A two weight inequality for Calderón-Zygmund operators on spaces of homogeneous type with applications
Xuan Thinh Duong, Ji Li, Eric T. Sawyer, Manasa N. Vempati, Brett D. Wick, Dongyong Yang
aa r X i v : . [ m a t h . C A ] J un A TWO WEIGHT INEQUALITY FOR CALDER ´ON–ZYGMUND OPERATORS ONSPACES OF HOMOGENEOUS TYPE WITH APPLICATIONS
XUAN THINH DUONG, JI LI, ERIC T. SAWYER, MANASA N. VEMPATI, BRETT D. WICK, AND DONGYONG YANG
Abstract.
Let (
X, d, µ ) be a space of homogeneous type in the sense of Coifman and Weiss, i.e. d is a quasimetric on X and µ is a positive measure satisfying the doubling condition. Suppose that u and v are twolocally finite positive Borel measures on ( X, d, µ ). Subject to the pair of weights satisfying a side condition,we characterize the boundedness of a Calder´on–Zygmund operator T from L ( u ) to L ( v ) in terms of the A condition and two testing conditions. For every cube B ⊂ X , we have the following testing conditions,with B taken as the indicator of B k T ( u B ) k L ( B,v ) ≤ T k B k L ( u ) , k T ∗ ( v B ) k L ( B,u ) ≤ T k B k L ( v ) . The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg,along with the pivotal side condition.
Contents
1. Introduction and Statement of Main Results 12. Applications of the Main Theorem 53. Preliminaries on Spaces of Homogeneous Type 104. First Reduction in the Proof of the Two-Weight Inequality 125. Main Decomposition 146. Stopping Cubes and Corona Decompositions 237. The Carleson Measure Estimates 288. The Paraproduct Terms 329. Appendix: Hilbert Space Valued Operators 37References 401.
Introduction and Statement of Main Results
The two weight conjecture for Calder´on–Zygmund operators T was first raised by Nazarov, Treil andVolberg on finding the necessary and sufficient conditions on the two weights u and v so that T is boundedfrom L ( u ) to L ( v ). The third author, in [Saw1] first introduced the testing conditions (which are referred toas the Sawyer-type testing conditions) into the two weight setting on the maximal function, and later [Saw2]on the fractional and Poisson integral operators; serving as motivation for the investigation by Nazarov, Treiland Volberg, see for example [NTV3].This conjecture in the special case that the pair of weights u and v do not share a common point masswas completely solved only recently when T is the Hilbert transform on R . This was supplied in two papers, Date : June 12, 2020.2010
Mathematics Subject Classification.
Primary: 42B20, 43A85.
Key words and phrases. two weight inequality, testing conditions, space of homogeneous type, Calder´on–Zygmund Operator,Haar Basis.X. Duong’s research supported by ARC DP 190100970.J. Li’s research supported by ARC DP 170101060.E. T. Sawyer’s research supported by NSERC.B. D. Wick’s research supported in part by NSF grant DMS-1800057 as well as ARC DP190100970.D. Yang’s research supported by NNSF of China
Lacey–Sawyer–Shen–Uriarte-Tuero [LaSaShUr], and Lacey [Lac] that built on pioneering work of Nazarov,Treil, and Volberg [NTV3]. The central question is providing a real-variable characterisation of the inequalitysup <α<β< ∞ k H α,β ( f v ) k L ( w ) ≤ N k f k L ( u ) , (1.1)where N is the best constant such that the above inequality holds, H α,β ( f v )( x ) is the standard truncationof the usual Hilbert transform, and u, v are non-negative Borel locally finite measures on R . The full solutionis as follows. Theorem A.
Suppose that for all x ∈ R , u ( { x } ) · v ( { x } ) = 0 for the pair of weights u and v . Define twopositive constants A and T as the best constants in the inequalities below, uniform over intervals I : P ( u, I ) · P ( v, I ) ≤ A ;(1.2) Z I H (1 I u ) dv ≤ T u ( I ) , Z I H (1 I v ) du ≤ T v ( I ) . (1.3) Then (1.1) holds if and only if both (1.2) and (1.3) hold, moreover,
N ≈ A / + T . In the theorem above, the term P ( u, I ) is the Poisson integral with respect to the measure u at the scalinglevel | I | and centred at x I , that is, P ( u, I ) := Z R | I | ( | I | + dist( x, I )) du ( x ) . The restriction regarding common point masses was removed by Hyt¨onen in [Hyt].The aim of this paper is to provide sufficient conditions for the two-weight inequality for general Calder´on–Zygmund operators on spaces of homogeneous type. Since we are working in a very general setting, the bestwe can hope for at the moment is to provide a collection of sufficient conditions on the weights that guaranteetwo weight estimates for Calder´on–Zygmund operators. Our main approach is a suitable version of stoppingcubes and corona decompositions originating in work of Nazarov, Treil and Volberg [NTV3], along with thepivotal side condition.Spaces of homogeneous type were introduced by Coifman and Weiss in the early 1970s, in [CW1], seealso [CW]. We say that ( X, d, µ ) is a space of homogeneous type in the sense of Coifman and Weiss if d isa quasi-metric on X and µ is a nonzero measure satisfying the doubling condition. A quasi-metric d on aset X is a function d : X × X −→ [0 , ∞ ) satisfying(i) d ( x, y ) = d ( y, x ) ≥ x , y ∈ X ;(ii) d ( x, y ) = 0 if and only if x = y ; and(iii) the quasi-triangle inequality : there is a constant A ∈ [1 , ∞ ) such that for all x , y , z ∈ X , d ( x, y ) ≤ A [ d ( x, z ) + d ( z, y )] . We say that a nonzero measure µ satisfies the doubling condition if there is a constant C µ such that forall x ∈ X and r > µ ( B ( x, r )) ≤ C µ µ ( B ( x, r )) < ∞ , where B ( x, r ) is the quasi-metric ball defined by B ( x, r ) := { y ∈ X : d ( x, y ) < r } for x ∈ X and r > n (the upper dimension of µ ) such that for all x ∈ X , m ≥ r > µ ( B ( x, mr )) ≤ C µ m n µ ( B ( x, r )) . Throughout this paper we assume that µ ( X ) = ∞ and that µ ( { x } ) = 0 for every x ∈ X .We now recall the definition of Calder´on–Zygmund operators on spaces of homogeneous type. As Yves Meyer remarked in his preface to [DH], “One is amazed by the dramatic changes that occurred in analysis duringthe twentieth century. In the 1930s complex methods and Fourier series played a seminal role. After many improvements,mostly achieved by the Calder´on–Zygmund school, the action takes place today on spaces of homogeneous type. No groupstructure is available, the Fourier transform is missing, but a version of harmonic analysis is still present. Indeed the geometryis conducting the analysis.”
WO WEIGHT INEQUALITY 3
Definition 1.
We say that T is a Calder´on–Zygmund operator on ( X, d, µ ) if T is bounded on L ( X ) andhas an associated kernel K ( x, y ) such that T ( f )( x ) = R X K ( x, y ) f ( y ) dµ ( y ) for any x supp f , and K ( x, y ) satisfies the following estimates: for all x = y , (1.5) | K ( x, y ) | ≤ CV ( x, y ) , and for d ( x, x ′ ) ≤ (2 A ) − d ( x, y ) , (1.6) | K ( x, y ) − K ( x ′ , y ) | + | K ( y, x ) − K ( y, x ′ ) | ≤ CV ( x, y ) ω (cid:18) d ( x, x ′ ) d ( x, y ) (cid:19) , where V ( x, y ) := µ ( B ( x, d ( x, y ))) , ω : [0 , → [0 , ∞ ) is continuous, increasing, subadditive, and ω (0) = 0 . Note that by the doubling condition we have that V ( x, y ) ≈ V ( y, x ). In the theorem below we have taken ω ( t ) = t κ for some κ ∈ (0 ,
1) for the kernel estimate (1.6) and we refer to κ as the smoothness parameter forthe kernel K ( x, y ).The main result of this paper provides sufficient conditions on a pair of weights u and v so that thefollowing two-weight norm inequality k T ( f · u ) k L ( v ) ≤ N k f k L ( u ) (1.7)holds for a Calder´on–Zygmund operator T on ( X, d, µ ), where N is the best constant (understood as theoperator norm).We also define l ( Q ) (in Section 3.1) to be the side-length of a dyadic cube Q , and for all Q = Q αk ∈ D k where {D k } k ∈ Z is a system of dyadic cubes with the parameter δ as given in Definition 8 below. In particularwe have l ( Q ) = δ k .We say that a set Q is a cube , or more precisely a ( c , C , δ )-cube, if there is z ∈ X such that B (cid:16) z, c δ k (cid:17) ⊂ Q kα ( ω ) ⊂ B (cid:16) z, C δ k (cid:17) , where c , C and δ are positive constants. The cubes appearing in the main theorem below are those with c , C and δ as in (3) of Theorem 7 below. Also for all cubes Q and x ∈ X \ Q we define dist( x, Q ) :=inf q ∈ Q { d ( q, x ) : q ∈ Q } where d is the quasi-metric on X . Recall that a measure u is locally finite if for anypoint in X there exists a neighborhood B about that point so that u ( B ) < ∞ .We need two quantities that control certain quantities in the proof that control certain informationuniformly over cubes Q ⊂ X . The first is a version of an A condition. Suppose the pair of weights u, v satisfies the following A condition for all cubes Q :(1.8) (cid:18) u ( Q ) l ( Q ) n K ( Q, v ) (cid:19) . A with K ( Q, v ) := R X (cid:16) l ( Q ) l ( Q )+dist( y,Q ) (cid:17) κ µ ( B ( x Q ,l ( Q )+dist( y,Q ))) dv ( y ), as well as the dual condition (cid:18) v ( Q ) l ( Q ) n K ( Q, u ) (cid:19) . A , where A is the best constant such that the above inequalities hold. Recall here that κ is the smoothnessparameter associated to the Calder´on–Zygmund kernel in Definition 1.The second is the pivotal condition:(1.9) sup Q = ∪ i ≥ S i X i ≥ Φ( S i , Q u ) ≤ V u ( Q ) , where Φ( Q, E u ) := v ( Q ) K ( Q, E u ) , as well as the dual version, in which u and v are interchanged. Here V is the best constant, and the supremum is over all r -good subpartitions { S i } i ≥ of Q where r is definedin Definition 16. An r -good subpartition consists of Q -dyadic subcubes { S i } of Q such that S i is r -good inany dyadic grid containing Q .With these preliminaries, our main result is the following: X. T. DUONG, J. LI, E. SAWYER, M. N. VEMPATI, B. D. WICK, AND D. YANG
Theorem 2.
Let T be a Calder´on–Zygmund operator with smoothness parameter κ . Let u and v be twolocally finite, positive Borel measures on X . Suppose that u ( { x } ) · v ( { x } ) = 0 for x ∈ X and that they satisfythe two weight condition with constant A and the pivotal condition with constant V . Then T : L ( u ) → L ( v ) is bounded if and only if the following testing conditions hold: for every cube Q ⊂ X , we have the followingtesting conditions, with Q taken as the indicator of Q (1.10) k T ( u Q ) k L ( Q,v ) ≤ T k Q k L ( u ) , (1.11) k T ∗ ( v Q ) k L ( Q,u ) ≤ T k Q k L ( v ) . Moreover, we have that N . A + T + V . Remark 3.
We would like to point out that we introduce a new version of a Poisson-type integral K ( Q, v ) in the main Theorem that plays the role of the standard Poisson integral as in (1.2) in Theorem A. The mainreason for providing such a condition is that, for some Calder´on–Zygmund operators in certain particularsetting, the typical Poisson integral in that setting is not linked directly to the study of two weight inequalityfor the Calder´on-Zygmund operator. We refer to Section 2.1 for a concrete example in the Bessel settingintroduced and studied by Muckenhoupt–Stein [MuSt] .We also remark that the choice of κ is flexible, but dictated by the smoothness of the Calder´on-Zygmundkernel. If one has a different kernel possessing a different smoothness, but satisfies the appropriate A ,testing and pivotal conditions, then one can have a version of Theorem 2. The choice of κ can be dictatedby the particular example at hand. It is immediate that the testing conditions are necessary and that T . N . The forward condition followsby testing (1.7) on an indicator function of a cube and restricting the region of integration. The dualcondition follows by testing the dual inequality (1.7) (obtained by interchanging the roles of u and v ) onthe indicator of a cube and then again restricting the integration. In the remainder of the paper we addresshow to show that these testing conditions are sufficient to prove (1.7) under the additional A and pivotalhypothesis. In the course of the proof we will also demonstrate that N . A + T + V . Throughout thepaper, we use the notation X . Y to denote that there is an absolute constant C so that X ≤ CY , where C may change from one occurrence to another. If we write X ≈ Y , then we mean that X . Y and Y . X .And, := means equal by definition.1.1. Extension to Hilbert space valued operators.
Following [Ste, Chapter II, Section 5], we considertwo Hilbert spaces H and H , and replace the scalar-valued expressions in the definition of singular integral T f ( x ) = Z X K ( x, y ) f ( y ) dµ ( y ) , with the appropriate Hilbert space valued expressions, namely with f : X → H and K : X × X → B ( H , H ), so that T f : X → H . Here B ( H , H ) is the Banach space of bounded linear operators L : H → H equipped with the usual operator norm. We refer to such an operator T as an H → H Calder´on–Zygmund operator if its kernel satisfies the usual size and smoothness conditions | K ( x, y ) | B ( H , H ) ≤ C CZ V ( x, y ) − , (1.12) | K ( x, y ) − K ( x ′ , y ) | B ( X × X,B ( H , H )) ≤ C CZ (cid:18) d ( x, x ′ ) d ( x, y ) (cid:19) κ V ( x, y ) − , d ( x, x ′ ) d ( x, y ) ≤ A , and if T is bounded from unweighted L H to unweighted L H . In the Appendix, Section 9, to this paperwe will fix two separable Hilbert spaces H and H , and describe in detail the definition and interpretationof standard fractional singular integrals, the weighted norm inequality, Poisson integrals and Muckenhouptconditions, Haar bases and pivotal conditions, which are for the most part routine. Theorem 4.
Let H and H be separable Hilbert spaces. Let T be a Calder´on–Zygmund operator taking L H to L H . Let u and v be two locally finite positive Borel measures on a space of homogeneous type ( X, d, µ ) .Suppose that u ( { x } ) · v ( { x } ) = 0 for x ∈ X . Suppose the above A and pivotal conditions hold. Suppose thefollowing testing conditions hold: for every cube Q ⊂ X , we have the following testing conditions, with Q taken as the indicator of Q : k T ( e Q u ) k L H ( v ) ≤ T k Q k L ( u ) , for all unit vectors e in H k T ∗ ( e Q v ) k L H ( u ) ≤ T k Q k L ( v ) , for all unit vectors e in H . Then there holds N . A + T + V . To see how this theorem follows from the scalar-valued Theorem 2, consider the scalar operators T e , e associated with T for every pair of unit vectors ( e , e ) ∈ H unit1 × H unit2 whose kernels K e , e ( x, y ) are givenby K e , e ( x, y ) := h K ( x, y ) e , e i H . It is easy to see that k T k L H ( u ) → L H ( v ) = sup ( e , e ) ∈H unit1 ×H unit2 k T e , e k L ( u ) → L ( v ) , with a similar equality for the testing conditions. Theorem 4 now follows immediately from Theorem 2.2. Applications of the Main Theorem
In this section we provide several typical examples of Calder´on–Zygmund operators arising from differentbackgrounds (including several complex variables, stratified Lie groups, and differential equations), whichare not the standard Euclidean setting, but fall into the scope of spaces of homogeneous type.2.1.
Bessel Riesz transforms.
