Adjacent dyadic systems and the L p -boundedness of shift operators in metric spaces revisited
AADJACENT DYADIC SYSTEMS AND THE L p -BOUNDEDNESS OF SHIFTOPERATORS IN METRIC SPACES REVISITED OLLI TAPIOLA
Abstract.
With the help of recent adjacent dyadic constructions by Hytönen and the author,we give an alternative proof of results of Lechner, Müller and Passenbrunner about the L p -boundedness of shift operators acting on functions f ∈ L p ( X ; E ) where < p < ∞ , X is ametric space and E is a UMD space. Introduction
During the last two decades, the highly influential T (1) theorem of G. David and J.-L. Journé[7] has been generalized to various settings by different authors (e.g. [10, 11]). One of thesegeneralizations was due to T. Figiel ([9, 8], different proof by T. Hytönen and L. Weis [18]) whoproved the theorem for UMD-valued functions f ∈ L p ( R d ; E ) and scalar-valued kernels using aclever observation that any Caldéron-Zygmund operator on R d can be decomposed into sums andproducts of Haar shifts (or rearragements), Haar multipliers and paraproducts. Not long ago,P.F.X Müller and M. Passenbrunner [25] extended this technique from the Euclidean setting tometric spaces to prove the T (1) theorem for UMD-valued functions f ∈ L p ( X ; E ) , where X is anormal space of homogeneous type (see Theorems 2 and 3 in [24]). One of the key elements oftheir (and Figiel’s) proof - the L p -boundedness of the shift operators - was revisited and simplifiedby R. Lechner and Passenbrunner in their recent paper [21] by proving the result in a more generalform with different techniques.Roughly speaking, a shift operator permutates the generating Haar functions in such a waythat if h Q (cid:55)→ h P , then the dyadic cubes P and Q are not too far away from each other and theybelong to the same generation of the given dyadic system. On the real line, this can be expressedin a very simple form: for every m ∈ Z , the shift operator T m is the linear extension of the map h I (cid:55)→ h I + m | I | . In [8, Theorem 1], Figiel showed that for UMD-valued functions f : [0 , → E andfor every p ∈ (1 , ∞ ) we have the norm estimate (cid:107) T m f (cid:107) p ≤ C log (2 + | m | ) α (cid:107) f (cid:107) p (1.1)where α < depends only on E and p , and the constant C depends on E , p and α (the sameresult was formulated for functions f : R d → E in [9, Lemma 1]). In [25, Sections 4.3 - 4.5], Müllerand Passenbrunner generalized the definition of shift operators for Christ-type dyadic systems [5]in quasimetric spaces and proved the corresponding L p -estimate for these generalized operators,among other things. Lechner and Passenbrunner then generalized the definition further and gavean alternative proof for this norm estimate by providing a way to modify the underlying dyadicsystem.In this paper, we revisit and improve some results related to the recent metric adjacent dyadicconstructions by Hytönen and the author [17] and give a proof for the estimate (1.1) for UMD-valued functions f : X → E as an application. Our central idea is that with the help of adjacentdyadic systems we can split a given dyadic system D into suitable subcollections D λ that give us acovenient way to approximate certain indicator functions by their conditional expectations. Thisapproximation technique combined with some classical results of UMD-valued analysis give us afairly straightforward proof of the L p estimate. Date : September 3, 2018.2010
Mathematics Subject Classification.
Key words and phrases. metric space, adjacent dyadic systems, shift operator, UMD. a r X i v : . [ m a t h . C A ] D ec knowledgements. This paper is part of the author’s PhD project written under the supervisionof Professor Tuomas Hytönen. The author is supported by the European Union through T.Hytönen’s ERC Starting Grant 278558 “Analytic-probabilistic methods for borderline singularintegrals” and he is part of Finnish Centre of Excellence in Analysis and Dynamics Research.2.
Dyadic cubes, conditional expectations and UMD spaces
Geometrically doubling metric spaces.
Let ( X, d ) be a geometrically doubling metricspace. That is, there exists a constant M such that every ball B ( x, r ) := { y ∈ X : d ( x, y ) < r } can be covered by at most M balls of radius r/ . In this subsection we do not assume anymeasurability of ( X, d ) but we note that if ( Y, d (cid:48) , µ ) is a doubling metric measure space, then ( Y, d (cid:48) ) is a geometrically doubling metric space.We use the following two standard lemmas repeatedly in different proofs without referring tothem every time we use them. Lemma 2.1 ([12, Lemma 2.3]) . The following properties hold for ( X, d ) : Any ball B ( x, r ) can be covered by at most (cid:98) M δ − log M (cid:99) balls B ( x i , δr ) for every δ ∈ (0 , . Any ball B ( x, r ) contains at most (cid:98) M δ − log M (cid:99) centres x i of pairwise disjoint balls B ( x i , δr ) for every δ ∈ (0 , . Lemma 2.2 ([17, Lemma 2.2]) . For any δ > there exists a countable maximal δ -separated set A δ ⊆ X : • d ( x, y ) ≥ δ for every x, y ∈ A δ , x (cid:54) = y • min x ∈ A δ d ( x, z ) < δ for every z ∈ X . Since the center points of dyadic cybes (see Theorem 2.5 below) form δ k -separated sets, thefollowing simple lemma is a convenient tool for splitting dyadic systems into smaller sparse systems.We will use the lemma later in Section 3. Lemma 2.3.
