Two-weight weak-type maximal inequalities for martingales

Acta Mathematica Scientia 2009,29B(2):402–408 http://actams.wipm.ac.cn

TWO-WEIGHT WEAK-TYPE MAXIMAL INEQUALITIES FOR MARTINGALES∗



Ren Yanbo (

)

Department of Mathematics & Physics, Henan University of Science and Technology, Luoyang 471003, China E-mail: [email protected]


)

Hou Youliang (

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China E-mail: [email protected]

Abstract In this article, some necessary and sufficient conditions are shown in order that the inequality of the form Φ1 (λ)Pu (f ∗ > λ) ≤ Ev (Φ2 (C|f∞ |)) holds with some constant C > 0 independent of martingale f = (fn )n≥0 and λ > 0, where Φ1 and Φ2 are a pair of Young functions, f ∗ = sup |fn | and f∞ = lim fn a.e. n≥0

n→∞

Key words Martingale, weight, weak-type inequality, Young function 2000 MR Subject Classification

1

60G42

Introduction

Weighted weak-type inequalities for Hardy-Littlewood maximal operator were studied by several authors under different conditions [1–3]. Pick [4] made considerable progress and proposed some necessary and sufficient conditions for the inequality Φ1 (λ)Pu (M f > λ) ≤ Ev (Φ2 (C|f |))

(1.1)

to hold, where M f is the Hardy-Littlewood maximal operator. Kikuchi [5] extended (1.1) to martingales when Φ1 = Φ2 . In this article, we consider the martingale analogue of inequality (1.1), and some necessary and sufficient conditions are obtained for Doob maximal function. Let (Ω, F , P) be a complete probability space and (Fn )n≥0 a sequence of nondecreasing  subalgebras of F satisfying F = σ( F n ). We denote by M the collection of all uniformly integrable martingales with respect to (Ω, F , P, (Fn )n≥0 ), and by T the collection of all stopping times with respect to (Fn )n≥0 . For f = (fn )n≥0 ∈ M, we denote f ∗ = sup |fn |, n≥0

∗ Received

f∞ = lim fn a.e. n→∞

November 28, 2006. Sponsored by the National NSFC (10671147)

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A weight is a measurable function that is positive almost everywhere. Let u be a weight. We denote by Pu the weighted measure udP, and by Eu the expectation relative to Pu . A t Young function is given by Φ(t) = 0 ϕ(s)ds, where ϕ is a nonnegative, nondecreasing, and right-continuous function on (0, ∞). We call Φ an N-function if ϕ satisfies the following three conditions: (i) ϕ(0) = lim ϕ(s) = 0; s→0+

(ii) 0 < s < ∞ ⇔ 0 < ϕ(s) < ∞; (iii) lim ϕ(s) = ∞. s→∞ Note that every N-function is strictly increasing and has its inverse function. The rightcontinuous inverse function of ϕ is given by ψ(t) = inf{s ∈ (0, ∞) : ϕ(s) ≥ t},

t ∈ (0, ∞).

It is clear that

1  ϕ ψ(t) ≤ t ≤ ϕ(ψ(t)) ∧ ψ(ϕ(t)), t ∈ (0, ∞). 2 The Young function given by  t Ψ(t) = ψ(s)ds, t ∈ (0, ∞)

(1.2)

0

is called the complementary function of Φ. Note that Ψ is an N-function if and only if so does Φ. Let us recall the Young inequality st ≤ Φ(s) + Ψ(t) and its important consequences: t ≤ Φ−1 (t)Ψ−1 (t) ≤ 2t, t ∈ (0, ∞); (1.3)  Φ(t)   Ψ(t)  ≤ Ψ(t), Ψ ≤ Φ(t), t ∈ (0, ∞). (1.4) Φ t t We say Φ ∈ Δ2 if Φ(2t) ≤ CΦ(t) for all t ∈ (0, ∞), where C > 0 is a constant. Note that Φ ∈ Δ2 if and only if there is a constant C  > 0 such that ϕ(2t) ≤ C  ϕ(t),

t ∈ (0, ∞).

We say Φ ∈ Δc2 if the complement Ψ ∈ Δ2 . Throughout this article, C, C  , and C  denote constants which may be different from one appearance to another.