As an application, we have a two-weight inequality for the Bessel Riesztransform, which is a Calder´on–Zygmund operator [MuSt]. In 1965, Muckenhoupt and Stein in [MuSt]introduced a notion of conjugacy associated with this Bessel operator ∆ λ , λ >
0, which is defined by∆ λ f ( x ) := − d dx f ( x ) − λx ddx f ( x ) , x > . They developed a theory in the setting of ∆ λ which parallels the classical one associated to standard Lapla-cian. For p ∈ [1 , ∞ ), R + := (0 , ∞ ) and dm λ ( x ) := x λ dx results on L p ( R + , dm λ )-boundedness of conjugatefunctions and fractional integrals associated with ∆ λ were obtained. Since then, many problems based onthe Bessel context were studied; see, for example, [AnKe, BeRuFaRo, BeHaNoVi, Ker, Vil]. In particular,the properties and L p boundedness (1 < p < ∞ ) of Riesz transforms R ∆ λ f := ∂ x (∆ λ ) − f related to ∆ λ have been studied extensively (see for example [AnKe, BeRuFaRo, BeFaBuMaTo, MuSt, Vil]).We recall that there is a standard Poisson integral in the Bessel setting. Let n P [ λ ] t o t> be the Poissonsemigroup n e − t √ ∆ λ o t> defined by P [ λ ] t f ( x ) := Z ∞ P [ λ ] t ( x, y ) f ( y ) y λ dy, where P [ λ ] t ( x, y ) = Z ∞ e − tz ( xz ) − λ + J λ − ( xz )( yz ) − λ + J λ − ( yz ) z λ dz and J ν is the Bessel function of the first kind and of order ν . Weinstein [Wei] established the followingformula for P [ λ ] t ( x, y ): t, x, y ∈ R + ,(2.1) P [ λ ] t ( x, y ) = 2 λtπ Z π (sin θ ) λ − ( x + y + t − xy cos θ ) λ +1 dθ. The two weight inequality for this Poisson operator was just established recently in [LiWi], that is, for ameasure µ on R , + := (0 , ∞ ) × (0 , ∞ ) and σ on R + : k P [ λ ] σ ( f ) k L ( R , + ; µ ) . k f k L ( R + ; σ ) , if and only if testing conditions hold for the Poisson operator and its adjoint. However, this two weightPoisson inequality does not relate directly to the two weight inequality for R ∆ λ . We have to link it to thePoisson type condition introduced in (1.8). X. T. DUONG, J. LI, E. SAWYER, M. N. VEMPATI, B. D. WICK, AND D. YANG
Bergman Projection.
As another application we look at the Bergman projection. Let D := { z ∈ C n : | z | < } be the unit ball in C n ; dµ a ( ζ ) := (1 − | ζ | ) a − dµ ( ζ ) where a > µ is Lebesgue measureon C n . Let us denote by L p ( dµ a ) the Lebesgue space related to µ a , 1 ≤ p ≤ ∞ . The Bergman projection T a f for f ∈ L ( dµ a ) is given by T a f ( z ) := Z D f ( ζ )(1 − z · ζ ) n + a dµ ( ζ ) . Here we have z · ζ := z ζ + · · · + z n ζ n where z := ( z , . . . , z n ) and ζ := ( ζ , . . . , ζ n ). The operator T a extendscontinuously on L p ( dµ a ) for 1 < p < ∞ and weakly continuously on L ( dµ a ). If D is provided with thepseudo-distance d then the triple ( D , d, µ a ) is a space of homogeneous type. Note that K a ( z, ζ ) := − z · ζ ) n + a is the kernel associated to the operator T a . We can observe that K a ( z, ζ ) satisfies the following smoothnessand size estimates: there are constants β, c B such that(2.2) | K a ( z, ζ ) − K a ( z, ζ ) | . [ d ( ζ, ζ )] β [ d ( z, ζ )] n + a + β for z, ζ, ζ such that d ( z, ζ ) > c B d ( ζ, ζ ). Here the pseudo-distance d is defined by d ( z, ζ ) := || z | − | ζ || + (cid:12)(cid:12)(cid:12)(cid:12) − z · ζ | z || ζ | (cid:12)(cid:12)(cid:12)(cid:12) . Also the following size estimate holds: for z, ζ in D with z = ζ , we have(2.3) | K a ( z, ζ ) | . d ( z, ζ ) n + a . So T a is a singular integral operator on ( D , d, µ a ). Using Theorem 2 we deduce a two weight inequalityfor the Bergman projection using A , testing conditions and the pivotal condition associated with the kernel K a .2.3. The Szeg˝o Projection on a Family of Unbounded Weakly Pseudoconvex Domains.
Recallthe family of weakly pseudoconvex domains { Ω k } ∞ k =1 defined in Greiner and Stein [GrSt] byΩ k := n ( z , z ) ∈ C : Im z > | z | k o ,∂ Ω k := n ( z , z ) ∈ C : Im z = | z | k o , which are naturally parameterized by z and Re z . We consider the points ζ, ω, ν in ∂ Ω k given by ζ := ( z , Re z ) := ( z, t ) , z = z ∈ C and t ∈ R , ω := ( w , Re w ) := ( w, s ) , w = w ∈ C and s ∈ R , ν := ( u , Re u ) := ( u, r ) , u = u ∈ C and r ∈ R .The Szeg˝o projection S on Ω k is the orthogonal projection from L ( ∂ Ω k ) to the Hardy space H (Ω k )of holomorphic functions on Ω k with L boundary values. The Szeg¨o kernel S ( ζ, ω ) is the unique kernelsatisfying S f ( ζ ) := Z ∂ Ω k S ( ζ, ω ) f ( ω ) dV ( ω ) , where dV ( ω ) := dV ( x, y, s ) = dxdyds with ω := ( w, s ) = ( x + iy, s ) is Lebesgue measure on the parameterspace R . Greiner and Stein [GrSt] have computed the Szeg¨o kernel with Lebesgue measure on the parameterspace via the formula S ( ζ, ω ) := 14 π i s − t ] + | z | k + | w | k µ + η ! k − z w − × i s − t ] + | z | k + | w | k µ + η ! k − , where µ := Im z − | z | k and η := Im w − | w | k . WO WEIGHT INEQUALITY 7
In [Dia], Diaz defined and analyzed a pseudometric d ( ζ, ω ), globally suited to the complex geometry of ∂ Ω k , which was arrived at by a study of the Szeg¨o kernel. This allows the treatment of the Szeg¨o kernel asa singular integral kernel: d ( ζ, ω ) := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i s − t ] + | z | k + | w | k µ + η ! k − z w (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Then the pseudometric balls are defined as B ζ ( δ ) = B dζ ( δ ) := { ω ∈ ∂ Ω k : d ( ζ, ω ) < δ } and the volume ofthe associated ball is V ( B ζ ( δ )) = 4 πδ sin k − (cid:0) πk (cid:1) | z | k − δ + 12 δ k ! , and it is shown that this volume measure is doubling. Thus S is a standard Calder´on–Zygmund operator inthe setting of the space of homogeneous type ( ∂ Ω k , d, V ). We can again deduce a two-weight inequality inthis setting from Theorem 2.2.4. Riesz Transforms Associated with the sub-Laplacian on Stratified Nilpotent Lie Groups.
Recall that a connected, simply connected nilpotent Lie group G is said to be stratified if its left-invariantLie algebra g (which is assumed real and of finite dimension) admits a direct sum decomposition g = ⊕ ki =1 V i , where [ V , V i ] = V i +1 for i ≤ k − . One identifies G and g via the exponential map exp : g → G , which is a diffeomorphism. We fix once and forall a (bi-invariant) Haar measure dg on G (which is just the lift of Lebesgue measure on g via exp). Thereis a natural family of dilations on g defined for r > δ r k X i =1 v i ! = k X i =1 r i v i , with v i ∈ V i . This permits the definition of a dilation on G , which we continue to denote by δ r . We choose a basis { X , ..., X n } for V and consider the sub-Laplacian △ := P nj =1 X j . Observe that X j , 1 ≤ j ≤ n , ishomogeneous of degree 1, and that △ is homogeneous of degree 2, with respect to dilations in the sense that X j ( f ◦ δ r ) = r ( X j f ) ◦ δ r , 1 ≤ j ≤ n , and δ r ◦ △ ◦ δ r = r △ for all r > Q denote the homogeneous dimension of G , namely Q = P ki =1 i dim V i . Let p h for h > e h △ on G . For convenience we set p h ( g ) = p h ( g, o ), which means that weidentify the integral kernel with the convolution kernel, and we set p ( g ) = p ( g ).Recall that (c.f. Folland and Stein [FoSt]) p h ( g ) = h − Q p (cid:16) δ √ h ( g ) (cid:17) for all h > g ∈ G . The kernel ofthe j th Riesz transform R j = X j ( −△ ) − , i ≤ j ≤ n , is written simply as K j ( g, g ′ ) = K j (cid:16) ( g ′ ) − g (cid:17) where K j ( g ) = 1 √ π Z ∞ h − X j p h ( g ) dh = 1 √ π Z ∞ h − Q − ( X j p ) (cid:16) δ √ h g (cid:17) dh. The standard metric d on G is defined as d ( g, g ′ ) := ρ (cid:16) ( g ′ ) − g (cid:17) where ρ is the homogeneous norm on G ([FoSt, Chapter 1, Section A]). The measure dg is then a doubling measure. It is well known that S is astandard Calder´on–Zygmund operator in the setting of the space of homogeneous type ( G , d, dg ), and oncemore we can deduce a two weight inequality from Theorem 2.2.5. Area Functions.
In the Euclidean homogeneous space ( R n , |·| , dx ), the Littlewood-Paley g -functionand the Lusin area function are both examples of H → H Calder´on–Zygmund operators with H = C .Indeed they are given by g ( f ) ( x ) = (cid:18)Z ∞ |∇ ( P t ∗ f ) ( x ) | tdt (cid:19) = | Gf ( x ) | G n , S f ( x ) = (cid:18)Z Z Γ |∇ ( P t ∗ f ) ( x − y ) | dtdyt n − (cid:19) = | Sf ( x ) | H n +1 , X. T. DUONG, J. LI, E. SAWYER, M. N. VEMPATI, B. D. WICK, AND D. YANG where Gf ( x ) ( t ) := ∇ ( P t ∗ f ) ( x ) , t ∈ (0 , ∞ ) , (2.4) Sf ( x ) ( t, y ) := ∇ ( P t ∗ f ) ( x − y ) , ( t, y ) ∈ Γ , and G n and H n +1 are Hilbert spaces with norms | g | G n := sZ ∞ | g ( t ) | tdt and | h | H n +1 := sZ Z Γ | h ( t, y ) | t − n dtdy < ∞ , and Γ is a fixed cone with vertex at the origin in R n +1 opening upward into the upper half space R n +1+ .In a general space of homogeneous type ( X, d, µ ), the notion of a Poisson kernel can be approached inseveral different ways. First, if D is a dyadic grid on X and { h Q : Q ∈ D} is the collection of Haar waveletsconstructed in [KLPW], then with the one-dimensional projection △ Q defined by △ Q f := h f, h Q i µ h Q , wehave(2.5) f = X Q ∈D △ Q f = X Q ∈D △ Q △ Q f, f ∈ L ( µ ) , where the orthonormal property and the self-adjointness of Haar projections gives the second sum P Q ∈D △ Q △ Q f , usually called a Calder´on reproducing formula for f . We can then define a discrete Poisson operator by P k f := X Q ∈D : l ( Q ) ≥ k △ Q △ Q f. However, the kernel of P k is not Lipschitz continuous, and thus fails to be a Calder´on–Zygmund kernel asdefined above. The following smoother construction of a Poisson kernel had been introduced much earlierby R. Coifman, see [DaJoSe] where this first appears.2.5.1. Coifman’s Construction of a Calder´on Reproducing Formula.
Start with a smooth function h : (0 , ∞ ) → (0 , ∞ ) that equals 1 on (cid:0) , (cid:3) and 0 on [2 , ∞ ). Let T k be the operator with kernel 2 k h (cid:0) k d ( x, y ) (cid:1) so that C ≤ T k ≤ C for some positive constant C . Let M k be the operator of multiplication by T k , and let W k be the operator of multiplication by T k (cid:16) Tk (cid:17) . Then the operator S k := M k T k W k T k M k , has kernel S k ( x, y ) that satisfies S k ( x, y ) = 0 if d ( x, y ) ≥ C k and k S k k ∞ ≤ C k , (2.6) | S k ( x, y ) − S k ( x ′ , y ) | + | S k ( y, x ) − S k ( y, x ′ ) | ≤ C k (cid:2) k d ( x, x ′ ) (cid:3) ε , Z X S k ( x, y ) dµ ( y ) := 1 and Z X S k ( x, y ) dµ ( x ) := 1 . Thus with D k := S k +1 − S k we have f = X k ∈ Z D k f, f ∈ L ( µ ) . Next fix N ∈ N so large that if T N := P k ∈ Z D Nk D k where D Nk := P | j |≤ N D k + j , then T N is invertible on L ( µ ). It follows that f = T N T − N f = X k ∈ Z D Nk D k T − N f = X k ∈ Z E k e E k f, f ∈ L ( µ ) , where E k := D Nk and e E k := D k T − N . This latter formula is usually called a Calder´on reproducing formula,and substitutes for the orthonormal wavelet formula (2.5). WO WEIGHT INEQUALITY 9
Discrete g -functions and Area functions. The function g ( f ) ( x ) := ∞ X k = −∞ | D k f ( x ) | = | Gf ( x ) | ℓ ( Z ) , Gf ( x ) := { D k f ( x ) } k ∈ Z plays the role of a g -function in a homogeneous space X . We could also replace D k by E k := D Nk in thisformula giving an alternative g -function. Property (2.6), together with an application of the Cotlar-Steinlemma, shows that in either case g is a Hilbert space valued Calder´on–Zygmund operator on X . The function S f ( x ) := ∞ X k = −∞ (cid:12)(cid:12)(cid:12) e E k f ( x ) (cid:12)(cid:12)(cid:12) = | Sf ( x ) | ℓ ( Z ) , Sf ( x ) := n e E k f ( x ) o k ∈ Z , plays the role of an area function in X , and [HaSa, Theorem 3.4] shows that the kernel of S satisfies (1.12),and the boundedness on L ( µ ) is proved in [DaJoSe]. Thus S is a Hilbert space valued Calder´on–Zygmundoperator on X , and Theorem 4 yields the following two weight norm inequality - a stronger result wasobtained in Euclidean space by Lacey and Li [LaLi], stronger in the sense that neither the dual testingcondition nor the dual pivotal condition was needed and they didn’t need to test over all unit vectors. Theorem 5.
Let u and v be two locally finite positive Borel measures on X . Suppose that u ( { x } ) · v ( { x } ) = 0 for x ∈ X . Suppose the above A and pivotal conditions hold. Suppose also that the following testingconditions hold for the area function S as defined above: for every ball B ⊂ X , we have the following testingconditions, with Q taken as the indicator of Q : k S ( u Q ) k L H ( v ) ≤ T k Q k L ( u ) , k S ∗ ( v e Q ) k L ( u ) ≤ T k Q k L ( v ) for all unit vectors e in H . Then there holds N . A + T + V . Classical Riesz transforms.
As an application to known work we can look at the two weight inequalityfor the Hilbert Transform. We define our Hilbert operator as below for a signed measure v on R : Hv ( x ) := p.v. Z R x − y v ( dy ) . For two weights u, v , the inequality we are interested in is k H ( uf ) k L ( v ) . k f k L ( u ) . Along with A for the pair of weights u, v given we have the following testing conditions holding uniformlyover intervals I : Z I | H (1 I u ) | v ( dx ) ≤ H u ( I ) , Z I | H (1 I v ) | u ( dx ) ≤ H v ( I ) . Here H denotes the smallest constants for which these inequalities are true uniformly over all intervals I .In beautiful series of papers, Nazarov, Treil and Volberg, [NTV1, NTV2, NTV3] have developed a sophis-ticated approach towards proving the sufficiency of these testing conditions combined with the improvementof the two weight A condition. The improvement is described below using a variant of Poisson integral. Foran interval I and measure v , P ( I, v ) := Z R | I | ( | I | + dist( x, I )) ω ( dx ) , sup I P ( I, v ) · P ( I, u ) := A < ∞ . We will refer to the last line above as the A condition. In [NTV3, Theorem 2.2] Nazarov, Treil and Volbergproved the sufficiency of the A and testing conditions above for the two weight inequality of the Hilberttransform in the presence of the pivotal condition given by ∞ X r =1 v ( I r ) P ( I r , I u ) ≤ V u ( I ) , and its dual, where the inequality is required to hold for all intervals I and decompositions { I r : r ≥ } of I into disjoint intervals I r ( I . We have taken inspiration from this proof and condition in providing theproof of our main result. In [Lac, Lac1, LaSaUr, LaSaShUr] the pivotal condition was removed as a sidecondition and replaced with an energy condition, ultimately yielding a characterization of the two weightinequality for the Hilbert transform.In higher dimension we look at two weight inequalities for Riesz transforms. Earlier work appears in[LaWi, SaShUr] where certain two weight inequalities for the Riesz transforms (and fractional versions) werestudied. Namely for two weights, nonnegative locally finite Borel measures u, v on R n , we are interested inthe following inequality for the d dimensional Riesz transform (cid:13)(cid:13)(cid:13)(cid:13)Z R n f ( y ) x − y | x − y | d +1 u ( dy ) (cid:13)(cid:13)(cid:13)(cid:13) L ( v ) ≤ N k f k L ( u ) . Here we take 0 < d = n − ≤ n and N is the best constant in the inequality above.The A type condition is expressed in terms of a Poisson type operator. For a cube Q ⊂ R n , we take P ( u, Q ) := Z R n | Q | d/n | Q | d/n + dist( x, Q ) d u ( dx ) . Using the A , testing conditions and pivotal condition we can obtain sufficient conditions for the two weightinequality for the d dimensional Riesz transform by the above Theorem 2 on ( R n , dx, | · | n ), viewed as spaceof homogeneous type where | · | n is a standard metric on R n .2.7. Riesz Transform Associated with Certain Schr¨odinger Operators.