Let D ≥ D > and let Z be a D -separated set of points in the space X . Then Z is a disjoint union of at most N D -separated sets where N depends only on M and D /D .Proof. First, notice that any ball of radius D can contain at most boundedly many, say M , pointsof Z by the second part of Lemma 2.1. By Lemma 2.2, we can choose a maximal D -separatedsubset Z from Z . By applying the same lemma M times, we can choose maximal D -separatedsubsets Z k ⊆ Z \ (cid:83) k − i =1 Z i for every k = 1 , , . . . , M . We claim that now Z \ (cid:83) M k =1 Z k = ∅ .For contradiction, suppose that there exists any point x ∈ Z \ (cid:83) M k =1 Z k . By maximality, B ( x, D ) ∩ Z k (cid:54) = ∅ for every k = 1 , , . . . , M since otherwise the point x would belong to one of thecollections Z k . Thus, the ball B ( x, D ) contains M + 1 points of Z , which is a contradiction. (cid:3) In the construction of metric dyadic cubes we need maximal δ k separated sets for every k ∈ Z .For this we can use Lemma 2.2 or the following stronger result: Theorem 2.4 ([17, Theorem 2.4]) . For every δ ∈ (0 , / there exist maximal nested δ k -separatedsets A k := { z kα : α ∈ N k } , k ∈ Z : • A k ⊆ A k +1 for every k ∈ Z ; • d ( z kα , z kβ ) ≥ δ k for α (cid:54) = β ; • min α d ( x, z kα ) < δ k for every x ∈ X and every k ∈ Z ,where N k = { , , . . . , n k } if the space ( X, d ) is bounded, and N k = N otherwise. Adjacent dyadic systems in metric spaces.
The following theorem is an improved versionof the famous constructions of (quasi)metric dyadic cubes by M. Christ [5] and E. Sawyer and R.L. Wheeden [27]. This version was proved by Hytönen and A. Kairema [15, Theorem 2.2] and ithas been adapted for different dyadic constructions in [17] (see [17, Theorem 2.9]) and Theorem2.6 below. heorem 2.5. Let ( X, d ) be a doubling metric space and δ ∈ (0 , be small enough. Then forgiven nested maximal sets of δ k -separated points { z kα : α ∈ A k } , k ∈ Z , there exist a countablecollection of dyadic cubes D := { Q kα : k ∈ Z , α ∈ A k } such that i) X = (cid:83) α Q kα for every k ∈ Z ; ii) P, Q ∈ D ⇒ P ∩ Q ∈ {∅ , P, Q } ; iii) B ( z kα , δ k ) ⊆ Q kα ⊆ B ( z kα , δ k ) ; iv) Q kα = (cid:83) β : Q k + mβ ⊆ Q kα Q k + mβ for every m ∈ N . For every dyadic system D and cube Q := Q jα ∈ D we use the following notation:lev ( Q ) := j, (level/generation of the cube Q ) D k := { Q kα ∈ D : α ∈ A k } , (cubes of level k ) B Q := B ( z jα , δ j ) , (ball containing cube Q ) x Q := z jα , (the center point of the cube Q ) . Like we mentioned earlier, the central idea of our techniques in Section 4 is to split a givendyadic system into suitable subcollections that help us approximate certain given indicators bytheir conditional expectations. For this we use adjacent dyadic systems which have turned out tobe a convenient tool for approximating arbitrary balls and other objects by cubes both in R n andmore abstract settings (see e.g. [20, 23]). In quasimetric spaces they were first constructed byHytönen and Kairema [15, Theorem 4.1] (based on the ideas of Hytönen and H. Martikainen [16])but by restricting ourselves to a strictly metric setting we can use systems with more powerfulproperties. The following theorem was proved recently by Hytönen and the author for n = 1 : Theorem 2.6.
Let ( X, d ) be a doubling metric space with a doubling constant M and let n ∈ N be fixed. Then for δ < / ( n · M ) there exist a bounded number of adjacent dyadic systems D ( ω ) , ω = 1 , , . . . , K = K ( δ ) , such that I) each D ( ω ) is a dyadic system in the sense of Theorem 2.5; II) for a fixed p ∈ N and fixed balls B , B , . . . , B n there exist ω ∈ { , , . . . , K } and cubes Q B , Q B , . . . , Q B n ∈ D ( ω ) such that for every i ∈ { , , . . . , n } we have i) B i ⊆ Q B i ; ii) (cid:96) ( Q B i ) ≤ δ − r ( B i ) ; iii) δ − p B i ⊆ Q ( p ) B i ,where (cid:96) ( Q ) = δ k if Q = Q kα , r ( B ) is the radius of the ball B and Q ( p ) B i is the unique dyadicancestor of Q B i of generation lev ( Q B i ) − p .Proof. In [17, Theorem 5.9] the case n = 1 was proved by showing that if B ( x, r ) is a ball suchthat δ k +2 < r ≤ δ k +1 , then P ω (cid:32)(cid:40) ω ∈ Ω : x ∈ (cid:32)(cid:91) α ∂ δ k − p +1 Q k − pα ( ω ) ∪ (cid:91) α ∂ δ k +1 Q kα ( ω ) (cid:33)(cid:41)(cid:33) ≤ M δ < (2.7)where P ω is the natural probability measure of the finite set Ω := { , , . . . , (cid:98) /δ (cid:99)} , Q ( ω ) is a cubeof the dyadic system D ( ω ) and ∂ ε A := { x ∈ A : d ( x, A c ) < ε } ∪ { x ∈ A c : d ( x, A ) < ε } . Given (2.7), the proof for general n ∈ N is simple. Let B , B , . . . , B n be balls and denote B i := B ( x i , r i ) , δ k i +2 < r i ≤ δ k i +1 . Then P ω (cid:32)(cid:40) ω ∈ Ω : x i ∈ (cid:32)(cid:91) α ∂ δ ki − p +1 Q k i − pα ( ω ) ∪ (cid:91) α ∂ δ ki +1 Q k i α ( ω ) (cid:33) for some i (cid:41)(cid:33) ≤ n · M δ < . Thus, there exists ω ∈ Ω such that x i / ∈ (cid:0)(cid:83) α ∂ δ ki − p +1 Q k i − pα ( ω ) ∪ (cid:83) α ∂ δ ki +1 Q k i α ( ω ) (cid:1) for every i = 1 , , . . . , n , which is enough to prove the claim. (cid:3) emark 2.8.
