### VOL. 71 1996 NO. 1

### AN EXTENSION OF AN INEQUALITY DUE TO STEIN AND LEPINGLE

BY

### FERENC W E I S Z (BUDAPEST)

### Hardy spaces consisting of adapted function sequences and generated by the q-variation and by the conditional q-variation are considered. Their dual spaces are characterized and an inequality due to Stein and Lepingle is extended.

### 1. Introduction. It is known that the dual of the martingale Hardy space H

_{1}

^{S}

^{2}

### generated by the quadratic variation is BMO

_{2}

^{−}

### and that of the Hardy space H

_{1}

^{s}

^{2}

### generated by the conditional quadratic variation is BMO

2### (see Garsia [4], Herz [5]). The first result is extended by Dellacherie and Meyer [3] to the space h

^{S}

_{1}

^{2}

### containing adapted function sequences. The inequality

### (1)

### X

^{∞}

n=0

### |E

_{F}

_{n−1}

### X

n### |

^{q}1/q

### p

### ≤ C

_{p}

### X

^{∞}

n=0

### |X

_{n}

### |

^{q}1/q

### p

### (1 < p < ∞) was proved by Stein [9] for q = 2 and by Asmar and Montgomery-Smith [1] for 1 ≤ q ≤ ∞, where X

n### (n ∈ N) are arbitrary measurable functions.

### Using the latter duality result Lepingle [8] verified (1) for p = 1, q = 2 and for adapted functions. The two-parameter analogue of Lepingle’s result can be found in Weisz [10].

### Lepingle [7] proved that the dual of the martingale Hardy space H

_{1}

^{s}

^{q}

### is BMO

_{q}0

### and more recently the author [12] verified that the dual of H

_{1}

^{S}

^{q}

### is BMO

_{q}

^{−}0

### , where 1 ≤ q < ∞, 1/q + 1/q

^{0}

### = 1 and s

q### (resp. S

q### ) denotes the conditional q-variation (resp. the q-variation).

### In this paper the Hardy spaces of adapted function sequences are em- bedded isometrically in martingale Hardy spaces and so the dual of h

^{S}p

^{q}

### generated by the q-variation and, moreover, the dual of h

^{s}p

^{q}

### generated by the conditional q-variation are characterized (1 ≤ p, q < ∞). Applying the

*1991 Mathematics Subject Classification: Primary 60G42; Secondary 42B30.*

### This research was supported by the Hungarian Scientific Research Funds (OTKA) No F4189.

[55]

### duality result with respect to h

^{S}

_{1}

^{q}

### we extend inequality (1) to 1 = p ≤ q < ∞.

### Moreover, if (F

n### ) is regular then (1) holds also for 0 < p < 1 ≤ q < ∞.

### 2. Preliminaries and notations. Let (Ω, A, P ) be a probability space and let (F

n### , n ∈ N) be a non-decreasing sequence of σ-algebras. For simplicity, we suppose that

### σ

### [

^{∞}

n=0

### F

_{n}

### :=

∞

### _

n=0

### F

_{n}

### = A.

### The expectation operator and the conditional expectation operator relative to a σ-algebra C are denoted by E and E

C### , respectively. We briefly write L

p### for the L

p### (Ω, A, P ) space with the norm (or quasinorm) kf k

p### := (E|f |

^{p}

### )

^{1/p}

### (0 < p ≤ ∞).

### In this paper we consider sequences X = (X

n### , n ∈ N) of integrable and adapted (i.e. X

n### is F

n### -measurable for all n ∈ N) functions. We always suppose that X

0### = 0. The q-variation S

q### (X) and the conditional q-variation s

q### (X) (0 < q < ∞) of X are defined by

### S

q### (X) := X

^{∞}

n=0

### |X

n### |

^{q}1/q

### and s

q### (X) := X

^{∞}

n=0

### E

_{F}

_{n−1}

### |X

n### |

^{q}1/q

### , respectively, while for q = ∞ we let

### S

∞### (X) := s

∞### (X) := sup

n∈N

### |X

_{n}

### |.

### Let us introduce the Hardy spaces h

^{S}p

^{q}

### and h

^{s}p

^{q}

### (0 < p, q ≤ ∞) consisting of the sequences X = (X

n### ) of adapted functions for which

### kXk

h^{Sq}p

### := kS

q### (X)k

p### < ∞ and kXk

_{h}sq

p

### := ks

q### (X)k

p### < ∞,

### respectively. Note that h

^{S}p

^{q}

### is a subspace of the well-known space L

p### (l

q### ) that contains sequences ξ = (ξ

n### , n ∈ N) of A-measurable functions and is equipped with the norm

### kξk

_{L}

_{p}

_{(l}

_{q}

_{)}

### := h E X

^{∞}

n=0

### |ξ

_{n}

### |

^{q}p/q

### i

1/p### .