Consider L := − ∆ + µ ,which is a Schr¨odinger operator with a non-negative Radon measure µ on R n for n ≥
3. We assume that µ satisfies the following conditions: there exists a positive constant σ ∈ (1 , ∞ ) such that µ ( B ( x, r )) . (cid:16) rR (cid:17) n − σ µ ( B ( x, R ))(2.7)and µ ( B ( x, r )) . (cid:8) µ ( B ( x, r )) + r n − (cid:9) (2.8)for all x ∈ R n and 0 < r < R , where B ( x, r ) denotes the open ball centered at x with radius r . As pointed in[Shen], condition (2.7) may be regarded as scale-invariant Kato-condition, and (2.8) says that the measure µ is doubling on balls satisfying µ ( B ( x, r )) ≥ cr n − . We will also assume that µ . When dµ = V ( x ) dx and V ≥ RH ) n , i.e.,(2.9) | B ( x, r ) | Z B ( x,r ) V ( y ) n dy ! /n ≤ C | B ( x, r ) | Z B ( x,r ) V ( y ) dy, then µ satisfies the conditions (2.7) and (2.8) for some σ > . However, in general, measures which satisfy(2.7) and (2.8) need not be absolutely continuous with respect to the Lebesgue measure on R n . For instance,when dµ = dσ ( x , x ) dx · · · dx n , where σ is a doubling measure on R , then µ satisfies (2.7) and (2.8) forsome σ > . It is well-known that the Riesz transform ∇ L − is bounded on L ( R n ). Moreover, let K ( x, y ) be thekernel of ∇ L − , the Riesz transforms associated to L . Then it was proved in [Shen] that | K ( x, y ) | . | x − y | n (2.10)and that for | x − x ′ | ≤ | x − y | , | K ( x, y ) − K ( x ′ , y ) | . (cid:16) | x − x ′ || x − y | (cid:17) σ − | x − y | n . (2.11)Here the implicit constants are independent of x and y .3. Preliminaries on Spaces of Homogeneous Type
Let (
X, d, µ ) be a space of homogeneous type as in Section 1.
WO WEIGHT INEQUALITY 11
A System of Dyadic Cubes.
We will recall from [HyKa] a construction of dyadic cubes, which is adeep elaboration of work by M. Christ [Chr], as well as that of Sawyer–Wheeden [SaWh]. We summarizethe dyadic construction of random dyadic systems from [HyMa] and [HyKa] in the following theorem. Firstwe need to define an appropriate notion of ‘reference points’ or ‘lattice points’ in X . Definition 6.
A set of points (cid:8) x kα (cid:9) k ∈ Z , α ∈A k ⊂ X is said to be a set of reference points if there existconstants < c ≤ C < ∞ and < δ < such that A C δ ≤ c and d (cid:0) x kα , x kβ (cid:1) ≥ c δ k , α = β, min α d (cid:0) x, x kα (cid:1) ≤ C δ k , x ∈ X. The following construction is from [HyKa, Theorems 5.1 and 5.6].
Theorem 7.
Given a set of reference points (cid:8) x kα (cid:9) k ∈ Z , α ∈A k with parameters c , C and δ , and sufficientlysmall < δ < e.g. A δ ≤ , there exists a probability space (Ω , P ) such that every ω ∈ Ω defines adyadic system D ( ω ) := (cid:8) Q kα ( ω ) (cid:9) k ∈ Z , α ∈A k related to new dyadic points (cid:8) z kα ( ω ) (cid:9) k ∈ Z , α ∈A k with the followinggeometric properties: for some c and C depending on c , C , A and δ , (1) If ℓ ≥ k , then either Q ℓβ ( ω ) ⊂ Q kα ( ω ) or Q ℓβ ( ω ) ∩ Q kα ( ω ) = ∅ ; (2) X = [ α Q kα ( ω ) for all k ∈ Z ; (3) B (cid:16) z kα ( ω ) , c δ k (cid:17) ⊂ Q kα ( ω ) ⊂ B (cid:16) z kα ( ω ) , C δ k (cid:17) =: B (cid:0) Q kα ( ω ) (cid:1) ; (4) If ℓ ≥ k and Q ℓβ ( ω ) ⊂ Q kα ( ω ) then B (cid:16) Q ℓβ ( ω ) (cid:17) ⊂ B (cid:0) Q kα ( ω ) (cid:1) ,and the following probabilistic property: There are positive constants C , η > such that for every x ∈ X, τ > and k ∈ Z , (3.1) P ( ω ∈ Ω : x ∈ [ α ∂ τδ k Q kα ( ω ) )! ≤ C τ η . We also have the following containment property: (3.2) B ( x kα , c δ k ) ⊆ Q kα ⊆ B ( x kα , C δ k ) =: B ( Q kα ); Definition 8.
We say that D ( ω ) = (cid:8) Q kα ( ω ) (cid:9) k ∈ Z , α ∈A k is a system of dyadic cubes if (1)—(4) hold inTheorem 7. Given a dyadic cube Q kα ( ω ) , we denote the quantity δ k by l ( Q kα ) , by analogy with the side lengthof a Euclidean cube. An Explicit Haar Basis on Spaces of Homogeneous Type.
Next we recall the explicit construc-tion in [KLPW] of a Haar basis { h ǫQ : Q ∈ D , ǫ = 1 , . . . , M Q − } for L p ( X, µ ), 1 < p < ∞ , associated to thedyadic cubes Q ∈ D as follows. Here M Q := H ( Q ) = { R ∈ D k +1 : R ⊆ Q } denotes the number of dyadicsub-cubes (which we will refer to as the “children”) the cube Q ∈ D k has; namely H ( Q ) is the collection ofdyadic children of Q . It is known in [KLPW] that sup Q ∈D M Q < ∞ . Theorem 9.
Let ( X, d, µ ) be a space of homogeneous type and suppose µ is a positive locally finite Borelmeasure on X . For < p < ∞ , for each f ∈ L p ( X, µ ) , we have f ( x ) = X Q ∈D M Q − X ǫ =1 h f, h ǫQ i µ h ǫQ ( x ) , where the sum converges (unconditionally) both in the L p ( X, µ ) -norm and pointwise µ -almost everywhere. The following theorem collects several basic properties of the functions h ǫQ . Theorem 10.
The Haar functions h ǫQ , Q ∈ D , ǫ = 1 , . . . , M Q − , have the following properties: (1) h ǫQ is a simple Borel-measurable real function on X ; (2) h ǫQ is supported on Q ; (3) h ǫQ is constant on each R ∈ H ( Q ) ; (4) R Q h ǫQ dµ = 0 (cancellation); (5) h h ǫQ , h ǫ ′ Q i = 0 for ǫ = ǫ ′ , ǫ , ǫ ′ ∈ { , . . . , M Q − } ; (6) the collection (cid:8) µ ( Q ) − / Q (cid:9) ∪ (cid:8) h ǫQ : ǫ = 1 , . . . , M Q − (cid:9) is an orthogonal basis for the vector space V ( Q ) of all functions on Q that are constant on each sub-cube R ∈ H ( Q ) ; (7) if h ǫQ = 0 then k h ǫQ k L p ( X,µ ) ≈ µ ( Q ) p − for ≤ p ≤ ∞ ;(8) k h ǫQ k L ( X,µ ) · k h ǫQ k L ∞ ( X,µ ) ≈ . We denote h Q := µ ( Q ) − / Q , which is a non-cancellative Haar function. Moreover, the martingaleassociated with the Haar functions are as follows: for Q ∈ D k , E Q f := h f, h Q i µ h Q and D Q f := M Q − X ǫ =1 D ǫQ f, where D ǫQ f := h f, h ǫQ i µ h ǫQ is the martingale operator associated with the ǫ -th subcube of Q . Also we have E k f = X Q ∈D k E Q f and D k f = E k +1 f − E k f. Hence, based on the construction of Haar system { h ǫQ } in [KLPW] we obtain that for each R ∈ D and η = 1 , . . . , M R − X Q : R ⊂ Q M Q − X ǫ =1 h f, h ǫQ i µ h ǫQ h ηR = X Q : R ⊂ Q D Q f · h ηR = E R f · h ηR = h f, h R i µ h R h ηR . The Carleson Embedding Theorem in Spaces of Homogeneous Type.
Now we will describethe familiar Carleson Embedding theorem, which will be crucial in the control of certain paraproduct terms.
Theorem 11.
Fix a weight u and consider nonnegative constants { a Q : Q ∈ D} . The following twoinequalities are equivalent: X Q ∈D a Q | E uQ f | ≤ c k f k L ( u ) , where E uQ f := h f, h Q i u h Q ; X Q ∈D : Q ⊂ S a Q ≤ Cu ( S ) . Taking c and C to be the best constants in these inequalities, we have c ≈ C . First Reduction in the Proof of the Two-Weight Inequality
We now begin to prove the two weight inequality in our setting. In this section we reduce to showing thatit suffices to prove Theorem 2 under the hypothesis that f and g are ‘good’ functions (as explained below).Let f ∈ L ( u ) and g ∈ L ( v ) be two functions. Without loss of generality, we can assume that thesetwo functions have compact support. Moreover, it is sufficient to assume that f and g are supported on acommon (large) cube Q , see for example [Vol]. From Theorem 9 we have f ( x ) = X Q ∈D M Q − X ǫ =1 h f, h ǫQ i u h ǫQ ( x ) . We write this sum in two parts as follows: f ( x ) = X Q ⊂ Q M Q − X ǫ =1 h f, h ǫQ i u h ǫQ ( x ) + X Q : Q ⊂ Q M Q − X ǫ =1 h f, h ǫQ i u h ǫQ ( x ) . Based on Theorem 10 and (3.3) we have that E Q f · h ηQ = h f, h Q i u h Q h ηQ = X Q : Q ⊂ Q M Q − X ǫ =1 h f, h ǫQ i u h ǫQ ,f = E Q f + X Q ⊂ Q M Q − X ǫ =1 h f, h ǫQ i u h ǫQ =: f + f . WO WEIGHT INEQUALITY 13
Similarly we write for g ∈ L ( v ): g = E Q g + X Q ⊂ Q M Q − X ǫ =1 h g, h ǫQ i v h ǫQ =: g + g . Here we have k f k L ( u ) = k f k L ( u ) + k f k L ( u ) ; similar formulas hold for the function g .Let T be a Calder´on–Zygmund Operator on ( X, d, µ ). Given these decompositions of f and g , let us beginthe proof by looking at their inner product h T ( uf ) , g i v = h T ( uf ) , g i v + h T ( uf ) , g i v + h T ( uf ) , g i v + h T ( uf ) , g i v := I + I + I + I . It is enough to obtain good estimates on each of the I j above. The first three terms are easy to control justusing the testing condition assumed on the operator T , the last term will then require substantial analysis.We now show how to control I , I and I just using the testing condition. First observe that I = h T ( uf ) , g i v = R Q f du · R Q gdvu ( Q ) · v ( Q ) h T ( u Q ) , Q i v . By Cauchy-Schwarz, applied to the function f and the function g , and in the inner product, and then usingthe testing conditions assumed on the operator T we have: | I | ≤ T k f k L ( u ) k g k L ( v ) . The terms I and I are symmetric I = h T ( uf ) , g i v = R Q f duu ( Q ) h T ( u Q ) , g i v . Using Cauchy-Schwarz, the testing conditions and the fact that k g k L ( v ) ≤ k g k L ( v ) we get the following: | I | ≤ T k f k L ( u ) k g k L ( v ) . An identical argument works for I .By the above it suffices to prove |h T ( uf ) , g i v | . k f k L ( u ) k g k L ( v ) when f and g have compact support in Q and R Q f du = 0 and R Q gdv = 0. We will now decompose theinner product h T ( uf ) , g i v using good-bad decomposition. Remark 12.
In order to use surgery to remove weak boundedness, we will need to work in the world of twoindependent systems of random grids.
The Good and Bad Parts of Functions.
We use the good-bad decomposition of test functions tosimplify the proof even further. Fix a number ǫ , 0 < ǫ <
1. Later the choice of ǫ will be dictated by theCalder´on–Zygmund properties of the operator T and the underlying measure µ . Also fix a sufficiently largeinteger r . The choice of r will be made in this section. Finally, we consider two grids D = (cid:8) Q kα ( ω ) (cid:9) k,α and D ′ = (cid:8) Q kα ( ω ′ ) (cid:9) k,α for ω, ω ′ ∈ Ω. Definition 13.
Take a dyadic cube Q ∈ D . We say that Q is r -good in D ′ for an integer r , if for every cube Q ∈ D ′ such that if δ k ≤ δ r δ n with k ≥ n + r , then either dist( Q, Q ) ≥ δ kǫ δ n (1 − ǫ ) or dist( Q, X \ Q ) ≥ δ kǫ δ n (1 − ǫ ) . Above we are letting l ( Q ) := δ k and l ( Q ) := δ n . If Q is not r -good we call it r -bad. We can now decompose f into good and bad parts as below. f = f good + f bad f bad := X Q ∈D ,Q is bad ∆ Q f. Theorem 14. [Vol, Theorem 17.1]
There holds on ( X, d, µ ) for f ∈ L ( u ) E ( k f bad k L ( u ) ) ≤ ε ( r ) k f k L ( u ) where ε ( r ) → as r → ∞ . A similar estimate holds for g bad ∈ L ( v ) . Proposition 15.
Consider the decompositions of f and g into bad and good parts on ( X, d, µ ) , where thesupport cubes of the Haar projections of f are good with respect to G , and the support cubes of the Haarprojections of g are good with respect to D . Let u and v be pairs of weights and suppose there holds uniformlyover all dyadic grids D and G for some finite constant C (4.1) E ( |h T ( uf good ) , g good i v | ) ≤ C k f k L ( u ) k g k L ( v ) , where E refers to expectation over the product probability space Ω × Ω . Then |h T ( uf ) , g i v | ≤ C k f k L ( u ) k g k L ( v ) , that is k T k L ( u ) → L ( v ) ≤ C. Proof.
Note that h T ( uf ) , g i v = h T ( uf good ) , g good i v + h T ( uf good ) , g bad i v + h T ( uf bad ) , g i v . So E ( |h T ( uf ) , g i v | ) ≤ E ( |h T ( uf good ) , g good i v | ) + E ( |h T ( uf good ) , g bad i v | ) + E ( |h T ( uf bad ) , g i v | ) . Using (4.1) and by Theorem 14 we have |h T ( uf ) , g i v | ≤ C k f k L ( u ) k g k L ( v ) + 2 k T k L ( u ) → L ( v ) ε ( r ) k f k L ( u ) k g k L ( v ) . Notice that k T f k L ( u ) → L ( v ) = sup |h T f, g i v | . Choose f , g and r sufficiently large such that(4.2) |h T ( uf ) , g i v | ≥ k T k L ( u ) → L ( v ) k f k L ( u ) k g k L ( v ) and ε ( r ) < . Then by absorbing the second term in (4.2) to left hand side, we get k T k L ( u ) → L ( v ) ≤ C. (cid:3) Definition 16.
