Two Weight Problems and Bellman Functions on filtered probability spaces by Jingguo Lai B. S., Fudan University; Shanghai, China, 2008 M. S., Michigan State University; East Lansing, MI, 2010 A Dissertation submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in the Department of Mathematics at Brown University Providence, Rhode Island May 2015 c Copyright 2015 by Jingguo Lai This dissertation by Jingguo Lai is accepted in its present form by the Department of Mathematics as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date Sergei Treil, Advisor Recommended to the Graduate Council Date Jill Pipher, Reader Date Brian Cole, Reader Approved by the Graduate Council Date Peter M. Weber, Dean of the Graduate School iii Curriculum Vita Jingguo Lai was born in Shenyang, Liaoning, P.R. China on Feburary 11, 1985 to Changbin Lai and Lijie Xue. He completed the B.S. at Fudan University on July 2008 and the M.S. at Michigan State University on July 2010. After graduating, he continued the study of mathematics at Brown University. He married Maggie Fong in the summer of 2013. Jingguo completed this thesis under the supervision of Sergei Treil. iv Dedicated to my beloved parents: Changbin Lai and Lijie Xue v Acknowledgements To my advisor, Prof. Sergei Treil for his invaluable mentoring over the past five years. He suggested these problems, taught me the right way of thinking process, and provided guidance and ideas for all the tough steps. To my readers, Prof. Jill Pipher and Prof. Brian Cole for their careful reading, gentle criticism, and insightful edits. To Prof. Justin Holmer for his help and support along the way. To the wonderful mathematics department staff, particularly Audrey Aguiar, Larry Larivee, and Doreen Pappas. To my parents Changbin Lai and Lijie Xue, and my wife Maggie Fong for their love and support. To my cat Gigi for bringing me so much fun. vi Contents Curriculum Vita iv Dedication v Acknowledgements vi Chapter 1. Introduction 1 1. Two Weight Problems 1 2. Bellman Functions on filtered probability spaces 5 3. Outline of the thesis 9 Chapter 2. Two weight estimates for a vector-valued positive operators 10 1. The case 1 < p ≤ q 10 2. The case q < p < ∞: a counterexample 11 3. The case q < p < ∞: necessity and sufficiency 12 Chapter 3. Two weight estimates for paraproducts 17 1. Construction of the stopping intervals 17 2. The case 1 < p ≤ 2 18 3. The case 2 < p < ∞: a counterexample 19 4. The case 2 < p < ∞: trilinear forms and necessity 21 5. The case 2 < p < ∞: from trilinear forms to shifted bilinear forms 22 6. The case 2 < p < ∞: sufficiency 24 6.1. A modified stopping interval construction 25 6.2. Estimation of T1 26 6.3. Estimation of T2 30 vii Chapter 4. Bellman functions on filtered probability spaces I: Burkh¨older’s hull and Super-solutions 33 1. Properties of the Bellman function B(F, f, M ; C) 33 2. Properties of the Super-solutions 34 2.1. The super-solutions and the dyadic Caleson Embedding Theorem 34 2.2. Further properties of B(F, f, M ; C) 35 2.3. Regularization of the super-solutions 36 2.4. The main inequality in its infinitesimal version 37 3. Finding a super-solution via the Burkh¨older’s hull 38 3.1. Burkh¨older’s hull and some reductions 38 3.2. The formula of the Burkh¨older’s hull and an explicit super-solution 40 Chapter 5. The Bellman functions on filtered probability spaces II: Remodeling and proof of the main theorems 42 1. Properties of the Bellman function BµF (F, f, M, C) 42 2. Remodeling of the Bellman function B(F, f, M ; C = 1) for an infinitely refining filtration 43 3. The Bellman function BµF (F, f, M ; C) of Theorem 1.13 46 3.1. BµF (F, f , M ) ≤ B(F, f , M ) 46 3.2. BµF (F, f , M ) = B(F, f , M ) for an infinitely refining filtration 47 4. The Bellman function BeµF (F, f) of the maximal operators 50 4.1. BeµF (F, f ) ≤ BµF (F, f , 1) 50 4.2. BeµF (F, f ) = BµF (F, f , 1) for an infinitely refining filtration 51 Bibliography 54 viii Abstract of ”Two Weight Problems and Bellman Funcfions on filtered probability spaces” by Jingguo Lai, Ph.D., Brown University, May 2015 Chapter 1 provides the necessary background and states the main results of the thesis. Two seperate topics are studied. The first topic is on two weight problems. The second topic is on Bellman functions on filtered probability spaces. Chapter 2 proves a two weight estimation for a vector-valued positive operator. We consider two different cases of this theorem. The easier case 1 < p ≤ q requires only one testing condition. However, we construct a counterexample showing this testing condtion alone is not sufficient for the case q < p < ∞. We apply the Rubio de Francia Algorithm to reduce our problem to the well-known two weight estimates for positive dyadic operators. Chapter 3 proves a two weight estimation for paraproducts. We again consider two different cases seperately. The first few steps of the proof proceeds exactly the same as in Chapter 2. However, for the harder case 2 < p < ∞, we need to characterize a two weight inequality for shifted bilinear forms, which takes up the majority of this chapter. Chapter 4 considers the celebrated Dyadic Carleson Embedding Theorem. We streamline a way of finding a super-solution of the Bellman function via the Burkh¨older’s hull. We give an explicit formula of the Burkh¨older’s hull and hence a super-solution in this chapter. Chapter 5 generalizes the Dyadic Carleson Embedding Theorem to the filtered probability spaces and proves the coincidence of the Bellman functions on an infinite refining filtration. The proof requires a remodeling of the Dyadic Carleson Embedding Theorem. Finally, we also consider the Bellman function of the Doob’s Martingale Inequality. CHAPTER 1 Introduction In this chapter we provide some useful background on the topics of this thesis. First we establish the general setup of two weight problems and raise the questions of interest. Then we introduce a well-known application of the Bellman function techniques and pose the questions we want to solve. Further we provide a brief outline of this thesis. 1. Two Weight Problems The original question about two weight estimates is to find a necessary and suf- ficient condition on the weights (non-negative locally integrable functions) w and v such that an operator T : Lp (w) → Lp (v) is bounded for all 1 < p < ∞, i.e. the inequality Z Z p p (1.1) |T f | vdx ≤ C · |f |p wdx, for f ∈ Lp (w). Let u = w−p/p . A symmetric formulation of (1.1), well-known from 80s, is Z Z (1.2) |T (uf )| vdx ≤ C · |f |p udx, for f ∈ Lp (u). p p (1.2) looks more natural than (1.1) in the two weight setting: in particular, if T is an integral operator, then the integration in the operator is performed with respect to the same measure udx as in the domain. Denote µ = udx and ν = vdx. Let T µ (f ) := T (µf ). We can rewrite (1.2) into Z Z (1.3) |T (f )| dν ≤ C · |f |p dµ, for f ∈ Lp (µ). µ p p Two weight problems are notoriously hard. The first few results are for T being • Hardy operator by Muckenhoupt [1]. • Maximal operators by Sawyer [2], where a testing condition is introduced. 1 • Fractional integrals by Sawyer and Wheeden [3] and [4]. For all these special operators, a characterization for all 1 < p < ∞ is given. In particlar, Fractional integrals are examples of positive operators which are relatively easier and more or less solved. To highlight some results of discrete positive operators, we have results for T being • Positive dyadic operators and p = 2 in [5]. • Positive dyadic operators and 1 < p < ∞ in [6]. • Vector-valued positive dyadic operators and 1 < p < ∞ in [9]. Several simplied proofs for the results listed above are also found. For example, • [7] and [8] simplifies the proof given in [6]. • [10] simplifies the proof given in [9]. In rescent years, there is a breakthrough on this problem for singular operators. Initiated by Nazarov, Treil and Volberg, and followed by Lacey, Sawyer, Uriarte-Tuero et al., we have the following results for T being • Haar multipliers in [5], which is the first case of discrete singular operators. • Well localized operators including Haar shifts in [11]. • Sufficient conditions for Calder´on-Zygmund singular integral operators, and necessary and sufficient conditions for Calder´on-Zygmund singular integral operators together with two maximal operators in [12]. • Hilbert Transform in [13] and [14]. • Cauchy Transform in [15]. • Riesz Transform in [16]. Two weight problems for general Calder´on-Zygmund singular integral operators re- main unsolved. Note the original Two Weight Problems (1.1), (1.2), (1.3) make sence for all 1 < p < ∞. However, rescent results in [5], [11]-[16] only consider for p = 2. The two weight problems we are interested in are both discrete ones. The first is a new two weight estimates for Vector-valued positive operators when 1 < p < ∞. 