and each one of them is a space of homogeneous type (see [2])In the classic euclidean context R n , the dyadic analysis leads to consider each quadrantseparately. Then, without loss of generality, we will assume from now on that X itself is aquadrant for D .Along this work, given a dyadic family D we denote by δ ( x, y ) the dyadic metric associatedto D for x, y ∈ X . That is δ is the function defined in X × X given by δ ( x, y ) = ( min { µ ( Q ) : x, y ∈ Q, Q ∈ ˜ D} if x = y x = y. (3.2)Now we state and prove the main result of this section. The proof follow the techniqueused in [13] where the authors prove that each quasi-metric space ( X, d ) is metrizable and that d is equivalent to ρ β , where ρ is a distance on X and β ≥
1. Moreover, they show that allspaces of homogeneous type (
X, d, µ ) can be normalized in the sense that there exists a metric ρ on X and two constants C y C such that C r ≤ µ ( B ρ ( x, r )) ≤ C r, (3.3)where B ρ ( x, r ) = { y ∈ X : ρ ( x, y ) < r } . In general, if ρ satisfies (3.3), we say that ( X, ρ, µ )is a normal space of homogeneous type or 1 − Ahlfors.
Lemma 3.3.
Let ( X, d, µ ) be a space of homogeneous type and let D be a dyadic family. Then ( X, δ, µ ) is a normal space of homogeneous type. Moreover, the characteristic functions ofdyadic cubes are Lipschitz functions in ( X, δ ) .Proof. For each z ∈ X we write Q j ( z ) to denote the unique dyadic cube Q ∈ ˜ D j such that z ∈ Q . Without loss of generality we can assume that X in not bounded. Thus, if x ∈ X , r >
0, and j is an integer in Z such that µ ( Q j ( x )) ≤ r < µ ( Q j − ( x )) , (3.4)then B δ ( x, r ) = Q j ( x ) . (3.5)In fact if y ∈ Q j ( x ) then x, y ∈ Q j ( x ) and therefore δ ( x, y ) ≤ µ ( Q j ( x )) ≤ r this implies that Q j ( x ) ⊆ B δ ( x, r ). On the other hand, let y ∈ B δ ( x, r ), if y / ∈ Q j ( x ) then Q j ( x ) ∩ Q j ( y ) = ∅ . Let n ∈ N be the first positive integer such that Q j ( y ) ⊆ Q j − n ( x ), then we get that δ ( x, y ) = µ ( Q j − n ( x )) ≥ µ ( Q j − ( x )) > r , this is a contradiction. Hence y ∈ Q j ( x ) andthen B δ ( x, r ) ⊆ Q j ( x ). In orden to prove that ( X, δ, µ ) is a normal space of homogeneoustype, observe that it is not difficult to see that (
X, δ ) is a metric space (see [2]) moreover, δ is an ultra-metric on X . Let x ∈ X be and r >
0, consider the number j given in (3.4).Since B δ ( x, r ) = Q j ( x ), we get that µ ( B δ ( x, r )) = µ ( Q j ( x )) ≤ r . On the other hand, since Q j ( x ) ⊆ L ( Q j − ( x )), by the doubling property of the measure (3.1) there exists a positiveconstant C such that µ ( Q j − ( x )) ≤ Cµ ( Q j ( x )), then from (3.4) and (3.5) we get that r < µ ( Q j − ( x )) ≤ Cµ ( Q j ( x )) = Cµ ( B δ ( x, r )) . Hence, rC < µ ( B δ ( x, r )) . Finally, for the last statement, let x, y ∈ X and Q ∈ ˜ D . If x, y ∈ Q or if y / ∈ Q , x / ∈ Q , then χ Q ( x ) − χ Q ( y ) = 0. If Q contain only the point x or the point y and Q ( x, y ) is the smallest dyadic cube such that x, y ∈ Q ( x, y ), then δ ( x, y ) = µ ( Q ( x, y )) ≥ µ ( Q ).Hence | χ Q ( x ) − χ Q ( y ) | = 1 ≤ µ ( Q ) δ ( x, y ).From now on we shall denote by Q ( x, y ) the smallest dyadic cube such that x, y ∈ Q ( x, y ).From each dyadic system D as above we can associate a Haar type systems that we borrowfrom ([3]). Definition 3.4.