We fix r > with ǫ ( r ) < and throughout the paper and abbreviate r -good as simply good. The upshot of the above is that if we manage to prove that for all f ∈ L ( u ) and g ∈ L ( v ) we have E ( |h T ( uf good ) , g good i v | ) ≤ C k f k L ( u ) k g k L ( v ) then we obtain (4.1). The remainder of the paper is devoted to proving (4.1). We remind the reader that E refers to expectation taken over the product probability space Ω × Ω. In fact, we will prove that (4.1) holdsfor all dyadic grids and so that statement regarding the expectation will then follow.5.
Main Decomposition
Fix a cube Q for the rest of the paper; this is the support of the functions f and g . Our goal is todemonstrate (4.1). This will require several additional reductions.Throughout the rest of this paper we assume the A conditions, the testing conditions and the pivotalconditions.5.1. Global to Local Reduction.
We will now try to control the bilinear form: h T ( uf ) , g i v = X Q ∈D ,S ∈G M Q − X ǫ =1 M S − X k =1 h f, h ǫQ i u h T ( uh ǫQ ) , h kS i v h g, h kS i v = X Q ∈D ,S ∈G l ( Q ) ≥ l ( S ) M Q − X ǫ =1 M S − X k =1 h f, h ǫQ i u h T ( uh ǫQ ) , h kS i v h g, h kS i v + X Q ∈D ,S ∈G l ( S ) >l ( Q ) M Q − X ǫ =1 M S − X k =1 h f, h ǫQ i u h T ( uh ǫQ ) , h kS i v h g, h kS i v =: A + A . WO WEIGHT INEQUALITY 15
Set A := { ( Q, S ) ∈ D × G : l ( Q ) ≥ l ( S ) } and A := { ( Q, S ) ∈ D × G : l ( S ) > l ( Q ) } and A ij := X ( Q,S ) ∈A ij M Q − X ǫ =1 M S − X k =1 h f, h ǫQ i u h T ( uh ǫQ ) , h kS i v h g, h kS i v . Here A is complementary to the sum A . The sums are estimated symmetrically. Hence it is enough to prove(4.1) for A . We will further decompose this term into a number of other bilinear forms A ij . The superscript i denotes the generation and subscript j counts the number of decompositions. To help understand thesedecompositions we can look at the flow chart where all the terms are listed. h T ( uf ) , g i v A A A A A A A A A A A A A A A A A A A Surgery A A A VT + V Paraproducts
VVV T + V V (1) The flow chart starts at the bilinear form in (4.1).(2) The hypothesis we used in controlling each bilinear form A , T , surgery and/or V is written on theedges of the chart.(3) To control the terms A , A , A and A we use the stopping cube arguments given in Section 6.(4) The edge leading to A has been labelled paraproduct as all the estimates below that use paraproductarguments to control them.Now let us begin by proving the estimate in (4.1) for the term A . We decompose A into the followingsets. Denote by A := { ( Q, S ) ∈ A : δ r l ( Q ) ≤ l ( S ) ≤ l ( Q ) , dist( Q, S ) ≤ l ( Q ) } ; A := { ( Q, S ) ∈ A : l ( S ) ≤ l ( Q ) , dist( Q, S ) ≥ l ( Q ) } ; A := { ( Q, S ) ∈ A : l ( S ) ≤ δ r l ( Q ) , dist( Q, S ) ≤ l ( Q ) } . We will show the following estimates below: | A | . (cid:16) C τ p A + T + τ η N (cid:17) k f k L ( u ) k g k L ( v ) | A | . A k f k L ( u ) k g k L ( v ) . The term A is easily controlled using a ‘weak boundedness property’, which we recall (even though itwill not be needed): Definition 17.
Weak Boundedness Condition: For a constant C W BP > , we have C W BP as the bestconstant in the inequality (cid:12)(cid:12)(cid:12)(cid:12)Z S T ( u Q ) dv (cid:12)(cid:12)(cid:12)(cid:12) ≤ C W BP u ( Q ) v ( S ) , where Q, S are cubes such that δ r l ( Q ) ≤ l ( S ) ≤ l ( Q ) and dist( Q, S ) ≤ l ( Q ) . We can avoid the weak boundedness property if we instead use ‘surgery’ to control the average over grids in terms of only the testing and A conditions, and a small multiple of the operator norm. It is clear thatthe lemma below provides control on the term A as desired with a small multiple of the norm that can beabsorbed by choosing the parameter τ appropriately small. Lemma 18.
The following estimate holds: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E D∈ Ω X ( Q,S ) ∈A M Q − X ǫ =1 M S − X k =1 h f, h ǫQ i u h T h ǫQ , h kS i v h g, h kS i v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (cid:16) C τ p A + T + τ η N (cid:17) k f k L ( u ) k g k L ( v ) . For the proof of Lemma 18 we recall the surgery estimate (3.1) for the random dyadic systems constructedin [HyMa] and [HyKa] with parameter δ >
0. There are positive constants C , η > x ∈ X, τ > k ∈ Z ,(5.1) P ( ω ∈ Ω : x ∈ [ α ∂ τδ k Q kα ( ω ) )! ≤ C τ η . We can now prove an extension to spaces of homogeneous type of the surgery lemma of Lacey and Wick in[LaWi, Lemma 8.5], by repeating their argument with obvious modifications. In the estimate of Lemma 18,the explicit Haar functions are used in the decompositions, whereas in the surgery lemma it is convenient touse instead the associated Haar projections △ uQ f = M Q − X ǫ =1 h f, h ǫQ i u h ǫQ and △ vS g = M S − X k =1 h g, h kS i v h kS . We say that two cubes Q and S are ρ -close if δ ρ ≤ l ( Q ) l ( S ) ≤ δ − ρ and d ( Q, S ) ≤ max { l ( Q ) , l ( S ) } . Lemma 19 (Surgery Lemma) . For < τ < and sufficiently large r we have (5.2) E D∈ Ω X ( Q,S ) ∈D good ×G good Q and S are ρ -close (cid:12)(cid:12)(cid:12)(cid:10) T (cid:0) u △ uQ f (cid:1) , △ vS g (cid:11) v (cid:12)(cid:12)(cid:12) . (cid:16) C τ p A + T + τ η N T α (cid:17) k f k L ( u ) k g k L ( v ) . Note that we can choose ρ to be r so Lemma 18 follows immediately from Lemma 19 since we are summingon a larger collection of cubes and we have pulled the absolute value inside the sum. Proof.
In order to prove (5.2), we invoke the surgery estimate using (5.1). Given 0 < λ < , define S λ := { x ∈ S : dist ( x, ∂S ) > λl ( S ) } . Then we write (cid:10) T (cid:0) u △ uQ f (cid:1) , △ vS g (cid:11) v = * T u X Q ′ ∈H ( Q ) Q ′ △ uQ f , X S ′ ∈H ( J ) S ′ △ vS g + v WO WEIGHT INEQUALITY 17 = X Q ′ ∈H ( Q ) X S ′ ∈H ( S ) (cid:0) E uQ ′ △ uQ f (cid:1) h T ( u Q ′ ) , S ′ i v ( E vS ′ △ vS g ) , and so X ( Q,S ) ∈D good ×G good Q and S are ρ -close (cid:12)(cid:12)(cid:12)(cid:10) T (cid:0) u △ uQ f (cid:1) , △ vS g (cid:11) v (cid:12)(cid:12)(cid:12) ≤ X ( Q,S ) ∈D good ×G good Q and S are ρ -close X Q ′ ∈H ( S ) X S ′ ∈H ( S ) (cid:12)(cid:12)(cid:0) E uQ ′ △ uQ f (cid:1) h T ( u Q ′ ) , S ′ ∩ Q ′ i v ( E vS ′ △ vS g ) (cid:12)(cid:12) + X ( Q,S ) ∈D good ×G good Q and S are ρ -close X Q ′ ∈H ( Q ) X S ′ ∈H ( S ) (cid:12)(cid:12)(cid:0) E uQ ′ △ uQ f (cid:1) (cid:10) T ( u Q ′ ) , S ′ \ Q ′ (cid:11) v ( E vS ′ △ vS g ) (cid:12)(cid:12) =: Term + Term . Now for convenience of notation and to shorten some displays we write z X := X ( Q,S ) ∈D good ×G good Q and S are ρ -close X Q ′ ∈H ( Q ) X S ′ ∈H ( S ) , then the inequality (cid:12)(cid:12) h T ( u Q ′ ) , S ′ ∩ Q ′ i v (cid:12)(cid:12) ≤ sZ Q ′ | T ( u Q ′ ) | dv p v ( S ′ ∩ Q ′ ) ≤ T p u ( Q ′ ) p v ( S ′ )shows thatTerm ≤ T z X (cid:16)(cid:12)(cid:12) E uQ ′ △ uQ f (cid:12)(cid:12) p u ( Q ′ ) (cid:17) (cid:16) | E vS ′ △ vS g | p v ( S ′ ) (cid:17) ≤ T s z X (cid:12)(cid:12)(cid:12) E uQ ′ △ uQ f (cid:12)(cid:12)(cid:12) u ( Q ′ ) s z X | E vS ′ △ vS g | v ( S ′ ) . T k f k L ( u ) k g k L ( v ) . To control Term we further decompose S ′ \ Q ′ into small and large parts, S ′ \ Q ′ := { ( S ′ \ Q ′ ) ∩ ∂ λ Q ′ } · [ { ( S ′ \ Q ′ ) \ ∂ λ Q ′ } := E · [ F, and estimate Term accordingly,Term ≤ z X (cid:12)(cid:12)(cid:0) E uQ ′ △ uQ f (cid:1) h T ( u Q ′ ) , E i v ( E vS ′ △ vS g ) (cid:12)(cid:12) + z X (cid:12)(cid:12)(cid:0) E uQ ′ △ uQ f (cid:1) h T ( u Q ′ ) , F i v ( E vS ′ △ vS g ) (cid:12)(cid:12) := Term + Term . Now F = ( S ′ \ Q ′ ) \ ∂ λ Q ′ is contained in S ′ and has distance at least cl ( Q ) from the cube Q ′ , so we cancontrol Term by the A condition using Cauchy-Schwarz as above,Term ≤ z X (cid:12)(cid:12) E uQ ′ △ uQ f (cid:12)(cid:12) (cid:16) A p u ( Q ′ ) p v ( S ′ ) (cid:17) | E vS ′ △ vS g | . A k f k L ( u ) k g k L ( v ) . Finally, we use the operator norm to control the average of B by E D∈ Ω Term ≤ E D∈ Ω z X (cid:12)(cid:12) E uQ ′ △ uQ f (cid:12)(cid:12) (cid:16) N p u ( Q ′ ) p v (( S ′ \ Q ′ ) ∩ ∂ λ Q ′ ) (cid:17) | E vS ′ △ vS g | . N E D∈ Ω k f k L ( u ) s z X |△ vS g | v (( J ′ \ Q ′ ) ∩ ∂ λ Q ′ ) . N k f k L ( u ) vuuuuut X S ∈G good |△ vS g | E D∈ Ω X Q ∈D good , Q ′ ∈H ( Q ) Q and S are ρ -close v ( S ∩ ∂ λ Q ′ ) . Now we use (5.1) to obtain E D∈ Ω X Q ∈D good , Q ′ ∈H ( Q ) Q and S are ρ -close v ( S ∩ ∂ τ Q ′ ) . τ η v ( S ) , which altogether gives E D∈ Ω Term . N k f k L ( u ) s X S ∈G good |△ vS g | τ η v ( S ) . τ η N k f k L ( u ) k g k L ( v ) . The proof of Lemma 19 is complete. (cid:3)
The next lemma controls A . Lemma 20.
The following estimate holds: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( Q,S ) ∈A M Q − X ǫ =1 M S − X k =1 h f, h ǫQ i u h T ( uh ǫQ ) , h kS i v h g, h kS i v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . A k f k L ( u ) k g k L ( v ) . Before we proceed to prove Lemma 20, let us collect a couple of auxiliary lemmas.
Lemma 21.
Let S ⊂ Q ′ ⊂ ˆ Q be three cubes with dist( ∂Q ′ , S ) ≥ l ( S ) . Let H S be a function supported on S and with v integral zero. Then we have (5.3) |h T ( u ˆ Q \ Q ′ ) , H S i v | . k H S k L ( v ) Φ( S, ˆ Q \ Q ′ u ) . Here Φ( S, ˆ Q \ Q ′ u ) := v ( S ) K (cid:16) S, ˆ Q \ Q ′ u (cid:17) where K ( S, ˆ Q \ Q ′ u ) := Z ˆ Q \ Q ′ (cid:18) l ( S ) l ( S ) + dist( y, S ) (cid:19) κ µ ( B ( x S , l ( S ) + dist( y, S ))) du ( y ) , and κ is as in Definition 1. The L formulation of (5.3) proves useful in many estimates below, in particular in several CarlesonEmbedding Theorem estimates, Theorem 29. We will apply (5.3) in the dual formulation. Namely, we have(5.4) k T ( u ˆ Q \ Q ′ ) − E vS T (1 ˆ Q \ Q ′ u ) k L ( S,v ) . Φ( S, ˆ Q \ Q ′ u ) . Proof.
This proof uses the standard computation in the Calder´on–Zygmund theory. We use cancellation ofa function to pull additional information onto the kernel of the operator. Using Fubini and the fact that H S has v integral zero we get: |h T ( u ˆ Q \ Q ′ ) , H S i v | = (cid:12)(cid:12)(cid:12)(cid:12)Z S H S ( x ) T ( u ˆ Q \ Q ′ ( x )) dv ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z S H S ( x ) Z ˆ Q \ Q ′ K ( x, y ) du ( y ) dv ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ˆ Q \ Q ′ Z S ( K ( x, y ) − K ( x S , y )) H S ( x ) dv ( x ) du ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z ˆ Q \ Q ′ Z S (cid:18) dist( x, x S )dist( y, x S ) (cid:19) κ µ ( B ( x S , dist( x S , y ))) | H S ( x ) | dv ( x ) du ( y ) ≤ Z ˆ Q \ Q ′ (cid:18) l ( S )dist( x S , y ) (cid:19) κ µ ( B ( x S , dist( x S , y ))) Z S | H S ( x ) | dv ( x ) du ( y ) . Here x S ∈ S , is the center of S and y ∈ ˆ Q \ Q ′ and so dist( x S , y ) ≈ dist( y, S ) + l ( S ). By the doublingproperty of the measure µ we have µ ( B ( x S , dist( x S , y ))) ≈ µ ( B ( x S , l ( S ) + dist( y, S ))). These estimates canthen be used to give: |h T ( u ˆ Q \ Q ′ , H S i v | WO WEIGHT INEQUALITY 19 . Z ˆ Q \ Q ′ (cid:18) l ( S ) l ( S ) + dist( y, S ) (cid:19) κ µ ( B ( x S , l ( S ) + dist( y, S ))) du ( y ) k H S k L ( v ) v ( S ) = k H S k L ( v ) Φ( S, u ˆ Q \ Q ′ ) completing the proof. (cid:3) The next Lemma is an extension to spaces of homogenous type of the Poisson inequality in [Vol]. Thislemma plays a crucial role in obtaining geometric decay from goodness in controlling A appearing below,and the neighbor and the stopping form appearing later in Section 6. Lemma 22.
Let < λ < κn + κ . If S ⊂ Q ⊂ ˆ Q and dist( S, e ( Q )) ≥ l ( S ) λ l ( Q ) − λ where e ( S ) := ∂S ∪{ ( center of S ) } then l ( S ) σ K (cid:16) S, ˆ Q \ Q u (cid:17) ≤ l ( Q ) σ K (cid:16) Q, ˆ Q \ Q u (cid:17) . Here σ := λ ( n + κ ) − κ with κ as in Definition 1. Proof.