1) In the previous theorem, the constant K is roughly /δ [17, Section 5.2].Thus, for a large n both the number of systems D ( ω ) and the change of length scalebetween two consecutive levels of cubes become large.2) We will use the previous theorem only for n = 2 in the following way. Let Q , Q ∈ D k and m > be fixed. Then by Theorem 2.6 there exists an index ω and cubes P , P ∈ D ( ω ) k − such that Q ⊆ B Q ⊆ P , Q ⊆ B Q ⊆ P , mB Q ⊆ P ( p m )1 for p m ∈ N such that mδ p m ≤ .2.3. Conditional expectations.
Conditional expectations are mostly used in the field of proba-bility theory but they have turned out to be extremely useful also with many questions related tomore classical analysis (see e.g. [13]). It is well known among specialists that most of the resultsrelated to conditional expectations remain true in more general measure spaces but, unfortunately,it is difficult to find a comprehensive presentation of this extended theory in the literature. Werefer to [28] for some basic properties of conditional expectations in σ -finite measure spaces and[29, Chapter 9] for a presentation of the classical probabilistic theory of conditional expectations.Let ( X, F , µ, d ) be a metric measure space such that µ is a doubling Borel measure, i.e. thereexists a constant D := D µ such that µ (2 B ) ≤ Dµ ( B ) < ∞ for every ball B . By construction we know that if D is a dyadic system given by Theorem 2.5,then D ⊆ Bor X . In particular, the σ -algebra generated by any subcollection of D is a subset of F .Let us denote G := { G ∈ G : µ ( G ) < ∞} for every σ -algebra G ⊆ F , and let L σ ( G ) be thespace of functions that are integrable over all G ∈ G . Definition 2.9.
Let G be σ -finite sub- σ -algebra of F and let f : X → E be a F -measurablefunction where E is a Banach space. Then a G -measurable function g is a conditional expectationof f with respect to G if ˆ G f dµ = ˆ G dµ for every G ∈ G .It is not difficult to prove that if the conditional expectation exists, it is unique a.e. Thus, wedenote E [ f | G ] := g if g is a conditional expectation of f with respect to G . Concerning existence,we only need the following elementary case in this paper. Lemma 2.10.
Let A := { A i : i ∈ N } ⊆ F be a countable partition of the space X such that µ ( A i ) < ∞ for every i ∈ N and let A be the σ -algebra generated by A . Then for every f ∈ L σ ( F ) we have E [ f | A ] = (cid:88) A ∈A A (cid:104) f (cid:105) A . Proof.
Let G ∈ A . Then there exist pairwise disjoint sets A G , A G , . . . ∈ A such that G = (cid:83) i A Gi .Now ˆ G f dµ = (cid:88) i ˆ A Gi (cid:32) A Gi f dµ (cid:33) dµ = ˆ G (cid:88) i A Gi (cid:32) A Gi f dµ (cid:33) dµ = ˆ G (cid:32) (cid:88) A ∈A A A f dµ (cid:33) dµ which proves the claim. (cid:3) UMD spaces; type and cotype of Banach spaces.
Let ( X, d, F , µ ) be a metric measurespace and let ( F k ) , k = 0 , , . . . , N , be a sequence of sub- σ -algebras of F such that F k ⊆ F k +1 for all k . For simplicity, let us denote (cid:107) · (cid:107) p := (cid:107) · (cid:107) L p ( X ; E ) where (cid:107) · (cid:107) L p ( X ; E ) is the L p -Bochner norm. efinition 2.11. A sequence of functions ( d k ) Nk =1 is a martingale difference sequence if d k is F k -measurable and E [ d k | F k − ] = 0 for every k . Definition 2.12.