2 The second is a two weight estimates for Paraproducts when 1 < p < ∞. Our setup follows from the one in [17], which is more general than the dyadic case. Definition 1.1. For a measurable space (X , T ), a lattice L ⊆ T is a collection of measurable subsets of X with the following properties (i) L is a union of generations Ln , n ∈ Z, where each generation is a collection of disjoint measurable sets (call them intervals), covering X . (ii) For each n ∈ Z, the covering Ln+1 is a countable refinement of the covering Ln , i.e. each interval I ∈ Ln is a countable union of disjoint intervals J ∈ Ln+1 . We allow the situation where there is only one such interval J, i.e. J = I; this means that I ∈ Ln also belongs to the generation Ln+1 . Definition 1.2. For an interval I ∈ L, let rk(I) be the rank of the interval I, i.e. the largest number n such that I ∈ Ln . For an interval I ∈ L, rk(I) = n, a child of I is an interval J ∈ Ln+1 such that J ⊆ I (actually, J $ I). The colletion of all children of I is denoted by child(I). Correspondingly, I is called the parent of J. Definition 1.3. For a positive measure µ on (X , T ) , define the averaging oper- ator as  Z  µ −1 (1.4) E f = hf i 1 = µ(I) f dµ 1 . I I,µ I I I where 1 is the indicator function of the interval I. The martingale difference operator I is then defined to be X (1.5) ∆µ f = −Eµ f + Eµ f. I I J J∈child(I) From now on, we assume (X , T ) is a measurable space, L ⊆ T is a lattice on X , and µ, ν are two positive measures. We denote the conjugate H¨older exponent of p by p0 , where 1/p + 1/p0 = 1. Here, and throughout the thesis, we use the notation A . B meaning that there exists an absolute constant C, such that A ≤ CB, and we write A ≈ B if A . B . A. 3 Definition 1.4. Let α = {α : I ∈ L} be non-negative constants associated to I a lattice L on (X , T ). Define a vector-valued operator (1.6) Tαµ f = {α · Eµ f } . I I I∈L Theorem 1.5 (Two weight estimates for a vector-valued positive operator [18]). Let 1 < p < ∞ and 1 ≤ q < ∞. Z "X #p q q Z µ p (1.7) α · E f dν ≤ C · |f |p dµ, I I X I∈L X holds if and only if (i) for the case 1 < p ≤ q, we have Z X p q (1.8) αq · 1 dν ≤ C1p · µ(J), J ∈ L I J I I∈L:I⊆J (ii) for the case q < p < ∞, we have both (1.8) and Z X ( p ) 0 ν(I) q q ( pq ) 0 q (1.9) α · ·1 dµ ≤ C2 · ν(J), J ∈ L. J I∈L:I⊆J I µ(I) I In particular, C ≈ C1 + C2 . Remark 1.6. Vector-valued positive operators in the thesis are viewed as a simple model of the paraproducts defined below. Definition 1.7. For a measurable function b, the paraproduct operator with sym- bol b is X   (1.10) πbµ f = Eµ f ∆ν b . I I I∈L Paraproducts play an important role in the investigation of the weighted inequal- ities for the singular integral operators. The L2 -boundedness of paraproducts is easy, a necessary and sufficient condition (1.12) follows immediately from the Carleson Embedding Theorem. This necessary and sufficient condition (1.12) can be stated as a testing condition, i.e. a paraproduct is bounded in L2 if and only if there is a uniform estimate on all 4 intervals. In the classical non-weighted situation, the L2 boundedness is equivalent to the boundedness of the paraproduct in all Lp , 1 < p < ∞. The weighted situation is much more interesting. It was shown in [17] that in the one weight situation, the testing condition (1.12) is still necessary and sufficient, but it now depends on p: the boundedness in Lp0 implies the boundedness in Lp with 1 < p ≤ p0 , but not in Lp with p0 < p < ∞. Two weight case becomes even more interesting: while it is not hard to show that for p ≤ 2 the testing condition is still sufficient for the boundedness, we will present a counterexample showing that for p > 2 the testing condition alone does not work. Theorem 1.8 (Two weight estimates for paraproducts [19]). (1.11) ||πbµ f ||p p ≤ C p · ||f ||p p , L (ν) L (µ) holds if and only if (i) for the case 1 < p ≤ 2, we have Z " X #p 2 2 dν ≤ C1p · µ(J), J ∈ L, ν (1.12) ∆ b J I I∈L,I⊆J (ii) for the case 2 < p < ∞, we have both (1.12) and (1.13)  ( p2 )0 ν(I 0 ) ν 0 Z   X X ν 2  2( p2 )  E ∆ b  dµ ≤ C2 · ν(J 0 ), J ∈ L, J 0 ∈ child(J). J0  I∈L I 0 ∈child(I) µ(I) I I  I 0 $J 0 In particular, C ≈ C1 + C2 . 2. Bellman Functions on filtered probability spaces Denote the Lebesgue measure of a set E by |E|, the average value of f on an interval I by hf i . The celebrated dyadic Carleson Embedding Theorem states I 5 Theorem 1.9 (Dyadic Carleson Embedding Theorem). Let D = {([0, 1) + j) · 2k : j, k ∈ Z} be the standard dyadic lattice on R, and let {α } be a sequence of non- I I∈D P negative numbers satisfying the Carleson condition that: α ≤ C|I| holds J∈D,J⊆I I for all dyadic intervals I ∈ D. Then the embedding X (1.14) α |hf i |p ≤ Cp · C||f ||p p holds f or all f ∈ Lp , where p > 1. I I L I∈D Moreover, the constant Cp = (p0 )p is sharp (cannot be replaced by a smaller one). An approach of proving Theorem 1.9 is the introduction of the Bellman function. Without loss of generality, we can assume f ≥ 0. Following [20] and [21], we define the Bellman function in three variables (F, f, M ) as (1.15) ( ) X B(F, f, M ; C) = sup |J|−1 α hf ip : f, {α } satisfy (i), (ii), (iii), and (iv) , I I I I∈D I∈D,I⊆J X X (i) hf p i = F ; (ii) hf i = f; (iii) |J|−1 α = M ; (iv) α ≤ C|J| for all J ∈ D. J J I I I⊆J I⊆J Note that the Bellman function B(F, f, M ; C) defined above dose not depend on the choice of the interval J. In [20], (1.14) was first proved using the Bellman function method for the case p = 2, and in [21], the sharpness for the case p = 2 was also claimed. Later, A. Melas found in [22] the exact Bellman function for all p > 1 in a tree-like setting using combinatorial and covering reasoning. In [23], an alternative way of finding the exact Bellman function based on Monge-Amp`ere equation was also established. The Bellman functions have deep connetions to the Stochastic Optimal Control theory [21]. Finding the exact Bellman functions is a difficult task. Both the com- binatorial methods in [22] and the methods of solving the Bellman PDE in [23] are quite complicated. Luckily, the proof of Theorem 1.9 only needs a super-solution instead of the exact Bellman function, see [20], [21]. In this thesis, we will present a way of calculating a super-solution via the Burkh¨older’s hull. 6 On the other hand, computation of the exact Bellman functions usually reflects deeper structure of the corresponding harmonic analysis problem. It is interesting to note that the exact Bellman function of Theorem 1.9 is not restricted to the standard dyadic lattice. In [22], it also works for the tree-like structure. Let us consider a more general situation here. Let (X , F, {Fn }n≥0 , µ) be a discrete-time filtered probability space. By a discrete- time filtration, we mean a sequence of non-decreasing σ-fields {∅, X } = F0 ⊆ F1 ⊆ ... ⊆ Fn ⊆ ... ⊆ F. = µ(E)−1 R We introduce notations fn = Eµ [f |Fn ] and hf i E f dµ. E,µ Definition 1.10. A sequence of non-negative random variables {αn }n≥0 is called a Carleson sequence, if each αn is Fn -measurable and " # X (1.16) Eµ αk |Fn ≤ C for every n ≥ 0. k≥n Definition 1.11. {Fn }n≥0 is called an infinitely refining filtration, if for every ε > 0, every n ≥ 0 and every set E ∈ Fn , there exists a real-valued Fk -measurable R (k > n) random variable h, such that: (i) |h1 | = 1 and (ii) E |hn |dµ ≤ ε. E E Theorem 1.12 (Martingale Carleson Embedding Theorem). If f ∈ Lp (X , F, µ) and {αn }n≥0 is a Carleson sequence, then " # X µ (1.17) E αn |fn | ≤ Cp · C · Eµ [|f |p ] . p n≥0 Moreover, if {Fn }n≥0 is an infinitely refining filtration, then the constant Cp = (p0 )p is sharp. Here, again without loss of generality, we can assume f ≥ 0. We define the Bellman function BµF (F, f, M ; C) in the martingale setting by (1.18) ( " # ) X BµF (F, f, M ; C) = sup Eµ αn fnp : f, {αn }n≥0 satisfy (i), (ii), (iii) and (iv) , n≥0 7 " # X (i) Eµ [f p ] = F ; (ii) Eµ [f ] = f; (iii) Eµ αn = M ; (iv) {αn }n≥0 satisfies (1.16). n≥0 Now, we are ready to state the first main theorem. Theorem 1.13 (Coincidence of the Bellman functions [24]). (1.19) BµF (F, f, M ; C) ≤ B(F, f, M ; C). Moreover, if {Fn }n≥0 is an infinitely refining filtration, then (1.20) BµF (F, f, M ; C) = B(F, f, M ; C). For the Doob’s martingale inequality, recall the definition of the maximal function associated to a discrete-time filtration {Fn }∞ n=0 (1.21) f ∗ (x) = sup |fn (x)|. n≥0 Theorem 1.14 (Doob’s Martingale Inequality). For every p > 1 and every f ∈ Lp (X , F, µ), we have (1.22) ||f ∗ ||p p ≤ (p0 )p · ||f ||p p . L (X ,F ,µ) L (X ,F ,µ) Moreover, if {Fn }n≥0 is an infinitely refining filtration, then the constant (p0 )p is sharp. The study of the Lp -norm of the maximal function was initiated from the cele- brated Doob’s martingale inequality, e.g. in [30]. The sharpness of this inequality was shown in [26] and [27] if one looks at all martingales. For particular martingales including the dyadic case, see [22] and [28]. Theorem 1.14 covers all these results. Assuming f ≥ 0, we define the Bellman function BeµF (F, f) associated to the Doob’s martingale inequality by (1.23) BeµF (F, f) = sup {Eµ [|f ∗ |p ] : Eµ [f p ] = F, Eµ [f ] = f} . The connection between the Carleson Embedding Theorem and the maximal the- ory has been known and exploited a lot, e.g. in [20] and [22]. Using this connection, we give a proof of the second main theorem. 8 Theorem 1.15 (The Bellman function of the maximal operators [24]). (1.24) BeµF (F, f) ≤ BµF (F, f, 1; C = 1). Moreover, if {Fn }n≥0 is an infinitely refining filtration, then (1.25) BeµF (F, f) = BµF (F, f, 1; C = 1). 3. Outline of the thesis In chapter 2, we prove Theorem 1.5. We discuss two cases 1 < p ≤ q and q < p < ∞ seperately. The theorem is actually equivalent to two weight estimates for positive dyadic operators. In chapter 3, we prove Theorem 1.8. Again, we discuss two cases 1 < p ≤ q and q < p < ∞ seperately. The sufficiency part of the case q < p < ∞ needs to be done in greater detail. In chapter 4, we find a super-solution of Theorem 1.9 via the Burkh¨older’s hull, which proves the existence of the Bellman function B(F, f, M ; C). In chapter 5, we present a remodeling of the Bellman function B(F, f, M ; C) and use this to prove the two main results Theorem 1.13 and Theorem 1.15. 