The Gundy-Stein decomposition with explicit constants

THE GUNDY –STEIN DECOMPOSITION WITH EXPLICIT CONST ANTS MAHDI HORMOZI AND JIE-XIANG ZHU A bstract . Let ( F n ) n ≥ 1 be a filtration and let f ≥ 0 belong to L 1 ( F ∞ ). For the martingale f n = E [ f | F n ] and each λ > 0 we prove a Gundy–Stein decomposition f = g + h + k with explicit numerical constants. In the positive closed case the thr ee parts satisfy explicit bounds, and the bounded part is bounded above by λ . W e also prove a one-parameter form for the bounded part and two-point sharpness results, includ- ing a joint sharpness statement for arbitrary decompositions under the condition 0 ≤ k ≤ λ . W e also obtain an exact four-term refinement of the decomposition, separating the bounded term into a stopped part and a conditional expectation term. As applications we obtain an explicit weak-type (1 , 1) estimate for truncated martingale multipliers and a John–Nirenberg inequality for martingale BMO on atomic α -regular filtrations. 1. I ntroduction Gundy’s decomposition is the martingale analogue of the Calder ´ on–Zygmund decomposition. It was introduced b y Gundy in his study of weak-type inequalities for martingales and was incorporated by Stein into martingale Littlewood–Paley theory; see [4, 8]. Closely related arguments or consequences appear in Davis [2], T suchikura [9], B ´ aez-Duarte [1], and Schipp [7]; for a non-commutative analogue see Parcet and Randrianantoanina [6]. For martingale BMO and John–Nirenber g type inequalities we also refer to Garsia [3, Ch. III] and Kazamaki [5]. Through- out the paper ( Ω , F , P ) is a complete probability space endowed with a discrete filtration ( F n ) n ≥ 1 , and F ∞ : = σ  [ n ≥ 1 F n  . For any sub- σ -field F ′ ⊆ F , we denote by L p ( F ′ ) the subspace of L p ( Ω , P ) consisting of F ′ -measurable functions. In this paper we work with terminal-value martingales f n = E [ f | F n ] , f ∈ L 1 ( F ∞ ) . Given f ∈ L 1 ( F ∞ ) and λ > 0, the Gundy–Stein decomposition seeks a splitting f = g + h + k , in which the martingale associated with g is localized on a set of small probability , the martingale di ff erences of h are absolutely summable, and k is bounded by a multiple of λ . This decomposition is one of the standard tools behind weak-type (1 , 1) estimates for martingale transforms and square functions. 2020 Mathematics Subject Classification. 60G44; 60G46; 46E30. Key words and phrases. Gundy–Stein decomposition; martingale transform; BMO martingale; John– Nirenber g inequality . 1 2 M. HORMOZI AND J.-X. ZHU In this paper we give a direct proof with explicit constants. The proof uses only optional stopping and elementary stopping-time constructions; no maximal- inequality argument is used. The modification of the classical proof is slight: the second stopping time is chosen so that, in the positive case, the conditional ex- pectation term disappears. This yields 0 ≤ k ≤ λ for the bounded part. W e also prove a one-parameter form for the bounded part, two-point sharpness examples, and a global two-point theorem showing that, under the condition 0 ≤ k ≤ λ , the coe ffi cients in the localized and absolutely summable pieces cannot be im- proved simultaneously for arbitrary decompositions. The two applications are an explicit weak-type (1 , 1) estimate for truncated martingale multipliers and a John–Nirenber g inequality for martingale BMO on atomic α -r egular filtrations. W e write E n : = E [ · | F n ] and set E 0 ≡ 0. Our main result is the following. Theorem 1.1. Let f ≥ 0 belong to L 1 ( F ∞ ) and let λ > 0 . Then one can write f = g + h + k with g , h , k ∈ L 1 ( F ∞ ) such that (a) P  sup n ≥ 1 | E n [ g ] | > 0  ≤ ∥ f ∥ 1 λ , ∥ g ∥ 1 ≤ 2 ∥ f ∥ 1 . (b)     ∞ X n = 1 | E n [ h ] − E n − 1 [ h ] |     1 ≤ 2 ∥ f ∥ 1 . In particular , ∥ h ∥ 1 ≤ 2 ∥ f ∥ 1 . (c) 0 ≤ k ≤ λ a.s. , ∥ k ∥ 1 ≤ ∥ f ∥ 1 , ∥ k ∥ 2 2 ≤ λ ∥ f ∥ 1 . For general f ∈ L 1 ( F ∞ ) one applies Theorem 1.1 to f + and f − separately . This yields the corresponding decomposition with doubled constants; see Corol- lary 2.13. The bounded term also admits a one-parameter variant: if one allows a condi- tional expectation term of size θλ , then one obtains a family of decompositions with bounded part k θ satisfying ∥ k θ ∥ ∞ ≤ (1 + θ ) λ . The sharpness discussion is divided into two parts. First, two-point examples show that the constants in Theorem 1.1 are attained by the construction, and they also give the parametric L ∞ bound for 0 ≤ θ < 1 (with asymptotic sharpness at θ = 1). Second, Second, Proposition 3.10 and Corollary 3.13 give a global two-point sharpness statement for arbitrary de- compositions. The paper is organized as follows. Section 2 contains the proof of Theorem 1.1, the extension to general f , and the counterexample showing that the assumption f ∈ L 1 ( F ∞ ) is needed in part (c). Section 3 tr eats the parametric variant and sharpness. Sections 4 and 5 contain the two applications. 