A Banach space ( E, (cid:107) · (cid:107) E ) is a UMD ( unconditional martingale difference ) space if for every p ∈ (1 , ∞ ) there exists a constant β p such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N (cid:88) i =1 ε i d i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ≤ β p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N (cid:88) i =1 d i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p for all E -valued L p -martingale difference sequences ( d i ) Ni =1 (i.e. ( d i ) is a martingale differencesequence such that d i ∈ L p ( X, F i ; E ) for every i ) and for all choices of signs ( ε i ) Ni =1 ∈ {− , +1 } N .UMD spaces are crucial in Banach space valued harmonic analysis due to their many goodproperties; for example, a Banach space E is a UMD space if and only if the Hilbert transform isbounded on L p ( R ; E ) [4, 3]. They give us a natural setting for analysis that is based on techniquesused in probability spaces in the following way. Let ( d i ) be a martingale difference sequence and let ( ε i ) be a sequence of random signs , i.e. independent random variables on some probability space (Ω , P ) , with distribution P ( ε i = −
1) = P ( ε i = +1) = 1 / . Then for every η ∈ Ω the sequence ( ε i ( η ) d i ) is a martingale difference sequence. In particular, the UMD property gives us (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N (cid:88) i =1 d i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p (cid:104) E ˆ Ω (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N (cid:88) i =1 ε i ( η ) d i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pp d P ( η ) /p =: (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N (cid:88) i =1 ε i d i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Ω ,p . (2.13)for every p ∈ (1 , ∞ ) .The following inequality by J. Bourgain is a standard tool in UMD valued analysis. Its originalscalar-valued version was due to E. Stein. Theorem 2.14 (See e.g. [6, Proposition 3.8]) . Let ( f k ) be a sequence of functions in L p ( X, F ; E ) and ( F k ) a sequence of σ -finite σ -algebras such that F k ⊆ F k +1 ⊆ F for every k ∈ N . Then forany sequence of random signs ( ε k ) we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) k ε k E [ f k | F k ] (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Ω ,p (cid:46) p,β p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) k ε k f k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Ω ,p . In our proofs we also need the following version of the well-known principle of contraction byJ.-P. Kahane. It holds in all Banach spaces.
Theorem 2.15 ( [19, Theorem 5 (Section 2.6)] ) . Suppose that ( ε i ) is a sequence of random signsand the series (cid:80) i ε i x i converges in E almost surely. Then for any bounded sequence of scalars ( c i ) the series (cid:80) i ε i c i x i converges in E almost surely and ˆ Ω (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) i ε i c i x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pE d P ≤ (cid:18) sup i | c i | (cid:19) p ˆ Ω (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) i ε i x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pE d P . Type and cotype of Banach spaces.
Definition 2.16.
Let ( E, (cid:107) · (cid:107) ) be a Banach space. We say that E has type t ∈ [1 , if there existsa constant C t > such that for every finite sequence ( x i ) in E and finite sequence ( ε i ) of randomsigns we have ˆ Ω (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) i ε i x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E d P ≤ C t (cid:32)(cid:88) i (cid:107) x i (cid:107) t (cid:33) /t . In a similar fashion, we say that E has cotype q ∈ [2 , ∞ ] if there exists a constant C q > suchthat (cid:32)(cid:88) i (cid:107) x i (cid:107) q (cid:33) /q ≤ C q ˆ Ω (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) i ε i x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E d P . he notion of type and cotype of Banach spaces was introduced by B. Maurey and G. Pisierin the 1970’s and it has become an important part of analysis on Banach spaces. Out of this richtheory, we need the following results:i) If Y is a σ -finite measure space and E is a Banach space of type r and cotype s , then L p ( X ; E ) has type min { p, r } and cotype max { p, s } .ii) If E is a UMD space, then E has a non-trivial type s > and non-trivial cotype t < ∞ .For proofs, see e.g. [22, Chapter 9] for i) and [2, Theorem 11.1.14], [26, Proposition 3] for ii).2.5. Structural constants.
We say that c is a structural constant if it depends only on thedoubling constant D , the UMD constant β p for a fixed p ∈ (1 , ∞ ) and the type and cotypeconstants C t and C q . We do not track the dependencies of our bounds on the structural constantsand thus, we use the notation a (cid:46) b if a ≤ cb for some structural constant c and a (cid:104) b if a (cid:46) b (cid:46) a .3. Embedding cubes into larger cubes
In this section we prove a decomposition result for dyadic systems using Theorem 2.6. Weformulate the result in such a way that it is easy to apply it in Section 4 but we note that it issimple to modify the proof for other similar decompositions.Let D be a dyadic system with δ < / (2 · M ) and { D ω } ω be adjacent dyadic systems forthe same δ given by Theorem 2.6. Let us fix a number m ≥ and an injective function τ : D → D such that τ ( Q ) ⊆ mB Q for every Q = Q kα ∈ D and τ D k ⊆ D k for every k ∈ Z . Proposition 3.1.