2

Modified Two-Weight Jensen Inequality

Here and in the section, and in the next, we assume that (Φi , Ψi ) is a pair of N-functions, where Ψi is the completementary function of Φi , the right-continuous derivatives of which are ϕi and ψi (i = 1, 2), respectively. Lemma 2.1 Let A be a sub-σ-algebra of F and C > 0 a constant. Then, the inequality    Φ (λ)E(u|A)  1 v|A ≤ Φ1 (λ)E(u|A) (2.1) E Ψ2 Cλv holds a.e. for all λ ∈ (0, ∞) if and only if the inequality    Φ (η)E(u|A)  1 v|A ≤ Φ1 (η)E(u|A) E Ψ2 Cην

(2.2)

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holds a.e. for all A-measurable random variables η such that 0 < η < ∞. Proof Since (2.2) ⇒ (2.1) is trivial, we just prove (2.1) ⇒ (2.2). Assume that (2.1) holds a.e. for all λ ∈ (0, ∞). Then, it is easy to verify that (2.2) holds a.e. for all simple A-measurable random variables η with 0 < η < ∞. If η is any positive bounded random variable, then there is a sequence (ηn ) of simple random variables such that ηn ↓ η. Hence, from Fatou’s lemma, we see that (2.2) holds for such η. Finally, using the monotone convergence theorem, we obtain (2.2) for any A-measurable random variable η (0 < η < ∞) a.e. This completes the proof. Theorem 2.1 Let A be as in Lemma 2.1. Then, (2.1) holds a.e. for all λ ∈ (0, ∞) if and only if the inequality Φ1 (E(x|A))E(u|A) ≤ E(Φ2 (C  x)v|A) (2.3) holds for any nonnegative random variable x. Proof Assume that (2.1) holds. Without loses of generality, we may assume that x ∈ L∞ and x > 0 a.e. Now let C  = 2C and set  E(Φ (C  x)v|A)  2 . η = Φ−1 1 E(u|A) Then, Φ1 (η)E(u|A) = E(Φ2 (C  x)v|A) a.e. By Lemma 2.1 and using the Young inequality, we have η Φ1 (η)E(u|A) E( · C  x · v|A) 2Φ1 (η)E(u|A) Cηv η Φ1 (η)E(u|A) ≤ {E(Ψ2 ( )v|A) + E(Φ2 (C  x)v|A)} 2Φ1 (η)E(u|A) Cηv η {Φ1 (η)E(u|A) + E(Φ2 (C  x)v|A)} ≤ η. ≤ 2Φ1 (η)E(u|A)

E(x|A) =

It follows that Φ1 (E(x|A))E(u|A) ≤ Φ1 (η)E(u|A) = E(Φ2 (C  x)v|A). Conversely, suppose that (2.3) holds a.e. for all λ ∈ (0, ∞). Given λ ∈ (0, ∞) and k ∈ (0, ∞), set  v  η Φ1 (λ)E(u|A) , x = Ψ2 χΛ , η=  2λ η Cv where Λ = {η ≤ kv} and χA denotes the characteristic function of set A. Then by (2.3),  2λE(Ψ ( η )vχ |A)  E(Φ2 (C  x)v|A) 2 Cv Λ = Φ1 (E(x|A)) ≤ . (2.4) Φ1 Φ1 (λ)E(u|A) E(u|A) On the other hand, by (1.4), Φ2 (C  x) = Φ2

 Cv η

Ψ2

 η    η  χ χΛ . ≤ Ψ Λ 2 C v Cv

Combining (2.4) and (2.5), we obtain that  E(Ψ ( η )vχ |A)  2λE(Ψ2 ( Cη v )vχΛ |A) 2 Cv Λ ≤ Φ−1 , 1 Φ1 (λ)E(u|A) E(u|A) which by (1.3) implies that 2λE(Ψ2 ( Cη v )vχΛ |A) −1  E(Ψ2 ( Cη v )vχΛ |A)  E(Ψ2 ( Cη v )vχΛ |A) Ψ1 ≤2 . Φ1 (λ)E(u|A) E(u|A) E(u|A)

(2.5)