Let D be a dyadic family on ( X, d, µ ). A system H of simple Borel measurablereal functions h on X is said to be a Haar system associated to D if it is an orthonormal basisof L ( X, µ ) such that (h.1) For each h ∈ H there exists a unique j ∈ Z and a cube Q ( h ) ∈ ˜ D j such that { x ∈ X : h ( x ) = 0 } ⊆ Q ( h ) , and this property does not hold for any cube in D j +1 .(h.2) For every Q ∈ ˜ D there exist exactly M Q = L ( Q )) − ≥ functions h ∈ H such that(h.1) holds. We denote with H ( Q ) the set of all these functions h .(h.3) For each h ∈ H we have that R X hdµ = 0 .(h.4) For each Q ∈ ˜ D let V Q denote the vector space of all functions on Q which are constanton each Q ′ ∈ L ( Q ) . Then the system { χ Q ( µ ( Q )) / } S H ( Q ) is an orthonormal basis for V Q .(h.5) There exists a positive constant C such that the inequality | h ( x ) | ≤ C | h ( y ) | holds foralmost every x and y in Q ( h ) and every h ∈ H . Observe also that from (d.7) , (h.4) and (h.5) we get that there exists two positive constants C and C such that C µ ( Q ( h )) / ≤ | h ( x ) | ≤ C µ ( Q ( h )) / , (3.6)for all h ∈ H and x ∈ Q ( h ). Let (
X, d, µ ) a space of homogeneous type, D and H the dyadic family of cubes and theHaar system associated given in Definitions 3.1 and 3.4 respectively. For simplicity we denoteby L = L ( X, µ ) of square integrable real functions defined on X . Since H is an orthonormalbasis for L , we have the resolution of the identity given by f = X h ∈H h f, h i h. The operators T η f ( x ) = X h ∈H η ( h ) h f, h i h ( x ) , (4.1)with η a bounded function defined on H , or more generally T η f ( x ) = X h ∈H η ( x, h ) h f, h i h ( x ) , (4.2)with η a bounded function defined on X × H , are bounded in L .With the heuristics described in the introduction we may think that the operator as in(4.1) is of convolution type while that in (4.2) is of non-convolution type singular. In thissection we give a sufficient condition on η ( x, h ) in such a way that T η defined by (4.2) becomesa Calder´on-Zygmund type operator in ( X, d, µ ).A bounded linear operator T : L −→ L is said to be of Calder´on-Zygmund type in( X, δ, µ ) if there exists K ∈ L loc ( X × X \ ∆), with ∆ the diagonal of X × X , such that(1) there exists a positive constant C such that | K ( x, y ) | ≤ Cδ ( x,y ) for x, y ∈ X with x = y ,(2) there exists two positive constants C and γ such that(2 .a ) | K ( x ′ , y ) − K ( x, y ) | ≤ C δ ( x ′ ,x ) γ δ ( x,y ) γ , if 2 δ ( x ′ , x ) ≤ δ ( x, y );(2 .b ) | K ( x, y ′ ) − K ( x, y ) | ≤ C δ ( y,y ′ ) γ δ ( x,y ) γ , if 2 δ ( y ′ , y ) ≤ δ ( x, y );(3) for ϕ, ψ ∈ S ( H ), the linear span of H , with suppϕ ∩ suppψ = ∅ , we have < T ( ϕ ) , ψ > = Z Z X × X K ( x, y ) ϕ ( x ) ψ ( y ) d ( µ × µ )( x, y ) . The main result of this section is contained in the following statement.
Theorem 4.1.
Let ( X, d, µ ) a space of homogeneous type, D a dyadic family, H a Haar systemand δ defined in (3.2) . Let η : X × H −→ R be a function such that is a measurable functionin x ∈ X for each h ∈ H and there exists a constant B > such that(a) | η ( x, h ) | ≤ B, for x ∈ X and h ∈ H (b) | η ( x ′ , h ) − η ( x, h ) | ≤ B δ ( x,x ′ ) µ ( Q ( h )) , for h ∈ H and x, x ′ ∈ X .Then the operator T η f ( x ) = X h ∈H η ( x, h ) h f, h i h ( x ) is of Calder´on-Zygmund type in the space of homogeneous type ( X, δ, µ ) . Hence T η is boundedon L p ( X ) (1 < p < ∞ ) and of weak type (1 , .Proof. The L boundedness of T η follows from ( a ) with k T η f k ≤ k η k ∞ k f k . By testing T η with simple function in S ( H ), we see that K ( x, y ) = X h ∈H η ( x, h ) h ( y ) h ( x )satisfies property (3) in the above definition of Calder´on-Zygmund kernel in the general setting.Let us prove (1) of the definition of Calder´on-Zygmund type operator. Let x = y in X and Q ( x, y ) in D such that µ ( Q ( x, y )) = δ ( x, y ). On the other hand for any cube strictly smallerthan Q ( x, y ) we must have h ( y ) = 0 or h ( x ) = 0. Hence from ( h. d.