The system D is a disjoint union of a bounded number of subcollections D λ ⊆ D , λ = ( i, j, ω ) , with the following property: for every Q ∈ D λ there exist cubes P Q , P τ ( Q ) ∈ D ( ω ) k − and P ∗ Q ∈ D ( ω ) k − − T , where mδ T ≤ , such that Q ⊆ P Q , τ ( Q ) ⊆ P τ ( Q ) , P Q ∪ P τ ( Q ) ∪ mB Q ⊆ P ∗ Q ; (3.2) if Q , Q ∈ D λ ∩ D k , Q (cid:54) = Q , then ( P Q ∪ P τ ( Q ) ) ∩ ( P Q ∪ P τ ( Q ) ) = ∅ ; (3.3) if Q , Q ∈ D λ , Q (cid:40) Q , then P ∗ Q ⊆ P Q . (3.4)In other words, we split the collection D into sparse subcollections D λ such that we can embedevery cube Q ∈ D λ and its image τ ( Q ) into some larger cubes P Q and P τ ( Q ) such that P Q and P τ ( Q ) belong to the same dyadic system and they have a mutual dyadic ancestor P ∗ Q .We form the sets D λ with the help of next technical lemma. Lemma 3.5.
The collection D is a disjoint union of L = L ( X ) subcollections Q i such that forevery k ∈ Z and Q , Q ∈ Q i ∩ D k we have δ − B R ∩ δ − B R = ∅ where R ∈ { Q , τ ( Q ) } and R ∈ { Q , τ ( Q ) } , R (cid:54) = R , and the number L is independent of m .Proof. Basically, we only need to use basic properties of geometrically doubling metric spaceswith the help of the observation that if
Q, P ∈ D k and d ( x ( Q ) , x ( P )) ≥ δ k − , then δ − B Q ∩ δ − B P = ∅ .Let k ∈ Z be fixed. For any subcollection Q ⊆ D k and any set A of center points of cubes, letus denote Y Q := { x ( Q ) : Q ∈ Q } , D A := { Q ∈ D : x ( Q ) ∈ A } . We split the set Y D k into smaller sets in three steps. To keep our notation simple, i is an indexwhose role may change from one occurence to the next.1) By Lemma 2.3, we can split the δ k -separated set Y D k into a bounded number of δ k − -separated subsets Y i,k .2) For every Q ∈ D Y i,k , the ball δ − B Q intersects at most a bounded number of balls δ − B τ ( P ) where P ∈ D Y i,k . Thus, we can split the set Y i,k into a bounded number ofsubsets Y i,k such that δ − B Q ∩ δ − B τ ( P ) = ∅ for every Q, P ∈ D Y i,k , Q (cid:54) = τ ( P ) . ) For every Q ∈ D Y i,k , the ball δ − B τ ( Q ) intersects at most a bounded number of balls δ − B τ ( P ) , P ∈ D Y i,k . Thus, we can split the set Y i,k into a bounded number of subsets Y i,k such that δ − B τ ( Q ) ∩ δ − B τ ( P ) = ∅ for every Q, P ∈ D Y i,k , Q (cid:54) = P .Now we can set Q i := (cid:83) k ∈ Z D Y i,k for every i . (cid:3) Let { Q i } i be the partition of D given by the previous lemma and let T ∈ N , T ≥ , be thesmallest number such that mδ T ≤ . Recall Theorem 2.6 and denote γ ( R ) := min (cid:110) ω : Q B R , Q B τ ( R ) ∈ D ( ω ) , δ − T B R ⊆ Q ( T ) B R (cid:111) for every cube R ∈ D and Q i,ω := { R ∈ Q i : γ ( R ) = ω } for every i = 1 , , . . . , L and ω = 1 , , . . . , K . Then the collections Q i,ω satisfy properties (3.2)and (3.3) but they are still not suitable for property (3.4). Thus, we split collections Q i,ω intosmaller collections whose cubes have large enough generation gaps: we set D i,j,ω := (cid:91) k ∈ Z (cid:0) Q i,ω ∩ D j +4 kT (cid:1) for every j = 0 , , . . . , T − . Notice that the indices i , j and ω are independent of each other. Proof of Proposition 3.1.
Clearly we only need to show the claim for the collections D i, ,ω =: D i .RecallNotice first that m · r ( B Q ) = 6 mδ kT ≤ δ − T δ kT = δ − T · r ( B Q ) for every Q := Q kTα ∈ D i . Thus, by Remark 2.8 and the definition of D i , for every cube Q ∈ D i there exist cubes P Q , P τ ( Q ) ∈ D ( ω ) kT − such that B Q ⊆ P Q , B τ ( Q ) ⊆ P τ ( Q ) , mB Q ⊆ P ( T ) Q =: P ∗ Q . Let us then show that the cubes P Q , P τ ( Q ) and P ∗ Q satisfy properties (3.2) - (3.4).(3.2) Since Q, τ ( Q ) ⊆ mB Q , we know that P Q ∩ P ∗ Q (cid:54) = ∅ and P τ ( Q ) ∩ P ∗ Q (cid:54) = ∅ . Thus, since D ( ω ) is a dyadic system and lev ( P ∗ Q ) < lev ( P Q ) = lev ( P τ ( Q ) ) , we have P Q ∪ P τ ( Q ) ⊆ P ∗ Q .(3.3) Since x ( Q ) ∈ P Q for every cube Q ∈ D , we have P Q ⊆ B ( x ( P Q ) , δ kT − ) ⊆ B ( x ( Q ) , δ kT − ) = 2 δ − B Q for every cube Q ∈ D . Thus, the property (3.3) follows directly from Lemma 3.5.(3.4) Suppose that R (cid:40) Q := Q kTα . Then lev(R) ≥ (4 k + 4) T and thus, lev ( P R ) ≥ (4 k + 4) T − and lev ( P ∗ R ) ≥ (4 k + 4) T − − T ≥ kT = lev ( Q ) ≥ lev ( P Q ) since T ≥ . In particular, P ∗ R ⊆ P Q since P ∗ R , P Q ∈ D ( ω ) and D ( ω ) is a dyadic system. (cid:3) L p -boundedness of shift operators In this section, we show that with the help of Proposition 3.1 we can give a straightforwardproof for the L p -boundedness of the shift operators in doubling metric measure spaces. We followsome ideas of [8] and [21] but mostly we rely on our own dyadic constructions.Let ( X, d ) be a metric space, µ a doubling Borel measure on X and ( E, (cid:107) · (cid:107) ) an UMD space.Since the doubling property of µ implies the geometrical doubling property of d , there exists afinite geometrical doubling constant M . Thus, we may fix a dyadic system D for δ < / (2 · M ) and adjacent dyadic systems { D ( ω ) } ω given by Theorem 2.5 for the same δ . .1. Haar functions.