### Now we introduce the corresponding bmo spaces. For 1 ≤ q < ∞, bmo

q### and bmo

^{−}

_{q}

### consist of all sequences X = (X

n### ) of adapted functions for which kXk

_{bmo}

_{q}

### =

### sup

n∈N

### E

Fn∞

### X

k=n+1

### |X

_{k}

### |

^{q}1/q

_{∞}

### < ∞ and

### kXk

_{bmo}− q

### =

### sup

n∈N

### E

_{F}

_{n}

∞

### X

k=n

### |X

_{k}

### |

^{q}1/q

### ∞

### < ∞,

### respectively. Furthermore, let bmo

∞### = bmo

^{−}

_{∞}

### = h

^{S}

_{∞}

^{∞}

### .

### Dellacherie and Meyer [3] showed that the dual of h

^{S}

_{1}

^{2}

### is bmo

^{−}

_{2}

### . It will be proved that the dual of h

^{S}

_{1}

^{q}

### is bmo

^{−}

_{q}0

### and the one of h

^{s}

_{1}

^{q}

### is bmo

q^{0}

### (1 ≤ q < ∞, 1/q + 1/q

^{0}

### = 1).

### 3. Duality results. We shall embed the spaces h

^{S}p

^{q}

### and h

^{s}p

^{q}

### in Hardy spaces of martingales, the duals of which are known. Let

### D

_{k}

### := σ(r

0### , . . . , r

k−1### ) = σ{[l2

^{−k}

### , (l + 1)2

^{−k}

### ) : 0 ≤ l < 2

^{k}

### }

### be the dyadic σ-algebras (see Weisz [11]), where r

k### is the Rademacher func- tion on [0, 1), i.e.

### r

k### (x) := 1 if x ∈ [2l/2

^{k+1}

### , (2l + 1)/2

^{k+1}

### ) (0 ≤ l < 2

^{k}

### ),

### −1 if x ∈ [(2l + 1)/2

^{k+1}

### , (2l + 2)/r2

^{k+1}

### ) (0 ≤ l < 2

^{k}

### ).

### Set

### A

_{n}

### := σ(F

n### × D

_{n}

### ) (n ∈ N).

### Consider the probability space (Ω ×[0, 1), σ(A×B), P ×λ) and the stochastic basis (A

n### , n ∈ N), where B denotes the Borel measurable sets and λ is Lebesgue measure.

### We investigate the martingales relative to (A

n### ), i.e. the sequences f = (d

n### f, n ∈ N) of adapted functions relative to (A

n### ) for which E

An−1### d

n### f = 0 (n ∈ N). The q-variation and conditional q-variation (0 < q < ∞) of a martingale f is given by

### S

q### (f ) := X

^{∞}

n=0

### |d

_{n}

### f |

^{q}1/q

### and s

q### (f ) := X

^{∞}

n=0

### E

An−1### |d

_{n}

### f |

^{q}1/q

### , respectively, and for q = ∞ we let

### S

∞### (f ) := s

∞### (f ) := sup

k∈N

### |d

_{k}

### f |.

### The martingale Hardy spaces H

p^{S}

^{q}

### and H

p^{s}

^{q}

### (0 < p, q ≤ ∞) containing martingales relative to (A

n### ) are defined with the help of the norms

### kf k

H_{p}

^{Sq}

### :=

### \

Ω 1

### \

0

### S

q### (f )

^{p}

### dP dλ

1/p### and kf k

_{H}

^{sq}

p

### :=

### \

Ω 1

### \

0

### s

q### (f )

^{p}

### dP dλ

1/p### , respectively. The corresponding dual spaces are equipped with the norms

### kf k

_{BMO}

_{q}

### = sup

n∈N

### E

_{A}

_{n}

∞

### X

k=n+1

### |d

k### f |

^{q}1/q

### ∞

### and

### kf k

_{BMO}− q

### =

### sup

n∈N

### E

An∞

### X

k=n

### |d

_{k}

### f |

^{q}1/q

_{∞}

### .

### Set

### BMO

_{∞}

### = BMO

^{−}

_{∞}

### = H

_{∞}

^{S}

^{∞}

### . It is easy to see that the operator

### (2) X 7→ f

^{X}

### := (X

n### r

n−1### , n ∈ N)