6) we get | K ( x, y ) | ≤ C k η k ∞ X Q ⊇ Q ( x,y ) X { h ∈H : Q ( h )= Q } µ ( Q ) ≤ C k η k ∞ M X Q ⊇ Q ( x,y ) µ ( Q ) , where M is as in ( d.
6) in Lemma 3.2. Notice that we are considering only the cubes in ˜ D . Thenif Q m is the m − th ancestor of Q ( x, y ) in ˜ D , the measure of this sequence grows geometrically,i.e. µ ( Q m ) ≥ (1 + ε ) m µ ( Q ( x, y )) (4.3)with a geometric constant ε >
0. Hence | K ( x, y ) | ≤ Cδ ( x, y )as desired. Let us now prove the smoothness properties of K . Notice first that, from (4.3) weget that X Q ∈ ˜ D Q ⊇ Q ( x,y ) µ ( Q )) = X m ∈ N µ ( Q m − )) ≤ X m ∈ N (cid:18) ε ) (cid:19) m − µ ( Q )) = 1( µ ( Q )) X m ∈ N (cid:18) ε ) (cid:19) m − = C ( µ ( Q )) , (4.4)where Q = Q ( x, y ) in ˜ D . On the other hand, notice that for h ∈ H if Q = Q ( h ) ∈ ˜ D , then h ( x ) = X Q ′ ∈L ( Q ) β Q ′ χ Q ′ ( x ) , where β Q ′ ∈ R . Thus, since the characteristic functions on dyadic cube are Lipschitz functionson ( X, δ ), from dyadic doubling property, ( d.
6) and (3.6) there exists a positive constant C such that if x, x ′ ∈ X we get that | h ( x ) − h ( x ′ ) | ≤ X Q ′ ∈L ( Q ( h )) | β Q ′ | (cid:12)(cid:12)(cid:12) χ Q ′ ( x ) − χ Q ′ ( x ′ ) (cid:12)(cid:12)(cid:12) ≤ X Q ′ ∈L ( Q ( h )) k h k ∞ (cid:12)(cid:12)(cid:12) χ Q ′ ( x ) − χ Q ′ ( x ′ ) (cid:12)(cid:12)(cid:12) ≤ C p µ ( Q ( h )) X Q ′ ∈L ( Q ( h )) (cid:12)(cid:12)(cid:12) χ Q ′ ( x ) − χ Q ′ ( x ′ ) (cid:12)(cid:12)(cid:12) ≤ C p µ ( Q ( h )) X Q ′ ∈L ( Q ( h )) δ ( x, x ′ ) µ ( Q ′ )0 ≤ C p µ ( Q ( h )) X Q ′ ∈L ( Q ( h )) δ ( x, x ′ ) µ ( Q ( h )) ≤ C δ ( x, x ′ )( µ ( Q ( h ))) L ( Q ( h )) ≤ M C δ ( x, x ′ )( µ ( Q ( h ))) ≤ C δ ( x, x ′ ) µ ( Q ( h )) . (4.5)Observe now that if x, y, x ′ ∈ X satisfy 2 δ ( x ′ , x ) ≤ δ ( x, y ) then x ′ ∈ Q ( x, y ) and moreover Q ( x, y ) = Q ( x ′ , y ) . In fact, if x ′ / ∈ Q ( x, y ) then δ ( x, x ′ ) > δ ( x, y ), which is a contradiction. On the other hand,since Q ( x, y ) ∈ ˜ D , there exists two different dyadic cubes Q ′ and ˆ Q in L ( Q ( x, y )) such that y ∈ Q ′ and x ∈ ˆ Q . So, if x ′ ∈ X satisfies 2 δ ( x ′ , x ) ≤ δ ( x, y ) and we suppose that x ′ / ∈ ˆ Q , then δ ( x, x ′ ) = µ ( Q ( x, y )) = δ ( x, y ) , which is again a contradiction. Then if 2 δ ( x ′ , x ) ≤ δ ( x, y ) we have Q ( x, y ) = Q ( x ′ , y ), thisimplies that δ ( x, y ) = δ ( x ′ , y ). Hence in such case, from the conditions ( a ) and ( b ) on η , (4.5),(4.4) and (3.6) we get that (cid:12)(cid:12)(cid:0) η ( x ′ , h ) h ( x ′ ) − η ( x, h ) h ( x ) (cid:1) h ( y ) (cid:12)(cid:12) = (cid:0)(cid:12)(cid:12) η ( x ′ , h ) − η ( x, h ) (cid:12)(cid:12) | h ( x ′ ) | + | η ( x, h ) | (cid:12)(cid:12) h ( x ′ ) − h ( x ) (cid:12)(cid:12)(cid:1) | h ( y ) |≤ (cid:18) CBδ ( x, x ′ )( µ ( Q ( h ))) / + B M C δ ( x, x ′ )( µ ( Q ( h ))) / (cid:19) | h ( y ) |≤ (cid:18) CBδ ( x, x ′ )( µ ( Q ( h ))) + B M C δ ( x, x ′ )( µ ( Q ( h ))) (cid:19) = C δ ( x, x ′ )( µ ( Q ( h ))) . Then from the above estimate and (4.4) we get that | K ( x ′ , y ) − K ( x, y ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X h ∈H (cid:0) η ( x ′ , h ) h ( x ′ ) − η ( x, h ) h ( x ) (cid:1) h ( y ) (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:12)(cid:12) X Q ∈ ˜ D Q ⊇ Q ( x,y ) X h ∈H Q ( h )= Q (cid:0) η ( x ′ , h ) h ( x ′ ) − η ( x, h ) h ( x ) (cid:1) h ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = C X Q ∈ ˜ D Q ⊇ Q ( x,y ) X h ∈H Q ( h )= Q δ ( x, x ′ )( µ ( Q ( h ))) ≤ C X Q ∈ ˜ D Q ⊇ Q ( x,y ) δ ( x, x ′ )( µ ( Q )) = C δ ( x, x ′ )( µ ( Q ( x, y ))) = C δ ( x, x ′ )( δ ( x, y )) , this complete the proof of (2 .a ). In a similar way we can prove (2 .b ). In this section we introduce Petermichl type operators P on spaces of homogeneous type.We prove, using Theorem 4.1, that this operator is a Calder´on-Zygmund type operator on asuitable space of homogeneous type and we show that P ∗ is almost the identity operator in asense that shall be made precise.Let ( X, d, µ ) be a space of homogeneous type, D a dyadic family, H a Haar system associatedto D and ( α h ) h ∈H a bounded sequence in R . For f ∈ L ( X, µ ) we consider the operator P defined as P f ( x ) = X Q ∈ ˜ D X h ∈H Q ( h )= Q < f, h > X ˜ h ∈H ( R ) R ∈L ( Q ) α ˜ h ˜ h ( x ) where we recall that H ( R ) is given in ( h. Proposition 5.1.
Let ( X, d, µ ) be a space of homogeneous type, D the dyadic family, H theHaar system associated to D and ( α h ) h ∈H a bounded sequence in R . Then the operator P satisfies the following properties(1) P f ( x ) = Z y ∈ X N ( x, y ) f ( y ) dµ ( y ) , where N ( x, y ) = X Q ∈ ˜ D X h ∈H Q ( h )= Q h ( y ) X ˜ h ∈H ( R ) R ∈L ( Q ) α ˜ h ˜ h ( x ) and f is a simple function in S ( H ) .(2) P ∗ f ( x ) = Z y ∈ X N ∗ ( x, y ) f ( y ) dµ ( y ) , where N ∗ ( z, w ) = N ( w, z ) and f is a simple function in S ( H ) . (3) P ∗ ( P f )( x ) = X h ∈H C ( Q ) < f, h > h ( x ) , with ≤ C ( Q ) ≤ M , with M as in ( d. in Lemma 3.2.Proof. In order to prove (1), we observe that for f in S ( H ) the sum in the definition of P f ( x )is finite and therefore we have that P f ( x ) = Z y ∈ X X Q ∈ ˜ D X h ∈H Q ( h )= Q h ( y ) X ˜ h ∈H ( R ) R ∈L ( Q ) α ˜ h ˜ h ( x ) f ( y ) dµ ( y )= Z y ∈ X N ( x, y ) f ( y ) dµ ( y ) , where N ( x, y ) = X Q ∈ ˜ D X h ∈H Q ( h )= Q h ( y ) X ˜ h ∈H ( R ) R ∈L ( Q ) α ˜ h ˜ h ( x ) . On the other hand, P ∗ f ( z ) = Z w ∈ X N ∗ ( z, w ) f ( w ) dµ ( w ) , for N ∗ ( z, w ) = N ( w, z ).Finally we compute the action of P ∗ on P . By Fubini’s theorem we get that P ∗ ( P f )( x ) = Z y ∈ X N ∗ ( x, y ) P f ( y ) dµ ( y )= Z y ∈ X N ∗ ( x, y ) Z z ∈ X N ( y, z ) f ( z ) dµ ( z ) dµ ( y )= Z y ∈ X N ( y, x ) Z z ∈ X N ( y, z ) f ( z ) dµ ( z ) dµ ( y )= Z z ∈ X (cid:18)Z y ∈ X N ( y, x ) N ( y, z ) dµ ( y ) (cid:19) f ( z ) dµ ( z )= Z z ∈ X U ( x, z ) f ( z ) dµ ( z ) , where U ( x, z ) = Z y ∈ X N ( y, x ) N ( y, z ) dµ ( y )= X Q ∈ ˜ D X h ∈H Q ( h )= Q X Q ′ ∈ ˜ D X h ′ ∈H Q ( h ′ )= Q h ( x ) h ′ ( z ) Z y ∈ X X ˜ h ∈H ( R ) R ∈L ( Q ) α ˜ h ˜ h ( y ) X ˆ h ∈H ( R ′ ) R ′∈L ( Q ′ ) α ˆ h ˆ h ( y ) dµ ( y ) . Z y ∈ X X ˜ h ∈H ( R ) R ∈L ( Q ) α ˜ h ˜ h ( y ) X ˆ h ∈H ( R ′ ) R ′∈L ( Q ′ ) α ˆ h ˆ h ( y ) dµ ( y ) = X ˜ h ∈H ( R ) R ∈L ( Q ) X ˆ h ∈H ( R ′ ) R ′∈L ( Q ′ ) α ˜ h α ˆ h Z y ∈ X ˜ h ( y )ˆ h ( y ) dµ ( y )= X ˜ h ∈H ( R ) R ∈L ( Q ) α h = ( L ( Q ))) ( L ( R ) − C ( Q ) . Therefore U ( x, z ) = Z y ∈ X N ( y, x ) N ( y, z ) dµ ( y )= X Q ∈ ˜ D X h ∈H Q ( h )= Q C ( Q ) h ( x ) h ( z ) . Thus P ∗ ( P f )( x ) = Z z ∈ X X Q ∈ ˜ D X h ∈H Q ( h )= Q C ( Q ) h ( x ) h ( z ) f ( z ) dµ ( z ) . with 1 ≤ C ( Q ) = ( L ( Q )) ( L ( R ) − ≤ M as desired.As an application of Theorem 4.1 we obtain the boundedness of these operators in Lebesguespaces. Theorem 5.2.
Let ( X, d, µ ) be a space of homogeneous type. Let D , H and δ be a dyadicfamily, a Haar systems associated to D and the dyadic metric induced by D respectively. Let ( α h ) h ∈H be a bounded sequence in R . Then the operator P f ( x ) = X Q ∈ ˜ D X h ∈H Q ( h )= Q < f, h > X ˜ h ∈H ( R ) R ∈L ( Q ) α ˜ h ˜ h ( x ) is a Calder´on-Zygmund type operator on the space ( X, δ, µ ) . Hence P is bounded in L p ( X )(1 < p < ∞ ) and of weak type (1 , . Proof.
By Theorem 4.1 it is enough to prove that the operator P can be written as P f ( x ) = X h ∈H η ( x, h ) h f, h i h ( x )for some function η : X × H −→ R satisfying the hypothesis in Theorem 4.1. In fact for h ∈ H with Q = Q ( h ) ∈ D we have that h ( x ) = X R ∈L ( Q ( h )) h R χ R ( x ) , where h R ∈ R . Thus, as h is different from zero on Q ( h ), we define for x ∈ X , η ( x, h ) = X R ∈L ( Q ( h )) X ˜ h ∈H ( R ) α ˜ h h R ˜ h ( x ) χ R ( x ) , which is a measurable function for x ∈ X . Then we get that η ( x, h ) h ( x ) = X ˜ h ∈H ( R ) R ∈L ( Q ) α ˜ h ˜ h ( x )and therefore P f ( x ) = X h ∈H η ( x, h ) h f, h i h ( x ) . Let us first prove that the function η satisfies condition ( a ) in the Theorem 4.1. Notice thatif h ∈ H and x / ∈ Q ( h ) then η ( x, h ) = 0 . On the other hand if x ∈ Q ( h ), from (3.6), doublingproperty on dyadic cubes ( d. d.