There are various different ways to construct Haar functions in metricspaces (see e.g. [1, Section 5]) and thus, we do not want to fix any particular construction. Wedo, however, refer to the construction in [14, Section 4] (with the choice b ≡ ) for a system ofHaar functions that satisfy the properties in the following definition. In [14] the construction isdone in R n for a non-doubling measure but it is simple to generalize the result for our setting. Definition 4.1.
A collection of functions h θQ : X → R , Q := Q kα ∈ D , θ = 1 , . . . , n ( Q ) ≤ Θ , is a system of Haar functions if it satisfies the following properties: for every Q and θ we have • supp h θQ ⊆ Q ; • h θQ is constant on every child cube Q k +1 β ⊆ Q ; • ´ h θQ = 0 = ´ h θQ h θ (cid:48) Q if θ (cid:54) = θ (cid:48) ; • (cid:107) h θQ (cid:107) = 1 ;and the space of finite linear combinations of the functions h θQ is dense in L ( X ; E ) .The number Θ in the previous definition depends only on M or, more precisely, the maximumnumber of child cubes Q k +1 β a cube Q kα can have. Henceforth, we fix some θ = θ ( Q ) for each Q ∈ D and drop the dependency on θ in the notation.Let h Q = (cid:80) k v k Q k be a Haar function, where Q k are the child cubes of Q . The followingproperties are straightforward consequences of the previous definition: (cid:107) h Q (cid:107) ∞ = max | v k | (cid:104) µ ( Q ) / ; (4.2) (cid:107) h Q (cid:107) (cid:104) µ ( Q ) / . (4.3)In particular, Q k ( x ) µ ( Q k ) / (cid:46) | h Q ( x ) | (cid:46) Q ( x ) µ ( Q ) / for every x ∈ Q and some Q k . (4.4)The previous properties give us the following lemma: Lemma 4.5.
For every p ∈ (1 , ∞ ) and finite collection of cubes Q we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) Q x Q h Q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p (cid:104) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) Q ε Q x Q Q µ ( Q ) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Ω ,p . Proof.
Let us denote (cid:80) Q x Q h Q = (cid:80) k (cid:80) α x Q kα h Q kα and let ( ε Q ) be a sequence of random signs.Then for every y ∈ X and k ∈ Z there exists at most one Q kα,y such that h Q kα,y ( y ) (cid:54) = 0 . Let σ yk ∈ {− , +1 } be such that σ yk h Q kα,y ( y ) = | h Q kα,y ( y ) | for every y ∈ X and k ∈ Z . Then, for afixed y ∈ X , ( σ yk ε Q kα,y ) k is a sequence of random signs. Since the functions h Q form a martingaledifference sequence and by (4.4) we know that | h Q | µ ( Q ) / (cid:46) for every Q , we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) Q x Q h Q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pp (cid:104) ˆ X ˆ Ω (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) k σ yk ε Q kα,y ( η ) x Q kα,y h Q kα,y ( y ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pE d P ( η ) dµ ( y )= ˆ X ˆ Ω (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) k ε Q kα,y ( η ) x Q kα,y µ ( Q kα,y ) / (cid:12)(cid:12)(cid:12) h Q kα,y ( y ) (cid:12)(cid:12)(cid:12) µ ( Q kα,y ) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pE d P ( η ) dµ ( y ) (cid:46) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) Q ε Q x Q Q µ ( Q ) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p Ω ,p . by the UMD property of E , Fubini’s theorem and Kahane’s contraction principle. Let us thendenote (cid:80) Q x Q h Q = (cid:80) Ni =1 x i h Q i where lev ( Q ) ≤ lev ( Q ) ≤ . . . ≤ lev ( Q N ) . Then by Lemma2.10 we have E [ | h Q i || F i ] = 1 Q (cid:104)| h Q |(cid:105) Q where F i be the σ -algebra generated by D lev ( Q i ) . Thus, ince / ( µ ( Q ) / (cid:104)| h Q |(cid:105) Q ) (cid:104) , the previous estimates, Stein’s inequality and Kahane’s contractionprinciple (in this order) give us (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N (cid:88) i =1 x i h Q i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pp (cid:104) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N (cid:88) i =1 ε i x i | h Q i | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p Ω ,p (cid:38) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N (cid:88) i =1 ε i x i E [ | h Q i || F i ] (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p Ω ,p = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N (cid:88) i ε i x i Q i µ ( Q i ) / µ ( Q i ) / (cid:104)| h Q i |(cid:105) Q i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p Ω ,p (cid:38) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N (cid:88) i ε i x i Q i µ ( Q i ) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p Ω ,p , which proves the claim. (cid:3) Shift operators.