2. P roof of T heorem 1.1 For f ∈ L 1 ( Ω , P ), set f n : = E n [ f ] , n ∈ N 0 . Then ( f n ) n ≥ 1 is the martingale associated with f . Since f ∈ L 1 ( F ∞ ), the martingale convergence theor em gives f n → f almost surely and in L 1 . THE GUNDY –STEIN DECOMPOSITION WITH EXPLICIT CONST ANTS 3 A stopping time for ( F n ) n ≥ 1 is a map τ : Ω → { 1 , 2 , . . . } ∪ {∞} such that { τ = n } ∈ F n for every n ≥ 1. Lemma 2.1. Let f ∈ L 1 ( F ∞ ) , let f n : = E n [ f ] for n ≥ 1 , and write f ∞ : = f . If τ is a stopping time, then f τ ∈ L 1 and E [ f τ ] = E [ f ] . Proof. Since { τ = j } ∈ F j for every j ≥ 1, E h | f j | 1 { τ = j } i ≤ E h E j [ | f | ] 1 { τ = j } i = E h | f | 1 { τ = j } i . Summing over j and adding the term on { τ = ∞} gives E [ | f τ | ] ≤ E [ | f | ] < ∞ . Likewise, E h f j 1 { τ = j } i = E h f 1 { τ = j } i , j ≥ 1 , so E [ f τ ] = ∞ X j = 1 E h f j 1 { τ = j } i + E h f 1 { τ = ∞} i = E [ f ] . ■ W e now fix f ≥ 0 in L 1 ( F ∞ ) and λ > 0, and write d f n : = f n − f n − 1 , n ≥ 1 . Define the first-passage time r : = inf { n ≥ 1 : f n > λ } , where inf ∅ = ∞ . Since f r > λ on { r < ∞} , Lemma 2.1 gives P ( r < ∞ ) ≤ ∥ f ∥ 1 λ . (2.2) Define the crossing incr ement ε n : = d f n 1 { r = n } , n ≥ 1 . If r = n , then f n − 1 ≤ λ < f n , hence d f n > 0 on { r = n } . Therefor e ε n ≥ 0 , and for each ω at most one ε n ( ω ) is nonzero. (2.3) Let Λ 0 : = 0 , Λ m : = m X k = 1 E h ε k + 1 | F k i , m ≥ 1 , and define s : = inf { m ≥ 1 : Λ m > 0 } , t : = r ∧ s . Then s and t are stopping times. Moreover , if r = ∞ then ε n = 0 for every n , hence Λ m = 0 for every m and ther efore s = t = ∞ . Since always t ≤ r , it follows that { t < ∞} = { r < ∞} . (2.4) The g term. Set g : = f − f t . Since E n [ f t ] = n X j = 1 f j 1 { t = j } + E n h f 1 { t > n } i = n X j = 1 f j 1 { t = j } + f n 1 { t > n } = f n ∧ t , 4 M. HORMOZI AND J.-X. ZHU one has E n [ g ] = f n − f n ∧ t . If t = ∞ , then g = 0 and E n [ g ] = 0 for every n . Thus, by (2.2) and (2.4), P  sup n ≥ 1 | E n [ g ] | > 0  ≤ P ( t < ∞ ) = P ( r < ∞ ) ≤ ∥ f ∥ 1 λ . Also, by Lemma 2.1, ∥ g ∥ 1 = E | f − f t | ≤ E [ f ] + E [ f t ] = 2 ∥ f ∥ 1 . This proves part (a) of Theor em 1.1. The splitting of the stopped martingale. Define γ n : = d f n 1 { r > n } , n ≥ 1 . Since t = r ∧ s , one has 1 { t ≥ j } = 1 { r ≥ j } 1 { s ≥ j } , and d f j 1 { r ≥ j } = ε j + γ j . Therefor e, f n ∧ t = n X j = 1 d f j 1 { t ≥ j } = n X j = 1 ( ε j + γ j ) 1 { s ≥ j } . (2.5) For n ≥ 1, set h n : = n X j = 1 ( ε j − E j − 1 [ ε j ]) 1 { s ≥ j } , k n : = n X j = 1 ( γ j + E j − 1 [ ε j ]) 1 { s ≥ j } . Then, by (2.5), h n + k n = f n ∧ t , n ≥ 1 . (2.6) Since dh n = ( ε n − E n − 1 [ ε n ]) 1 { s ≥ n } , 1 { s ≥ n } ∈ F n − 1 , one has E n − 1 [ dh n ] = 0. Hence ( h n ) n ≥ 1 is a martingale, and then so is ( k n ) n ≥ 1 by (2.6). The h term. Lemma 2.7. One has ∞ X n = 1 ∥ dh n ∥ 1 ≤ 2 ∥ f ∥ 1 . Consequently P ∞ n = 1 | dh n | < ∞ almost surely and h n converges almost sur ely and in L 1 to h : = ∞ X n = 1 dh n ∈ L 1 ( F ∞ ) , ∥ h ∥ 1 ≤ 2 ∥ f ∥ 1 . Proof. By (2.3), | dh n | ≤ ε n 1 { s ≥ n } + E n − 1 [ ε n ] 1 { s ≥ n } . For n ≥ 2, since 1 { s ≥ n } ∈ F n − 1 , E h E n − 1 [ ε n ] 1 { s ≥ n } i = E h ε n 1 { s ≥ n } i , so ∥ dh n ∥ 1 ≤ 2 E [ ε n 1 { s ≥ n } ] ≤ 2 E [ ε n ] , n ≥ 2 . THE GUNDY –STEIN DECOMPOSITION WITH EXPLICIT CONST ANTS 5 For n = 1, one has dh 1 = ε 1 because E 0 ≡ 0 and { s ≥ 1 } = Ω . Thus again ∥ dh 1 ∥ 1 ≤ 2 E [ ε 1 ] . Hence ∥ dh n ∥ 1 ≤ 2 E [ ε n ] , n ≥ 1 . On the other hand, ∞ X n = 1 ε n = ∞ X n = 1 ( f n − f n − 1 ) 1 { r = n } ≤ ∞ X n = 1 f n 1 { r = n } = f r 1 { r < ∞} . T aking expectations and applying Lemma 2.1 with τ = r , we obtain ∞ X n = 1 E [ ε n ] ≤ E [ f r ] = E [ f ] = ∥ f ∥ 1 . Therefor e, ∞ X n = 1 ∥ dh n ∥ 1 ≤ 2 ∞ X n = 1 E [ ε n ] ≤ 2 ∥ f ∥ 1 . T onelli’s theorem gives P n ≥ 1 | dh n | < ∞ almost surely , and the stated convergence in L 1 follows from the summability of P n ∥ dh n ∥ 1 . ■ Since h n → h in L 1 and ( h n ) n ≥ 1 is a martingale, one has h n = E n [ h ] for every n ≥ 1. Hence dh n = E n [ h ] − E n − 1 [ h ] , and part (b) of Theorem 1.1 follows fr om Lemma 2.7. The k term. Set k : = f t − h . Then f = g + h + k . Since E n [ f t ] = f n ∧ t and h n = E n [ h ], we also have k n = E n [ k ] , n ≥ 1 . Lemma 2.8. One has 0 ≤ k ≤ λ almost surely . Proof. Since k n → k almost surely and k n is given by (2.6), we obtain k = ∞ X j = 1 ( γ j + E j − 1 [ ε j ]) 1 { s ≥ j } a.s. (2.9) Now ∞ X j = 1 γ j 1 { s ≥ j } = ∞ X j = 1 d f j 1 { r > j } 1 { s ≥ j } . If r < ∞ , set r − : = r − 1; if r = ∞ , set r − : = ∞ . Then ∞ X j = 1 d f j 1 { r > j } 1 { s ≥ j } = s ∧ r − X j = 1 d f j = f s ∧ r − . If r < ∞ , then f m ≤ λ for every m < r , hence f s ∧ r − ≤ λ . If r = ∞ , then f n ≤ λ for every n , so f ≤ λ almost surely and also s = ∞ . Therefor e, 0 ≤ ∞ X j = 1 γ j 1 { s ≥ j } ≤ λ a.s. (2.10) 6 M. HORMOZI AND J.