### maps h

^{S}p

^{q}

### in H

p^{S}

^{q}

### isometrically (0 < p ≤ ∞, 1 ≤ q ≤ ∞). Indeed, the function d

n### f

^{X}

### := X

n### r

n−1### is A

n### -measurable and integrable, of course. On the other hand,

### E

An−1### (d

n### f

^{X}

### ) = E

Fn−1### (X

n### )E

Dn−1### (r

n−1### ) = 0

### because the Rademacher functions are independent. Since S

q### (f

^{X}

### ) = S

q### (X), our statement is proved. As E

An−1### |d

_{n}

### f

^{X}

### |

^{q}

### = E

Fn−1### |X

_{n}

### |

^{q}

### , we have s

q### (f

^{X}

### ) = s

q### (X), and so (2) is isometric from h

^{s}p

^{q}

### to H

p^{s}

^{q}

### (0 < p ≤ ∞, 1 ≤ q ≤ ∞).

### Similarly, we can show that (2) is an isometry from bmo

q### to BMO

q### and from bmo

^{−}

_{q}

### to BMO

^{−}

_{q}

### (1 ≤ q ≤ ∞).

### We can prove in the same way as Theorem 14 in Weisz [12] that the dual of h

^{S}p

^{q}

### is h

^{S}

_{p}0

^{q0}

### , where 1 < p, q < ∞ or 1 = q ≤ p < ∞ and 1/p + 1/p

^{0}

### = 1/q + 1/q

^{0}

### = 1. The following result, due to Dellacherie and Meyer [3] for q = 2, extends this result to p = 1.

### Theorem 1. The dual of h

^{S}1

^{q}

### is bmo

^{−}

_{q}0

### whenever 1 ≤ q < ∞ and 1/q + 1/q

^{0}

### = 1.

### P r o o f. Since the proof is similar to that of Theorem 1 in Weisz [10], we sketch it only. For Y ∈ bmo

^{−}

_{q}0

### consider the functional

### l

Y### (X) := E

### X

^{∞}

n=0

### X

n### Y

n### (X ∈ h

^{S}

_{q}

^{q}

### ).

### Notice that h

^{S}q

^{q}

### is dense in h

^{S}

_{1}

^{q}

### . We verified in [12] that the dual of H

_{1}

^{S}

^{q}

### is BMO

^{−}

_{q}0

### with the same assumption on q and q

^{0}

### as in the theorem. Using this we conclude that

### |l

_{Y}

### (X)| =

### \

Ω 1

### \

0

∞

### X

n=0

### d

n### f

^{X}

### d

n### f

^{Y}

### dP dλ

### ≤ Ckf

^{X}

### k

H_{1}^{Sq}

### kf

^{Y}

### k

_{BMO}− q0

### = CkXk

h^{Sq}_{1}

### kY k

_{bmo}− q0

### , which yields that l

Y### is bounded on h

^{S}

_{1}

^{q}

### .

### Conversely, if l is in the dual of h

^{S}

_{1}

^{q}

### then it is also in the dual of h

^{S}q

^{q}

### . Consequently, there exists Y ∈ h

^{S}

_{q}0

^{q0}

### such that

### (3) l(X) = E

### X

^{∞}

n=0

### X

n### Y

n### (X ∈ h

^{S}

_{q}

^{q}

### ).

### On the other hand, l can be extended preserving its norm onto H

_{1}

^{S}

^{q}

### . There- fore there exists g ∈ BMO

^{−}

_{q}0

### such that

### (4) l(X) = l(f

^{X}

### ) =

### \

Ω 1

### \

0

∞

### X

n=0

### X

n### r

n−1### d

n### g dP dλ (X ∈ h

^{S}

_{q}

^{q}

### ) and

### kgk

_{BMO}− q0

### ≤ Cklk.

### It follows from (3) and (4) that Y

n### (ω) =

1

### \

0

### r

n−1### (x)d

n### g(ω, x) dλ(x).