Let us fix the number m ≥ and let τ : D → D be an injective functionsuch that1) τ D k ⊆ D k for every k ∈ Z ;2) for every Q ∈ D we have τ ( Q ) ⊆ mB Q ;3) the measures of cubes Q and τ ( Q ) are approximately the same: µ ( Q ) (cid:104) µ ( τ ( Q )) . (4.6)Let { h Q } Q ∈ D be a system of Haar functions. Then we can define the shift operator T := T τ asthe linear extension of the operator ˆ T , ˆ T h Q = h τ ( Q ) . It is easy to see that without condition (4.6) an estimate of the type (1.1) is out of reach for all p ∈ (1 , ∞ ) . More precisely: by property (4.4) we have (cid:107) h Q (cid:107) p (cid:104) µ ( Q ) /p − / for every cube Q andthus, without condition (4.6) the estimate cannot hold simultaneously for all p ∈ (1 , and for all q ∈ (2 , ∞ ) . We note that the condition (4.6) is automatically valid in metric measure spaces thatsatisfy an Ahlfors-regularity type condition.4.3. L p -boundedness of shift operators. Using Proposition 3.1 and Lemma 4.5 we can nowprove the following theorem quite easily.
Theorem 4.7.
Let p ∈ (1 , ∞ ) and f ∈ L p ( X ; E ) . Then (cid:107) T f (cid:107) p ≤ C (log(2 m ) + 1) α (cid:107) f (cid:107) p where C = C ( p, X, E, α ) , α = 1 / min { t E , p } − / max { q E , p } < and t E and q E are the type andcotype of the space E .Proof. Suppose that f ∈ L p ( X ; E ) . Then, by the properties of the Haar functions and Proposition3.1, we may assume that the function f is of the form f = L (cid:88) i =1 4 T − (cid:88) j =0 K (cid:88) ω =1 (cid:88) Q ∈ D i,j,ω x Q h Q where x Q (cid:54) = 0 only for finitely many Q . Thus, we can denote f = (cid:80) i,j,ω (cid:80) nk =1 x k h Q k wherelev ( Q ) ≤ lev ( Q ) ≤ . . . ≤ lev ( Q n ) .For every k = 1 , , . . . , n , let F k be the σ -algebra generated by F k := D ( ω ) lev ( Q k ) − \ (cid:91) l =1 ,...,n lev ( Q l )= lev ( Q k ) (cid:8) P Q l , P τ ( Q l ) (cid:9) ∪ (cid:91) l =1 ,...,n lev ( Q l )= lev ( Q k ) (cid:8) P Q l ∪ P τ ( Q l ) (cid:9) . otice that if lev ( Q k ) = lev ( Q k ) , then F k = F k . By property (3.3) we know that F k is apartition of the space X and by property (3.4) we know that the sequence ( F k ) is nested. Thus,for every k = 1 , , . . . , n we have E [1 Q k | F k ] . = 1 P Qk ∪ P τ ( Qk ) (cid:104) Q k (cid:105) P Qk ∪ P τ ( Qk ) (4.6) (cid:104) P Qk ∪ P τ ( Qk ) µ ( Q k ) µ ( P Q k ) (cid:104) P Qk ∪ P τ ( Qk ) . (4.8)In particular, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) k x k h τ ( Q k ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p . (cid:104) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) k ε k x k µ ( τ ( Q k )) / τ ( Q k ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Ω ,p . (4.6) (cid:46) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) k ε k x k µ ( Q k ) / P Qk ∪ P τ ( Qk ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Ω ,p (4.8) (cid:104) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) k ε k x k µ ( Q k ) / E [1 Q k | F k ] (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Ω ,p . (cid:46) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) k ε k x Q k µ ( Q k ) / Q k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Ω ,p . Hence, since by Section 2.4.1 the space L p ( X ; E ) has a non-trivial type t > and a non-trivialcotype q < ∞ , we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) i,j,ω (cid:88) k x k h τ ( Q k ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p (cid:46) (cid:88) i,j,ω (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) k ε k x Q k µ ( Q k ) / Q k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) t Ω ,p /t ≤ (4 T KL ) /t − /q (cid:88) ω,i,j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) k ε k x Q k µ ( Q k ) / Q k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) q Ω ,p /q (cid:46) T /t − /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) ω,i,j (cid:88) k ε k x Q k µ ( Q k ) / Q k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Ω ,p . (cid:46) (log(2 m ) + 1) /t − /q (cid:107) f (cid:107) p by Hölder’s inequality. (cid:3) References [1] Hugo Aimar, Ana Bernardis, and Bibiana Iaffei. Multiresolution approximations and unconditional bases onweighted Lebesgue spaces on spaces of homogeneous type.
J. Approx. Theory , 148(1):12–34, 2007.[2] Fernando Albiac and Nigel J. Kalton.
Topics in Banach space theory , volume 233 of
Graduate Texts in Math-ematics . Springer, New York, 2006.[3] Jean Bourgain. Some remarks on Banach spaces in which martingale difference sequences are unconditional.