To begin with, recall that for each Q ∈ D , the containment in (3.2) holds and the outer ball thatcontains S is denoted by B ( S ).By decomposing the space X into annuli based on B ( S ), we have that K (cid:16) S, ˆ Q \ Q u (cid:17) = Z B ( S ) (cid:18) l ( S ) l ( S ) + dist( y, S ) (cid:19) κ µ ( B ( x S , l ( S ) + dist( y, S ))) 1 ˆ Q \ Q ( y ) du ( y )+ ∞ X k =1 Z δ − k B ( S ) \ δ − k B ( S ) (cid:18) l ( S ) l ( S ) + dist( y, S ) (cid:19) κ µ ( B ( x S , l ( S ) + dist( y, S ))) 1 ˆ Q \ Q ( y ) du ( y )= κ X k = κ Z δ − k B ( S ) \ δ − k B ( S ) (cid:18) l ( S ) l ( S ) + dist( y, S ) (cid:19) κ µ ( B ( x S , l ( S ) + dist( y, S ))) 1 ˆ Q \ Q ( y ) du ( y ) . Here κ and κ are determined by the conditions that for k < κ δ − k B ( S ) ∩ ( ˆ Q \ Q ) = ∅ , and for k > κ , we have δ − k B ( S ) ∩ ( ˆ Q \ Q ) = ∅ . Hence, for κ ≤ k ≤ κ , we have 12 l ( S ) λ l ( Q ) − λ ≤ dist( S, e ( Q )) . δ − k l ( S ) , and thus δ k . (cid:18) l ( S ) l ( Q ) (cid:19) − λ . We now estimate K (cid:16) S, ˆ Q \ Q u (cid:17) . To begin with, we give a partition between κ and κ to create a newcollection of integers ˜ κ , ˜ κ , . . . , ˜ κ L such that ˜ κ = κ , ˜ κ L = κ and that ˜ κ satisfies δ − ˜ κ l ( S ) ≈ l ( Q ), ˜ κ satisfies δ − ˜ κ l ( S ) ≈ δl ( Q ), . . . , ˜ κ L satisfies δ − ˜ κ L l ( S ) ≈ δ − L l ( Q ). In fact, we have ˜ κ = κ , ˜ κ > ˜ κ ,˜ κ = ˜ κ + 1, and so on. Then we get that κ X k = κ = L X ℓ =1 ˜ κ ℓ X k =˜ κ ℓ − . Hence, K (cid:16) S, ˆ Q \ Q u (cid:17) . L X ℓ =1 ˜ κ ℓ X k =˜ κ ℓ − Z δ − k B ( S ) \ δ − k B ( S ) (cid:18) l ( S ) l ( S ) + dist( y, S ) (cid:19) κ µ ( B ( x S , l ( S ) + dist( y, S ))) 1 ˆ Q \ Q ( y ) du ( y ) . L X ℓ =1 ˜ κ ℓ X k =˜ κ ℓ − Z ( δ − k B ( S ) \ δ − k B ( S )) ∩ ( ˆ Q \ Q ) (cid:18) l ( S ) δ − k l ( S ) (cid:19) κ µ ( B ( x S , δ − k l ( S ))) du ( y ) . ˜ κ X k =˜ κ δ κk Z ( δ − k B ( S ) \ δ − k B ( S )) ∩ ( ˆ Q \ Q ) µ ( B ( x S , δ − k l ( S ))) du ( y )+ L X ℓ =2 δ κ (˜ κ + ℓ ) Z ( δ − (˜ κ ℓ ) B ( S ) \ δ (1 − ˜ κ − ℓ ) B ( S )) ∩ ( ˆ Q \ Q ) µ ( B ( x S , δ − ˜ κ − ℓ l ( S ))) du ( y )=: Term + Term . For the first term, by using the doubling property, we have thatTerm . ˜ κ X k =˜ κ δ κk µ ( B ( x S , l ( Q ))) µ ( B ( x S , δ − k l ( S ))) 1 µ ( B ( x S , l ( Q ))) u h ( δ − k B ( S ) \ δ − k B ( S )) ∩ ( ˆ Q \ Q ) i . ˜ κ X k =˜ κ δ κk (cid:18) l ( Q ) δ − k l ( S ) (cid:19) n µ ( B ( x Q , l ( Q ))) u h ( δ − k B ( S ) \ δ − k B ( S )) ∩ ( ˆ Q \ Q ) i . ˜ κ X k =˜ κ δ κk δ kn (cid:18) l ( Q ) l ( S ) (cid:19) n µ ( B ( x Q , l ( Q ))) u h ( δ − k B ( S ) \ δ − k B ( S )) ∩ ( ˆ Q \ Q ) i . ˜ κ X k =˜ κ δ κk (cid:18) l ( Q ) l ( S ) (cid:19) nλ µ ( B ( x Q , l ( Q ))) u h ( δ − k B ( S ) \ δ − k B ( S )) ∩ ( ˆ Q \ Q ) i . δ κ ˜ κ (cid:18) l ( Q ) l ( S ) (cid:19) nλ µ ( B ( x Q , l ( Q ))) u h B ( Q ) ∩ ( ˆ Q \ Q ) i . (cid:18) l ( Q ) l ( S ) (cid:19) λ ( n + κ ) − κ Z B ( Q ) µ ( B ( x Q , l ( Q ))) 1 ˆ Q \ Q ( y ) du ( y ) . For the second term, we get thatTerm . δ κ ˜ κ L X ℓ =2 δ κℓ µ ( B ( x S , δ − ℓ l ( Q ))) u h ( δ − (˜ κ + ℓ ) B ( S ) \ δ − ˜ κ − ℓ B ( S )) ∩ ( ˆ Q \ Q ) i . δ κ ˜ κ L X ℓ =2 δ κℓ µ ( B ( x Q , δ − ℓ l ( Q ))) u h ( δ − ℓ B ( Q ) \ δ − ℓ B ( Q )) ∩ ( ˆ Q \ Q ) i . δ κ ˜ κ L X ℓ =2 Z δ − ℓ B ( Q ) \ δ − ℓ B ( Q ) (cid:18) l ( Q ) δ − ℓ l ( Q ) (cid:19) κ µ ( B ( x Q , δ − ℓ l ( Q ))) 1 ˆ Q \ Q ( y ) du ( y ) . Next, we note that K (cid:16) Q, ˆ Q \ Q u (cid:17) = Z B ( Q ) (cid:18) l ( Q ) l ( Q ) + dist( y, Q ) (cid:19) κ µ ( B ( x Q , l ( Q ) + dist( y, Q ))) 1 ˆ Q \ Q ( y ) du ( y )+ j X j =1 Z δ − j B ( Q ) \ δ − j B ( Q ) (cid:18) l ( Q ) l ( Q ) + dist( y, Q ) (cid:19) κ µ ( B ( x Q , l ( Q ) + dist( y, Q ))) 1 ˆ Q \ Q ( y ) du ( y ) ≈ Z B ( Q ) µ ( B ( x Q , l ( Q ))) 1 ˆ Q \ Q ( y ) du ( y )+ j X j =1 Z δ − j B ( Q ) \ δ − j B ( Q ) (cid:18) l ( Q ) δ − j l ( Q ) (cid:19) κ µ ( B ( x Q , δ − j l ( Q ))) 1 ˆ Q \ Q ( y ) du ( y ) . WO WEIGHT INEQUALITY 21
Combining the estimates of Term and Term , and the equality above, we see that K (cid:16) S, ˆ Q \ Q u (cid:17) . (cid:18) l ( Q ) l ( S ) (cid:19) λ ( n + κ ) − κ K (cid:16) Q, ˆ Q \ Q u (cid:17) . The proof of Lemma 22 is complete. (cid:3)
Let us begin with the proof of Lemma 20.
Proof of Lemma 20.
Recall that the pairs of cubes (
Q, S ) ∈ A satisfy l ( S ) ≤ l ( Q ) and dist( Q, S ) ≥ l ( Q ).We can apply Lemma 21 to h T ( uh ǫQ ) , h kS i v . To see this note that h ǫQ is constant on each child Q ǫ where ǫ ∈ { , ...., M Q − } .Take a child Q ǫ and apply Lemma 21 with the largest cube ˆ Q taken to be ˆ Q := hull h Q ǫ , ( l ( Q ) l ( S ) ) − λ S i .Here ρS means the cube with same centre as S and length equal to ρl ( S ) and by hull above we mean ˆ Q isthe smallest cube containing both cubes Q ǫ and (cid:16) l ( Q ) l ( S ) (cid:17) − λ S .The two cubes Q ǫ and (cid:16) l ( Q ) l ( S ) (cid:17) − λ S are disjoint. We take Q ′ ⊂ ˆ Q so that ˆ Q \ Q ′ = Q ǫ . Then using Lemma21 we get the following estimate β ( Q, S ) := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X ǫ h T (1 Q ǫ h ǫQ u ) , h kS i v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X ǫ | E uQ ǫ ( h ǫQ ) ||h T (1 Q ǫ u ) , h kS i v | . Here E uQ ǫ ( h ǫQ ) := u ( Q ǫ ) R Q ǫ h ǫQ ( x ) du ( x ) . Observe | E uQ ǫ ( h ǫQ ) | ≤ u ( Q ǫ ) .We have K ( S, Q ǫ u ) . l ( S ) κ ( l ( S )+dist( Q i ,S )) κ + n u ( Q ǫ ), where n is the upper dimension of µ and we have used thedoubling property of the measure µ . Hence we get β ( Q, S ) . X ǫ v ( S ) / u ( Q ǫ ) / l ( S ) κ ( l ( S ) + dist( Q, S )) κ + n . To continue, we may assume that k f k L ( u ) = k g k L ( v ) = 1. We then estimate | A | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( Q,S ) ∈A M Q − X ǫ =1 M S − X k =1 h f, h ǫQ i u h T ( uh ǫQ ) , h kS i v h g, h kS i v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) which can be written as follows | A | ≤ X Q X S : l ( S ) ≤ l ( Q )dist( Q,S ) ≥ l ( Q ) M Q − X ǫ =1 M S − X k =1 |h f, h ǫQ i u | β ( Q, S ) |h g, h kS i v | . X Q X S : l ( S ) ≤ l ( Q )dist( Q,S ) ≥ l ( Q ) M Q − X ǫ =1 M S − X k =1 |h f, h ǫQ i u | u ( Q ) / l ( S ) κ ( l ( S ) + dist( Q, S )) κ + n v ( S ) / |h g, h kS i v | . X Q M Q − X ǫ =1 |h f, h ǫQ i u | X S : l ( S ) ≤ l ( Q )dist( Q,S ) ≥ l ( Q ) (cid:18) l ( S ) l ( Q ) (cid:19) − σ u ( Q ) / l ( S ) κ ( l ( S ) + dist( Q, S )) κ + n v ( S ) / + X S M S − X k =1 |h g, h kS i v | X Q : l ( S ) ≤ l ( Q )dist( Q,S ) ≥ l ( Q ) (cid:18) l ( S ) l ( Q ) (cid:19) σ u ( Q ) / l ( S ) κ ( l ( S ) + dist( Q, S )) κ + n v ( S ) / =: A + A , where in the last inequality we have inserted the gain and loss term (cid:16) l ( S ) l ( Q ) (cid:17) ± σ with 0 < σ < We first consider the term A . For each fixed Q we have X S : l ( S ) ≤ l ( Q )dist( Q,S ) ≥ l ( Q ) (cid:18) l ( S ) l ( Q ) (cid:19) σ u ( Q ) / l ( S ) κ ( l ( S ) + dist( Q, S )) κ + n v ( S ) / . u ( Q ) / ∞ X i =0 δ iσ X S : l ( S )= δ i l ( Q )dist( Q,S ) ≥ l ( Q ) l ( S ) κ (dist( Q, S )) κ + n v ( S ) / X S : l ( S )= δ i l ( Q )dist( Q,S ) ≥ l ( Q ) l ( S ) κ (dist( Q, S )) κ + n . ∞ X i =0 δ iσ (cid:18) u ( Q ) l ( Q ) n K ( Q, v ) (cid:19) . A , where the last inequality follows from the fact that σ >
0. Consider the term A . For each fixed S we have X Q : l ( S ) ≤ l ( Q )dist( Q,S ) ≥ l ( Q ) (cid:18) l ( S ) l ( Q ) (cid:19) − σ u ( Q ) / l ( S ) κ ( l ( S ) + dist( Q, S )) κ + n v ( S ) / . v ( S ) ∞ X j =0 δ j (1 − σ ) X Q : l ( S )= δ j l ( Q )dist( Q,S ) ≥ l ( Q ) l ( Q ) κ (dist( Q, S )) κ + n u ( Q ) . v ( S ) ∞ X j =0 δ j (1 − σ ) X Q : l ( S )= δ j l ( Q )dist( Q,S ) ≥ l ( Q ) l ( Q ) κ (dist( Q, S )) κ + n u ( Q ) X Q : l ( S )= δ j l ( Q )dist( Q,S ) ≥ l ( Q ) l ( Q ) κ (dist( Q, S )) κ + n , which is bounded by v ( S ) ∞ X j =0 δ j (1 − σ ) K ( δ j/n S, u ) l ( δ j/n S ) n ! . A , where the last inequality follows from the fact that σ <
1. Thus, with any fixed 0 < σ < | A | . A X Q M Q − X ǫ =1 |h f, h ǫQ i u | + A X S M S − X k =1 |h g, h kS i v | = (cid:16) k f k L ( u ) + k g k L ( v ) (cid:17) A = 2 A . The proof of Lemma 20 is complete. (cid:3)
We have now reduced matters to the case of considering A . We further decompose A into A := { ( Q, S ) ∈ A : l ( S ) ≤ δ r l ( Q ) , Q ∩ S = ∅ , dist( Q, S ) ≤ l ( Q ) } , A := { ( Q, S ) ∈ A : l ( S ) ≤ δ r l ( Q ) , Q ∩ S = ∅ , dist( Q, S ) ≤ l ( Q ) } . We are now going to prove in this section that | A | ≤ A k f k L ( u ) k g k L ( v ) . WO WEIGHT INEQUALITY 23
Fix p ≥ r . Observe that A ( p ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X Q X S : δ − p l ( S )= l ( Q )dist( Q,S ) ≤ l ( Q ) Q ∩ S = ∅ M Q − X ǫ =1 M S − X k =1 h f, h ǫQ i u h T ( uh ǫQ ) , h kS i v h g, h kS i v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . X Q M Q − X ǫ =1 (cid:12)(cid:12) h f, h ǫQ i u (cid:12)(cid:12) X Q M Q − X ǫ =1 X S : δ − p l ( S )= l ( Q )dist( Q,S ) ≤ l ( Q ) Q ∩ S = ∅ M S − X k =1 (cid:12)(cid:12) h T ( uh ǫQ ) , h kS i v (cid:12)(cid:12) (cid:12)(cid:12) h g, h kS i v (cid:12)(cid:12) . Λ( p ) k f k L ( u ) k g k L ( v ) . The last two inequalities follow from the Cauchy-Schwarz inequality and we use Fubini to get k g k L ( v ) aboveand δ − p in the expression below. HereΛ( p ) := δ − p sup Q X S : δ − p l ( S )= l ( Q )dist( Q,S ) ≤ l ( Q ) Q ∩ S = ∅ M Q − X ǫ =1 M S − X k =1 |h T ( uh ǫQ ) , h kS i v | . But S is good, so that Lemma 21 applies to each child of Q and S yieldingΛ( p ) . sup Q δ − p X ǫ X k X S : δ − p l ( S )= l ( Q )dist( Q,S ) ≤ l ( Q ) Q ∩ S = ∅ v ( S k ) u ( Q ǫ ) K ( S k , Q ǫ u ) . Hence we have the following, using Lemma 22Λ( p ) . sup Q δ − p X ǫ X k X S : δ − p l ( S )= l ( Q )dist( Q,S ) ≤ l ( Q ) Q ∩ S = ∅ v ( S k ) u ( Q ǫ ) . (cid:18) l ( S ) l ( Q ) (cid:19) − σ · K ( Q, Q ǫ u ) . sup Q δ − p δ − pσ X ǫ u ( Q ǫ ) l ( Q ) n X S : δ − p l ( S )= l ( Q )dist( Q,S ) ≤ l ( Q ) Q ∩ S = ∅ X k v ( S k ) . δ − p (1+2 σ ) A . Above, we have used that K ( S, Q i u ) ≤ l ( S ) κ ( l ( S ) + dist( Q i , S )) κ + n u ( Q i ) . This is clearly summable in p ≥ r as − σ > (as we can fix λ ∈ (0 ,
1) such that λ < κn + κ ) so the proof iscomplete. 6. Stopping Cubes and Corona Decompositions
Our focus is now on the “short range terms” given by A . Here we will use our pivotal condition. In thissection we will decompose A further and estimate each piece.Recall A := { ( Q, S ) ∈ A : l ( S ) ≤ δ r l ( Q ) , Q ∩ S = ∅ , dist( Q, S ) ≤ l ( Q ) } and A := X ( Q,S ) ∈A M Q − X ǫ =1 M S − X k =1 h f, h ǫQ i u h T ( uh ǫQ ) , h kS i v h g, h kS i v . Denote ∆ uQ f = M Q − P i =1 h f, h iQ i u h iQ . Then observe that: h T (∆ uQ f ) , ∆ uS g i v = h T ((1 Q − Q S )∆ uQ f ) , ∆ uS g i v + h T (1 Q S ∆ uQ f ) , ∆ uS g i v = h T ((1 Q \ Q S )∆ uQ f ) , ∆ uS g i v (6.1) + E uQ S (∆ uQ f ) h T (1 e Q ) , ∆ uS g i v (6.2) − E uQ S (∆ uQ f ) h T (1 e Q \ Q S ) , ∆ uS g i v . (6.3)First observe that 1 e Q − e Q \ Q S = 1 Q S and secondly1 Q S ∆ uQ f = M Q − X i =1 h f, h iQ i u h iQ · Q S and so E uQ S (∆ uQ f ) = 1 u ( Q S ) Z Q S ∆ uQ f ( x ) du ( x )= 1 u ( Q S ) Z X M Q − X i =1 h f, h iQ i u h iQ ( x ) · Q S ( x ) du ( x )= 1 u ( Q S ) M Q − X i =1 h f, h iQ i u Z X h iQ ( x ) · Q S ( x ) du ( x ) . Here Q S is child of Q containing S and e Q the parent of Q S .6.1. The Decomposition of the Short Range Term.