9 CHAPTER 2 Two weight estimates for a vector-valued positive operators In this chapter, we prove Theorem 1.5. We first discuss the easier case 1 < p ≤ q. Then we present a counterexample that (1.8) itself dose not imply (1.7). Enventually, we reduce Theorem 1.5 to the well-known two weight estimates for positive dyadic operators and complete our proof. A comparison of our theorem and the main results in [9] and [10] is also given. 1. The case 1 < p ≤ q We will see in this section that when 1 < p ≤ q, (1.8) is equivalent to (1.7). On one hand, (1.8) can be deduced from (1.7) by setting f = 1 . On the other hand, J consider the maximal function (2.1) Mµ f (x) := sup |EµI f (x)| . x∈I,I∈L The celebrated Doob’s martingale inequality asserts (2.2) ||Mµ f || ≤ p0 · ||f || . Lp (µ) Lp (µ) Let Ek := {x ∈ X : Mµ f (x) > 2k } and let Ek := {I ∈ L : I ∈ Ek }. Note that Ek is a disjoint union of maximal intervals in Ek , maximal in the sense of inclusion. Denote these disjoint maximal intervals by Ek∗ . Hence, Ek = ∪ J. J∈Ek∗ 10 #p  p Z "X q q XZ X q q µ αI · E f dν ≤ αI · Eµ f  dν, 1 < p ≤ q  X I Ek I I∈L k I∈Ek \Ek+1   pq X Z X ≤ 2(k+1)p  αq · 1  dν I I k Ek I∈Ek \Ek+1 " # pq X XZ X ≤ 2(k+1)p αq · 1 dν I I k J∈Ek∗ J I∈L:I⊆J X ≤ C1p · 2(k+1)p · µ(Ek ), (1.8) k . C1p · ||Mµ f ||p p L (µ) ≤ C1p · (p0 )p · ||f ||p p , (2.2). L (µ) 2. The case q < p < ∞: a counterexample In this section, we see that (1.8) itself is not sufficient for (1.7) for the case q < p < ∞. Consider the real line R with the Borel σ-algebra B(R). Let the lattice be all the tri-adic intervals. We specify the positive measures µ, ν, the non-negative constants α = {α : I ∈ L}, and the functions f in the following way. I Let C = ∩n≥0 Cn be the 1/3-Cantor set, where C0 = [0, 1), C1 = [0, 1/3) ∪ [2/3, 1) n o and, in general, Cn = ∪ [x, x + 3−n ) : x = nj=1 εj 3−j , εj ∈ {0, 2} . P (i) The measure µ is the Lebsgue measure restricted on [0, 1) and the measure ν is the Cantor measure, i.e. ν(I) = 2−n for each I belongs to a connect component of Cn . (ii) Define α = (2/3)n/p for each I belongs to a connect component of Cn . I (iii) For the function f , consider the gap of C, i.e. [0, 1) \ C. This is a disjoint union of tri-adic intervals. Let f = (3/2)n/p · n−r for each I ∈ [0, 1) \ C with length of I equals 3−n , where r is to be chosen later. Claim 2.1. The construction gives a counterexample with properly chosen r. 11 Proof. We begin with checking (1.8). It suffices to check for every J belongs to a connected component of Cn , and thus µ(J) = 3−n . Note that p p   X  2  qkp q q n X q 2 α · 1 ≤  . 3 3 I I I∈L:I⊆J k≥n Hence, Z X p  n q 2 q α · 1 dν . · ν(J) = µ(J). 3 I J I I∈L:I⊆J Next, we show that (1.7) fails. This requires a careful choice of r in the definition of f . Picking r > p1 , we have Z 1 X  3 n 1 X ||f || p = p |f | dx = n−pr · · 2n = n−pr < ∞. Lp (µ) 0 n≥1 2 3n n≥1 1 Since q < p < ∞, we can pick r such that p < r < 1q . Note that for every I belongs to a connected component of Cn , we have   n+1 1 3 p Eµ f ≥ (n + 1)−r . I 3 2 Hence, consider In = {I : I is tri-adic with length less than or equal to 3−n }, 1 q X q X 1  3  p X −r µ α I · E f · 1C ≥ (k + 1) & (k + 1)−qr . 3 2 I n I∈I n k≤n k≤n And so, Z "X #p " # pq q q X αI · Eµ f dν & (k + 1)−qr · ν(Cn ) → ∞ as n → ∞. I I∈In k≤n We can see that the condition q < p < ∞ is crutial in our construction.  3. The case q < p < ∞: necessity and sufficiency We discuss the case q < p < ∞ of Theorem 1.5 in this section. In particular, we see that both (1.8) and (1.9) are testing conditions on some families of special functions. 12 To start, since Z "X q # (2.3) ||Tαµ f ||q p = sup αI · Eµ f gdν, L (lq ,ν) ||g|| =1 X I 0 I∈L L(p/q) (ν) we can write Z "X q # (2.4) ||Tαµ ||q p = sup sup αI · Eµ f gdν. L (µ)→Lp (lq ,ν) ||f || =1 ||g|| =1 X I Lp (µ) 0 I∈L L(p/q) (ν) Without loss of generality, we assume that both f and g are non-negative. The following lemma reduces us to the scalar-valued case. Lemma 2.2. Z "X # (2.5) ||Tαµ ||q p ≈ sup sup q µ α · E (f ) gdν. q L (µ)→Lp (lq ,ν) ||f || =1 ||g|| =1 X I I Lp (µ) (p/q)0 I∈L L (ν) An easy application of H¨older’s inequality shows that the LHS of (2.5) is no more than its RHS. The other half of this lemma depends on the following famous Rubio de Francia Algorithm. Lemma 2.3 (Rubio de Francia Algorithm). For every q < p < ∞ and f ∈ Lp (µ), there exists a function F ∈ Lp (µ), such that f ≤ F , ||F || ≈ |f || and Lp (µ) Lp (µ) Z −1 µ(I) F q dµ . inf F q (x), I ∈ L. I x∈I Proof. Consider the maximal operator Mµ defined in (2.1). Doob’s martingale inequality (2.2) implies  0 p (2.6) ||Mµ || ≤ . Lp/q (µ)→Lp/q (µ) q (0) (1) (k) (k−1) Denote Mµ = Id, Mµ = Mµ and Mµ = Mµ ◦ Mµ . Define the function F by " −k # 1q X (2.7) F = 2||Mµ || Mµ(k) (f q ) . Lp/q (µ)→Lp/q (µ) k≥0 First we check the validity of the definition for F . Note that 13  " −k # pq  pq Z X   ||F ||q p = 2||Mµ || p/q (k) q Mµ (f ) dµ L (µ)  X L (µ)→Lp/q (µ)  k≥0 X −k Z  pq (k) q pq ≤ 2||Mµ || Mµ (f ) dµ , Minkowski inequality Lp/q (µ)→Lp/q (µ) X k≥0 X −k  k ≤ 2||Mµ || ||Mµ || ||f ||q p = 2||f ||q p . Lp/q (µ)→Lp/q (µ) Lp/q (µ)→Lp/q (µ) L (µ) L (µ) k≥0 Hence, F is the Lp/q (µ)-limit of the partial sums and thus well-defined. Moreover, we have also proved that ||F || . ||f || . Lp (µ) Lp (µ) Considering only k = 0 in the definition for F , we have F ≥ f . And so ||F || ≈ Lp (µ) ||f || . Finally, note that Lp (µ) Z −1 (2.8) µ(I) F q dµ ≤ inf Mµ (F q )(x) I x∈I and X −k q Mµ (F ) = 2||Mµ || Mµ(k+1) (f q ) Lp/q (µ)→Lp/q (µ) k≥0 = 2||Mµ || (F q − f q ) . F q . Lp/q (µ)→Lp/q (µ) Therefore, we deduce Z −1 µ(I) F q dµ . inf F q (x), I ∈ L. I x∈I  Applying Rubio de Francia Algorithm, we obtain Z "X # Z "X # αq · Eµ (f q ) gdν ≤ αq · Eµ (F q ) gdν X I I X I I I∈L I∈L Z "X  q # . αq · Eµ (F ) gdν X I I I∈L ≤ ||Tαµ ||q p · ||F || · ||g|| 0 , (2.4) L (µ)→Lp (lq ,ν) Lp (µ) L(p/q) (ν)   . ||Tαµ ||q p , ||f || = ||g|| 0 =1 . L (µ)→Lp (lq ,ν) Lp (µ) L(p/q) (ν) 14 Now that our problem is reduced to determine a necessary and sufficient condition of Z X p q Z p q µ p (2.9) α · E (f ) dν . C |f | q dµ, X I I X I∈L we may consult to the scalar-valued Theorem 2.4 below. Therefore, Theorem 1.8 follows from Theorem 2.4 for free, and both (1.8) and (1.9) are testing conditions with respect to this derived scalar-valued problem. Consider the linear operator defined by X (2.10) Tαµ f := α · Eµ f. I I I∈L Theorem 2.4. Let 1 < p < ∞ and let 1/p + 1/p0 = 1. Tαµ : Lp (µ) → Lp (ν) if and only if Z X p (2.11) α · 1 dν ≤ C1p · µ(J), J ∈ L I I J I∈L:I⊆J Z X p 0 ν(I) 0 (2.12) α · · 1 dµ ≤ C2p · ν(J), J ∈ L. I µ(I) I J I∈L:I⊆J In particular, ||Tαµ || ≈ C1 + C2 . Lp (µ)→Lp (ν) Remark 2.5. Theorem 2.4 is originally proved for the dyadic case. This general version is explained in [7]. Remark 2.6. In [9] and [10], to obtain the two testing conditions, they first rewrite (1.7) into X (2.13) α · Eµ f · Eν g · ν(I) ≤ C||f || · ||{g } || 0 . I I I I Lp (µ) I I∈L Lp (lq ,ν) I∈L Setting f = 1 , one deduces (1.8). For the second testing condition, one turns to J consider the family of functions {g } supported on J ∈ L with L∞ (lq , ν)-norm I I∈L equal to 1. This gives Z X p 0 ν(I) 0 (2.14) α · · E g dµ ≤ C p · ν(J), J ∈ L. ν µ(I) I I I J I∈L:I⊆J 15 Compare Theorem 1.5 with the main results in [9] and [10]. We have a very different condition (1.9) than (2.14) with seemingly ’wrong’ exponents. However, both (1.8) and (1.9) are testing conditions on some families of special functions as we have shown in this chapter. 16 CHAPTER 3 Two weight estimates for paraproducts In this chapter, we prove Theorem 1.8. We first follow the line of chapter 2. But for the sufficient part of the case 2 < p < ∞, we need to be more careful. We start with some useful reductions. Obviously, it sufficies to constider only for non-negative functions f ≥ 0 in Theorem 1.8. Moreover, recall the following version of Littlewood-Paley theorem Theorem 3.1 (Littlewood-Paley). If a function f has a Littlewood-Paley decom-  1 ν 2 2 ν ν P P position f = I∈L ∆ f , and define its square function to be S f = I∈L ∆ f , I I ν then ||f ||  ||S f || for all 1 < p < ∞. Lp (ν) Lp (ν) Proof. See [17].  Applying this theorem, (1.11) is equivalent to Z "X #p 2 Z µ 2 ν 2 (3.1) ||S ν (πbµ f )||p p = E f ∆ b dν . C p |f |p dµ. L (ν) X I I X I∈L We will consider (3.1) instead of (1.11) in the following. 1. Construction of the stopping intervals Let us construct a collection G ⊆ F ⊆ L of stopping intervals as follows. Given a non-negative function f ≥ 0. For J ∈ F, let G ∗ (J) be the collection of maximal intervals I ⊆ F, I ∈ J such that hf i > 2hf i . I,µ J,µ In case there are more than one such intervals I, we choose the one from the smallest generation. Note that intervals from G ∗ (J) are pairwisely disjoint. Let F(J) = {I ∈ 17 F : I ⊆ J} and let G(J) = ∪ I. Define also E(J) = F(J) \ ∪ F(I). Then I∈G ∗ (J) I∈G ∗ (J) we have the following properties (i) For any I ∈ E(J), hf i ≤ 2hf i , I,µ J,µ (ii) µ(G(J)) < 21 µ(J). To construct a collection G, fix some large integer N ∈ Z and consider all maximal intervals J from {Lk }k≥−N and J ∈ F. These intervals form the first generation G1∗ of stopping intervals. Inductively define the (n+1)-th generation of stopping intervals by ∗ Gn+1 =∪ ∗ G ∗ (I) and we define the collection of stopping intervals by G = ∪n≥1 Gn∗ . I∈Gn Property (ii) implies that the collection G of stopping intervals satisfies the famous Carleson measure condition X (3.2) µ(I) < 2µ(J), J ∈ L. I∈G,I⊆J A special form of the Martingale Carleson Embedding Theorem 1.12 says Theorem 3.2. Let µ be a measure on (X , T ) and let α ≥ 0, I ∈ L satisfy the I Carleson measure condition X (3.3) α ≤ C · µ(J). I I∈G,I⊆J Then for any measurable function f and any 1 < p < ∞ X p (3.4) α hf i ≤ C · (p0 )p · ||f ||p p . I I,µ L (X ,T ,µ) I∈L 2. The case 1 < p ≤ 2 We will see in this section that when 1 < p ≤ 2, (1.12) is equivalent to (3.1). On one hand, (1.12) can be deduced from (3.1) by setting f = 1 . On the other hand, J we can apply the stopping intervals with F = L constructed in the previous section to obtain 18  p  p Z X 2 2 2 Z X X 2 2 2 µ ν  µ ν   E f ∆ b dν =  E f ∆ b dν, X I I X I I I∈L,I⊆L−N J∈G I⊆E(J)   p2 Z 2 X X ≤  4hf i2 ∆ν b  dν, 1 < p ≤ 2 X J,µ I J∈G I⊆E(J)  p X Z X 2 2 ≤ 2p hf ip ∆ν b  dν, (1.12)  J,µ J I J∈G I⊆E(J) X ≤ 2p hf ip · C1p · µ(J), (3.4) J,µ J∈G ≤ C1p · 2p+1 · (p0 )p · ||f ||p p . L (X ,T ,µ) Letting N → ∞, we prove exactly (3.1). We can see 1 < p ≤ 2 plays an important role in this argument. There is no analogue for the case 2 < p < ∞. 