-X. ZHU For the compensator term, since E 0 ≡ 0, ∞ X j = 1 E j − 1 [ ε j ] 1 { s ≥ j } = ∞ X j = 2 E j − 1 [ ε j ] 1 { s ≥ j } . If s = m < ∞ , this equals m X j = 2 E j − 1 [ ε j ] = Λ m − 1 = 0 , since m is the first index with Λ m > 0. If s = ∞ , then Λ m = 0 for every m ≥ 1, so the same sum is again zero. Thus ∞ X j = 1 E j − 1 [ ε j ] 1 { s ≥ j } = 0 a.s. (2.11) Combining (2.9), (2.10), and (2.11) proves the claim. ■ Lemma 2.12. One has ∥ k ∥ 1 ≤ ∥ f ∥ 1 and ∥ k ∥ 2 2 ≤ λ ∥ f ∥ 1 . Proof. Since k ≥ 0 and k = f t − h , ∥ k ∥ 1 = E [ k ] = E [ f t ] − E [ h ] . By Lemma 2.1, E [ f t ] = E [ f ] = ∥ f ∥ 1 . Moreover , E [ h ] = E [ h 1 ] = E [ ε 1 ] ≥ 0, because h 1 = ε 1 . Hence ∥ k ∥ 1 ≤ ∥ f ∥ 1 . Since 0 ≤ k ≤ λ almost surely by Lemma 2.8, ∥ k ∥ 2 2 ≤ λ ∥ k ∥ 1 ≤ λ ∥ f ∥ 1 . ■ This completes the proof of Theor em 1.1. For general f ∈ L 1 ( F ∞ ), write f = f + − f − . Applying Theorem 1.1 separately to f + and f − gives the following consequence. Corollary 2.13. Let f ∈ L 1 ( F ∞ ) and λ > 0 . Then one can write f = g + h + k with g , h , k ∈ L 1 ( F ∞ ) such that (a ′ ) P  sup n ≥ 1 | E n [ g ] | > 0  ≤ 2 ∥ f ∥ 1 λ , ∥ g ∥ 1 ≤ 4 ∥ f ∥ 1 . (b ′ )     ∞ X n = 1 | E n [ h ] − E n − 1 [ h ] |     1 ≤ 4 ∥ f ∥ 1 . In particular , ∥ h ∥ 1 ≤ 4 ∥ f ∥ 1 . (c ′ ) ∥ k ∥ ∞ ≤ λ, ∥ k ∥ 1 ≤ 2 ∥ f ∥ 1 , ∥ k ∥ 2 2 ≤ 2 λ ∥ f ∥ 1 . Remark 2.14 . The same argument applies, without change of constants, to finite filtrations ( F n ) M n = 1 and terminal values in L 1 ( F M ). Remark 2.15 . The probability normalization is not essential. The proof uses only conditional expectations and stopping times, so the ar gument extends verbatim to σ -finite measure spaces. THE GUNDY –STEIN DECOMPOSITION WITH EXPLICIT CONST ANTS 7 The next example shows that the assumption f ∈ L 1 ( F ∞ ) is needed for part (c) of Theorem 1.1. Counterexample 2.16. Let Ω = [0 , 1] with Lebesgue measure and the trivial filtra- tion F n = { ∅ , Ω } . Fix λ = 1 and set f = 100 1 [0 , 0 . 01] . Then ∥ f ∥ 1 = 1 and f n = E n [ f ] = 1 for all n ≥ 1. Hence r = ∞ , so ε n = 0 for all n , Λ m = 0, s = ∞ , and t = ∞ . Consequently g = 0, h = 0, and k = f . Thus ∥ k ∥ ∞ = 100 > λ . 3. V ariants and sharpness W e retain the notation of Section 2. Formula (2.9) may be written as k = k st + k pr , k st : = ∞ X j = 1 γ j 1 { s ≥ j } , k pr : = ∞ X j = 1 E j − 1 [ ε j ] 1 { s ≥ j } . The second term disappears when the stopping time s is defined by the condition Λ m > 0. If one allows the conditional expectation term to have size at most θλ , then the same proof gives a one-parameter family of decompositions. Recall that Λ 0 : = 0 , Λ m : = m X k = 1 E [ ε k + 1 | F k ] , m ≥ 1 . For θ ≥ 0, define s θ : = inf { m ≥ 1 : Λ m > θλ } , t θ : = r ∧ s θ . (3.1) Replacing s , t by s θ , t θ in the construction of Section 2 gives functions g θ , h θ , k θ . Proposition 3.2. Assume the hypotheses of Theorem 1.1. Let λ > 0 and θ ≥ 0 . Then one can write f = g θ + h θ + k θ where g θ and h θ satisfy the same estimates as in Theorem 1.1 (a)–(b) , and 0 ≤ k θ ≤ (1 + θ ) λ a.s. , ∥ k θ ∥ 1 ≤ ∥ f ∥ 1 , ∥ k θ ∥ 2 2 ≤ (1 + θ ) λ ∥ f ∥ 1 . (3.3) More pr ecisely , k θ = k st θ + k pr θ , where k st θ : = ∞ X j = 1 γ j 1 { s θ ≥ j } , k pr θ : = ∞ X j = 1 E j − 1 [ ε j ] 1 { s θ ≥ j } , and 0 ≤ k st θ ≤ λ, 0 ≤ k pr θ ≤ θλ a.s. Proof. The proofs of the estimates for g θ and h θ are unchanged, since they only use that s θ is a stopping time and that s θ = ∞ whenever r = ∞ . Repeating the proof of Lemma 2.8 with s r eplaced by s θ gives k θ = X j ≥ 1 γ j 1 { s θ ≥ j } + X j ≥ 1 E j − 1 [ ε j ] 1 { s θ ≥ j } = k st θ + k pr θ . 8 M. HORMOZI AND J.-X. ZHU As before, k st θ = f s θ ∧ r − , hence 0 ≤ k st θ ≤ λ . For the conditional expectation part, set B θ : = ∞ X j = 2 E j − 1 [ ε j ] 1 { s θ ≥ j } . If s θ = m < ∞ , then B θ = m X j = 2 E j − 1 [ ε j ] = Λ m − 1 ≤ θλ. If s θ = ∞ , then Λ m ≤ θλ for every m , so again B θ ≤ θλ . This proves the pointwise bound in (3.3). The L 1 and L 2 estimates are the same as in Lemma 2.12: ∥ k θ ∥ 1 = E [ k θ ] = ∥ f ∥ 1 − E [ ε 1 ] ≤ ∥ f ∥ 1 , and therefor e ∥ k θ ∥ 2 2 ≤ (1 + θ ) λ ∥ k θ ∥ 1 ≤ (1 + θ ) λ ∥ f ∥ 1 . ■ For θ ≥ 0, Proposition 3.2 yields a decomposition f = g θ + h θ + k θ , 0 ≤ k θ ≤ (1 + θ ) λ. The next proposition provides a further decomposition of k θ into a stopped part and a conditional expectation part, which will also be useful in other problems. Proposition 3.4. Let f ≥ 0 belong to L 1 ( F ∞ ) , let λ > 0 , and let θ ≥ 0 . Retain the notation of Section 2 and (3.1) ; thus r : = inf { n ≥ 1 : f n > λ } , ε n : = d f n 1 { r = n } , γ n : = d f n 1 { r > n } , Λ 0 : = 0 , Λ m : = m X k = 1 E k [ ε k + 1 ] , s θ : = inf { m ≥ 1 : Λ m > θλ } , t θ : = r ∧ s θ . Also set r − : = r − 1 on { r < ∞} , r − : = ∞ on { r = ∞} . Define g θ : = f − f t θ , h θ, n : = n X j = 1  ε j − E j − 1 [ ε j ]  1 { s θ ≥ j } , n ≥ 1 , k st θ, n : = n X j = 1 γ j 1 { s θ ≥ j } , k pr θ, n : = n X j = 1 E j − 1 [ ε j ] 1 { s θ ≥ j } , n ≥ 1 . Then h θ, n converges almost sur ely and in L 1 to a limit h θ ∈ L 1 ( F ∞ ) , and the sequences k st θ, n , k pr θ, n converge almost sur ely and in L 1 to limits k st θ , k pr θ ∈ L 1 ( F ∞ ) . Moreover , f = g θ + h θ + k st θ + k pr θ . (a) P  sup n ≥ 1 | E n [ g θ ] | > 0  ≤ ∥ f ∥ 1 λ , ∥ g θ ∥ 1 ≤ 2 ∥ f ∥ 1 . THE GUNDY –STEIN DECOMPOSITION WITH EXPLICIT CONST ANTS 9 (b)     ∞ X n = 1    E n [ h θ ] − E n − 1 [ h θ ]        1 ≤ 2 ∥ f ∥ 1 . In particular , ∥ h θ ∥ 1 ≤ 2 ∥ f ∥ 1 . (c) k st θ = f s θ ∧ r − , 0 ≤ k st θ ≤ λ a.s. (d) k pr θ = Λ s θ − 1 1 { s θ < ∞} +  sup m ≥ 1 Λ m  1 { s θ = ∞} , 0 ≤ k pr θ ≤ θλ a.s. (e) E [ h θ ] = E [ ε 1 ] , ∥ k st θ ∥ 1 + ∥ k pr θ ∥ 1 = ∥ f ∥ 1 − E [ ε 1 ] ≤ ∥ f ∥ 1 , ∥ k st θ ∥ 2 2 ≤ λ ∥ k st θ ∥ 1 , ∥ k pr θ ∥ 2 2 ≤ θλ ∥ k pr θ ∥ 1 , and ∥ k st θ + k pr θ ∥ 2 2 ≤ (1 + θ ) λ ∥ f ∥ 1 . Proof. Since r = ∞ implies ε n = 0 for every n , one has Λ m = 0 for every m on { r = ∞} , hence s θ = ∞ on { r = ∞} . Because always t θ ≤ r , it follows that { t θ < ∞} = { r < ∞} . As before, E n [ f t θ ] = f n ∧ t θ , E [ f t θ ] = E [ f ] = ∥ f ∥ 1 . Therefor e, E n [ g θ ] = f n − f n ∧ t θ . If t θ = ∞ , then g θ = 0 and E n [ g θ ] = 0 for all n . Hence P  sup n ≥ 1 | E n [ g θ ] | > 0  ≤ P ( t θ < ∞ ) = P ( r < ∞ ) ≤ ∥ f ∥ 1 λ by (2.2). Also, ∥ g θ ∥ 1 = E [ | f − f t θ | ] ≤ E [ f ] + E [ f t θ ] = 2 ∥ f ∥ 1 . This proves (a). Next, f n ∧ t θ = n X j = 1 d f j 1 { t θ ≥ j } = n X j = 1 d f j 1 { r ≥ j } 1 { s θ ≥ j } . Since d f j 1 { r ≥ j } = ε j + γ j , we obtain f n ∧ t θ = n X j = 1 ( ε j + γ j ) 1 { s θ ≥ j } = h θ, n + k st θ, n + k pr θ, n . (3.5) Set dh θ, n : = h θ, n − h θ, n − 1 =  ε n − E n − 1 [ ε n ]  1 { s θ ≥ n } , h θ, 0 : = 0 . Since 1 { s θ ≥ n } ∈ F n − 1 , one has E n − 1 [ dh θ, n ] =  E n − 1 [ ε n ] − E n − 1 [ ε n ]  1 { s θ ≥ n } = 0 , 10 M. HORMOZI AND J.-X. ZHU so ( h θ, n ) n ≥ 1 is a martingale. By (2.3), | dh θ, n | ≤ ε n 1 { s θ ≥ n } + E n − 1 [ ε n ] 1 { s θ ≥ n } . For n ≥ 2, E h E n − 1 [ ε n ] 1 { s θ ≥ n } i = E h ε n 1 { s θ ≥ n } i , while for n = 1, dh θ, 1 = ( ε 1 − E 0 [ ε 1 ]) 1 { s θ ≥ 1 } = ε 1 . Hence, for every n ≥ 1, ∥ dh θ, n ∥ 1 ≤ 2 E [ ε n ] . Arguing as in Lemma 2.7, one has ∞ X n = 1 E [ ε n ] ≤ ∥ f ∥ 1 . Therefor e ∞ X n = 1 ∥ dh θ, n ∥ 1 ≤ 2 ∥ f ∥ 1 , and T onelli’s theorem yields     ∞ X n = 1 | dh θ, n |     1 = ∞ X n = 1 ∥ dh θ, n ∥ 1 ≤ 2 ∥ f ∥ 1 . Hence P n ≥ 1 | dh θ, n | < ∞ almost surely , so h θ, n converges almost sur ely and in L 1 to h θ : = ∞ X n = 1 dh θ, n ∈ L 1 ( F ∞ ) . Since ( h θ, n ) n ≥ 1 is an L 1 -bounded martingale converging to h θ , one has h θ, n = E n [ h θ ] , E n [ h θ ] − E n − 1 [ h θ ] = dh θ, n . Consequently ,     ∞ X n = 1    E n [ h θ ] − E n − 1 [ h θ ]        1 =     ∞ X n = 1 | dh θ, n |     1 ≤ 2 ∥ f ∥ 1 , and ∥ h θ ∥ 1 ≤     ∞ X n = 1 | dh θ, n |     1 ≤ 2 ∥ f ∥ 1 . This proves (b). For the stopped part, k st θ, n = n X j = 1 γ j 1 { s θ ≥ j } = n X j = 1 d f j 1 { r > j } 1 { s θ ≥ j } = n ∧ s θ ∧ r − X j = 1 d f j = f n ∧ s θ ∧ r − , because f 0 = E 0 [ f ] = 0. Hence k st θ, n → f s θ ∧ r − = : k st θ a.s. and in L 1 . If r < ∞ , then f m ≤ λ for every m < r , so 0 ≤ f s θ ∧ r − ≤ λ. THE GUNDY –STEIN DECOMPOSITION WITH EXPLICIT CONST ANTS 11 If r = ∞ , then f n ≤ λ for every n ≥ 1, and since f n → f almost surely , 0 ≤ f ≤ λ a.s. Thus k st θ = f s θ ∧ r − , 0 ≤ k st θ ≤ λ a.s. This proves (c). For the conditional expectation part, ε n ≥ 0 implies E n − 1 [ ε n ] ≥ 0, and therefore 0 ≤ k pr θ, n ≤ k pr θ, n + 1 , n ≥ 1 . Since E 0 ≡ 0, k pr θ, n = n X j = 2 E j − 1 [ ε j ] 1 { s θ ≥ j } . If s θ = m < ∞ , then for every n ≥ m , k pr θ, n = m X j = 2 E j − 1 [ ε j ] = m − 1 X k = 1 E k [ ε k + 1 ] = Λ m − 1 ≤ θλ. If s θ = ∞ , then k pr θ, n = n X j = 2 E j − 1 [ ε j ] = Λ n − 1 ↑ sup m ≥ 1 Λ m ≤ θλ. Hence k pr θ, n → Λ s θ − 1 1 { s θ < ∞} +  sup m ≥ 1 Λ m  1 { s θ = ∞} = : k pr θ almost surely , and 0 ≤ k pr θ ≤ θλ a.s. Since 0 ≤ k pr θ, n ≤ θλ, dominated convergence gives k pr θ, n → k pr θ in L 1 . This proves (d). Now (3.5), together with the L 1 convergence of h θ, n , k st θ, n , and k pr θ, n , yields f t θ = h θ + k st θ + k pr θ . Therefor e, f = ( f − f t θ ) + h θ + k st θ + k pr θ = g θ + h θ + k st θ + k pr θ . Since h θ, 1 = ε 1 and h θ, n → h θ in L 1 , E [ h θ ] = E [ h θ, 1 ] = E [ ε 1 ] . Also, k st θ ≥ 0 , k pr θ ≥ 0 , so ∥ k st θ ∥ 1 + ∥ k pr θ ∥ 1 = E [ k st θ + k pr θ ] = E [ f t θ ] − E [ h θ ] = ∥ f ∥ 1 − E [ ε 1 ] ≤ ∥ f ∥ 1 . Finally , 0 ≤ k st θ ≤ λ, 0 ≤ k pr θ ≤ θλ, 0 ≤ k st θ + k pr θ ≤ (1 + θ ) λ. Hence ∥ k st θ ∥ 2 2 ≤ λ ∥ k st θ ∥ 1 , ∥ k pr θ ∥ 2 2 ≤ θλ ∥ k pr θ ∥ 1 , 12 M. HORMOZI AND J.