To estimate A and conclude this section, wecombine the splitting into (6.1), (6.2), (6.3) and the following Corona decomposition. Namely select thecubes e Q that appear in (6.1)-(6.3) according to the stopping rule below. Recall the set A . Using the factthat S is good we can make this set more explicit, i.e. S ⊂ Q and l ( S ) < δ r l ( Q ) A := { ( Q, S ) ∈ A : l ( S ) ≤ δ r l ( Q ) , S ⊂ Q } . The Corona Decomposition.
We are going to define the ‘stopping cubes’ and the ‘Corona decomposi-tion’. Let us first define the following functionals:Φ( Q, E u ) := v ( Q ) K ( Q, E u ) , Ψ( Q, E u ) := sup Q = ∪ i ≥ Q i X i ≥ Φ( Q i , E u ) , where Q i are dyadic subcubes of Q , hence lie in some dyadic grid as Q and where the supremum is over all r -good dyadic subpartitions { Q i } i ≥ of Q . We have the following pivotal condition(6.4) X r ≥ Φ( Q r , Q u ) ≤ V u ( Q ) . We will use certain key properties of Ψ to estimate the term A defined below. Definition 23.
Given any cube Q o , we will set S ( Q o ) to be the maximal D u strict subcubes S ⊂ Q o suchthat (6.5) Ψ( S, Q o u ) ≥ V u ( S ) . The collection S ( Q o ) can be empty. WO WEIGHT INEQUALITY 25
We will be able to now recursively define S := { Q o } and S j +1 := ∪ S ∈S j S ( S ). The collection of S := ∪ ∞ j =1 S j is the collection of stopping cubes. Let us define ρ : S 7→ N by ρ ( S ) := j for all S ∈ S j , so that ρ ( S )denotes the generation in which S occurs in the construction of S .Let us now discuss the associated Corona Decomposition. Definition 24.
For S ′ ∈ S , we are going to set P ( S ′ ) to be all the pairs of cubes ( Q, S ) such that (1) Q ∈ D u , S ∈ D v , S ⊂ Q and l ( S ) ≤ δ r l ( Q ) ; (2) S ′ is the S parent of Q S which is the child of Q containing S . Observe that we can write A = ∪ S ′ ∈S P ( S ′ ), where A is as defined above. We next define the Coronasassociated to f and g . Definition 25.
Let C u ( S ′ ) be all those Q ∈ D u such that S ′ is a minimal member of S that contains a D u child of Q . The definition of C v ( S ′ ) is similar but not symmetric: all those S ∈ D v such that S ′ is thesmallest member of S that contains S and satisfies l ( S ) ≤ δ r l ( S ′ ) together with all those S ∈ D v such thatfor some S ′′ ∈ S ( S ) we have S ∈ C v ( S ′′ ) with S ⊂ S ′′ with l ( S ) ≥ δ r l ( S ′′ ) . The collections {C u ( S ′ ) : S ′ ∈ S} and {C v ( S ′ ) : S ′ ∈ S} are referred to as the Corona Decompositions (the collection C v is called the shiftedCorona in the literature). We will now define the projection operators associated to these Coronas P uS ′ f := X Q ∈C u ( S ′ ) M Q − X ǫ =1 h f, h ǫQ i u h ǫQ . Similarly we can define P vS ′ g . Observe that P vS ′ g projects only cubes S with l ( S ) ≤ δ r l ( S ′ ).We have the estimate below which we will use in the proofs below X S ′ ∈S kP uS ′ f k L ( u ) ≤ sup Q ∈D u M Q k f k L ( u ) where M Q is the number of children of Q in the grid D u . Recall that we also assume throughout the papersup Q ∈D u M Q < ∞ . We have a similar inequality for P vS ′ .Observe in the definition of stopping cubes we are using the functional Ψ associated with hypothesis (6.4).So the stopping cubes can be viewed as the enemy of verifying (6.4). Definition 26.
Given a pair ( Q, S ) ∈ A , choose e Q ∈ S to be the unique stopping cube such that Q S ∈ C u ( e Q ) ,where Q S is the S child of e Q . Equivalently, e Q ∈ S is determined by the requirement ( Q, S ) ∈ P ( e Q ) . Note that if Q S / ∈ S , then Q ⊂ e Q , while if Q S ∈ S , then e Q is the child of Q containing S . With the choiceof e Q in the splitting of (6.1)-(6.3) we obtain | A | ≤ P j =1 | A j | where A := X ( Q,S ) ∈A T (1 Q \ Q S u ∆ uQ f ) , ∆ uS g i v , (6.6) A := X S ′ ∈S X ( Q,S ) ∈P ( S ′ ) E uQ S (∆ uQ f ) h T (1 S ′ u ) , ∆ uS g i v , (6.7) A := X S ′ ∈S X ( Q,S ) ∈P ( S ′ ) E uQ S (∆ uQ f ) h T (1 S ′ \ Q S u ) , ∆ uS g i v . (6.8)The three terms above are referred to as the neighbor , paraproduct and stopping term respectively.The paraproduct term A is further decomposed and estimates on this term will be handled in Section 8,while in the remainder of this section we will prove: | A | . A k f k L ( u ) k g k L ( v ) , (6.9) | A | . Vk f k L ( u ) k g k L ( v ) . (6.10) Control of the Neighbor Term A . The neighbor terms are defined in (6.6) and we are to prove(6.9). Recall we have Q ∈ D u , S ∈ G v contained in Q , with l ( S ) ≤ δ r l ( Q ) and Q S is the child of Q containing S . Fix a child θ ∈ { , , ..., M Q − } and an integer s ′ ≥ r . Here we use that Q \ Q θ = ∪ M Q − i =1 i = θ Q i .We are now going to estimate the inner product as that in (6.1): h T (1 Q \ Q θ u ∆ uQ f ) , ∆ uS g i v = M Q − X i =0 i = θ h T (1 Q i u ∆ uQ f ) , ∆ uS g i v = M Q − X i =1 i = θ E uQ i (∆ uQ f ) h T (1 Q i u ) , ∆ uS g i v . Here we will use k ∆ vS g k L ( v ) = (cid:18) M S − P k =1 |h g, h kS i| v (cid:19) and l ( S ) l ( Q θ ) = δ s ′ in Lemma 21 with S ⊂ Q θ ⊂ Q to obtain |h T (1 Q i u ) , ∆ uS g i v | . v ( S ) M S − X k =1 |h g, h kS i| v ! M Q − X i =0 i = θ K ( S, Q i u ) . v ( S ) M S − X k =1 |h g, h kS i| v ! δ − ( s ′ σ ) M Q − X i =0 i = θ K ( Q θ , Q i u ) . Here we applied Lemma 21 and Lemma 22 to S ⊂ Q \ ∪ M Q − i =1 i = θ Q i .For the sum below we keep the lengths of the cubes S fixed and we are under the assumption that S ⊂ Q θ .Define Λ( Q, θ, s ′ ) := X S : l ( S )= δ s ′ l ( Q ) S ⊂ Q θ M S − X k =1 |h g, h kS i| v ! . Then we have the following estimate using Cauchy-Schwarz A ( Q, θ, s ′ ) := X S : l ( S )= δ s ′ l ( Q ) S ⊂ Q θ |h T (1 Q \ Q θ u ∆ uQ f ) , ∆ uS g i v |≤ δ − ( s ′ σ ) M Q − X i =1 i = θ | E uQ i (∆ uQ f ) | K ( Q θ , M Q − X i =1 i = θ Q i u ) X S : l ( S )= δ s ′ l ( Q ) S ⊂ Q θ v ( S ) M S − X k =1 |h g, h kS i| v ! ≤ δ − ( s ′ σ ) M Q − X i =1 i = θ | E uQ i (∆ uQ f ) | K Q θ , M Q − X i =1 i = θ Q i u v ( Q θ ) Λ( Q, θ, s ′ ) . We will now use the following to estimate A ( Q, θ, s ′ ): | E uQ i (∆ uQ f ) | ≤ M Q − X j =1 |h f, h jQ i u | u ( Q i ) − . Substituting this into the above we find:
WO WEIGHT INEQUALITY 27 A ( Q, θ, s ′ ) . δ − ( s ′ σ ) M Q − X i =1 |h f, h iQ i| u Λ( Q, θ, s ′ ) M Q − X i =1 i = θ M Q − X j =1 j = θ u ( Q i ) − K ( Q θ , Q j u ) v ( Q θ ) . A δ − ( s ′ σ ) M Q − X i =1 |h f, h iQ i| u Λ( Q, θ, s ′ ) . We can then sum A ( Q, θ, s ′ ) in Q and apply Cauchy-Schwarz to show that X Q A ( Q, θ, s ′ ) . A δ − ( s ′ σ ) k f k L ( u ) k g k L ( v ) ;this last estimate is then summable in the parameter s ′ as we have − σ > as mentioned in Section 5. Thiscompletes the proof of the estimate (6.9).6.3. Control of the Stopping Term A . To control (6.8) we want to prove (6.10). Here we will use thehypothesis (6.5).We first define for S ′ ∈ S and s ≥ A ( S ′ , s ) := X ( Q,S ) ∈P ( S ′ ) l ( S )= δ s l ( Q ) | E uQ S (∆ uQ f ) h T (1 S ′ \ Q S u ) , ∆ uS g i v | . δ − σ s VF ( S ′ )Λ( S ′ , s ) , where F ( S ′ ) := X Q ∈C u ( S ′ ) M Q − X ǫ =1 |h f, h ǫQ i u | , Λ ( S ′ , s ) := X Q ∈C u ( S ′ ) X S :( Q,S ) ∈P ( S ′ ) l ( S )= δ s l ( Q ) M S − X k =1 |h g, h kS i v | . Using Cauchy-Schwarz in the variable Q variable above, and appealing to the following inequality | E uQ i (∆ uQ f ) | ≤ M Q − X j =1 |h f, h jQ i u | u ( Q i ) − , to continue the estimate for A ( S ′ , s ), A ( S ′ , s ) ≤ F ( S ′ ) X Q ∈C u ( S ′ ) X S :( Q,S ) ∈P ( S ′ ) l ( S )= δ s l ( Q ) u ( Q S ) |h T (1 S ′ \ Q S u ) , ∆ uS g i v | . We can now estimate the terms in the square bracket above by X Q ∈C u ( S ′ ) X ( Q,S ) ∈P ( S ′ ) l ( S )= δ s l ( Q ) M S − X k =1 |h g, h kS i v | × X ( Q,S ) ∈P ( S ′ ) l ( S )= δ s l ( Q ) M S − X k =1 u ( Q S ) |h T (1 S ′ \ Q S u ) , h kS i v | . Λ ( S ′ , s ) A ( S ′ , s ) , where A ( S ′ , s ) := sup Q ∈C u ( S ′ ) X ( Q,S ) ∈P ( S ′ ) l ( S )= δ s l ( Q ) M S − X k =1 u ( Q S ) |h T (1 S ′ \ Q S u ) , h kS i v | . We will now estimate the term A ( S ′ , s ). We will denote the children of Q by Q θ for θ ∈ { , ...., M Q − } and denote the children of S by S k for k ∈ { , , ..., M S − } . Also observe that h kS is supported on S and h kS = M S − P k =1 C S k S k such that k h kS k L ( u ) = M S − P k =1 C S k = 1. So using (5.3) we have the following: A ( S ′ , s ) . sup Q ∈C u ( S ′ ) sup θ ∈{ , ...M Q − } X ( Q,S ) ∈P ( S ′ ): Q S = Q θ l ( S )= δ s l ( Q ) M S − X k =1 u ( Q θ ) Φ( S k , S ′ \ Q θ u ) . sup Q ∈C u ( S ′ ) sup θ ∈{ , ...M Q − } X ( Q,S ) ∈P ( S ′ ): Q S = Q θ l ( S )= δ s l ( Q ) M S − X k =1 u ( Q θ ) v ( S k ) K ( S k , S ′ \ Q θ u ) . sup Q ∈C u ( S ′ ) sup θ ∈{ , ...M Q − } X ( Q,S ) ∈P ( S ′ ): Q S = Q θ l ( S )= δ s l ( Q ) M S − X k =1 u ( Q θ ) v ( Q θ ) (cid:18) l ( Q θ ) l ( S k ) (cid:19) σ K ( Q θ , S ′ \ Q θ u ) . sup Q ∈C u ( S ′ ) sup θ ∈{ , ...M Q − } X ( Q,S ) ∈P ( S ′ ): Q S = Q θ l ( S )= δ s l ( Q ) M S − X k =1 u ( Q θ ) δ − σ ( s +1) Φ( Q θ , S ′ \ Q θ u ) . sup Q ∈C u ( S ′ ) sup θ ∈{ , ...M Q − } u ( Q θ ) X ( Q,S ) ∈P ( S ′ ): Q S = Q θ l ( S )= δ s l ( Q ) M S − X k =1 δ − σ ( s +1) Φ( Q θ , S ′ u ) . sup Q ∈C u ( S ′ ) sup θ ∈{ , ...M Q − } u ( Q θ ) δ − σ ( s +1) u ( S ′ ) V . δ − σ ( s +1) V . Here we used Lemma 22 in the third inequality, used (6.4) in the second to last line and also that the numberof children of any cube is uniformly finite. Here we have also used that (
Q, S ) ∈ P ( S ′ ), so we have that S ′ is the S -parent of Q S hence Q S is not a stopping cube, so (6.5) does not hold, hence giving the estimateabove.We can observe that X S ′ ∈S F ( S ′ ) . k f k L ( u ) . And we have the following | A | ≤ X S ′ ∈S ∞ X s =0 A ( S ′ , s ) . Vk f k L ( u ) k g k L ( v ) . The Carleson Measure Estimates
In this section we will prove Carleson measure estimates useful for the analysis on the paraproduct term A . In the lemma below we use the stopping time definition. For a cube S ∈ D v let˜ P vS ( g ) := X S ′ ∈D v : S ′ ⊂ S M S ′ − X k =1 h g, h kS ′ i h kS ′ . Observe that the projection ˜ P vS is onto the span of all Haar functions h ′ S supported in the D v cube S . Incontrast P vS projects onto the span of all Haar functions h S with S in the corona C v ( S ) where S is a stoppingcube in the D u grid. Lemma 27.
Fix a cube Q ∈ D u and let ˆ Q ∈ S be its S -parent. Let { Q m : m ≥ } ⊂ D u be a strictsubpartition of Q . Suppose that Q m is good for m ≥ . Let { S m,s ′ : s ′ ≥ } ⊂ D v be a subpartition of Q m WO WEIGHT INEQUALITY 29 with l ( S m,s ′ ) < δ r l ( Q m ) for all m, s ′ ≥ . We then have the following (7.1) X m,s ′ ≥ k ˜ P vS m,s ′ T (1 ˆ Q \ Q m u ) k L ( v ) . V u ( Q ) . Proof.