3. The case 2 < p < ∞: a counterexample In this section, we see that (1.12) itself is not sufficient for (3.1) for the case 2 < p < ∞. Consider the real line R with the Borel σ-algebra B(R). Let the lattice be all the tri-adic intervals. We specify the admissible measures µ, ν and the functions b, f in the following way. Let C = ∩n≥0 Cn be the 1/3-Cantor set, where C0 = [0, 1), C1 = [0, 1/3) ∪ [2/3, 1) n o and, in general, Cn = ∪ [x, x + 3−n ) : x = nj=1 εj 3−j , εj ∈ {0, 2} . P (i) The measure µ is the Lebsgue measure restricted on [0, 1) and the measure ν is the Cantor measure, i.e. ν(I) = 2−n for each I belongs to a connect component of Cn . (ii) For the function b, we specify its martingale differences ∆ν b. Let |∆ν b| = I I (2/3)n/p for each I belongs to a connect component of Cn such that R (∆ν b)dν = 0. I I 19 (iii) For the function f , consider the gap of C, i.e. [0, 1) \ C. This is a disjoint union of tri-adic intervals. Let f = (3/2)n/p · n−r for each I ∈ [0, 1) \ C with length of I equals 3−n , where r is to be chosen later in the proof. Claim 3.3. The above construction gives a counterexample. Proof. We begin with checking (1.12). It suffices to check for every J belongs to a connected component of Cn , and thus µ(J) = 3−n . Note that " #p " 2 2   2kp # p2  n X X 2 2 ∆ν b ≤ . . I⊆J I k≥n 3 3 Hence, Z "X #p  n  n  n 2 2 2 2 1 ν ∆ b dν . ν(J) = · = µ(J). J I⊆J I 3 3 2 Next, we show that (3.1) fails. This requires a careful choice of r in the definition of f . Picking r > p1 , we have Z 1 X  3 n 1 X ||f || p = p |f | dx = n−pr · n · 2n = n−pr < ∞. Lp (µ) 0 n≥1 2 3 n≥1 1 Since 2 < p < ∞, we can pick r such that p < r < 12 . Note that for every I belongs to a connected component of Cn , we have   n+1 1 3 p µ E f≥ (n + 1)−r . I 3 2 Hence, consider In = {I : I is tri-adic with length less than or equal to 3−n },  1 2 X 2 2 X 1 3 p X −r µ ν f ∆ b · 1 ≥ (k + 1) & (k + 1)−2r . E 3 2 Cn I I I∈In k≤n k≤n And so, Z "X #p " # p2 2 2 2 X (k + 1)−2r · ν(Cn ) → ∞ as n → ∞. µ ν E f ∆ b dν & I I I∈In k≤n  20 4. The case 2 < p < ∞: trilinear forms and necessity We discuss the necessity of Theorem 1.8 in this section. In particular, we see that both (1.12) and (1.13) are testing conditions on some families of special functions. To make our explanations more clear and also for later purpose, we generalize Theorem 1.8 to a trilinear form. (1.12) is a simple testing condtion on functions of the form f = 1 , but (1.13) is J not that clear. To deduce (1.13) from (3.1), we first note that is constant on each ∆ν b  2 I I 0 ∈ child(I). Let β 0 = µ(I)−1 ∆ν b · 1 0 for each I 0 ∈ child(I). (3.1) becomes II I I   p2 Z X X Z 2 Z p (3.5)  β f dµ 1 0  dν . C |f |p dµ. X II 0 I I X I∈L I 0 ∈child(I) Consider the following generalization of Theorem 1.8 to a trilinear form. Theorem 3.4 (Two weight estimates for a trilinear form). For every sequence of n o non-negative constants β 0 , define the trilinear operator II I∈L,I 0 ∈child(I) X X Z  Z  Z  (3.6) Π(f, g, h) = β f dµ gdµ hdν . II 0 I I I0 I∈L I 0 ∈child(I) (3.7) Π(f, g, h) ≤ C||f || ||g|| ||h|| p 0 Lp (µ) Lp (µ) L( 2 ) (ν) holds if and only if (i)   p2 Z X X p (3.8)  β · µ(I)2 · 1 0  dν ≤ C12 · µ(J), J ∈ L, J II 0 I I∈L,I⊆J I 0 ∈child(I) (ii) (3.9)  ( p2 )0 Z 0 X X (p)  β · µ(I) · ν(I 0 ) · 1  dµ ≤ C2 2 · ν(J 0 ), J ∈ L, J 0 ∈ child(J). J0 II 0 I I∈L I 0 ∈child(I),I 0 $J 0 In particular, C ≈ C1 + C2 . 21 Remark 3.5. Note that Theorem 3.4 is written in duality form. In Theorem 3.4,  2 if we choose g = f and β 0 = µ(I)−1 ∆ν b · 1 0 , and take care of the powers of II I I the constants, then we recover Theorem 1.8. All amounts to deduce (3.9) from (3.7). The argument depends on again the Rubio de Francia Algorithm Lemma 2.3. Let f = g = F in (3.7), we obtain X X Z 2 Z  Π(F, F, h) = β 0 F dµ hdν II I I0 I∈L I 0 ∈child(I)   Z X X Z 2 =  β F dµ · 1 0  hdν ≤ C||F ||2 p ||h|| . II 0 I p 0 X I∈L I 0 ∈child(I) I L (µ) L( 2 ) (ν) By Lemma 2.3 with q = 2, we have ||F || ≈ ||f || and Lp (µ) Lp (µ) Z 2 Z Z 2 2 2 F dµ ≥ µ(I) · inf F (x) & µ(I) · F dµ ≥ µ(I) · f 2 dµ, I x∈I I I thus we deduce   Z X X Z  (3.10)  β · µ(I) f 2 dµ · 1 0  hdν . C||f ||2 p ||h|| , II 0 I p 0 X I∈L I 0 ∈child(I) I L (µ) L( 2 ) (ν) which implies   Z X X Z   β · µ(I) hdν · 1  f 2 dν . C||f ||2 p ||h|| . II 0 I p 0 X I∈L I 0 ∈child(I),I 0 $J 0 I0 L (µ) L( 2 ) (ν) Testing on h = 1 0 , we get exactly (3.9). J 5. The case 2 < p < ∞: from trilinear forms to shifted bilinear forms In this section, we give an equivalent statement of Theorem 3.4 in terms of a shifted positive operator. Based on this, we will prove the sufficiency in the next section. We start to understand Theorem 3.4 by two claims. Claim 3.6. Π(f, g, h) ≤ C||f || ||g|| ||h|| p 0 Lp (µ) Lp (µ) L( 2 ) (ν) 22 is equivalent to Π(f, f, h) ≤ C||f ||2 p ||h|| p 0 . L (µ) L( 2 ) (ν) Proof. Only need to see the later implies the former. Since by definition (3.6), we have   2Π(f, g, h) ≤ Π(f, f, h) + Π(g, g, h) ≤ C ||f ||2 p + ||g|| 2 ||h|| p 0 L (µ) Lp (µ) L( 2 ) (ν) Hence, by homogeniety, for every t > 0,   1 2 2 1 2 Π(f, g, h) = Π(tf, g, h) ≤ C t ||f || p + 2 ||g|| p ||h|| p 0 . t L (µ) t L (µ) L( 2 ) (ν) Taking t2 = ||g|| /||f || , we conclude that Lp (µ) Lp (µ) Π(f, g, h) ≤ C||f || ||g|| ||h|| p 0 . Lp (µ) Lp (µ) L( 2 ) (ν)  Claim 3.7. Π(f, f, h) ≤ C||f ||2 p ||h|| p 0 holds, if and only if L (µ) L( 2 ) (ν) (3.11) X X Z  Z  2 β · µ(I) f dµ hdν ≤ C||f 2 || p ||h|| holds. II 0 L( 2 ) (µ) p 0 I∈L I 0 ∈child(I) I I0 L( 2 ) (ν) Proof. In the last section, we have deduced that Π(f, f, h) ≤ C||f ||2 p ||h|| p 0 L (µ) L( 2 ) (ν) R implies (3.10), which is equivalent to (3.11). On the other hand, since µ(I) · I f 2 dµ ≥ R 2 I f dµ , we know (3.11) implies Π(f, f, h) ≤ C||f ||2 p ||h|| p 0 . L (µ) L( 2 ) (ν)  Because of the two claims, if we supress notation α =β · µ(I) in (3.11) and II 0 II 0 instead of assuming 2 < p < ∞ and considering p/2, we still let 1 < p < ∞ and consider p. Theorem 3.4 can be restated into the following. 23 Theorem 3.8 (Two weight estimates for shifted positive operator). For every n o sequence of non-negative constants α 0 , define the shifted positive op- II I∈L,I 0 ∈child(I) erator X X Z  Z  (3.12) Tα (f, g) = α f dµ gdν . II 0 I I0 I∈L I 0 ∈child(I) (3.13) Tα (f, g) ≤ C||f || ||g|| 0 Lp (µ) Lp (ν) holds if and only if (i)  p Z X X (3.14)  α · µ(I) · 1 0  dν ≤ C1p · µ(J), J ∈ L, J II 0 I I∈L,I⊆J I 0 ∈child(I) (ii) (3.15)  p0 Z X X 0  α · ν(I 0 ) · 1  dµ ≤ C2p · ν(J 0 ), J ∈ L, J 0 ∈ child(J). J0 II 0 I I∈L I 0 ∈child(I),I 0 $J 0 In particular, C ≈ C1 + C2 . 6. The case 2 < p < ∞: sufficiency This section is dedicated to prove Theorem 3.8 and hence the sufficiency of The- orem 1.8 for the case 2 < p < ∞. The idea of the proof is from [7] with some new twists. It suffices to consider only for f ≥ 0 and g ≥ 0. We split the estimate into two parts according to the following splitting condition: L = A ∪ B, where n 0 o (3.16) A = I ∈ L : hf ip · µ(I) ≥ hgip · ν(I) and B = L \ A. I,µ I,ν Standard approximation reasoning allows us to assume that only finitely many terms α are non-zero, so all the sums are finite. II 0 24 For an interval I ∈ L, let Ib denote its parents. Using the splitting condition (3.16), we can write Tα (f, g) = T1 + T2 , where X Z  Z  (3.17) T1 = α f dµ gdν , II b Ib I I∈A X Z  Z  (3.18) T2 = α f dµ gdν . II b Ib I I∈B 6.1. A modified stopping interval construction. To estimate T1 we need to modify a bit the construction of stopping intervals from Section 1. The main feature of the construction is that the stopping intervals well be the intervals I ∈ A, but the stopping criterion will be checked on their parents I. b We start with some interval J (not necessarily in A). For the interval J we define the primary preliminary stopping intervals to be the maximal by inclusion intervals Ib ⊆ J, I ∈ A, such that (3.19) hf i b > 2hf iJ,µ . I,µ Note that different I ∈ A can give the same I, b but this Ib is counted only once. It is obvious that these primary preliminary stopping intervals are disjoint and their total µ-measure is at most µ(J)/2. For each such preliminary stopping interval pick all its children L that belong to A (there is at least one such L), and declare these children to be the stopping intervals. For the children K ∈ / A we continue the process: we will find the maximal by inclusion intervals Ib ⊆ K, I ∈ A satisfying (3.19), and declare these Ib to be the secondary preliminary stopping intervals (note that in the stopping criterion (3.19) we still compare with the average over the original interval J). For these secondary preliminary stopping intervals we add their children L ∈ A to the stopping intervals, and for the children K ∈ / A we continue the precess (again, still comparing the averages with the average over the original interval J). We assumed that the collection A is finite, so at some point the process will stop (no I ∈ A, Ib ⊆ K). We end up with the disjoint collection G ∗ (J) of stopping intervals. 