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 E(Ψ2 ( η )vχΛ |A)  C v ≤ Φ1λ(λ) . It follows Since E(Ψ2 ( Cη v )vχΛ |A) ≤ Ψ2 ( Ck )E(v|A) < ∞ a.e., Ψ−1 1 E(u|A) from (1.4) and the definition of η that    Φ (λ)E(u|A)  1 vχ({η ≤ kv})|A ≤ Φ1 (λ)E(u|A). E Ψ2 2C  λv Now let k → ∞, we get (2.1). The proof is completed. Remark It is clear that if Φ1 = Φ2 , u = v ≡ 1, and C  = 1 in (2.3) is just the Jensen inequality. Theorem 2.1 is valid if Φ1 = Φ2 ∈ Δ2 , and it is a weighted generalization of Jensen inequality. Theorem 2.2 Let A be as in Lemma 2.1 and suppose that Φ1 ∈ Δ2 ∩ Δc2 . Then, (2.1) holds a.e. for all λ ∈ (0, ∞) if and only if   1 (2.6) sup εϕ1 E(ψ2 ( )|A) E(u|A) ≤ K εv ε∈(0,∞) a.e. for some constant K > 0. Proof Suppose that inequality (2.6) holds a.e. Then, by (1.2),   Kη   η , η ∈ (0, ∞). E ψ2 ( )|A ≤ ψ1 v E(u|A) As in the proof of Lemma 2.1, we can show the same inequality holds for any nonnegative A-measurable random variable η. Since Φ1 ∈ Δc2 , if η is nonnegative and A-measurable, then     Kη   η η E Ψ2 ( )v|A ≤ E ηψ2 ( )|A ≤ ηψ1 v v E(u|A)  Cη  C  Kη  ≤ Ψ1 E(u|A) ≤ Ψ1 E(u|A). K E(u|A) E(u|A) Setting η = (Cλ)−1 Φ1 (λ)E(u|A) and using (1.4), we get   Φ (λ)    Φ (λ)E(u|A)  1 1 v|A ≤ Ψ1 E(u|A) ≤ Φ1 (λ)E(u|A) E Ψ2 Cλv λ for all λ ∈ (0, ∞). Conversely, suppose that (2.1) holds a.e. for all λ ∈ (0, ∞). Then, (2.2) holds for all A-measurable random variables η such that 0 < η < ∞ a.e. Given ε ∈ (0, ∞), set  4C  . η = 2ψ1 εE(u|A) Then, by (1.2) and the inequality tϕ1 (t) ≤ Φ1 (2t), we have 4C 2 η ≤ ϕ1 ( ) ≤ Φ1 (η) εE(u|A) 2 η and hence

Φ1 (η)E(u|A) 2 ≤ . (2.7) εv Cvη On the other hand, since Φ1 ∈ Δ2 , using (1.2) and the inequality Φ1 (t) ≤ tϕ1 (t), we obtain η η η Φ1 (η) ≤ C  Φ1 ( ) ≤ C  ϕ1 ( ) 4 4 4 C   1  4C   4C  ≤ ϕ1 ψ1 ψ1 2 2 εE(u|A) εE(u|A)    C 4C ≤ ψ1 . (2.8) εE(u|A) εE(u|A)

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Therefore, combining (2.1), (2.7), and (2.8) yield      Φ (η)E(u|A)  2 1 v|A E Ψ2 ( )v|A ≤ E Ψ2 εv Cvη C   4C  ≤ Φ1 (η)E(u|A) ≤ ψ1 . ε εE(u|A) It follows from the inequality

1 1 εv ψ2 ( εv )

2 ≤ Ψ2 ( εv ) that

  4C   1 . E ψ2 ( )|A ≤ C  ψ1 εv εE(u|A) Since Φ1 ∈ Δ2 , there is a constant C  > 0 such that ϕ1 (C  t) ≤ C  ϕ1 (t) for all t ∈ (0, ∞). Therefore, by (1.2),     4C  1 ϕ1 E(ψ2 ( )|A) ≤ ϕ1 C  ψ1 εv εE(u|A)  1  4C  C ≤ C  ϕ1 ψ1 ≤ , 2 εE(u|A) εE(u|A) from which we complete the proof. Corollary 2.1 Let (Φ, Ψ) be a pair of complementary N-functions and (u, v) a pair of weights. If Φ ∈ Δ2 ∩ Δc2 , then the following statements are equivalent: (i) Two-weight modified Jensen inequality holds, that is, there is a constant C > 0 such that Φ(E(x|A))E(u|A) ≤ CE(Φ(x)v|A). (ii)

There is a constant C > 0, independent of λ ∈ (0, ∞), such that   Φ(λ)E(u|A)   E Ψ v|A ≤ Φ(λ)E(u|A). Cλv

(iii)

3

There is a constant K > 0 such that   1 sup εϕ E(ψ( )|A) E(u|A) ≤ K. εv ε∈(0,∞)

Weighted Weak-Type Maximal Inequalities for Martingales

Theorem 3.1 Let (Φ1 , Ψ1 ) and (Φ2 , Ψ2 ) be two pairs of N-functions and (u, v) a pair of weights. Then, the following statements are equivalent: (i) There is a constant C > 0, independent of f = (fn )n≥0 ∈ M, such that sup Φ1 (λ)Pu (f ∗ > λ) ≤ Eν (Φ2 (C|f∞ |)) a.e.

λ∈(0,∞)

(ii)

There is a constant C > 0, independent of f = (fn )n≥0 ∈ M and n ∈ N, such that Φ1 (|fn |)E(u|Fn ) ≤ E(Φ2 (C|f∞ |)v|Fn )

(iii)

a.e.

There is a constant C > 0, independent of f = (fn )n≥0 ∈ M and τ ∈ T , such that Φ1 (|fτ |)E(u|Fτ ) ≤ E(Φ2 (C|f∞ |)v|Fτ ) a.e.