-X. ZHU and ∥ k st θ + k pr θ ∥ 2 2 ≤ (1 + θ ) λ ∥ k st θ + k pr θ ∥ 1 ≤ (1 + θ ) λ ∥ f ∥ 1 . This proves (e) and completes the pr oof. ■ Remark 3.6 . Pr oposition 3.4 refines Proposition 3.2 by writing the bounded term as k θ = k st θ + k pr θ = f s θ ∧ r − + ∞ X j = 2 E j − 1 [ ε j ] 1 { s θ ≥ j } . When θ = 0, one has k pr 0 = 0 , k st 0 = k , and Proposition 3.4 r educes to Theor em 1.1. W e now turn to sharpness. The next examples concern the stopping-time con- struction above. Example 3.7. Let Ω = E ⊔ E c with P ( E ) = p ∈ (0 , 1), and let F n = σ ( E ) for all n ≥ 1. Fix λ > 0 and set f = (1 + δ ) λ 1 E with δ > 0. Then f n = f for all n , so r = 1 on E and r = ∞ on E c . Hence P ( r < ∞ ) = p , ∥ f ∥ 1 λ = (1 + δ ) p . Letting δ ↓ 0 shows that the coe ffi cient 1 in (2.2), and therefor e in Theorem 1.1(a), is sharp. Example 3.8. Let F 1 = { ∅ , Ω } and F 2 = F ∞ = { ∅ , E , E c , Ω } . Define f : = p − 1 1 E , so that ∥ f ∥ 1 = 1 and f 1 = E [ f ] = 1. Fix λ ∈ (0 , 1). Then f 1 > λ , so r = t = 1 and f t ≡ 1. Hence g = f − f t = p − 1 1 E − 1 . Therefor e, ∥ g ∥ 1 = p     1 p − 1     + (1 − p ) | 0 − 1 | = 2(1 − p ) = 2(1 − p ) ∥ f ∥ 1 . Letting p ↓ 0 shows that the constant 2 in Theorem 1.1(a) is sharp. Example 3.9. Let F 1 = { ∅ , Ω } and F 2 = F ∞ = { ∅ , E , E c , Ω } . Fix λ > 0 and define f : = λ p 1 E . Then ∥ f ∥ 1 = λ , f 1 = λ , and f 2 = f . Hence r = 2 on E and r = ∞ on E c . The only nonzero cr ossing incr ement is ε 2 = ( f 2 − f 1 ) 1 { r = 2 } =  λ p − λ  1 E . Since F 1 is trivial, Λ 1 = E [ ε 2 ] = (1 − p ) λ. Sharpness of the constant 2 in Theorem 1.1 (b) . If θ ≥ 1 − p , then Λ 1 ≤ θλ , so s θ = ∞ and t θ = r . The construction gives g θ = 0 and h θ = ε 2 − E [ ε 2 ] =  λ p − (2 − p ) λ  1 E − (1 − p ) λ 1 E c . THE GUNDY –STEIN DECOMPOSITION WITH EXPLICIT CONST ANTS 13 Since E 1 [ h θ ] = 0 and E 2 [ h θ ] = h θ , X n ≥ 1 | E n [ h θ ] − E n − 1 [ h θ ] | = | h θ | . Hence     X n ≥ 1 | E n [ h θ ] − E n − 1 [ h θ ] |     1 = ∥ h θ ∥ 1 = p  λ p − (2 − p ) λ  + (1 − p ) 2 λ = 2(1 − p ) 2 λ = 2(1 − p ) 2 ∥ f ∥ 1 − − − → p → 0 2 ∥ f ∥ 1 . Thus the coe ffi cient 2 in Theorem 1.1(b) is sharp. The parametric L ∞ bound for k θ . Still assuming θ ≥ 1 − p , one finds k θ = f − h θ = (2 − p ) λ 1 E + (1 − p ) λ 1 E c , so ∥ k θ ∥ ∞ = (2 − p ) λ . If 0 ≤ θ < 1, choose p = 1 − θ . Then Λ 1 = θλ , s θ = ∞ , and ∥ k θ ∥ ∞ = (1 + θ ) λ. Thus the coe ffi cient 1 + θ in Proposition 3.2 is sharp for 0 ≤ θ < 1. For θ = 1, the same example gives ∥ k θ ∥ ∞ = (2 − p ) λ → 2 λ as p ↓ 0, so the coe ffi cient 2 cannot be improved. Likewise, k st θ = λ 1 E , k pr θ = (1 − p ) λ. Hence ∥ k st θ ∥ ∞ = λ , and for 0 ≤ θ < 1 choosing p = 1 − θ gives ∥ k pr θ ∥ ∞ = θλ . Thus the individual bounds for k st θ and k pr θ are also sharp in this range; at θ = 1 one again obtains asymptotic sharpness. The preceding examples only concern the stopping-time decomposition. The next proposition is global: on the two-point filtration it establishes a lower-bound dichotomy for arbitrary decompositions we consider . Proposition 3.10. Let Ω = E ⊔ E c with P ( E ) = p ∈ (0 , 1 2 ] , and define F 1 = { ∅ , Ω } , F n = F ∞ = { ∅ , E , E c , Ω } , n ≥ 2 . Fix λ > 0 and let f : = λ p 1 E , ∥ f ∥ 1 = λ. Assume that f = g + h + k with g , h , k ∈ L 1 ( F ∞ ) and 0 ≤ k ≤ βλ a.s. for some β ∈ [0 , p − 1 ] . Then one of the following alternatives holds: (i) P  sup n ≥ 1 | E n [ g ] | > 0  = 1; (ii) g = 0 almost surely and     ∞ X n = 1 | E n [ h ] − E n − 1 [ h ] |     1 ≥        (3 − β − 2 p ) λ, 0 ≤ β ≤ 1 , 2(1 − p β ) λ, 1 ≤ β ≤ p − 1 . (3.11) 14 M. HORMOZI AND J.-X. ZHU Both lower bounds are sharp. Proof. W rite q : = 1 − p . Every u ∈ L 1 ( F ∞ ) takes only two values, and hence u = u E 1 E + u E c 1 E c . Since the filtration stabilizes at n = 2, E 1 [ u ] = E [ u ] = pu E + qu E c , E n [ u ] = u ( n ≥ 2) . Therefor e, n sup n ≥ 1 | E n [ u ] | > 0 o = { u , 0 } ∪ { E [ u ] , 0 } . If E [ u ] , 0, then P  sup n ≥ 1 | E n [ u ] | > 0  = 1 . If E [ u ] = 0, then pu E + qu E c = 0. Hence either u E = u E c = 0, in which case u = 0 a.s., or u E and u E c have opposite signs, so u , 0 on both E and E c , and again P  sup n ≥ 1 | E n [ u ] | > 0  = 1 . Applying this to u = g shows that P  sup n ≥ 1 | E n [ g ] | > 0  < 1 = ⇒ g = 0 a.s. Thus alternative (i) fails only when g = 0. Assume from now on that g = 0. W rite k = a 1 E + b 1 E c , 0 ≤ a , b ≤ βλ. Then h =  λ p − a  1 E − b 1 E c . Since E 0 ≡ 0, E 1 [ h ] = E [ h ], and E n [ h ] = h for n ≥ 2, ∞ X n = 1 | E n [ h ] − E n − 1 [ h ] | = | E [ h ] | + | h − E [ h ] | . Set x : = λ p − a , y : = − b . Then h = x 1 E + y 1 E c and E [ h ] = px + q y = λ − pa − qb . Moreover , x − E [ h ] = q ( x − y ) , y − E [ h ] = − p ( x − y ) , so E [ | h − E [ h ] | ] = pq | x − y | + qp | x − y | = 2 pq | x − y | . Therefor