25 Since all the stopping intervals are inside the primary preliminary stopping inter- vals, we can conclude that X 1 (3.20) µ(I) < µ(J). 2 I∈G ∗ (J) S S Let G(J) := I∈G ∗ (J) I. Define E(J) = A(J) \ I∈G ∗ (J) A(I), where A(J) = {I ∈ A : I ⊆ J} . It easily follows from the construction that for any I ∈ E(J) (3.21) hf iI,µ ≤ 2hf iJ,µ . To construct a collection G, we start with G0 of disjoint intervals covering the S b For each J ∈ G0 we run the stopping intervals construction to get the set I∈A I. collection G ∗ (J). The union J∈G0 G ∗ (J) give us the first generation of stopping S ∗ intervals G1∗ . Define inductively Gk+1 = J∈G ∗ G ∗ (J) and put G = k≥1 Gk∗ . S S k Note that the condition (3.20) implies that the collection G satisfies the following Carleson measure condition X (3.22) µ(I) < 2µ(J), J ∈ L. I∈G,I⊆J we also can replace G by G ∪ G0 here, and still have the same estimate. 6.2. Estimation of T1 . We start with the estimation of T1 . Using the modified stopping intervals constructed in the previous subsection and remember that J ∈ G0 is chosen such that J ∈/ A, we obtain X Z  Z  X X Z  Z  α f dµ gdν = α f dµ gdν II b Ib I II b Ib I I∈A J∈G∪G0 I∈E(J) = A + B X X Z  Z  A = α f dµ gdν II b Ib I J∈G∪G0 I∈E(J),I6=J X Z  Z  B = α f dµ gdν II b Ib I I∈G 26 For piece , A by (3.21), we have X X Z  A ≤ 2hf i α · µ(I) b · gdν J,µ II I b J∈G∪G0 I∈E(J),I6=J   X Z X = 2hf i  α · µ(I) b · 1  gdν J,µ II I J b J∈G∪G0 I∈E(J),I6=J = 1 + 2   X Z X 1 = 2hf i  α · µ(I) b · 1  gdν, J,µ II I J\G(J) b J∈G∪G0 I∈E(J),I6=J   X Z X 2 = 2hf i  α · µ(I) b · 1  gdν. J,µ II I J∩G(J) b J∈G∪G0 I∈E(J),I6=J To estimate , 1 since the sets J \ G(J) are pairwisely disjoint,  p  1 Z p Z  10 p p0 X X 1 ≤ 2hf i  α · µ(I) b · 1 dν  |g| dν , (3.14) J,µ I J I∈E(J),I6=J II J\G(J) b J∈G∪G0 Z  10 1 p p0 X ≤ 2hf i · C1 · µ(J) · p |g| dν , H¨older’s inequality J,µ J∈G∪G0 J\G(J) " # p1 " # 10 X Z p p0 X ≤ C1 · 2 hf ip · µ(J) |g| dν , (3.4) and disjointness J,µ J\G(J) J∈G∪G0 J∈G∪G0 1+ p1 ≤ C1 · 2 · p0 · ||f || ||g|| 0 . Lp (µ) Lp (ν) 2 since the sets from G ∗ (J) are pairwisely disjoint and G(J) = To estimate , ∪ K, K∈G ∗ (J)   X X Z X 2 = 2hf i  α · µ(I) b · 1  gdν. J,µ II I K b J∈G K∈G ∗ (J) I∈E(J),I6=J 27 Note that the integrand is constant on every K ∈ G ∗ (J). Hence, we obtain    X Z X X 2 = 2hf i  α · µ(I) b · 1  hgi · 1  dν J,µ II I K,ν K J b J∈G∪G0 I∈E(J),I6=J K∈G ∗ (J)  p  1  p0  p10 Z p Z X X X ≤ 2hf i α · µ(I) b · 1 dν  hgi · 1 dν    J,µ I  K,ν K  J I∈E(J),I6=J II J K∈G ∗ (J) b J∈G∪G0  p  1   10 Z p p 0 X X X = 2hf i  α · µ(I) b · 1 dν   hgip · ν(K) . J,µ II I K,ν J I∈E(J),I6=J b J∈G∪G0 K∈G ∗ (J) Using the splitting condition (3.16) for the definition A, we can estimate  p  1   10 Z p p X X X p 2 ≤ 2hf i  α · µ(I) b · 1 dν   hf i · µ(K) J,µ I K,µ J I∈E(J),I6=J II b J∈G∪G0 ∗ K∈G (J)   10 p X 1 X p ≤ 2hf i · C1 · µ(J) ·  p hf i · µ(K) J,µ K,µ J∈G∪G0 K∈G ∗ (J) # p1   10 " p X X X p p ≤ C1 · 2 hf i · µ(J)  hf i · µ(K) , (3.4) J,µ K,µ J∈G∪G0 J∈G∪G0 K∈G ∗ (J) ≤ C1 · 4(p0 )p · ||f ||p p . L (µ) Combine the estimation of 1 and . 2 We conclude 1 1+ A ≤ C1 · 2 p · p0 · ||f || ||g|| 0 + C1 · 4(p0 )p · ||f ||p p . Lp (µ) Lp (ν) L (µ) 28 For piece , B note that X B = α · µ(I) b · ν(I) · hf i · hgi II b I,µ b I,ν I∈G " # p1 " # 10 p 0 X X ≤ αp · µ(I) b p · ν(I) · hf ip p hgi · ν(I) , (3.16) II b I,µ b I,ν I∈G I∈G " # p1 " # 10 X X p ≤ αp · µ(I) b p · ν(I) · hf ip p hf i · µ(I) , (3.4) II b I,µ b I,ν I∈G I∈G " # p1 1 X ≤2 p0 · p · ||g|| 0 · αp · µ(I) b p · ν(I) · hf ip . Lp (ν) II b I,µ b I∈G To finish, we need the following lemma. Lemma 3.9. The sequence {α } , I I∈L X α = αp 0 · µ(I)p · ν(I 0 ) I II I 0 ∈child(I) satisfies the Carleson measure condition. Proof. For J ∈ L, we have   p1 p X X X α =  αp 0 · µ(I)p · 1 0  , || · || p ≤ || · || 1 I II I l l I∈L:I⊆J I⊆J I 0 ∈G,I 0 ∈child(I) Lp (ν) p X X ≤ α 0 · µ(I) · 1 0 I⊆J I 0 ∈G,I 0 ∈child(I) II I Lp (ν) p Z X X = α 0 · µ(I) · 1 0 dν, (3.14) J I⊆J I 0 ∈G,I 0 ∈child(I) II I ≤ C1p · µ(J).  Hence, we can estimate 1 B ≤ C1 · 2 p0 · p · p0 · ||f || ||g|| 0 , Lp (µ) Lp (ν) 29 which, together with the estimation of , A imply that 1 T1 ≤ C1 · 21+ p · p0 · ||f || ||g|| 0 + C1 · 4(p0 )p · ||f ||p p Lp (µ) Lp (ν) L (µ) 1 + C1 · 2 p0 · p · p0 · ||f || ||g|| 0 . Lp (µ) Lp (ν) 6.3. Estimation of T2 . Now we take care of the estimation of T2 . The estimation proceeds similar as in subsection 6.2. Using the stopping intervals constructed in section 1 with F = B, we obtain X Z  Z  X X Z  Z  α f dµ gdν = α f dµ gdν II b Ib I II b Ib I I∈B J∈G I∈E(J) = A + B X X Z  Z  A = α f dµ gdν , II b Ib I J∈G I∈E(J),I6=J X Z  Z  B = α f dµ gdν . JJ b Jb J J∈G Note here I 6= J, we can write   X Z X A ≤ 2hgi ·  α · ν(I) · 1  f dµ = 1 + 2 J,ν II Ib J b J∈G I∈E(J),I6=J   X Z X 1 = 2hgi ·  α · ν(I) · 1  f dµ, J,ν II Ib J\G(J) b J∈G I∈E(J),I6=J   X Z X 2 = 2hgi ·  α · ν(I) · 1  f dµ. J,ν II Ib J∩G(J) b J∈G I∈E(J),I6=J 30 To estimate , 1 again since the sets J \ G(J) are pairwisely disjoint,  p0  p10 X Z Z  p1 X 1 ≤ 2hgi α · ν(I) · 1 dν  |f |p dµ , (3.15)   J,ν  J I∈E(J),I6=J II Ib J\G(J) b J∈G " # 10 " # p1 p XZ 0 X p ≤ C2 · 2 hgi · ν(J) |f |p dµ , (3.4) and disjointness J,ν J\G(J) J∈G J∈G 1+ p10 ≤ C2 · 2 · p · ||f || ||g|| 0 . Lp (µ) Lp (ν) 2 note again that the sets from G ∗ (J) are pairwisely disjoint and To estimate , G(J) = ∪ K. Hence, K∈G ∗ (J)   X X Z X 2 = 2hgi  α · ν(I) · 1  f dµ. J,ν II Ib K b J∈G K∈G ∗ (J) I∈E(J),I6=J Since the integrand is constant on K, we have    X Z X X 2 = 2hgi ·  α · ν(I) · 1   hf i · 1  dµ J,ν II Ib K,µ K J b J∈G I∈E(J),I6=J K∈G ∗ (J)  p  1 Z p X 1 X ≤ 2hgi · C2 · ν(J) p0 · hf i · 1 dµ , disjointness J,ν K,µ K J∈G J K∈G ∗ (J)   p1 X 1 X = 2hgi · C2 · ν(J) p0 ·  hf ip · ν(K) , splitting condition (3.16) J,ν K,µ J∈G K∈G ∗ (J) " # 10   p1 p 0 0 X X X ≤ C2 · 2 hgip · ν(J)  hgip · ν(K) , (3.4) J,ν K,ν J∈G J∈G K∈G ∗ (J) 0 0 ≤ C2 · 4(p)p · ||g||p 0 . Lp (ν) Combine the estimation of 1 and , 2 we have 1+ p10 0 0 A ≤ C2 · 2 · p · ||f || ||g|| 0 + C2 · 4(p)p · ||g||p . Lp (µ) Lp (ν) 0 Lp (ν) 31 Finally, to estimate , B note again X B = α · µ(J) b · ν(J) · hf i · hgi JJ b J,µ b J,ν J∈G " # p1 " # 10 p p0 X X p b p · ν(J) · hf ip ≤ α · µ(J) hgi · ν(J) , (3.4) JJ b J,µ b J,ν J∈G J∈G " # p1 1 X ≤2 p0 · p · ||g|| 0 · αp · µ(J) b p · ν(J) · hf ip , Lemma 3.9 Lp (ν) JJ b J,µ b J∈G 1 ≤ C1 · 2 p0 · p · p0 · ||f || ||g|| 0 . Lp (µ) Lp (ν) Hence, we deduce that 1+ p10 0 0 T2 ≤ C2 · 2 · p · ||f || ||g|| 0 + C2 · 4(p)p · ||g||p Lp (µ) Lp (ν) 0 Lp (ν) 1 + C1 · 2 p0 · p · p0 · ||f || ||g|| 0 . Lp (µ) Lp (ν) Eventually, we conclude that Tα (f, g) = T1 + T2 1 ≤ C1 · 21+ p · p0 · ||f || ||g|| 0 + C1 · 4(p0 )p · ||f ||p p Lp (µ) Lp (ν) L (µ) 1+ p10 0 0 + C2 · 2 · p · ||f || ||g|| 0 + C2 · 4(p)p · ||g||p Lp (µ) Lp (ν) 0 Lp (ν) 1+ p10 + C1 · 2 · p · p0 · ||f || ||g|| 0 . Lp (µ) Lp (ν) By homogeniety, for every t > 0,   1 p p 1 p0 Tα (f, g) = Tα (tf, g) . (C1 + C2 ) · t ||f || p + ||f || p ||g|| 0 + p0 ||g|| 0 . t L (µ) L (µ) Lp (ν) t Lp (ν) 1 1 Taking t = ||g|| p 0 /||f || p0p , we prove exactly Lp (ν) L (µ) Tα (f, g) . (C1 + C2 ) · ||f || ||g|| 0 . Lp (µ) Lp (ν) 32 CHAPTER 4 Bellman functions on filtered probability spaces I: Burkh¨ older’s hull and Super-solutions In this chapter, we find explicitly a super-solution of the dyadic Carleson Embed- ding Theorem 1.9 via the Burkh¨older’s hull. We start with some properties of the Bellman function B(F, f, M ; C), in particular, we prove the main inequality. Then, we define and discuss the super-solutions in detail. In the last section, we introduce the Burkh¨older’s hull and solve for a super-solution of Theorem 1.9 via the Burkh¨older’s hull. This chapter proves the existence of the Bellman function B(F, f, M ; C). 1. Properties of the Bellman function B(F, f, M ; C) Proposition 4.1 (Properties of the Bellman function B(F, f, M ; C)). (i) Domain: fp ≤ F and 0 ≤ M ≤ C. (ii) Range: 0 ≤ B(F, f, M ; C) ≤ Cp · C · F . (iii) The main inequality: For all triples (F, f , M ) and (F± , f ± , M± ) belong to the domain f p ≤ F , 0 ≤ M ≤ C, with F = 12 (F+ + F− ), f = 12 (f + + f − ) and M = ∆M + 21 (M+ + M− ), where 0 ≤ ∆M ≤ M , we have 1 (4.1) B(F, f , M ; C) ≥ {B(F+ , f + , M+ ; C) + B(F− , f − , M− ; C)} + ∆M · f p . 2 Proof. (i) follows from the H¨older’s inequality and that {α } is a Carleson I I∈D sequence. (ii) holds if we assume Theorem 1.9 is true. We explain (iii) in more detail. Split the sum in the definition (1.15) of B(F, f, M ) into three pieces X 1 X 1 X |I|−1 α hf ip = |I+ |−1 α hf ip + |I− |−1 α hf ip + |I|−1 α hf ip , J⊆I J J 2 J⊆I J J 2 J⊆I J J I I + − where I± means the right and left halves of I, respectively. 33 Now, we choose f ± on the interval I± that almost give the supremum in the definition (1.15) of B(F± , f± , M± ), i.e. for small ε > 0, X ε |I± |−1 α hf ± ip ≥ B(F± , f± , M± ; C) − , J⊆I± J J 2 and note that |I|−1 α hf ip = ∆M · f, we conclude I I X 1 |I|−1 α hf ip ≥ {B(F+ , f+ , M+ ; C) + B(F− , f− , M− ; C)} − ε + ∆M · fp , J⊆I J J 2 which yields exactly (4.1).  