6) and ( h.
2) we get | η ( x, h ) | ≤ X R ∈L ( Q ( h )) X ˜ h ∈H ( R ) | α ˜ h || h R | | ˜ h ( x ) | | χ R ( x ) |≤ X R ∈L ( Q ( h )) X ˜ h ∈H ( R ) k ( α ˜ h ) k ∞ p µ ( Q ( h )) C C q µ ( Q (˜ h )) ≤ k ( α ˜ h ) k ∞ √ C C C X R ∈L ( Q ( h )) X ˜ h ∈H ( R ) ≤ M k ( α ˜ h ) k ∞ √ C C C = B, (5.1)where M is as in ( d.
6) in Lemma 3.2.In order to prove that the function η satisfies ( b ) in Theorem 4.1, take h ∈ H with Q = Q ( h ) ∈D as above h ( x ) = P R ∈L ( Q ( h )) h R χ R ( x ) . We split the proof in five cases.
Case 1. x, x ′ / ∈ Q ( h ). Then | η ( x, h ) − η ( x ′ , h ) | = 0 . Case 2. x, x ′ ∈ Q ′ for some Q ′ ∈ L ( R ) and some R ∈ L ( Q ( h )). Then, since in such case˜ h ( x ) = ˜ h ( x ′ ) for every ˜ h ∈ H ( R ), we have that | η ( x, h ) − η ( x ′ , h ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X R ∈L ( Q ( h )) X ˜ h ∈H ( R ) α ˜ h h R ˜ h ( x ) χ R ( x ) − X ˜ h ∈H ( R ) α ˜ h h R ˜ h ( x ′ ) χ R ( x ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ˜ h ∈H ( R ) α ˜ h h R ˜ h ( x ) − α ˜ h h R ˜ h ( x ′ ) ! χ R ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 Case 3. x ∈ Q and x ′ ∈ Q ′ with Q, Q ′ ∈ L ( R ) and R ∈ L ( Q ( h )). Then, from (4.5),(3.6), doubling property on dyadic cubes ( d. d.
6) and ( h.
2) we get that | η ( x, h ) − η ( x ′ , h ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X R ∈L ( Q ( h )) X ˜ h ∈H ( R ) α ˜ h h R ˜ h ( x ) χ R ( x ) − X ˜ h ∈H ( R ) α ˜ h h R ˜ h ( x ′ ) χ R ( x ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ˜ h ∈H ( R ) α ˜ h h R ˜ h ( x ) − X ˜ h ∈H ( R ) α ˜ h h R ˜ h ( x ′ ) χ R ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ˜ h ∈H ( R ) α ˜ h h R (cid:16) ˜ h ( x ) − ˜ h ( x ′ ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k ( α ˜ h ) k ∞ | h R | M C δ ( x, x ′ ) X ˜ h ∈H ( R ) µ ( Q (˜ h ))) / ≤ k ( α ˜ h ) k ∞ ( µ ( Q ( h ))) / C M C δ ( x, x ′ ) X ˜ h ∈H ( R ) ( µ ( Q ( h ))) / ( µ ( Q ( h ))) / ( µ ( Q (˜ h ))) / ≤ k ( α ˜ h ) k ∞ M C / C δ ( x, x ′ ) µ ( Q ( h )) X ˜ h ∈H ( R ) ≤ k ( α ˜ h ) k ∞ M C / C δ ( x, x ′ ) µ ( Q ( h )) . Case 4. x ∈ Q ( h ) and x ′ / ∈ Q ( h ) then η ( x ′ , h ) = 0, also δ ( x, x ′ ) > µ ( Q ( h )). Hence, from(5.1) we obtain that | η ( x, h ) − η ( x ′ , h ) | = | η ( x, h ) |≤ B ≤ B δ ( x, x ′ ) µ ( Q ( h )) . Case 5. x ∈ R and x ′ ∈ R with R , R ∈ L ( Q ( h ))different. Then δ ( x, x ′ ) = µ ( Q ( h ))and hence from (5.1) we get that | η ( x, h ) − η ( x ′ , h ) | ≤ | η ( x, h ) | + | η ( x ′ , h ) | ≤ B = 2 B δ ( x, x ′ ) µ ( Q ( h )) . as desired. References [1] H. Aimar,
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