e,     ∞ X n = 1 | E n [ h ] − E n − 1 [ h ] |     1 = | λ − pa − qb | + 2 pq     λ p − a + b     . (3.12) THE GUNDY –STEIN DECOMPOSITION WITH EXPLICIT CONST ANTS 15 Since 0 ≤ a , b ≤ βλ and β ≤ p − 1 , λ p − a + b ≥ λ  1 p − β  ≥ 0 , so the second absolute value in (3.12) can be dropped. If 0 ≤ β ≤ 1, then λ − pa − qb ≥ λ − βλ ≥ 0 , so both absolute values can be removed and     ∞ X n = 1 | E n [ h ] − E n − 1 [ h ] |     1 = λ − pa − qb + 2 pq  λ p − a + b  . Because p ≤ 1 2 , the right-hand side is decreasing in both a and b , hence minimized at a = b = βλ . This gives     ∞ X n = 1 | E n [ h ] − E n − 1 [ h ] |     1 ≥ (1 − β ) λ + 2(1 − p ) λ = (3 − β − 2 p ) λ. Equality holds for g = 0 , k = βλ 1 Ω , h = f − k . Assume next that 1 ≤ β ≤ p − 1 . Define D + : = { ( a , b ) : 0 ≤ a , b ≤ βλ, pa + qb ≤ λ } , D − : = { ( a , b ) : 0 ≤ a , b ≤ βλ, pa + qb ≥ λ } . If ( a , b ) ∈ D + , then Φ ( a , b ) : = | λ − pa − qb | + 2 pq  λ p − a + b  = λ − pa − qb + 2 pq  λ p − a + b  . For fixed a , the map b 7→ Φ ( a , b ) is decreasing on the corresponding slice of D + , so its minimum is attained on the boundary pa + qb = λ . If ( a , b ) ∈ D − , then Φ ( a , b ) = pa + qb − λ + 2 pq  λ p − a + b  , and for fixed a the map b 7→ Φ ( a , b ) is increasing on the corresponding slice of D − . Hence the global minimum is attained on the same boundary line pa + qb = λ . Along this boundary one has Φ ( a , b ) = 2 pq  λ p − a + b  = 2( λ − pa ) , which is decreasing in a . Since a ≤ βλ , the minimum is attained at a = βλ, b = λ − p βλ q . This choice is admissible because β ≥ 1 and β ≤ p − 1 . Therefore,     ∞ X n = 1 | E n [ h ] − E n − 1 [ h ] |     1 ≥ 2( λ − p βλ ) = 2(1 − p β ) λ. Equality holds for g = 0 , k = βλ 1 E + 1 − p β 1 − p λ 1 E c , h = f − k . 16 M. HORMOZI AND J.-X. ZHU This proves (3.11). ■ Corollary 3.13. In the setting of Proposition 3.10, take β = 1 . Then every decomposition f = g + h + k , 0 ≤ k ≤ λ, satisfies P  sup n ≥ 1 | E n [ g ] | > 0  = 1 or     ∞ X n = 1 | E n [ h ] − E n − 1 [ h ] |     1 ≥ 2(1 − p ) ∥ f ∥ 1 . Consequently , no pair of constants c 1 < 1 and c 2 < 2 can simultaneously replace the coe ffi cients in Theorem 1.1 (a)–(b) under the same condition 0 ≤ k ≤ λ . Proof. The first assertion is Proposition 3.10 with β = 1 and ∥ f ∥ 1 = λ . Now let c 1 < 1 and c 2 < 2. Choose p ∈ (0 , 1 2 ] so small that c 2 < 2(1 − p ). If there were a decomposition such that P  sup n ≥ 1 | E n [ g ] | > 0  ≤ c 1 ∥ f ∥ 1 λ = c 1 < 1 and     ∞ X n = 1 | E n [ h ] − E n − 1 [ h ] |     1 ≤ c 2 ∥ f ∥ 1 , then Proposition 3.10 would for ce c 2 ∥ f ∥ 1 ≥ 2(1 − p ) ∥ f ∥ 1 , contrary to the choice of p . ■ Remark 3.14 . Cor ollary 3.13 is a global statement concerning all decompositions f = g + h + k on the two-point filtration. While one may suppress the localized term at the cost of enlarging h , the two coe ffi cients cannot be improved simultaneously under the constraint 0 ≤ k ≤ λ . The same proposition also applies to the parametric condition 0 ≤ k ≤ (1 + θ ) λ : one simply takes β = 1 + θ . Thus, whenever 0 < p ≤ (1 + θ ) − 1 and P  sup n ≥ 1 | E n [ g ] | > 0  < 1 , one has     ∞ X n = 1 | E n [ h ] − E n − 1 [ h ] |     1 ≥ 2  1 − (1 + θ ) p  ∥ f ∥ 1 . In particular , lim inf p ↓ 0 1 ∥ f ∥ 1     ∞ X n = 1 | E n [ h ] − E n − 1 [ h ] |     1 ≥ 2 . Thus enlarging the L ∞ allowance on the bounded term by any fixed factor does not remove the asymptotic cost 2 ∥ f ∥ 1 in the h term when one insists on eliminating the localized part. THE GUNDY –STEIN DECOMPOSITION WITH EXPLICIT CONST ANTS 17 4. A weak - type (1 , 1) estima te for trunca ted martingale mul tipliers Let a = ( a n ) n ≥ 1 ∈ ℓ ∞ . For N ≥ 1 and f ∈ L 1 ( Ω , P ), define the truncated martingale multiplier T N ( a ; f ) : = N X n = 1 a n d f n , d f n : = E n [ f ] − E n − 1 [ f ] . Theorem 4.1. For every N ≥ 1 , every f ∈ L 1 ( Ω , P ) , and every λ > 0 , P  | T N ( a ; f ) | > λ  ≤ 16 ∥ a ∥ ℓ ∞ ∥ f ∥ 1 λ . Proof. By homogeneity , it is enough to assume ∥ a ∥ ℓ ∞ = 1. Replacing f by E [ f | F ∞ ] does not change the martingale di ff erences and does not increase the L 1 norm, so we may assume f ∈ L 1 ( F ∞ ). Apply Corollary 2.13 to f at level λ/ 2, obtaining f = g + h + k . Then P  | T N ( a ; f ) | > λ  ≤ P  | T N ( a ; g ) | > 0  + P  | T N ( a ; h ) | > λ/ 2  + P  | T N ( a ; k ) | > λ/ 2  . (4.2) If sup n ≥ 1 | E n [ g ] | = 0, then all martingale di ff erences of g vanish and therefor e T N ( a ; g ) = 0. Hence, by Corollary 2.13(a ′ ), P  | T N ( a ; g ) | > 0  ≤ P  sup n ≥ 1 | E n [ g ] | > 0  ≤ 4 ∥ f ∥ 1 λ . (4.3) Next, | T N ( a ; h ) | ≤ N X n = 1 | dh n | ≤ ∞ X n = 1 | dh n | , so Markov’s inequality and Corollary 2.13(b ′ ) give P  | T N ( a ; h ) | > λ/ 2  ≤ 2 λ     ∞ X n = 1 | dh n |     1 ≤ 8 ∥ f ∥ 1 λ . (4.4) Finally , since k n = E n [ k ] and martingale di ff erences ar e orthogonal in L 2 , ∥ T N ( a ; k ) ∥ 2 2 = N X n = 1 | a n | 2 ∥ E n [ k ] − E n − 1 [ k ] ∥ 2 2 ≤ ∥ k ∥ 2 2 . Therefor e, by Markov’s inequality and Cor ollary 2.13(c ′ ), P  | T N ( a ; k ) | > λ/ 2  ≤ 4 ∥ T N ( a ; k ) ∥ 2 2 λ 2 ≤ 4 ∥ k ∥ 2 2 λ 2 ≤ 4 ∥ f ∥ 1 λ . (4.5) Combining (4.2), (4.3), (4.4), and (4.5) completes the proof. ■ Remark 4.6 . Standard Marcinkiewicz interpolation, together with the estimate ∥ T N ( a ; f ) ∥ 2 ≤ ∥ a ∥ ℓ ∞ ∥ f ∥ 2 , yields the usual L p boundedness of martingale transforms 18 M. HORMOZI AND J.