Remark 4.2. From (1.15), we know that the Bellman function B(F, f, M ; C) exists and 0 ≤ B(F, f, M ; C) ≤ Cp · C · F if and only if Theorem 1.9 is true. The sharpness is explained as B(F, f, M ; C) (4.2) sup = (p0 )p . fp ≤F, 0≤M ≤C C ·F 2. Properties of the Super-solutions 2.1. The super-solutions and the dyadic Caleson Embedding Theorem. Definition 4.3. A function satisfies Propositon 4.1 is called a super-solution. We denote a super-solution by B(F, f, M ; C). We have seen that the dyadic Carleson Embedding Theorem 1.9 gives rise to a super-solution B(F, f, M ; C). On the other hand, to prove (1.14) and actually Theorem 1.9, it suffices to find any super-solution. Indeed, pick f ≥ 0 and {α } satisfying the Carleson condition. For every I I∈D dyadic interval I ∈ D, let F , f , M be the corresponding averages I I I X F = hf p i , f = hf i , M = |I|−1 α . I I I I I J J⊆I Note that F = 12 (F + F ), f = 12 (f + f ) and M = ∆M + 12 (M + M ), I I+ I− I I+ I− I I I+ I− −1 where 0 ≤ ∆M = |I| α ≤ M . For the interval I, the main inequality (4.1) I I I implies α hf ip ≤ |I|B(F , f , M ; C) − |I+ |B(F , f , M ; C) − |I− |B(F , f , M ; C). I I I I I I+ I+ I+ I− I− I− 34 Going n levels down, we get the inequality X X α hf ip ≤ |I|B(F , f , M ; C) − |J|B(F , f , M ; C) J J I I I J J J J⊆I,|J|>2−n |I| J⊆I,|J|=2−n |I| Z ≤ |I|B(F , f , M ; C) ≤ Cp · C · |I|F = Cp · C · f p. I I I I I Applying the above estimate for the intervals [−2n , 0) and [0, 2n ) and taking the limit as n → ∞, we prove exactly (1.14). Remark 4.4. To prove Theorem 1.9, all amounts to finding a super-solution B(F, f, M ; C). We will see in section 3 that the least possible constant for which B(F, f, M ; C) exists is Cp = (p0 )p . 2.2. Further properties of B(F, f, M ; C). We start with the following cele- brated theorem in convex analysis. We will give a proof for the sake of completeness, for more details, see [29]. Theorem 4.5. Let f : Ω → R be a locally bounded function defined on some f (x)+f (y) convex domain Ω ∈ Rn and f satisfies the midpoint concavity: f ( x+y 2 )≥ 2 for all x, y ∈ Ω. Then f is concave and locally Lipschitz. Proof. For concavity: If f is not concave, then there exist two points a, b ∈ Ω, as well as the line segment connecting them [a, b] = {λa + (1 − λ)b : 0 ≤ λ ≤ 1} ⊆ Ω, such that the function ϕ(λ) = f (λa + (1 − λ)b) − λf (a) − (1 − λ)f (b) verifies −∞ < C = inf{ϕ(λ) : 0 ≤ λ ≤ 1} < 0. Note that we have used Ω being convex and f being locally bounded here. Fur- thermore, ϕ(0) = ϕ(1) = 0 and a direct computation shows that ϕ is also midpoint concave. Take 0 < δ < − C2 and let 0 ≤ λ0 ≤ 1, such that ϕ(λ0 ) ≤ C + δ, without loss of generality, further assuming 0 < λ0 < 12 , hence we have ϕ(0) = 0 and ϕ(2λ0 ) ≥ C, however C ϕ(0) + ϕ(2λ0 ) ϕ(λ0 ) ≤ C + δ < = , a contradiction! 2 2 35 For locally Lipschitz continuity: Given a ∈ Ω, we can find a ball B(a, 2r) ⊆ Ω on which f is bounded by a constant M . For x 6= y in B(a, 2r), put z = y + ( αr )(y − x), r α where α = ||y − x||. Clearly, z ∈ B(a, 2r). Moreover, since y = r+α x + r+α z, from r α the concavity of f we infer that f (y) ≥ r+α f (x) + r+α f (z). So |f (y) − f (x)| ≤ α ||y−x|| r+α |f (z) − f (x)| ≤ r · 2M .  In the case of our main inequality (4.1), first put F = 21 (F+ + F− ), f = 12 (f+ + f− ) and M = 12 (M+ + M− ) (i.e. ∆M = 0) and assume all triples (F, f, M ), (F± , f± , M± ) are in the convex domain: fp ≤ F, 0 ≤ M ≤ C, then we obtain the midpoint concavity of B(F, f, M ; C). Apply Theorem 4.5 to the function B, so B is itself concave and locally Lipschitz. In particular, B is a continuous function. Now let 0 ≤ λ ≤ 1 and F = λF+ (1 − λ)F− , f = λf+ (1 − λ)f− , M = ∆M + λM+ + (1 − λ)M− . The main inequality (4.1) and concavity of B imply that ∆M · fp ≤ B(F, f, M ; C) − B(F, f, M − ∆M ; C) ≤ B(F, f, M ; C) − {λB(F+ , f+ , M+ ; C) + (1 − λ)B(F− , f− , M− ; C)} . Hence, the Bellman function B(F, f, M ) is continuous and satisfies (4.3) B(F, f, M ; C) ≥ λB(F+ , f+ , M+ ; C) + (1 − λ)B(F− , f− , M− ; C) + ∆M · fp . 2.3. Regularization of the super-solutions. As we have seen, the Bellman function B is concave and locally Lipschitz, and thus continuous, but hardly any better than that. Fortunately, we know that the proof of Theorem 1.9 boils down to finding just a super-solution B. We recall the trick of regularization of the super-solutions from [25]. Given a super-solution B(F, f, M ; C) satisfying Proposition 4.1. Let φε , ψε : (0, ∞) → [0, ∞) be any two nonnegative C ∞ functions, such that supp(φε ) ⊆ R∞ R∞ [1, (1 + ε)p ], supp(ψε ) ⊆ [1 + ε, 1 + 2ε] and 0 φε (t) dtt = 0 ψε (t) dtt = 1. Define ZZZ   F f M dudvdw Bε (F, f, M ; C) = B , , ; C φε (u)ψε (v)φε (w) (0,∞)3 u v w uvw ZZZ       F f M dudvdw = B(u, v, w; C)φε ψε φε (0,∞)3 u v w uvw 36 Note that the second representation shows Bε ∈ C ∞ . Since B is continuous, the family of smooth functions {Bε : ε > 0} converges to B pointwisely as ε → 0. To check Proposition 4.1 for Bε . Note that the supports of φε and ψε guarantee that Bε is well-defined in the region {fp ≤ F, 0 ≤ M ≤ C} and an easy calculation shows that 0 ≤ Bε ≤ Cp · C · F . For the main inequality, the first representation and (4.1) imply that 1 Bε (F, f, M ; C) − {Bε (F+ , f+ , M+ ; C) + Bε (F− , f− , M− ; C)} 2 Z (1+ε)p Z 1+2ε Z (1+ε)p p 1 dudvdw ≥ ∆M · f p φε (u)ψε (v)φε (w) 1 1+ε 1 v w uvw 1 ≥ ∆M · fp → ∆M · fp as ε → 0. (1 + 2ε)p (1 + ε)p Hence, the proof of (1.14) given in subsection 2.1 works for the smooth function Bε (F, f, M ; C) as well. In what follows, it suffices to consider only for smooth super- solutions B(F, f, M ; C). 2.4. The main inequality in its infinitesimal version. For a smooth super- solution B(F, f, M ; C), being concave means the second differential d2 B ≤ 0. By the main inequality (4.1), we have: B(F, f, M ) − B(F, f, M − ∆M ) ≥ ∆M · fp , and thus ∂B ∂M ≥ fp . Therefore, the main inequality (4.1) implies the following two infinitesimal ones ∂B (4.4) d2 B(F, f, M ; C) ≤ 0 and (F, f, M ; C) ≥ fp . ∂M Actually, (4.4) is equivalent to the main inequality (4.1). Since by (4.4), we can deduce ∆M · fp ≤ B(F, f, M ; C) − B(F, f, M − ∆M ; C) 1 ≤ B(F, f, M ; C) − {B(F+ , f+ , M+ ; C) + B(F− , f− , M− ; C)} . 2 37 3. Finding a super-solution via the Burkh¨ older’s hull 3.1. Burkh¨ older’s hull and some reductions. Assume B(F, f , M ; C) is a smooth super-solution. In this section, we present an explicit function B(F, f , M ; C) with the help of the Burkh¨older’s hull. Definition 4.6. The Burkh¨older’s hull of B(F, f, M ; C) is defined by (4.5) u(f , M ; C) = sup{B(F, f, M ; C) − Cp · C · F }, f ≥ 0, 0 ≤ M ≤ C. F Remark 4.7. This trick of eliminating one variable is due to D. Burkh¨older [27]. It follows from the definition (1.15) of B(F, f, M ; C) that (4.6) B(F, f, M ; C) = C · B(F, f, M/C; 1). Scaling Property Thus, it suffices to consider only for C = 1. We adopt the notations B(F, f , M ) = B(F, f , M ; C = 1), B(F, f , M ) = B(F, f , M ; C = 1) and u(f , M ) = u(f , M ; C = 1). Proposition 4.8. The Bukh¨older’s hull u(f, M ) satisfies the following properties ∂u (i) (f, M ) ≥ fp and (ii) u(f, M ) is concave. ∂M Proof. The proof follows from the definition (4.5). (i) From ∂B ∂M (F, f, M ) ≥ fp , we conclude that B(F, f, M + ∆M ) − B(F, f, M ) ≥ ∆M · fp . Choose F0 that almost gives the supremum in the definition of u(f, M ), i.e. for small ε > 0, B(F0 , f, M ) − Cp · F0 > u(f, M ) − ε, then u(f, M + ∆M ) − u(f, M ) ≥ [B(F0 , f, M + ∆M ) − Cp · F0 ] − [B(F0 , f, M ) − Cp · F0 + ε] = [B(F0 , f, M + ∆M ) − B(F0 , f, M )] − ε ≥ ∆M · fp − ε. Letting ε → 0, so ∂u ∂M (f, M ) ≥ fp . (ii) We need the following simple lemma. 38 Lemma 4.9. Let ϕ(x, y) be a convex function and let Φ(x) = supy ϕ(x, y), then Φ(x) is also a concave function. Proof. We need to see Φ(λx1 + (1 − λ)x2 ) ≥ λΦ(x1 ) + (1 − λ)Φ(x2 ) for all x1 , x2 and 0 ≤ λ ≤ 1. Again choose y1 and y2 in the definition of Φ(x), such that for small ε > 0, ϕ(x1 , y1 ) > Φ(x1 ) − ε and ϕ(x2 , y2 ) > Φ(x2 ) − ε. Then λΦ(x1 ) + (1 − λ)Φ(x2 ) < λϕ(x1 , y1 ) + (1 − λ)ϕ(x2 , y2 ) + ε ≤ ϕ(λx1 + (1 − λ)x2 , λy1 + (1 − λ)y2 ) + ε ≤ Φ((λx1 + (1 − λ)x2 ) + ε, which proves the lemma.  A direct application of this lemma gives (ii).  Remark 4.10. From Proposition 4.8 and (4.5), if the dyadic Carleson Embedding Theorem 1.9 holds with constant Cp , then there exists a concave function u(f, M ) satisfying ∂u ∂M (f, M ) ≥ fp and −Cp · fp ≤ u(f, M ) ≤ 0. On the other hand, if such a u(f, M ) exists, then we can define B(F, f, M ) = u(f, M )+Cp ·F for F ≥ fp , 0 ≤ M ≤ 1, and so B is a super-solution that proves the dyadic Carleson Embedding Theorem with the same constant Cp . Hence, the best constant in the dyadic Carleson Embedding Theorem is exactly the best constant for which the fuction u(f, M ) exists. Now, note the definition (1.15) of B(F, f , M ; C) implies that (4.7) B(tp F, tf , M ; C) = tp · B(F, f , M ; C) for all t ≥ 0. Homogeniety Hence, u(tf, M ) = tp · u(f, M ), which means u(f, M ) can be represented as u(f, M ) = fp · ϕ(M ). For such a function u(f, M ), the Hessian equals   p−2 p−1 0 p(p − 1)f ϕ(M ) pf ϕ (M )  , pfp−1 ϕ0 (M ) fp ϕ00 (M ) 39 so the concavity of u(f, M ) is equivalent to the following two inequalities 00 ϕ(M ) ≤ 0 and ϕϕ − (p0 )(ϕ0 )2 ≥ 0 for 0 ≤ M ≤ 1. The inequality ∂u ∂M (f, M ) ≥ fp means ϕ0 (M ) ≥ 1 and ϕ(M ) also satisfies −Cp ≤ ϕ(M ) ≤ 0. Hence, our task is to find ϕ(M ), such that (i) 0 ≤ M ≤ 1 (ii) −Cp ≤ ϕ(M ) ≤ 0 (iii) ϕ0 (M ) ≥ 1 00 (iv) ϕϕ − (p0 )(ϕ0 )2 ≥ 0, and the least possible constant is Cp = inf ϕ sup {−ϕ(M )}. 