We are now going to use the L formulation (5.3) of Lemma 21 to deduce (7.1). We begin with the L formulation to obtain X m,s ′ ≥ k ˜ P vS m,s ′ T (1 ˆ Q \ Q m u ) k L ( v ) . X m,s ′ ≥ Φ( S m,s ′ , ˆ Q \ Q m u ) . X m,s ′ ≥ Φ( S m,s ′ , ˆ Q \ Q u ) + X m,s ′ ≥ Φ( S m,s ′ , Q \ Q m u ) . The last inequality follows from the definition of Φ and K ( S, ˆ Q \ Q m u ) = K ( S, ˆ Q \ Q u ) + K ( S, Q \ Q m u ) . If Q = ˆ Q , we estimate the sum involving ˆ Q \ Q using the fact that { S m,s ′ } m,s ′ ≥ is a m -good subpartitionof Q . We use Lemma 22 to get the first inequality below and given the fact that (6.5) fails and using (6.4)we obtain the following when 0 < t < r X m,s ′ ≥ Φ( S m,s ′ , ˆ Q \ Q u ) . X m,s ′ ≥ (cid:20) l ( Q m ) l ( S m,s ′ ) (cid:21) σ Φ( Q m , ˆ Q u ) . X m,b ′ ≥ X S m,b ′ ∈D v l ( S m,b ′ )= δ t l ( Q m ) (cid:20) l ( Q m ) l ( S m,b ′ ) (cid:21) σ Φ( Q m , ˆ Q u ) . X m,b ′ ≥ X t We have the following Carleson measure estimates for S ′ ∈ S and K ∈ D u : X J ′ ∈S ( S ′ ) u ( J ′ ) ≤ u ( S ′ ) and X S ′ ∈S : S ′ ( K u ( S ′ ) . u ( K );(7.2) X J ∈C v ( S ′ ): J ⊂ K,l ( J ) <δ r l ( K ) |h T (1 S ′ u ) , h vJ i v | . ( V + T ) u ( K ) . (7.3) Proof. For the second part of the inequality in (7 . K = S ∈ S . And then thecase we are interested in follows from recursive application of the estimate from the first half of the inequality(7.2) to the cube S and all of its children in S .We now prove the first half of (7.2). The cubes in the collection S ( S ) = { S ′ m : m ≥ } given in theDefinition 23 are pairwise disjoint and strictly contained in Q . Each of them satisfies (6.5), so we can applythat estimate along with (6.4) to get X J ′ ∈S ( S ) u ( J ′ ) = X r ≥ u ( S ′ m ) ≤ V X m ≥ Ψ( S ′ m , S u ) ≤ u ( S ) . We will now prove (7.3). First we will fix S ′ ∈ S and K , which can be assumed to be a subset of S ′ . Wewill apply the operator T to u K as opposed to u S ′ , and we can then use the testing condition for T to get X J ∈C v ( S ′ ): J ⊂ K,l ( J ) <δ r l ( K ) |h T (1 K u ) , h vJ i v | ≤ Z K | T (1 K u ) | dv ≤ T u ( K ) . Now we will apply T to u S ′ \ K and show X J ∈C v ( S ′ ): J ⊂ K,l ( J ) <δ r l ( K ) |h T (1 S ′ \ K u ) , h vJ i v | . V u ( K ) . We can assume that K ( S ′ and there is some J ∈ C v ( S ′ ) with J ⊂ K . From this we can say that K is nota stopping cube. Therefore the cube K must fail (6.5).Let J denote the maximal cubes J ∈ C v ( S ′ ) with J ⊂ K and l ( J ) ≤ δ r l ( K ). Using the definition of˜ P vS ( g ), we can use (7.2), with ˆ Q = S ′ and J ∈ J . It gives X J ∈J k ˜ P vJ T (1 S ′ \ K u ) k L ( v ) . X J ∈J Φ( J, S ′ \ K u ) . V u ( K ) . The second inequality uses the fact K fails (6.4). This proves (7.3). (cid:3) The following Carleson measure estimate uses hypothesis (6.4) in the proof. It will provide the decay inthe parameter t in Theorem 29. For all integers t ≥ 0, we define for S ∈ S , which are not maximal α t ( S ) := X S ′ : π tS ( S ′ )= S kP vS ′ T ( u π S ( S ) \ S ) k L ( v ) . Here π t D u ( S ) is the t -ancestor of S in D u . Also we are taking the projection T ( u π S ( S ) \ S ) associated to partsof the corona decomposition which are ‘far below’ S . We have the following off-diagonal estimate. Theorem 29. The following Carleson measure estimate holds: (7.4) X S : π D u ( S ) ⊂ K α t ( S ) . δ − σ t V u ( K ) , K ∈ D u . The implicit constant is independent of the choices of the cube K and t ≥ . In the estimate (7.4), we need to observe the fact that the dyadic parent π D v ( S ) of S appears. In factthe role of dyadic parents is revealed in the next proof. We use the negation of (6.5) when π D u ( S ) / ∈ S , andotherwise we use (6.4). Proof. We will first show that X S ∈S ( ˆ S ) α t ( S ) ≤ δ − σ t V u ( ˆ S ) , ˆ S ∈ S . For this proof, we will set S t ( S ) := { S ′ ∈ S : π tS ( S ′ ) = S } , using this notation for S ∈ S ( ˆ S ). We apply the L formulation estimate (5.4) of Lemma 21 to the expression α t . S ( S ′ ) := { J ∈ C v ( S ) : J is maximal with J ⊂ S ′ , l ( J ) < δ r l ( S ′ ) } . (7.5)From the definition above we have l ( J ) < δ r l ( S ′ ) for all J ∈ S ( S ′ ) and as all Haar functions have meanzero, we can apply the L formulation (5.4) of Lemma 21. Using this, we see that α t ( S ) . X S ′ ∈S t ( S ) X J ∈S ( S ′ ) Φ( J, ˆ S \ S u ) . And by using (6.4) we get(7.6) X S ∈S ( ˆ S ) α t ( S ) . X S ∈S ( ˆ S ) X S ′ ∈S t ( S ) X J ∈S ( S ′ ) Φ( J, ˆ S \ S u ) . δ − σ t V X S ∈S ( ˆ S ) u ( S ) . δ − σ t V u ( ˆ S ) . The last inequality follows from hypothesis (6.4).Now fix K as in (7.4) and let ˆ S ∈ S be the stopping cube such that K ∈ C u ( ˆ S ) . Let G := { S i } i be themaximal cubes from S that are strictly contained in K . Inductively we define the ( k + 1) st generation G k +1WO WEIGHT INEQUALITY 31 to consist of the maximal cubes from S that are strictly contained in some k th generation cube S ∈ G k .Inequality (7.6) shows that X S ∈G k +1 α t ( S ) . δ − σ t V X S ∈G k +1 u ( S ) . We have from (7.2) that ∞ X k =1 X S ∈G k u ( S ) . X S ∈G u ( S ) . u ( K ) . This will be all we need for the case K = ˆ S . For the case K = ˆ S we will use Lemma 27 to control the firstgeneration of cubes S in G : X S ∈G α t ( S ) . δ − σ t u ( K ) . Indeed we will apply Lemma 27 with ˆ Q = ˆ S , Q = K , { Q r } r ≥ = G and { J r,s } s ≥ = [ S ′ ∈S t ( S ′ ) S ( S ′ ).When K = ˆ S we have X S ∈S : π D u ( S ) ⊂ K α t ( S ) = X S ∈G α t ( S ) + ∞ X k =1 X S ∈G k +1 α t ( S ) . δ − σ t u ( K ) + δ − σ t V ∞ X k =1 X S ∈G k u ( S ) . δ γt V u ( K ) . If K = ˆ S we have G = { ˆ S } and we get the estimate X S ∈S : π D u ( S ) ⊂ ˆ S α t ( S ) = X S ∈G α t ( S ) + ∞ X k =1 X S ∈G k +1 α t ( S ) . δ − σ t V ∞ X k =1 X S ∈G k u ( S ) . δ − σ t V u ( ˆ S ) . The proof of Theorem 29 is complete. (cid:3) We need a Carleson measure estimate that is a common variant of (7.2) and (7.4). We define β ( S ) := k P vS T ( u π D u ( S ) ) k L ( v ) . Theorem 30. We have the following Carleson measure estimate X S ∈S : π D u ( S ) ⊂ K β ( S ) . ( T + V ) u ( K ) . Proof. Using the decomposition π D u ( S ) = S ∪ { π D u ( S ) \ S } , we write β ( S ) ≤ β ( S ) + β ( S )) where β ( S ) := k P vS T ( u S ) k L ( v ) and β ( S ) := k P vS T ( u π D u ( S ) \ S ) k L ( v ) . We have by the testing condition β ( S ) ≤ T u ( S ), so now by (7.2), we need only consider the Carlesonmeasure norm of the terms β ( S ).Now we will fix an cube K of the form K = π D u ( S ) for some S ∈ S . Let R be the maximal cubes ofthe form π D u ( S ) ( K and for R ∈ R , let S ( R ) be all cubes S ∈ S with S ⊂ R and S is maximal. Now byusing the definition of ˜ P vS ( g ) and (7.5), we can estimate X R ∈R X S ∈S ( R ) β ( S ) . X R ∈R X S ∈S ( R ) X J ∈S ( S ) k ˜ P vJ T ( u π D u ( S ) \ S ) k L ( v ) . V u ( K ) . By careful arrangement of the collections R , S ( R ) and S ( S ) we have applied (7.1) in the last step. Here weuse the same strategy as we used in the proof of Theorem 29.We argue this inequality is enough to conclude the Theorem. Suppose that S ′ ∈ S , with S ′ ⊂ K , but S ′ is not in any collection S ( R ) for R ∈ R . It follows that S ′ ( S for some S ∈ S ( R ) and R ∈ R . This impliesthat the Carleson measure estimate (7.2) completes the proof. (cid:3) We collect one last Carleson measure estimate. Define γ ( S ) := k P vS T ( u π S ( S ) \ π D u ( S ) k L ( v ) . Theorem 31. We have the estimate X S ∈S : π D u ( S ) ⊂ K γ ( S ) . V u ( K ) . Proof. We can take K = π D u ( S ) for some S ∈ S , and we can assume that K / ∈ S as otherwise we areapplying the T to the zero function. We then repeat the argument as in the previous proof. We use a similarconstruction for the proof here as in Theorem 30. (cid:3) The Paraproduct Terms We are going to prove bounds on the paraproduct term A now. Before we prove the bounds, we willreorganize the sum in (6.2) according to the corona decomposition. We need to observe that for S ∈ C v ( S ′ )and S ⊂ Q , we need not have Q ∈ C u ( S ′ ). It could be the case that Q ∈ C u ( π t S ( S ′ )) for some ancestor π t S ( S ′ ) of S ′ . Remember the ancestor π t S ( S ′ ) is defined only for 1 ≤ t ≤ ρ ( S ′ ).Now we will split the sum into two parts A = A + A where A := X S ′ ∈S X ( Q,S ) ∈P ( S ′ ) S ∈C v ( S ′ ) E uQ S (∆ uQ f ) h T (1 S ′ u ) , ∆ uS g i v ;(8.1) A := X S ′ ∈S\ Q ρ ( S ′ ) X t =1 X ( Q,S ) ∈P ( π t S ( S ′ )) S ∈C v ( S ′ ) E uQ S (∆ uQ f ) h T (1 π t S ( S ′ ) u ) , ∆ uS g i v . (8.2)Observe that in A we consider the case where both Q and S are controlled by the same stopping cube.Whereas in A , ( Q, S ) ∈ P ( π t S ( S ′ )), where π t S ( S ′ ) is t -fold parent of S ′ in the grid S .We will now show | A | . ( T + V ) k f k L ( u ) k g k L ( v ) . We will have to further decompose A .8.1. The first Paraproduct Term A . Fix S ′ ∈ S and S ∈ C v ( S ′ ). Observe that we have the followingtelescoping indentity:(8.3) X Q :( Q,S ) ∈P ( S ′ ) E uQ S (∆ uQ f ) = E uQ S, ∗ f − E uπ D ( S ′ ) f. Here Q S, ∗ is the minimal member of C u ( S ′ ) that contains S and l ( S ) < δ r l ( Q ). As S is good, such cubesexist. So we have A = X S ′ ∈S A ( S ′ );(8.4) A ( S ′ ) := X S ∈C v ( S ′ ) ( E uQ S, ∗ f − E uπ D ( S ′ ) f ) h T (1 S ′ u ) , ∆ uS g i v . (8.5)We now give our first paraproduct estimate. Lemma 32. We have the following estimate A ( S ′ ) . ( T + V ) kP uS ′ f − S ′ E uπ D ( S ′ ) f k L ( u ) kP vS ′ g k L ( v ) , S ′ ∈ S . We have defined the projections appearing on the righthand side in Section 6.Proof. For Q ∈ C u ( S ′ ), let us define L vQ g := X S ∈C v ( S ′ ): Q S, ∗ = Q ∆ uS g. Now using Cauchy-Schwarz and the fact that E uQ f = E uQ P uS ′ f and L vQ g = L vQ P vS ′ g , then we have by re-indexing WO WEIGHT INEQUALITY 33 | A ( S ′ ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈C u ( S ′ ) ( E uQ P uS ′ f − E uπ D ( S ′ ) f ) h T (1 S ′ u ) , L vQ P vS ′ g i v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X Q ∈C u ( S ′ ) | E uQ P uS ′ f − E uπ D ( S ′ ) f | k L vQ T (1 S ′ u ) k L ( v ) X Q ∈C u ( S ′ ) k L vQ P vS ′ g k L ( v ) ≤ kP vS ′ g k L ( v ) X Q ∈C u ( S ′ ) | E uQ ( P uS ′ f − E uπ D ( S ′ ) f ) | k L vQ T (1 S ′ u ) k L ( v ) . By the Carleson Embedding Theorem, Theorem 11, this last factor is at most k ( P uS ′ f − E uπ D ( S ′ ) f ) k L ( u ) times the Carleson measure norm of the coefficients {k L vQ T (1 S ′ u ) k L ( v ) : Q ∈ C u ( S ′ ) } . Using (7.3) in Theorem 28 we know this at most a constant multiple of T + V , so the proof is complete. (cid:3) Now using Lemma 32 we get the desired estimate on A | A | . ( T + V ) X S ′ ∈S kP uS ′ f − S ′ E uπ D ( S ′ ) f k L ( u ) kP vS ′ g k L ( v ) . ( T + V ) X S ′ ∈S kP uS ′ f − S ′ E uπ D ( S ′ ) f k L ( u ) X S ′ ∈S kP vS ′ g k L ( v ) ! . ( T + V ) k f k L ( u ) k g k L ( v ) . That is because we have P uS ′ f = X Q ∈C u ( S ′ ) M Q − X ǫ =1 h f, h ǫQ i u h ǫQ and P vS ′ g = X S ∈C v ( S ′ ) M S − X k =1 h g, h kS i v h kS , kP vS ′ g k L ( v ) ≤ k g k L ( v ) and kP uS ′ f k L ( u ) ≤ k f k L ( u ) . Also we have X S ′ ∈S kP uS ′ f − S ′ E uπ D ( S ′ ) f k L ( u ) ≤ X S ′ ∈S kP uS ′ f k L ( u ) + X S ′ ∈S k S ′ E uπ D ( S ′ ) f k L ( u ) and X S ′ ∈S k S ′ E uπ D ( S ′ ) f k L ( u ) = X S ′ ∈S u ( S ′ ) | E uπ D ( S ′ ) f | ≤ X S ′ ∈S u ( S ′ )( E uπ S ′ | f | ) . k M u f k L ( u ) . k f k L ( u ) . Here we are using (7.2) to conclude that the maximal function M u dominates the sum above. This completesthe proof.8.2. Telescoping Arguments. We will now use telescoping arguments as in (8.3) for A . For S ′ ∈ S \{ Q } ,fixing a S ∈ C v ( S ′ ) then summing over Q , we get A ( S ′ ) := ρ ( S ′ ) X t =1 X ( Q,S ) ∈P ( π t S ( S ′ )) S ∈C v ( S ′ ) E uQ S (∆ uQ f ) h T (1 π t S ( S ′ ) u ) , ∆ uS g i v = ρ ( S ′ ) X t =1 X S ∈C v ( S ′ ) ( E uπ D u ( π t − S ( S ′ )) ( f ) − E uπ D u ( π t S ( S ′ )) ( f )) h T (1 π t S ( S ′ ) u ) , ∆ uS g i v . (8.6)This is because with S ∈ C v ( S ′ ) fixed, the sum over Q such that ( Q, S ) ∈ P ( π t S ( S ′ )) is a function of S ′ and t . The smallest cube that contributes to the sum is π D u ( π t − S ( S ′ )) which is the second parent of π t − S ( S ′ )and the largest cube that contributes to the sum is π D u ( π t S ( S ′ )). Observe the sum over S is independent ofthe sum over t in (8.6). Below we will further decompose A ( S ′ ) by adding and subtracting a cancellativeterm