-X. ZHU for 1 < p < ∞ . W e now adopt a di ff erent point of view and fix f ∈ L 2 ( Ω , P ). Then T N ( a ; f ) is the martingale transform of f associated with a , and ∥ T N ( a ; f ) ∥ 2 2 = N X n = 1 a 2 n E h | d f n | 2 i , (4.7) which may be viewed as a discrete analogue of It ˆ o’s isometry . Interpolating The- orem 4.1 with (4.7), and viewing T N ( a ; f ) as an operator acting on a , we obtain for 1 < p ≤ 2, ∥ T N ( a ; f ) ∥ p ≤ C p ∥ f ∥ 2 p − 1 1  N X n = 1 | a n | p ′ E h | d f n | 2 i 1 / p ′ . Unlike the usual L p boundedness of martingale transforms, which depends on ∥ f ∥ p , this estimate is expressed in terms of the quadratic ener gies E [ | d f n | 2 ]. 5. A martingale J ohn –N irenberg inequality John–Nirenber g inequalities for martingale BMO are classical; see Garsia [3, Ch. III] and Kazamaki [5]. In this section we prove an explicit form on atomic α -regular filtrations. Throughout this section we assume that each F n is generated by a countable measurable partition A n of Ω into atoms, and that P ( A ) > 0 for every A ∈ A n . For A ∈ A n and f ∈ L 1 ( Ω , P ), write f A : = 1 P ( A ) E [ f 1 A ] . Then f n = E n [ f ] = X A ∈A n f A 1 A . Definition 5.1. For f ∈ L 1 ( F ∞ ), define ∥ f ∥ BMO : = sup n ≥ 1 sup A ∈A n 1 P ( A ) E h | f − f A | 1 A i = sup n ≥ 1    E n [ | f − f n | ]    ∞ . (5.2) W e write f ∈ BMO( Ω , F • ) if ∥ f ∥ BMO < ∞ . Definition 5.3. An atomic filtration ( F n ) n ≥ 1 is called α -regular if there exists α ∈ (0 , 1] such that for every n ≥ 2, every parent atom P ∈ A n − 1 , and every child atom C ∈ A n with C ⊆ P , P ( C ) ≥ α P ( P ) . (5.4) Lemma 5.5. Assume that ( F n ) n ≥ 1 is an α -regular atomic filtration. Let f ∈ L 1 ( Ω , P ) with f ≥ 0 , and set f n = E n [ f ] . Then for every n ≥ 2 , f n ≤ α − 1 f n − 1 a.s. (5.6) Proof. Fix n ≥ 2 and let ω ∈ C ⊆ P , where P ∈ A n − 1 and C ∈ A n . Then f n ( ω ) = 1 P ( C ) E [ f 1 C ] ≤ α − 1 P ( P ) E [ f 1 P ] = α − 1 f n − 1 ( ω ) . ■ THE GUNDY –STEIN DECOMPOSITION WITH EXPLICIT CONST ANTS 19 Lemma 5.7. Assume that ( F n ) n ≥ 1 is an α -regular atomic filtration. Fix N ≥ 1 and an atom A ∈ A N . Let g ∈ L 1 ( F ∞ ) and λ > 0 . If 1 P ( A ) E [ | g | 1 A ] ≤ λ, (5.8) then there exists a finite or countable collection of pairwise disjoint atoms { A j } j ∈ J contained in A such that (i) λ < 1 P ( A j ) E [ | g | 1 A j ] ≤ α − 1 λ, j ∈ J ; (5.9) (ii) | g ( ω ) | ≤ λ for a.e. ω ∈ A \ [ j ∈ J A j ; (5.10) (iii) X j ∈ J P ( A j ) ≤ 1 λ E [ | g | 1 A ] . (5.11) Proof. Define the nonnegative martingale X m : = E m [ | g | 1 A ] , m ≥ N , with terminal value X ∞ : = | g | 1 A . Let r : = inf { m ≥ N + 1 : X m > λ } . For m ≥ N + 1, set B m : = { r = m } ∈ F m . Since the filtration is atomic, each B m ∩ A is a union of atoms in A m contained in A . Collecting these atoms over all m gives a finite or countable family { A j } j ∈ J of pairwise disjoint atoms such that [ j ∈ J A j = A ∩ { r < ∞} . On { r < ∞} one has X r > λ , so λ P ( r < ∞ ) ≤ E [ X r ] . Lemma 2.1, applied to the closed martingale ( X m ) m ≥ N , yields E [ X r ] = E [ X ∞ ] = E [ | g | 1 A ] . Since X m = 0 on A c , one has r = ∞ on A c , hence P ( r < ∞ ) = X j ∈ J P ( A j ) , which proves (5.11). If ω ∈ A and r ( ω ) = ∞ , then X m ( ω ) ≤ λ for every m ≥ N . As X m → X ∞ = | g | 1 A almost surely , this yields (5.10). Finally , let A j ∈ A m with m ≥ N + 1. Then r = m on A j , so X m > λ on A j . Moreover , X m − 1 ≤ λ on the par ent atom in A m − 1 containing A j : for m ≥ N + 2 this follows from the definition of r , while for m = N + 1 it follows from (5.8). Lemma 5.5 gives X m ≤ α − 1 X m − 1 ≤ α − 1 λ on A j . Since X m is constant on A j and equals P ( A j ) − 1 E [ | g | 1 A j ], this proves (5.9). ■ 20 M. HORMOZI AND J.