### Applying this we obtain kY k

_{bmo}−

q0

### = sup

n∈N

### E

Fn∞

### X

k=n

### |Y

k### |

^{q}

^{0}1/q

^{0}

### ∞

### ≤ sup

n∈N

### E

_{A}

_{n}

∞

### X

k=n

### |d

k### g|

^{q}

^{0}1/q

^{0}

### ∞

### = kgk

_{BMO}− q0

### and the theorem is proved.

### The following theorem can be proved similarly.

### Theorem 2. The dual of h

^{s}

^{p}

^{q}

### is h

^{s}

_{p}

^{q0}0

### , where 1 < p ≤ q < ∞ or p ≥ q ≥ 2 and 1/p+1/p

^{0}

### = 1/q +1/q

^{0}

### = 1. Moreover , the dual of h

^{s}

_{1}

^{q}

### is bmo

q^{0}

### provided that 1 ≤ q < ∞ and 1/q + 1/q

^{0}

### = 1.

### It is interesting to note that the duals of bmo

q^{0}

### and bmo

^{−}

_{q}0

### are not h

^{s}

_{1}

^{q}

### and h

^{S}

_{1}

^{q}

### , respectively. However, a kind of special subspaces of bmo

q^{0}

### and bmo

^{−}

_{q}0

### can be defined, having duals h

^{s}

_{1}

^{q}

### and h

^{S}

_{1}

^{q}

### , respectively.

### Let vmo

q### (resp. vmo

^{−}

_{q}

### ) contain all elements X ∈ bmo

q### (resp. X ∈ bmo

^{−}

_{q}

### ) for which

n→∞

### lim

### E

Fn∞

### X

k=n+1

### |X

_{k}

### |

^{q}1/q

### ∞

### = 0

### resp. lim

n→∞

### E

Fn∞

### X

k=n

### |X

_{k}

### |

^{q}1/q

_{∞}

### = 0 .

### With the method used in Weisz [12] one can show that if every σ-algebra

### F

n### is generated by finitely many atoms then the dual of vmo

q^{0}

### is h

^{s}

_{1}

^{q}

### and

### the dual of vmo

^{−}

_{q}0

### is h

^{S}

_{1}

^{q}

### whenever 1 < q

^{0}

### < ∞ and 1/q + 1/q

^{0}

### = 1.

### 4. Inequalities. It follows from the convexity and concavity lemma (see Garsia [4], pp. 113–114) that

### (5)

### X

^{∞}

n=0

### E

Fn−1### |X

_{n}

### |

^{q}1/q

### p

### ≤ C

_{p}

### X

^{∞}

n=0

### |X

_{n}

### |

^{q}1/q

### p

### (q ≤ p < ∞), and

### X

^{∞}

n=0

### |X

_{n}

### |

^{q}1/q

### p

### ≤ C

_{p}

### X

^{∞}

n=0

### E

Fn−1### |X

_{n}

### |

^{q}1/q

### p

### (0 < p ≤ q), where (X

n### ) is a sequence of A-measurable functions. Note that by H¨ older’s inequality (1) follows from (5) for q ≤ p < ∞.

### In case there exists a constant R > 0 such that for all f ∈ L

1### one has E

Fn### |f | ≤ RE

_{F}

_{n−1}

### |f | (n ∈ N), the stochastic basis (F

n### ) is said to be regular . Since the sequence of dyadic σ-algebras is regular, it can easily be seen that whenever (F

n### ) is regular, so is (A

n### ). It is proved in [12] that in this case the spaces H

p^{s}

^{q}

### and H

p^{S}

^{q}

### are equivalent (0 < p < ∞, 1 ≤ q < ∞). Hence h

^{s}p

^{q}

### and h

^{S}p

^{q}

### are also equivalent. This means, amongst other things, that if (F

n### ) is regular then (5) also holds for 0 < p < ∞ and 1 ≤ q < ∞ when (X

n### ) is an adapted function sequence. Consequently, under these conditions we obtain (1) for the parameters 0 < p < ∞ and 1 ≤ q < ∞.

### If (F

n### ) is not regular then (1) is not true for p = 1 (see Lepingle [8]).

### However, if we take again adapted sequences then it holds for p = 1, too.

### The case q = 2 can also be found in Lepingle [8].

### Theorem 3. If (X

^{n}

### , n ∈ N) is a sequence of adapted functions and 1 ≤ q < ∞ then

### X

^{∞}

n=0

### |E

_{F}

_{n−1}

### X

n### |

^{q}1/q

### 1

### ≤ C

### X

^{∞}

n=0

### |X

_{n}

### |

^{q}1/q

### 1

### .

### P r o o f. Since the dual of L

1### (l

q### ) is L

∞### (l

q^{0}

### ) (1 ≤ q < ∞, 1/q + 1/q

^{0}

### = 1) we have

### E X

^{∞}

n=0

### |E

_{F}

_{n−1}

### X

n### |

^{q}1/q

### = sup

Y ∈L_{∞}(l_{q0})
kY k_{L∞(lq0 )}≤1

### E h X

^{∞}

n=0

### (E

Fn−1### X

n### )Y

n### i .

### By Theorem 1,

### E

### h X

^{∞}

n=0

### (E

Fn−1### X

n### )Y

n### i

### ≤ CkXk

h^{Sq}_{1}

### k(E

_{F}

_{n−1}

### Y

n### , n ∈ N)k

bmo^{−}

q0

### .

### The inequality kY k

_{L}

_{∞}

_{(l}

_{q0}

_{)}

### ≤ 1 implies E

Fn∞

### X

k=n

### |E

_{F}

_{k−1}

### Y

k### |

^{q}

^{0}

### ≤ |E

_{F}

_{n−1}

### Y

n### |

^{q}

^{0}

### + E

Fn∞

### X

k=n+1

### |Y

_{k}

### |

^{q}

^{0}

### ≤ 2, which shows that

### k(E

_{F}

_{n−1}

### Y

n### , n ∈ N)k

_{bmo}

^{−}

q0

### ≤ 2

^{1/q}

^{0}

### . The proof of the theorem is complete.

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