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There is a constant C > 0, independent of λ ∈ (0, ∞) and n ∈ N, such that   Φ (λ)E(u|F )   1 n v|Fn ≤ Φ1 (λ)E(u|Fn ) a.e. E Ψ2 Cλv If, in addition, Φ1 , Φ2 ∈ Δ2 ∩ Δc2 , then each of (i), (ii), (iii), and (iv) is equivalent to (v) There is a constant C > 0, independent of f = (fn )n≥0 ∈ M, such that (iv)

sup Φ1 (λ)Pu (f ∗ > λ) ≤ CEν (Φ2 (|f∞ |)) a.e.

λ∈(0,∞)

(vi)

There is a constant K > 0, such that   1 sup εϕ1 E(ψ2 ( )|Fn ) E(u|Fn ) ≤ K εv ε∈(0,∞),n∈N

a.e.

Proof By Theorem 2.1, we know (ii)⇔(iv). (i)⇒(ii). Suppose that (i) holds, and let fn ∈ M. For arbitrary Λ ∈ F n and λ ∈ (0, ∞), we have Φ1 (λ)Eu (χ({|fn | > λ} Λ)) = Φ1 (λ)Pu ({|fn | > λ} Λ) ≤ Φ1 (λ)Pu ({E(|f∞ ||Fn ) > λ} Λ) ≤ Φ1 (λ)Pu (sup E(|f∞ |χ(Λ)|Fn ) > λ) n

≤ Ev (Φ2 (C|f∞ |χ(Λ))). Hence, Φ1 (λ)χ({|fn | > λ})E(u|Fn ) ≤ E(Φ2 (C|f∞ |)v|Fn )

a.e.,

which follows Φ1 (|fn |)E(u|Fn ) =

sup Φ1 (λ)χ({|fn | > λ})E(u|Fn )

λ∈(0,∞)

≤ E(Φ2 (C|f∞ |)v|Fn ) a.e. (ii)⇒(iii). It follows from (ii) that Φ1 (|f∞ |)u ≤ Φ2 (C|f∞ |)v. Therefore, Φ1 (|fτ |)E(u|Fτ ) = ≤

∞

n=0 ∞

Φ1 (|fn |)E(u|Fn )χ(τ = n) + Φ1 (|f∞ |)uχ(τ = ∞) E(Φ2 (C|f∞ |)v|Fn )χ(τ = n) + Φ2 (C|f∞ |)vχ(τ = ∞)

n=0

= E(Φ2 (C|f∞ |)v|Fτ ). (iii)⇒(i). Let λ ∈ (0, ∞), and define τ = inf{n ∈ N : |fn | > λ} ∈ T ,

inf ∅ = ∞.

Then, {τ < ∞} = {f ∗ > λ} and |fτ | > λ on {τ < ∞}. Using (iii), we obtain Φ1 (λ)Pu (f ∗ > λ) ≤ Eu (Φ1 (|fτ |)χ(τ < ∞)) = E(Φ1 (|fτ |)E(u|Fτ )χ(τ < ∞)) ≤ E(Φ2 (C|f∞ |)vχ(τ < ∞)) ≤ Ev (Φ2 (C|f∞ |)).

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If Φ1 , Φ2 ∈ Δ2 ∩ Δc2 , it is clear that (i) is equivalent to (v). By Theorem 2.2, we know (iv)⇔(vi). The proof is completed. References [1] Lai Qinsheng. Two weight Φ-inequalities for the Hardy operator,Hardy-Littlewood maximal operator, and fractional integrals. Proc Amer Math Soc, 1993, 118(1): 129–142 [2] Gogatishvili A, Kokilashvili V. Necessary and sufficient conditions for weighted Orlicz class inequalities for maximal functions and singular integrals. I. Georgian Math J, 1995, 2(4): 361–384 [3] Bloom S, Kerman R. Weighted Orlicz space integral inequalities for the Hardy-Littlewood maximal operator. Studia Math, 1994, 110(2): 149–167 [4] Pick L. Two-weight weak type maximal inequalities in Orlicz classes. Studia Math, 1991, 100(3): 207–218 [5] Kikuchi M. On weighted weak type maximal inequalities for martingales. Math Inequalities Appl, 2003, 6(1): 163–175 [6] Bloom S, Kerman R. Weighted LΦ integral inequalities for operators of Hardy type. Studia Math, 1994, 110(1): 36–52 [7] Weisz F. Martingale Hardy Spaces and their Applications in Fourier Analysis. Lecture Notes in Math, Vol.1568. Spring-Verlag, 1994 [8] Long R L. Martingale spaces and inequalities. Peking University Press, 1993 [9] Zuo Hongliang, Liu Peide. Weighted inequalities for the geometric maximal operator on martingale spaces. Acta Mathematica Scientia, 2008, 28(1): 81–85