0≤M ≤1 3.2. The formula of the Burkh¨ older’s hull and an explicit super-solution. We first introduce φ(M ) = −ϕ(M ) ≥ 0, then φ(M ) satisfies 00 (i) 0 ≤ M ≤ 1; (ii) 0 ≤ φ(M ) ≤ Cp ; (iii) φ0 (M ) ≤ −1; (iv) φφ − (p0 )(φ)2 ≥ 0, and we need to consider Cp = inf sup {φ(M )}. φ 0≤M ≤1 00 0 0 p0 +1 0 0 Rewrite φφ − (p0 )(φ)2 ≥ 0 as φ · φ /φp ≥ 0 or equivalently φ0 /φp ≥ 0. 0 0 0 RM Let φ0 /φp = g(M ) ≥ 0 and denote G(M ) = 0 g, we can solve " #p−1 p−1 φ(M ) = RM , C2 M + C1 − 0 G RM where C1 and C2 are some constants, such that C2 M +C1 − 0 G ≥ 0 for 0 ≤ M ≤ 1. h ip−1 Note that φ0 (M ) ≤ −1, so sup φ(M ) = φ(0) = p−1 C1 . All we need to do 0≤M ≤1 h ip−1 now is to minimize p−1 C1 among all possible φ(M ). To this end, we compute " #p p−1 φ0 (M ) = − RM · [C2 − G(M )] , C2 M + C1 − 0 G and use again φ0 (M ) ≤ −1 with M = 1, which yields Z 1 1 C1 ≤ −C2 + G + (p − 1) · [C2 − G(1)] p . 0 40 R1 Remember that G0 (M ) = g(M ) ≥ 0, thus G(M ) is increasing, in particular, 0 G≤ 1 G(1), so C1 ≤ − [C2 − G(1)] + (p − 1) · [C2 − G(1)] . An easy calculation gives the p 0 0 maximum of the right hand side equals (p − 1) · (p0 )−p when C2 = G(1) + (p0 )−p , h ip−1 0 −p0 therefore, C1 is at most (p − 1) · (p ) and thus p−1 C1 ≥ (p0 )p . 0 To write down an explicit super-solution, simply take G(M ) = 0, C2 = (p0 )−p 0 and C1 = (p − 1) · (p0 )−p , then " #p−1 p−1 pp φ(M ) = RM = , C2 M + C1 − 0 G (p − 1) · [M + (p − 1)]p−1 and recall the relation B(F, f, M ) = u(f, M ) + Cp F = (p0 )p F − fp · φ(M ), we obtain (pf)p (4.8) u(f, M ) = − , (p − 1) · [M + (p − 1)]p−1 (pf)p (4.9) B(F, f, M ) = (p0 )p F − . (p − 1) · [M + (p − 1)]p−1 In the general case, we have u(f , M ; C) = C · u(f , M/C) and B(F, f , M ; C) = C · B(F, f , M/C). Therefore, we have proved the following theorem. Theorem 4.11. The Burkh¨older’s hull of the dyadic Carleson Embedding Theorem 1.9 is given by C · (pf )p (4.10) u(f , M ; C) = − . (p − 1) · [ M C + (p − 1)]p−1 A super-solution that gives the sharpness Cp = (p0 )p is C · (pf )p (4.11) B(F, f , M ; C) = (p0 )p F − . (p − 1) · [ M C + (p − 1)]p−1 Remark 4.12. Now that the dyadic Carleson Embedding Theorem 1.9 is proved, the Bellman function B(F, f, M ; C) exists with Cp = (p0 )p . However, the super- solution B(F, f , M ; C) obtained above is not the real Bellman function, since on the boundary F = fp the real Bellman function must satisfy the boundary condition B(F, f, M ; C) = M fp = M F , but the function we constructed does not satisfy this condition. So, this super-solution only touches the real one along some set. For the exact Bellman function B(F, f, M ; C), see [22] and [23]. 41 CHAPTER 5 The Bellman functions on filtered probability spaces II: Remodeling and proof of the main theorems In this chapter, we prove the two main results: Theorem 1.13 and Theorem 1.15. The proof depends on a remodeling of the Bellman function B(F, f, M ; C = 1) for an infinitely refining filtration. 1. Properties of the Bellman function BµF (F, f, M, C) The Bellman function BµF (F, f, M ; C) associated to the martingale Carleson Em- bedding Theorem 1.13 does not formally have the main inequality. But it still satisfies the following properties. Proposition 5.1 (Properties of the Bellman function BµF (F, f, M ; C)). (i) Domain: fp ≤ F and 0 ≤ M ≤ C. (ii) Range: 0 ≤ BµF (F, f, M ; C) ≤ Cp · C · F . (iii) Homogeniety: BµF (tp F, tf, M ; C) = tp · BµF (F, f, M ; C) for all t ≥ 0. (iv) Scaling Property: BµF (F, f, M ; C) = C · BµF (F, f, M/C; 1). (v) BµF (F, f, M ; C) ≥ BµF (F, f, M − ∆M ; C) + ∆M · fp for 0 ≤ ∆M ≤ M . In particular, BµF (F, f, M ; C) is increasing in M . Proof. (i) follows from the H¨older’s inequality and that {αn }n≥0 is a Carleson sequence. (ii) holds if we assume Theorem 1.12 is true. (iii) and (iv) are obtained directly from definition (1.18). We explain (v) in more detail. Choose f ≥ 0 and {αn }n≥0 that almost give the supremum in the definition (1.18), i.e. for small ε > 0, " # X Eµ αn fnp ≥ BµF (F, f, M − ∆M ; C) − ε, n≥0 42 P  P  where Eµ [f p ] = F , Eµ [f ] = f, Eµ n≥0 αn = M − ∆M and Eµ k≥n αk |Fn ≤ C for every n ≥ 0. Since 0 ≤ M ≤ C, if we increase α0 to α0 + ∆M then everything is P  retained except we have now Eµ n≥0 α n = M and " # X Eµ αn fnp ≥ BµF (F, f, M − ∆M ; C) − ε + ∆M · fp . n≥0 Letting ε → 0, we obtain BµF (F, f, M ; C) ≥ BµF (F, f, M − ∆M ; C) + ∆M · fp .  2. Remodeling of the Bellman function B(F, f, M ; C = 1) for an infinitely refining filtration In this section, we present a remodeling of the Bellman function B(F, f, M ; C = 1) for an infinitely refining filtration, which is central to the proof of Theorem 1.13 and Theorem 1.15. We use the notation B(F, f, M ) = B(F, f, M ; C = 1) in this and later sections. Consider the unit interval I = [0, 1] ∈ D, let {Ijk : 1 ≤ j ≤ 2k } be its k-th generation desendant by subdividing I into 2k congruent dyadic intervals and denote I10 = I. Starting from the definition (1.15) of the Bellman function B(F, f, M ), we can find a function f ≥ 0 with hf p i = F , hf i = f and a sequence {α } , I I J J⊆I P α = M satisfying the Carleson condition with constant C = 1, such that J⊆I J α hf ip (almost) attains B(F, f, M ). P the sum J⊆I J J To proceed, we further assume that the sequence {α } has only finitely many J J⊆I non-zero terms. Hence, the indices of {α } belong to the collection {Ijk : 1 ≤ k ≤ J J⊆I N, 1 ≤ j ≤ 2k } for some fixed integer N , i.e. for all J ∈ / {Ijk : 1 ≤ k ≤ N, 1 ≤ j ≤ 2k }, we have α = 0. As a consequence, we can think the function f being piecewise J constant on all {IjN : 1 ≤ j ≤ 2N }. Now, let us do the remodeling. Fix a small ε, 0 < ε < 1. Consider a discrete-time filtered probability space (X , F, {Fn }n≥0 , µ). The initial construction is X10 = X , and this is Fn0 -measurable, where n0 = 0. Assume that the Fnk -measurable sets Xjk , 1 ≤ j ≤ 2k are constructed. We want to inductively construct Fnk+1 -measurable 43 sets Xjk+1 , 1 ≤ j ≤ 2k+1 . Take a Fnk -measurable set Xjk . Our construction consists of two steps. The first step is a modification of the set Xjk . For the given ε > 0 and Xjk ∈ Fnk , Definition 1.11 guarantees the existence of a real-valued Fnkj -measurable random 2 variable h (nkj > nk ), such that: (i) |h1 | = 1 and (ii) X k |hnk |dµ ≤ ε4 µ(Xjk ). The R E E j condition (ii) is chosen in such a way that n ε o ε (5.1) µ x ∈ Xjk : |hnk | > ≤ µ(Xjk ). 2 2 fk = X k \ x ∈ X k : |h | > ε/2 . So we can conclude |h | ≤ ε/2 on X Let X fk , and j j j nk nk j j fk ) ≤ µ(X k ). moreover, (1 − ε/2) µ(X k ) ≤ µ(X j j k+1 fk ∩{h = 1} and X k+1 = X fk ∩{h = −1}. Since In the second step, we set X2j−1 =X j 2j j R fk ), which gives µ(X k+1 ) − µ(X k+1 ) ≤ ε µ(X fk hdµ ≤ fk |hn |dµ ≤ ε µ(X R fk ) ≤ X X k 2 j 2j−1 2j 2 j j j ε 2 µ(Xjk ), we have ( ) k+1 1 µ(X2j−1 ) µ(X2jk+1 ) 1 (5.2) (1 − ε) ≤ max , ≤ (1 + ε). 2 µ(Xjk ) µ(Xjk ) 2 Do this for all Xjk , 1 ≤ j ≤ 2k and let nk+1 = max{nkj : 1 ≤ j ≤ 2k }. Hence, we construct Fnk+1 -measurable sets Xjk+1 , 1 ≤ j ≤ 2k+1 . Our construction stops when k = N. Now that we have constructed {Xjk : 0 ≤ k ≤ N, 1 ≤ j ≤ 2k }. We can define a new sequence {αn }n≥0 on the space (X , F, µ) as   µ(Xjk )−1 α , if x ∈ Xjk  k Ij αnk =  0, if x ∈ X \ S2k X k  j=1 j and αn = 0 for all n’s different from nk , 1 ≤ k ≤ N . Finally, set the new function fe as fe1 N = f 1 N , 1 ≤ j ≤ 2N , and set fe = 0 on Xj Ij S2N N X \ j=1 Xj . Note that the function fe is also piecewise constant on all {Xjk : 0 ≤ k ≤ N, 1 ≤ j ≤ 2k }. 44 P  P Remark 5.2. This construction guarantees that Eµ n≥0 αn = α =M h i J⊆I J and Eµ fe = hf i = f. Later in subsection 2.2 and subsection 3.2, we use a slightly I modified version of this construction. We will frequently consult to the following proposition.  k+1 k+1  1 µ(X2j−1 ) µ(X2j ) Proposition 5.3. (i) 2 (1 − ε) ≤ max µ(Xjk ) , µ(Xjk ) ≤ 21 (1 + ε). (ii) For every subset E ∈ Fnk and µ(E ∩ Xjk ) > 0, we have ( ) k+1 µ(E ∩ X2j−1 ) µ(E ∩ X2jk+1 ) 1 (5.3) max , ≤ (1 + ε). µ(E ∩ Xjk ) µ(E ∩ Xjk ) 2 Combined with (i), we have ( ) k+1 µ(E ∩ X2j−1 ) µ(E ∩ X2jk+1 ) 1 + ε µ(E ∩ Xjk ) (5.4) max k+1 , ≤ · . µ(X2j−1 ) µ(X2jk+1 ) 1−ε µ(Xjk ) µ(Xjk ) (iii) (1 − ε)k ≤ |Ijk | ≤ (1 + ε)k for all 0 ≤ k ≤ N, 1 ≤ j ≤ 2k . (iv) (1−ε)k hf i ≤ hfei ≤ (1+ε)k hf i for all 0 ≤ k ≤ N, 1 ≤ j ≤ 2k . IjN −k XjN −k ,µ IjN −k Proof. (i) This is (5.2) from our construction. (ii) This is an important extension of (i). But we only have the upper bound es- Recall that our construction gives |hnk | ≤ timation in this general case. fk , so R R ε fk ε/2 on X j E∩Xj fk hdµ ≤ fk |hnk |dµ ≤ 2 µ(E ∩ Xj ), which is E∩Xj µ(E ∩ X k+1 ) − µ(E ∩ X k+1 ) ≤ ε µ(E ∩ Xfk ) ≤ ε µ(E ∩ X k ). So we obtain j 2j−1 2j 2 j 2 (5.3). (5.4) follows from (5.3) and (i). (iii) We prove this by induction. For k = 0, we have µ(X10 ) = |I10 | = 1. Assuming (iii) holds for k, by (i) we can estimate for X2jk+1 (same for X2j−1 k+1 ) that k+1 µ(X2jk+1 ) µ(X2jk+1 ) µ(Xjk ) (1 − ε) ≤ k+1 =2· k · k ≤ (1 + ε)k+1 . |I2j | µ(Xj ) |Ij | (iv) Again by induction, for k = 0, since fe1 N = f 1 N , 1 ≤ j ≤ 2N and fe = 0 Xj Ij S2N N on X \ j=1 Xj , we have hf i N = hf i N . Assuming (iv) holds for k, by e Xj ,µ Ij 45 (i) we have (1 − ε)k+1 hf i N −(k+1) ≤ hfei N −(k+1) Ij Xj ,µ N −k µ(X2j−1 ) µ(X2jN −k ) = N −(k+1) hfei N −k + N −(k+1) hfeiN −k µ(Xj ) X2j−1 ,µ µ(Xj ) X2j ,µ ≤ (1 + ε)k+1 hf i N −(k+1) . Ij  3. The Bellman function BµF (F, f, M ; C) of Theorem 1.13 3.1. BµF (F, f , M ) ≤ B(F, f , M ). We show (1.19) for the case C = 1 and the general case follows from the scaling property. Take the Bellman function B(F, f, M ) of the dyadic Carleson Embedding Theorem. Consider an arbitrary function f ≥ 0 and an arbitrary Carleson sequence {αn }n≥0 with C = 1. Set for every n ≥ 0, " #! X X n = (F n , fn , M n ) = Eµ [f p |Fn ] , Eµ [f |Fn ] , Eµ αk |Fn . k≥n Fix the initial step " #! X 0 µ p µ µ X = E [f ], E [f ], E αn = (F, f, M ). n≥0 By (1.16), we have 0 ≤ M n ≤ 1. Also, fn = fn and when n ≥ 1, F n , fn and M n are random variables. Lemma 5.4. For every n ≥ 0, we have Eµ [B(X n )] − Eµ B(X n+1 ) ≥ Eµ [αn fnp ] ,   where B(X n ) = B(F n , fn , M n ). Proof. Recall that the Bellman function B(F, f, M ) satisfies (4.3). Note also we have X n = Eµ X n+1 |Fn + (0, 0, αn ).   