A ( S ′ ) = ρ ( S ′ ) X t =1 ( E uπ D u ( π t − S ( S ′ )) ( f ) − E uπ D u ( π t − S ( S ′ )) ( f )+ E uπ D u ( π t − S ( S ′ )) ( f ) − E uπ D u ( π t S ( S ′ )) ( f )) h T (1 π t S ( S ′ ) u ) , ∆ uS g i v = A ( S ′ ) + A ( S ′ ) , where we have defined: A ( S ′ ) := ρ ( S ′ ) X t =1 ( E uπ D u ( π t − S ( S ′ )) ( f ) − E uπ D u ( π t − S ( S ′ )) ( f )) h T (1 π t S ( S ′ ) u ) , ∆ uS g i v , (8.7) A ( S ′ ) := ρ ( S ′ ) X t =1 ( E uπ D u ( π t − S ( S ′ )) ( f ) − E uπ D u ( π t S ( S ′ )) ( f )) h T (1 π t S ( S ′ ) u ) , ∆ uS g i v . (8.8)Observe here that A is a telescoping term in itself, so we can sum by parts to get the following A ( S ′ ) = E uπ D u ( S ′ ) ( f ) h T (1 π t S ( S ′ ) u ) , ∆ uS g i v + ρ ( S ′ ) X t =1 E uπ D u ( π t S ( S ′ )) ( f ) T (1 π t +1 S ( S ′ ) \ π t S ( S ′ ) u ) , P vS ′ g i v = A ( S ′ )+ A ( S ′ ) . In the sum above for the missing term E uπ ρ ( S ′ ) S ( S ′ ) f = E uQ f , where Q is the largest cube that was fixed, weare going to assume the expectation is zero.Combining the steps above we can now decompose A as A = A + A + A where A i = X S ′ ∈S\ Q A i ( S ′ ) for i = 1 , , , and A ( S ′ ) := ρ ( S ′ ) X t =1 ( E uπ D u ( π t − S ( S ′ )) ( f ) − E uπ D u ( π t − S ( S ′ )) ( f )) h T (1 π t S ( S ′ ) u ) , P vS ′ g i v ;(8.9) A ( S ′ ) := ( E uπ D u ( S ′ ) f ) h T (1 π t S ( S ′ ) u ) , P vS ′ g i v ;(8.10) A ( S ′ ) := ρ ( S ′ ) X t =1 ( E uπ D u ( π t S ( S ′ )) f ) h T (1 π t +1 S ( S ′ ) \ π t S ( S ′ ) u ) , P vS ′ g i v . (8.11)Observe the expression A has cancellative terms in both f and g , so it is not a paraproduct term, while A is a paraproduct similar to A . The third term is also a paraproduct term. We will now prove the followingestimates for each of these terms: | A | . ( T + V ) k f k L ( u ) k g k L ( v ) ;(8.12) | A | . ( T + V ) k f k L ( u ) k g k L ( v ) ;(8.13) | A | . Vk f k L ( u ) k g k L ( v ) . (8.14)We will begin the proof of these estimates above starting with the term A . WO WEIGHT INEQUALITY 35 The Paraproduct Term A . Let us fix t and define A ( S ′ , t ) := ( E uπ D u ( π t S ( S ′ )) f ) h T (1 π t +1 S ( S ′ ) \ π t S ( S ′ ) u ) , P vS ′ g i v , S ′ ∈ S , ρ ( S ′ ) ≥ t ; A ( t ) := X S ′ ∈S : ρ ( S ′ ) ≥ t A ( S ′ , t ) . We can see that the t -fold parent of S ′ is defined by imposing the restriction ρ ( S ′ ) ≥ t . We want to show | A ( t ) | . δ σt Vk f k L ( u ) k g k L ( v ) t ≥ . Here the constant σ = − σ > 0, hence we get (8.14) when we will sum over t ≥ P vS ′ are orthogonal, we get the following using Cauchy-Schwarz | A ( t ) | ≤ k g k L ( v ) X S ′ ∈S : ρ ( S ′ ) ≥ t | E uπ D u ( π t S ( S ′ )) f | kP vS ′ T (1 π t +1 S ( S ′ ) \ π t S ( S ′ ) u ) k L ( v ) . Using the definition of α t ( S ) given in Section 6, the sum above is " X S ′ ∈S α t ( S ′ ) | E uπ D u ( S ′ ) f | . Now using Theorem 29 we have our desired estimate on A ( t ) as the Carleson measure norm of the coefficients { α t ( S ′ ) : S ′ ∈ S} is at most Cδ σt V .8.4. The paraproduct term A . We have π D u ( S ′ ) ⊂ π S ( S ′ ), so we will now decompose term A into twoterms by writing π S ( S ′ ) = π D u ( S ′ ) ∪ { π S ( S ′ ) \ π D u ( S ′ ) } to give us, | A | := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X S ′ ∈S ( E uπ D u ( S ′ ) f ) h T (1 π D u ( S ′ ) u ) , P vS ′ g i v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . ( T + V ) k f k L ( u ) k g k L ( v ) ;(8.15) | A | := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X S ′ ∈S ( E uπ D u ( S ′ ) f ) h T (1 π S ( S ′ ) \ π D u ( S ′ ) u ) , P vS ′ g i v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Vk f k L ( u ) k g k L ( v ) . (8.16)Using these we get (8.13). Now we will prove these inequalities. Remembering the definition of β ( S ), wecan now estimate |h T (1 π D u ( S ′ ) u ) , P vS ′ g i v | = |hP vS ′ T (1 π D u ( S ′ ) u ) , P vS ′ g i v | ≤ β ( S ′ ) kP vS ′ g k L ( v ) . As the projections are mutually orthogonal we have, summing over S ′ | A | ≤ " X S ′ ∈S β ( S ′ ) | ( E uπ D u ( S ′ ) f ) | k g k L ( v ) . ( T + V ) k f k L ( u ) k g k L ( v ) . We have used the estimate on β ( S ) in Theorem 30 to get the estimate in the last step.We can use a similar approach to prove (8.15). Using the definition of γ ( S ), we have |h T (1 π S ( S ′ ) \ π D u ( S ′ ) u ) , P vS ′ g i v | = |hP vS ′ T (1 π S ( S ′ ) \ π D u ( S ′ ) u ) , P vS ′ g i v | ≤ γ ( S ′ ) kP vS ′ g k L ( v ) . So we have the following estimate using Theorem 31, | A | ≤ " X S ′ ∈S γ ( S ′ ) | ( E uπ D u ( S ′ ) f ) | k g k L ( v ) . Vk f k L ( u ) k g k L ( v ) . The term A . Observe in the definition of A ( S ′ ) we can write the following for the difference ofexpectations, E uπ D u ( π t − S ( S ′ )) ( f ) − E uπ D u ( π t − S ( S ′ )) ( f ) = − E uπ D u ( π t − S ( S ′ )) ∆ uπ D u ( π t − S ( S ′ )) f, π t − S ( S ′ ) ∈ S . By re-indexing the sum A ( S ′ ) defined above we get A ( S ′ ) = ρ ( S ′ ) X t =1 ( E uπ D u ( π t − S ( S ′ )) ∆ uπ D u ( π t − S ( S ′ )) f ) h T (1 π t S ( S ′ ) u ) , P vS ′ g i v = A ( S ′ ) + A ( S ′ );where A ( S ′ ) := ρ ( S ′ ) X t =1 ( E uπ D u ( π t − S ( S ′ )) ∆ uπ D u ( π t − S ( S ′ )) f ) h T (1 π t S ( S ′ ) \ π t − S ( S ′ ) u ) , P vS ′ g i v , (8.17) A ( S ′ ) := ρ ( S ′ ) X t =1 ( E uπ D u ( π t − S ( S ′ )) ∆ uπ D u ( π t − S ( S ′ )) f ) h T (1 π t − S ( S ′ ) u ) , P vS ′ g i v . (8.18)We will now show that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X S ′ ∈S\{ S } A ( S ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Vk f k L ( u ) k g k L ( v ) ;(8.19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X S ′ ∈S\{ S } A ( S ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Vk f k L ( u ) k g k L ( v ) . (8.20)We can prove (8.19) using a similar approach as we did for (8.14) as there is orthogonality present with Haardifferences applied to f in (8.17).We will start the proof of (8.20) by re-indexing the sum. We have A ( S ′ , t ) := ( E uπ D u ( S ′ ) ∆ uπ D u ( S ′ ) f ) X J ∈S π t − ( J )= S ′ h T (1 S ′ u ) , P vS ′ g i v and we will prove that:(8.21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X S ′ ∈S\{ S } A ( S ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X S ′ ∈S A ( S ′ , t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . δ − σ t Vk f k L ( u ) k g k L ( v ) t ≥ . (Here the decay in t is slightly worse in comparison to the previous estimates.) We will exploit the implicitorthogonality in the sum above. Note that we have X S ′ ∈S M π D u ( S ′ ) − X i =1 |h f, h iπ S ( S ′ )) i v | ≤ k f k L ( u ) , X S ′ ∈S (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X J ∈S : π t − ( J )= S ′ P vJ g (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( v ) ≤ k g k L ( v ) , and combining these facts we get (8.21) from the estimate(8.22) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X J ∈S : π t − ( J )= S ′ P vJ T (1 π t − S ( S ′ ) u ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( v ) . δ − σ t ( T + V ) u ( π D u ( S ′ )) S ′ ∈ S , t ≥ . WO WEIGHT INEQUALITY 37 Let us now prove (8.22). We use the geometric decay in (7.2) and apply hypothesis (6.4). Fix a S ′ ∈ S and an integer w := t − . Let us denote by S ′ w , all the cubes J ∈ S with π w S ( J ) = S ′ . We have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X J ∈S : π t − ( J )= S ′ P vJ T (1 π t − S ( S ′ ) u ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( v ) = X J ∈S w B ( J ) , where B ( J ) := (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X J ′ ∈S : π t − − w ( J ′ )= S ′ P vJ ′ T (1 π t − S ( S ′ ) u ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( v ) . We now decompose B ( J ) as B ( J ) = B ( J ) + B ( J ), where B ( J ) := (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X J ′ ∈S : π t − − w ( J ′ )= S ′ P vJ ′ T (1 π t − S ( S ′ ) \ J u ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( v ) ,B ( J ) := (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X J ′ ∈S : π t − − w ( J ′ )= S ′ P vJ ′ T (1 J u ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( v ) . Using the testing condition we have X J ∈S w B ( J ) ≤ T X J ∈S w u ( J ) ≤ δ w T u ( π D u ( S ′ )) . Here we used the Carleson measure property of u on stopping cubes (7.2) to deduce the last line. Now usingthe notation of (7.5) and applying (7.1) we can see that X J ∈S w B ( J ) = X J ∈S w X J ′ ∈S : π t − − w ( J ′ )= J X I ∈J ( J ′ ) kP vI T (1 π t − S ( S ′ ) \ J u ) k L ( v ) ≤ V δ − σ t u ( π D u ( S ′ )) . This completes the proof of (8.22).9. Appendix: Hilbert Space Valued Operators Here we make precise the definitions arising in the setting of weighted norm inequalities for Hilbert spacevalued singular integrals, beginning with a Calder´on–Zygmund kernel. We define a standard B ( H , H )-valued Calder´on–Zygmund kernel K ( x, y ) to be a function K : X × X → B ( H , H ) satisfying the followingfractional size and smoothness conditions of order δ for some δ > 0: For x = y , | K ( x, y ) | B ( H , H ) ≤ C CZ V ( x, y ) , (9.1) |∇ K ( x, y ) − ∇ K ( x ′ , y ) | B ( X × X,B ( H , H )) ≤ C CZ (cid:18) d ( x, x ′ ) d ( x, y ) (cid:19) δ V ( x, y ) , d ( x, x ′ ) d ( x, y ) ≤ A , and the last inequality also holds for the adjoint kernel in which x and y are interchanged.We now turn to a precise definition of the weighted norm inequality(9.2) k T σ f k L H ( ω ) ≤ N k f k L H ( σ ) , f ∈ L ( σ ) , where σ and ω are locally finite positive Borel measures on X , and L H ( σ ) is the Hilbert space consistingof those functions f : X → H for which k f k L H ( σ ) := sZ X | f ( x ) | H dσ ( x ) < ∞ , equipped with the usual inner product. A similar definition holds for L H ( ω ). For a precise definitionof (9.2) we suppose that K is a standard B ( H , H )-valued Calder´on–Zygmund kernel, and we introduce a family n η αδ,R o <δ We say that a singular integral operator T = (cid:16) K , (cid:8) η δ,R (cid:9) <δ 1. Indeed,this follows from (9.3) and Lebesgue’s differentiation theorem for cubes. We also have the following usefulestimate. If I ′ is any of the 2 n D -children of I , and a ∈ Γ n , then(9.4) | E µI ′ h µ,aI | ≤ q E µI ′ | h µ,aI | ≤ p µ ( I ′ ) . Finally, let E µQ f ( x ) := (cid:26) E µQ f if x ∈ Q x / ∈ Q be projection onto the subspace H Q of constant H -valued functions on Q , and note that we have f = X Q ∈D △ µ H ; Q f, with convergence in L H ( µ ) since △ µ H ; Q f = X Q ′ ∈H ( Q ) E µQ ′ f − E µQ f = X Q ′ ∈H ( Q ) Q ′ (cid:16) E µQ ′ f − E µQ f (cid:17) , and the Hilbert space valued version of the dyadic Lebesgue differentiation theorem giveslim Q ց x E µQ f = f ( x ) , for µ -a.e. x ∈ X. Caution: While the scalar identity Q ′ △ µ H ; Q f = Q ′ E µQ ′ △ µQ f extends readily to the Hilbert spacesetting, the operator identity T σ (cid:16) Q ′ E µQ ′ △ µ H ; Q f (cid:17) = (cid:16) E µQ ′ △ µ H ; Q f (cid:17) T σ ( Q ′ ) in the scalar setting , fails in the Hilbert space setting where T σ : L H → L , loc H since T σ (cid:0) Q ′ E σQ ′ △ σ H ; Q f (cid:1) is a vector in H , while E σQ ′ △ σ H ; Q f is a vector H . Even if H = H = H , an operator T σ does not typicallycommute with an element in H unless H is the scalar field.Nevertheless, we can indeed take the H -norm of the element E µQ ′ △ µ H ; Q f outside the operator T σ , i.e. T σ (cid:16) Q ′ E µQ ′ △ µ H ; Q f (cid:17) = (cid:12)(cid:12)(cid:12) E µQ ′ △ µ H ; Q f (cid:12)(cid:12)(cid:12) H T σ ( Q ′ e ) , e := E µQ ′ △ µ H ; Q f (cid:12)(cid:12)(cid:12) E µQ ′ △ µ H ; Q f (cid:12)(cid:12)(cid:12) H . 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Soc., (1948), 342–354.5 Xuan Thinh Duong, Department of Mathematics, Macquarie University, NSW, 2109, Australia. E-mail address : [email protected] Ji Li, Department of Mathematics, Macquarie University, NSW, 2109, Australia. E-mail address : [email protected] Eric T. Sawyer, Department of Mathematics, McMaster University, Hamilton, Ontario, Canada. E-mail address : [email protected] Manasa N. Vempati, Department of Mathematics, Washington University – St. Louis, One Brookings Drive, St.Louis, MO USA 63130-4899 E-mail address : [email protected] Brett D. Wick, Department of Mathematics, Washington University – St. Louis, One Brookings Drive, St.Louis, MO USA 63130-4899 E-mail address : [email protected] Dongyong Yang, Department of Mathematics, Xiamen University, Xiamen 361005, China. E-mail address ::