-X. ZHU Theorem 5.12. Assume that ( F n ) n ≥ 1 is an α -regular atomic filtration. Let f ∈ L 1 ( F ∞ ) with ∥ f ∥ BMO < ∞ . Set c 1 : = e , c 2 : = α e . Then for every N ≥ 1 , every atom A ∈ A N , and every t > 0 , P  { ω ∈ A : | f ( ω ) − f A | > t }  ≤ c 1 exp − c 2 t ∥ f ∥ BMO ! P ( A ) . (5.13) Proof. Fix A ∈ A N and write B : = ∥ f ∥ BMO . The case B = 0 is trivial, so assume B > 0. Let s > B be a parameter . Apply Lemma 5.7 to ( f − f A ) 1 A at level λ = s . Since 1 P ( A ) E [ | f − f A | 1 A ] ≤ B < s , we obtain pairwise disjoint atoms { A i 1 } i 1 ∈ N ⊆ A such that s < 1 P ( A i 1 ) E [ | f − f A | 1 A i 1 ] ≤ α − 1 s , (5.14) | f ( ω ) − f A | ≤ s for a.e. ω ∈ A \ [ i 1 A i 1 . (5.15) Moreover , X i 1 P ( A i 1 ) ≤ 1 s E [ | f − f A | 1 A ] ≤ B s P ( A ) . (5.16) For each such atom, | f A i 1 − f A | ≤ 1 P ( A i 1 ) E [ | f − f A | 1 A i 1 ] ≤ α − 1 s . (5.17) Proceed inductively . Suppose that for some k ≥ 1 we have constructed pairwise disjoint atoms { A i 1 ,..., i k } inside A , and set E k : = [ i 1 ,..., i k A i 1 ,..., i k . For each parent P = A i 1 ,..., i k , apply Lemma 5.7 to ( f − f P ) 1 P at level s . The defining property of ∥ f ∥ BMO implies 1 P ( P ) E [ | f − f P | 1 P ] ≤ B < s . Summing (5.11) over all parents yields P ( E k ) ≤  B s  k P ( A ) , k ≥ 1 . (5.18) Fix k ≥ 1 and let ω ∈ A \ E k . Construct nested atoms A = P 0 ⊇ P 1 ⊇ · · · ⊇ P ℓ by letting P j be the j th generation atom containing ω while ω ∈ E j , and stop when ω < E ℓ + 1 . Then ℓ ≤ k − 1. Using (5.15) at the last stage and (5.17) at each previous THE GUNDY –STEIN DECOMPOSITION WITH EXPLICIT CONST ANTS 21 stage, we obtain | f ( ω ) − f A | ≤ | f ( ω ) − f P ℓ | + ℓ X j = 1 | f P j − f P j − 1 | ≤ s + ℓα − 1 s ≤ ( ℓ + 1) α − 1 s ≤ k α − 1 s . (5.19) Let t > 0. If t ≥ α − 1 s , choose k ∈ N such that k α − 1 s ≤ t < ( k + 1) α − 1 s . (5.20) Then (5.19) implies { ω ∈ A : | f ( ω ) − f A | > t } ⊆ E k , so that by (5.18), P ( { ω ∈ A : | f ( ω ) − f A | > t } ) ≤  B s  k P ( A ) = exp  − k log s B  P ( A ) . (5.21) Since t < ( k + 1) α − 1 s , one has k > α t / s − 1, and therefore P ( { ω ∈ A : | f ( ω ) − f A | > t } ) ≤ s B exp  − α t s log s B  P ( A ) . (5.22) For 0 < t < α − 1 s , the trivial estimate by P ( A ) is still dominated by the right-hand side of (5.22). Finally choose s = eB . Then s / B = e and log( s / B ) = 1, so (5.22) becomes exactly (5.13). ■ Remark 5.23 . The proof yields a one-parameter family of tail bounds. If one chooses s = u ∥ f ∥ BMO with any u > 1 in (5.22), then P ( { ω ∈ A : | f ( ω ) − f A | > t } ) ≤ u exp − α log u u t ∥ f ∥ BMO ! P ( A ) . The factor α log u / u is maximized at u = e , which r ecovers Theorem 5.12. Corollary 5.24. Assume the hypotheses of Theorem 5.12. For every N ≥ 1 , every atom A ∈ A N , and every β ∈ (0 , α/ e ) , 1 P ( A ) E " exp β | f − f A | ∥ f ∥ BMO ! 1 A # ≤ 1 + e 2 β α − e β . (5.25) Proof. W rite B : = ∥ f ∥ BMO . The case B = 0 is trivial. Assume B > 0 and set Y : = | f − f A | on A . By Theorem 5.12, P ( { ω ∈ A : Y > t } ) ≤ e exp  − α e t B  P ( A ) , t > 0 . For K > 0, the layer-cake identity gives E h ( e KY − 1) 1 A i = Z ∞ 0 Ke Kt P ( { ω ∈ A : Y > t } ) dt . Choose K = β/ B . Then E  e β Y B 1 A  ≤ P ( A ) + e β B P ( A ) Z ∞ 0 exp  −  α e − β  t B  dt = 1 + e 2 β α − e β ! P ( A ) . Dividing by P ( A ) gives (5.25). ■ 22 M. HORMOZI AND J.-X. ZHU Remark 5.26 . In the probabilistic literatur e one often defines martingale BMO by a stopping-time norm such as sup T     E h | M ∞ − M T | p | F T i 1 / p     ∞ < ∞ , and proves conditional exponential integrability for M ∞ − M T ; see, for instance, Kazamaki [5]. For discrete martingales, Garsia [3, Ch. III] established a John– Nirenber g inequality without any regularity assumption on the filtration, using a di ff erent BMO seminorm. The estimate obtained here is of a di ff erent kind: it is proved for atomic α -r egular filtrations and all constants ar e explicit. R eferences [1] L. B ´ aez-Duarte, On the convergence of martingale transforms , Z. W ahrscheinlichkeitstheorie verw . Gebiete 19 (1971), no. 4, 319–322. [2] B. Davis, A comparison test for martingale inequalities , Ann. Math. Statist. 40 (1969), no. 2, 505–508. [3] A. M. Garsia, Martingale Inequalities: Seminar Notes on Recent Pr ogress , Mathematics Lectur e Note Series, W . A. Benjamin, Inc., Reading, MA, 1973. [4] R. F . Gundy , A decomposition for L 1 -bounded martingales , Ann. Math. Statist. 39 (1968), no. 1, 134–138. [5] N. Kazamaki, Continuous Exponential Martingales and BMO, Lecture Notes in Mathematics, vol. 1579, Springer , Berlin, 1994. [6] J. Parcet and N. Randrianantoanina, Gundy’ s decomposition for non-commutative martingales and applications , Proc. Lond. Math. Soc. (3) 93 (2006), no. 1, 227–252. [7] F . Schipp, On L p -norm convergence of series with r espect to pr oduct systems , Anal. Math. 2 (1976), no. 1, 49–64. [8] E. M. Stein, T opics in Harmonic Analysis Related to the Littlewood–Paley Theory , Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, NJ, 1970. [9] T . T suchikura, Sample properties of martingales and their arithmetic means , T ˆ ohoku Math. J. (2) 20 (1968), no. 3, 400–415. (M. Hormozi) B eijing I nstitute of M a thema tical S ciences and A pplica tions (BIMSA), B eijing 101408, C hina Email address : hormozi@bimsa.cn (J.-X. Zhu) D ep artment of M a thema tics , S hanghai N ormal U niversity , S hanghai 200234, C hina Email address : zhujx@shnu.edu.cn