46 By (4.3) and the Jensen’s inequality, we deduce B(X n ) ≥ B Eµ X n+1 |Fn + αn fnp ≥ Eµ B(X n+1 )|Fn + αn fnp .     Taking expectation, we prove exactly Eµ [B(X n )] − Eµ B(X n+1 ) ≥ Eµ [αn fnp ] .    Summing up, we get the inequality " # X X Eµ αn fnp ≤ Eµ [B(X n )] − Eµ [B(X n+1 )] ≤ B(X 0 ).  n≥0 n≥0 Hence, we conclude that BµF (F, f , M ) ≤ B(F, f , M ). 3.2. BµF (F, f , M ) = B(F, f , M ) for an infinitely refining filtration. To show (1.20), again we consider C = 1. Note first that on the boundary fp = F , we have BµF (F, f , M ) = B(F, f , M ) = M F . For the case fp < F , we need to apply the remodeling from section 1. For technical issues, we slightly modify our remodeling here. First, by the conti- nuity of B, there exists δ1 > 0, such that fp < F − δ1 and B(F − δ1 , f , M ) is close to B(F, f , M ). Next, by the definition of B, we can find a non-negative function f on the unit interval I = [0, 1] with hf p i = F − δ1 , hf i = f and a sequence {α } , I I J J⊆I P α = M satisfying the Carleson condition with constant C = 1, such that the J⊆I J α hf ip (almost) equals B(F, f, M ). Moreover, by again the continuity, we P sum J⊆I J J P can choose a finite subset of {α } such that α = M − δ2 for some δ2 > 0 J J⊆I J⊆I J α hf ip still (almost) equals B(F, f, M ). For simplicity, we assume exactly P and J⊆I J J X (5.5) α hf ip = B(F, f, M ). J J J⊆I Let the indices of {α } belong to the collection {Ijk : 1 ≤ k ≤ N, 1 ≤ j ≤ 2k } J J⊆I for some fixed integer N . Choose ε > 0, such that F − δ1 ≤ F/(1 + ε)N . We do the remodeling with this ε > 0 to construct {Xjk : 0 ≤ k ≤ N, 1 ≤ j ≤ 2k }, {αn }n≥0 and fe on the space (X , F, µ). To proceed, we observe that 47 Lemma 5.5. h i (5.6) Eµ fep ≤ (1 + ε)N hf p i . I Proof. By (iii) of Proposition 5.3, 2N h i X 2 X N µ p E f = e hfep i µ(XjN ) ≤ hf p i · (1 + ε)N |IjN | = (1 + ε)N hf p i . XjN ,µ IjN I j=1 j=1  h i So (5.6) and hf p i = F − δ1 ≤ F/(1 + ε)N imply that Eµ fep ≤ F . Also recall I h i from the remodeling, we know Eµ fe = hf i = f. Let us further modify the function I fe in the following way. Note that we are working on an infinitely refining filtration (see definition 1.11). There exists a simple function g behaving like a Haar function, such that g is supported on X1N , hgi = 0 and 0 < Eµ [|g|p ] < ∞. Consider the X1N ,µ continuous function h p i a(t) = Eµ fe + tg . Thus, a(0) ≤ F and limt→∞ a(t) = ∞. Hence, we can find t0 ≥ 0, such that h p i h p i Eµ fe + t0 g = F . Update fe to fe + t0 g. We have then Eµ fe = F and h i Eµ fe = f. Note here the updated function fe might be negative, however, all the relevant average values we will use are still non-negative. Now, let us discuss the properties of the Carleson sequence {αn }n≥0 . Directly P  from the remodeling, we know Eµ P n≥0 α n = α = M − δ2 . Moreover, we J⊆I J can prove Lemma 5.6. The non-negative sequence {αn }n≥0 satisfies each αn is Fn -measurable and " # X (1 + ε)N (5.7) Eµ αk |Fn ≤ for every n ≥ 0. k≥n (1 − ε)2N Proof. From the construction, it is clear that each αn is non-negative and Fn - measurable. So we need to show for every Fn -measurable set E, we have " # X (1 + ε)N Eµ αk 1 ≤ · µ(E). k≥n E (1 − ε)2N 48 hP i h hP i i Denote by k0 = min{k : nk ≥ n}. Since Eµ k≥n αk 1 = Eµ Eµ α k≥k0 nk |Fnk0 1 , E E it suffices to show " # X (1 + ε)N Eµ αnk |Fnk0 ≤ , k≥k0 (1 − ε)2N or equivalently, for every Fnk0 -measurable set E, we have " # µ X (1 + ε)N E αnk 1 ≤ · µ(E). k≥k0 E (1 − ε)2N Now the explicit computation shows " # 2 k µ X XX µ(E ∩ Xjk ) E αnk 1 = αk . k≥k0 E k≥k0 j=1 Ij µ(Xjk ) An iteration of (5.4) gives µ(E ∩ Xjk ) (1 + ε)N µ(E ∩ Xlk0 ) ≤ · , whenever Xjk ⊆ Xlk0 . µ(Xjk ) (1 − ε)N k0 µ(Xl ) So we can estimate " # k 2 0 µ X X (1 + ε)N µ(E ∩ Xlk0 ) X E αnk 1 ≤ · α k , {α } Carleson sequence k≥k0 E l=1 (1 − ε)N µ(Xlk0 ) k Ij I k,j:Xjk ⊆Xl 0 2 0 k (1 + ε)N X µ(E ∩ Xlk0 ) k0 ≤ · Il , Proposition 5.3 (iii) (1 − ε)N l=1 µ(Xlk0 ) 2 0 k (1 + ε)N X k0 (1 + ε)N ≤ µ(E ∩ X l ) ≤ · µ(E). (1 − ε)2N l=1 (1 − ε)2N  To finish, we need one final lemma. Lemma 5.7. " # X p X (5.8) Eµ αn fn ≥ (1 − ε)pN α hf ip . e J J n≥0 J⊆I 49 Proof. 2k " # " # X p X p XX D p E Eµ αn fn = Eµ αnk fnk = αIjk fenk e e n≥0 k≥0 k≥0 j=1 Xjk ,µ 2 k 2 k XX XX ≥ α k hfenk ip = α k hfeip , Proposition 5.3 (iv) Ij Xjk ,µ Ij Xjk ,µ k≥0 j=1 k≥0 j=1 XX 2k X ≥ (1 − ε)pN αIjk hf ip = (1 − ε)pN α hf ip . Ijk J J k≥0 j=1 J⊆I  Summarizing, we have constructed a function fe and a Carleson sequence {αn }n≥0 h p i h i P  satisfying (5.7) with E f = F , Eµ fe = f and Eµ µ e P n≥0 α n = α = J⊆I J M − δ2 . By (5.5) and (5.8), we deduce (1 + ε)N   X BµF F, f, M − δ2 ; C = ≥ (1 − ε)pN α hf ip = (1 − ε)pN B (F, f, M ) . (1 − ε)2N J⊆I J J And Proposition 5.1 (iv) and (v) imply that (1 + ε)N (1 + ε)N F (1 − ε)2N     BµF F, f, M − δ2 ; C = = B F, f, (M − δ2 ) (1 − ε)2N (1 − ε)2N µ (1 + ε)N (1 + ε)N F ≤ B (F, f, M ) . (1 − ε)2N µ Letting ε → 0, we prove exactly BµF (F, f , M ) ≥ B(F, f , M ). The other inequality is proved in the subsection 3.1. 4. The Bellman function BeµF (F, f) of the maximal operators 4.1. BeµF (F, f ) ≤ BµF (F, f , 1). Let us relate the maximal function (1.21) to the Bellman function BµF (F, f , M ). Define a sequence of sets En = {x ∈ X : n is the smallest non-negative integer, such that f ∗ (x) = |fn (x)|}. 50 Obviously, {En }n≥0 forms a disjoint partition of X . We can compute " # X ||f ∗ ||p p = Eµ [|f ∗ |p ] = Eµ |fn |p 1 L (X ,F ,µ) En n≥0 " # X = Eµ Eµ [1 |Fn ] · |fn |p . En n≥0 Let αn = Eµ [1 |Fn ], n ≥ 0. The connection between the maximal function (1.21) En and BµF (F, f , M ) relies on the following simple fact. Lemma 5.8. {αn }n≥0 is a Carleson sequence with C = 1. (see Definition 1.10). Proof. It is clear that each αn is non-negative and Fn -measurable. Moreover, hP i hP i for every set E ∈ Fn , we have Eµ k≥n αk 1 = Eµ k≥n 1 ≤ µ(E). So we E Ek ∩E prove the claim.  To prove (1.24), fix Eµ [f p ] = F and Eµ [f ] = f. Since {αn }n≥0 is a Carleson eF F P  sequence with C = 1 and Eµ n≥0 αn = 1, we conclude that Bµ (F, f ) ≤ Bµ (F, f , 1). 4.2. BeµF (F, f ) = BµF (F, f , 1) for an infinitely refining filtration. Again, we appeal to the modified remodeling from subsection 2.2, but only with M = 1. Note that we have 2k " # " # X p X p XX D p E µ µ αn fn = E αnk fnk = αIjk fenk . e E e n≥0 k≥0 k≥0 j=1 Xjk ,µ To proceed, we observe that Lemma 5.9. For every 0 ≤ k ≤ N and 1 ≤ j ≤ 2k , we have (1 + ε)k e (5.9) fn 1 ≤ hf i N −k . e N −k XjN −k (1 − ε)k Xj ,µ Proof. First note that fen 1 = fen 1 for every 0 ≤ k ≤ N N −k XjN −k N −k XjN −k and 1 ≤ j ≤ 2k . Induction on k, for k = 0, the construction of fe immediately gives fen 1 = hfei , 1 ≤ j ≤ 2N . Assuming (5.9) holds for k, then for every N XjN XjN ,µ 51 N −(k+1) Fn -measurable set E, E ⊆ Xj and µ(E) > 0, we can estimate N −(k+1) Z Z Z Z fen 1 N −(k+1) dµ = fedµ = fen dµ + fen dµ E N −(k+1) Xj N −(k+1) E∩Xj N −k E∩X2j−1 N −k N −k E∩X2j N −k (1 + ε)k e   N −k N −k ≤ hf i N −k µ(E ∩ X2j−1 ) + hf i N −k µ(E ∩ X2j ) . e (1 − ε)k X2j−1 ,µ X2j ,µ And hence, we deduce Z −1 N −(k+1) µ(E) fen 1 N −(k+1) dµ, (E ⊆ Xj ) E N −(k+1) Xj " # N −k (1 + ε)k e µ(E ∩ X2j−1 ) µ(E ∩ X2jN −k ) ≤ hf i N −k N −k+1 + hfei N −k N −k+1 , (5.3) (1 − ε)k X2j−1 ,µ µ(E ∩ Xj ) X2j ,µ µ(E ∩ Xj ) 1 (1 + ε)k+1 e   ≤ hf i N −k + hfei N −k , Proposition 5.3 (i) 2 (1 − ε)k X2j−1 ,µ X2j ,µ (1 + ε)k+1 e ≤ hf i N −(k+1) . (1 − ε)k+1 Xj ,µ N −(k+1) Since this is true for every Fn -measurable set E, E ⊆ Xj and µ(E) > 0, N −(k+1) we prove (5.9) for k + 1.  Applying (5.9), we have 2k k " # 2 µ X p XX D p E (1 + ε)pN X X αn f n = αIjk fenk ≤ αIjk hfeip . E e pN n≥0 k≥0 j=1 Xjk ,µ (1 − ε) k≥0 j=1 Xjk ,µ And note that Proposition 5.3 (iii) implies X 1 ≤ Ijk00 ≤ µ(Xjk00 ) for every 0 ≤ k0 ≤ N, 1 ≤ j0 ≤ 2k0 . α k Ijk (1 − ε)N k,j:Ijk ⊆Ij 0 0 Now, let us recall a useful lemma established in [22], formulated in our language, Lemma 5.10. Suppose α ≥ 0, where 0 ≤ k ≤ N, 1 ≤ j ≤ 2k , satisfies Ijk X (5.10) α ≤ Cµ(Xjk00 ) Ijk k k,j:Ijk ⊆Ij 0 0 for some constant C > 0, then we can choose pairwise disjoint measurable Akj ⊆ X such that Akj ⊆ Xjk and α = Cµ(Akj ). Ijk 52 Proof. Without loss of generality, we can assume C = 1. We start at the level k = N . Since (5.10) with C = 1 implies α ≤ µ(XjN ) for every 1 ≤ j ≤ 2N , we can IjN choose AN N j ⊆ Xj such that α = µ(AN j ). Assuming that we have chosen pairwise IjN disjoint measurable Akj for all k ≥ k0 + 1 and 1 ≤ j ≤ 2k , note that (5.10) with C = 1 gives   X [ α + α ≤ µ(Xjk00 ), so α ≤ µ Xjk00 \ Akj  ,   k Ijk k Ij 0 k Ij 0 k 0 0 k,j:Ijk $Ij 0 k,j:Ijk $Ij 0 0 0 Akj00 Xjk00 Akj , S and thus we can choose measurable set ⊆ \ k k,j:Ijk $Ij 0 such that α k = 0 Ij 0 0 µ(Akj00 ). Continue this process for all the indices. This proves the lemma.  By Lemma 5.10, we can estimate 2k " # pN X X µ X p (1 + ε) E αn fen ≤ pN αIjk hfeip n≥0 (1 − ε) k≥0 j=1 Xjk ,µ 2 k (1 + ε)pN X X 1 = µ(Akj )hfenk ip (1 − ε) k≥0 j=1 (1 − ε)N pN Xjk ,µ 2k (1 + ε)pN X X µ e∗ p   ≤ E f 1 k , disjointness (1 − ε)(p+1)N k≥0 j=1 Aj (1 + ε)pN h p i µ e∗ ≤ E f . 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