Ark. Mat. , 21(2):163–168, 1983.[4] Donald L. Burkholder. A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions. In
Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago,Ill., 1981) , Wadsworth Math. Ser., pages 270–286. Wadsworth, Belmont, CA, 1983.[5] Michael Christ. A T ( b ) theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. ,60/61(2):601–628, 1990.[6] P. Clément, B. de Pagter, F. A. Sukochev, and H. Witvliet. Schauder decomposition and multiplier theorems.
Studia Math. , 138(2):135–163, 2000.[7] Guy David and Jean-Lin Journé. A boundedness criterion for generalized Calderón-Zygmund operators.
Ann.of Math. (2) , 120(2):371–397, 1984.[8] Tadeusz Figiel. On equivalence of some bases to the Haar system in spaces of vector-valued functions.
Bull.Polish Acad. Sci. Math. , 36(3-4):119–131 (1989), 1988.[9] Tadeusz Figiel. Singular integral operators: a martingale approach. In
Geometry of Banach spaces (Strobl,1989) , volume 158 of
London Math. Soc. Lecture Note Ser. , pages 95–110. Cambridge Univ. Press, Cambridge,1990.[10] M. Frazier, Y.-S. Han, B. Jawerth, and G. Weiss. The T theorem for Triebel-Lizorkin spaces. In Harmonicanalysis and partial differential equations (El Escorial, 1987) , volume 1384 of
Lecture Notes in Math. , pages168–181. Springer, Berlin, 1989.
11] Y.-S. Han and Steve Hofmann. T1 theorems for Besov and Triebel-Lizorkin spaces. Trans. Amer. Math. Soc. ,337(2):839–853, 1993.[12] Tuomas Hytönen. A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa.
Publ. Mat. , 54(2):485–504, 2010.[13] Tuomas Hytönen. The sharp weighted bound for general Calderón-Zygmund operators.
Ann. of Math. (2) ,175(3):1473–1506, 2012.[14] Tuomas Hytönen. The vector-valued nonhomogeneous
T b theorem.
Int. Math. Res. Notices IMRN , online,2012.[15] Tuomas Hytönen and Anna Kairema. Systems of dyadic cubes in a doubling metric space.
Colloq. Math. ,126(1):1–33, 2012.[16] Tuomas Hytönen and Henri Martikainen. Non-homogeneous
T b theorem and random dyadic cubes on metricmeasure spaces.
J. Geom. Anal. , 22(4):1071–1107, 2012.[17] Tuomas Hytönen and Olli Tapiola. Almost Lipschitz-continuous wavelets in metric spaces via a new random-ization of dyadic cubes.
J. Approx. Theory , 185:12–30, 2014.[18] Tuomas Hytönen and Lutz Weis. A T theorem for integral transformations with operator-valued kernel. J.Reine Angew. Math. , 599:155–200, 2006.[19] Jean-Pierre Kahane.
Some random series of functions , volume 5 of
Cambridge Studies in Advanced Mathe-matics . Cambridge University Press, Cambridge, second edition, 1985.[20] Anna Kairema. Two-weight norm inequalities for potential type and maximal operators in a metric space.
Publ. Mat. , 57(1):3–56, 2013.[21] Richard Lechner and Markus Passenbrunner. Adaptive deterministic dyadic grids on spaces of homogeneoustype.
Bull. Pol. Acad. Sci. Math. , 62(2):139–160, 2014.[22] Michel Ledoux and Michel Talagrand.
Probability in Banach spaces , volume 23 of
Ergebnisse der Mathematikund ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] . Springer-Verlag, Berlin, 1991.Isoperimetry and processes.[23] Kangwei Li and Wenchang Sun. Characterization of a two weight inequality for multilinear fractional maximaloperators.
Houston Journal of Mathematics (to appear) , 2015. arxiv: 1312.7707.[24] Roberto A. Macías and Carlos Segovia. Lipschitz functions on spaces of homogeneous type.
Adv. in Math. ,33(3):257–270, 1979.[25] Paul F. X. Müller and Markus Passenbrunner. A decomposition theorem for singular integral operators onspaces of homogeneous type.
J. Funct. Anal. , 262(4):1427–1465, 2012.[26] José L. Rubio de Francia. Martingale and integral transforms of Banach space valued functions. In
Probabilityand Banach spaces (Zaragoza, 1985) , volume 1221 of
Lecture Notes in Math. , pages 195–222. Springer, Berlin,1986.[27] E. Sawyer and R. L. Wheeden. Weighted inequalities for fractional integrals on Euclidean and homogeneousspaces.
Amer. J. Math. , 114(4):813–874, 1992.[28] Hitoshi Tanaka and Yutaka Terasawa. Positive operators and maximal operators in a filtered measure space.
J. Funct. Anal. , 264(4):920–946, 2013.[29] David Williams.
Probability with martingales . Cambridge Mathematical Textbooks. Cambridge UniversityPress, Cambridge, 1991.
Olli Tapiola, Department of Mathematics and Statistics, P.O.B. 68 (Gustaf Hällströmin katu2b), FI-00014 University of Helsinki, Finland
E-mail address : [email protected]@helsinki.fi