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

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

generated by the quadratic variation is BMO

and that of the Hardy space H

generated by the conditional quadratic variation is BMO

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

containing adapted function sequences. The inequality

(1)

 X

n=0

|E

X

|



≤ C

 X

n=0

|X

|



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

(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

is BMO

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

is BMO

, where 1 ≤ q < ∞, 1/q + 1/q

= 1 and s

(resp. S

) 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

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

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

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

Moreover, if (F

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

σ

 [

n=0

F

 :=

∞

_

n=0

F

= A.

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

, respectively. We briefly write L

for the L

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

:= (E|f |

)

(0 < p ≤ ∞).

In this paper we consider sequences X = (X

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

is F

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

= 0. The q-variation S

(X) and the conditional q-variation s

(X) (0 < q < ∞) of X are defined by

S

(X) :=  X

n=0

|X

|



and s

(X) :=  X

n=0

E

|X

|



, respectively, while for q = ∞ we let

S

(X) := s

(X) := sup

n∈N

|X

|.

Let us introduce the Hardy spaces h

and h

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

) of adapted functions for which

kXk

:= kS

(X)k

< ∞ and kXk

p

:= ks

(X)k

< ∞,

respectively. Note that h

is a subspace of the well-known space L

(l

) that contains sequences ξ = (ξ

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

kξk

:= h E  X

n=0

|ξ

|



i

.

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

and bmo

consist of all sequences X = (X

) of adapted functions for which kXk

=

sup

n∈N

 E

∞

X

k=n+1

|X

|



< ∞ and

kXk

=

sup

n∈N

 E

∞

X

k=n

|X

|



< ∞,

respectively. Furthermore, let bmo

= bmo

= h

.

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

is bmo

. It will be proved that the dual of h

is bmo

and the one of h

is bmo

(1 ≤ q < ∞, 1/q + 1/q

= 1).

3. Duality results. We shall embed the spaces h

and h

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

D

:= σ(r

, . . . , r

) = σ{[l2

, (l + 1)2

) : 0 ≤ l < 2

}

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

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

r

(x) :=  1 if x ∈ [2l/2

, (2l + 1)/2

) (0 ≤ l < 2

),

−1 if x ∈ [(2l + 1)/2

, (2l + 2)/r2

) (0 ≤ l < 2

).

Set

A

:= σ(F

× D

) (n ∈ N).

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

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

We investigate the martingales relative to (A

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

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

) for which E

d

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

S

(f ) :=  X

n=0

|d

f |



and s

(f ) :=  X

n=0

E

|d

f |



, respectively, and for q = ∞ we let

S

(f ) := s

(f ) := sup

k∈N

|d

f |.

The martingale Hardy spaces H

and H

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

) are defined with the help of the norms

kf k

:=

 \

Ω 1

\

0

S

(f )

dP dλ



and kf k

p

:=

 \

Ω 1

\

0

s

(f )

dP dλ



, respectively. The corresponding dual spaces are equipped with the norms

kf k

= sup

n∈N

 E

∞

X

k=n+1

|d

f |



and

kf k

=

sup

n∈N

 E

∞

X

k=n

|d

f |



.

Set

BMO

= BMO

= H

. It is easy to see that the operator

(2) X 7→ f

:= (X

r

, n ∈ N)

maps h

in H

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

f

:= X

r

is A

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

E

(d

f

) = E

(X

)E

(r

) = 0

because the Rademacher functions are independent. Since S

(f

) = S

(X), our statement is proved. As E

|d

f

|

= E

|X

|

, we have s

(f

) = s

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

to H

(0 < p ≤ ∞, 1 ≤ q ≤ ∞).

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

to BMO

and from bmo

to BMO

(1 ≤ q ≤ ∞).

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

is h

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

= 1/q + 1/q

= 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

is bmo

whenever 1 ≤ q < ∞ and 1/q + 1/q

= 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

consider the functional

l

(X) := E

 X

n=0

X

Y



(X ∈ h

).

Notice that h

is dense in h

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

is BMO

with the same assumption on q and q

as in the theorem. Using this we conclude that

|l

(X)| =   

\

Ω 1

\

0

∞

X

n=0

d

f

d

f

dP dλ   

≤ Ckf

k

H₁^Sq

kf

k

= CkXk

h^Sq₁

kY k

, which yields that l

is bounded on h

.

Conversely, if l is in the dual of h

then it is also in the dual of h

. Consequently, there exists Y ∈ h

such that

(3) l(X) = E

 X

n=0

X

Y



(X ∈ h

).

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

. There- fore there exists g ∈ BMO

such that

(4) l(X) = l(f

) =

\

Ω 1

\

0

∞

X

n=0

X

r

d

g dP dλ (X ∈ h

) and

kgk

≤ Cklk.

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

(ω) =

1

\

0

r

(x)d

g(ω, x) dλ(x).

Applying this we obtain kY k

q0

= sup

n∈N

 E

∞

X

k=n

|Y

|



≤ sup

n∈N

 E

∞

X

k=n

|d

g|



= kgk

and the theorem is proved.

The following theorem can be proved similarly.

Theorem 2. The dual of h

is h

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

= 1/q +1/q

= 1. Moreover , the dual of h

is bmo

provided that 1 ≤ q < ∞ and 1/q + 1/q

= 1.

It is interesting to note that the duals of bmo

and bmo

are not h

and h

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

and bmo

can be defined, having duals h

and h

, respectively.

Let vmo

(resp. vmo

) contain all elements X ∈ bmo

(resp. X ∈ bmo

) for which

n→∞

lim

 E

∞

X

k=n+1

|X

|



= 0



resp. lim

n→∞

 E

∞

X

k=n

|X

|



= 0  .

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

F

is generated by finitely many atoms then the dual of vmo

is h

and

the dual of vmo

is h

whenever 1 < q

< ∞ and 1/q + 1/q

= 1.

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

(5)

 X

n=0

E

|X

|



≤ C

 X

n=0

|X

|



(q ≤ p < ∞), and

 X

n=0

|X

|



≤ C

 X

n=0

E

|X

|



(0 < p ≤ q), where (X

) 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

one has E

|f | ≤ RE

|f | (n ∈ N), the stochastic basis (F

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

) is regular, so is (A

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

and H

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

and h

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

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

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

If (F

) 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) is a sequence of adapted functions and 1 ≤ q < ∞ then

 X

n=0

|E

X

|



≤ C

 X

n=0

|X

|



.

P r o o f. Since the dual of L

(l

) is L

(l

) (1 ≤ q < ∞, 1/q + 1/q

= 1) we have

E  X

n=0

|E

X

|



= sup

Y ∈L_∞(l_q0) kY k_{L∞(lq0 )}≤1

   E h X

n=0

(E

X

)Y

i   .

By Theorem 1, 

  E

h X

n=0

(E

X

)Y

i 

 ≤ CkXk

h^Sq₁

k(E

Y

, n ∈ N)k

q0

.

The inequality kY k

≤ 1 implies E

∞

X

k=n

|E

Y

|

≤ |E

Y

|

+ E

∞

X

k=n+1

|Y

|

≤ 2, which shows that

k(E

Y

, n ∈ N)k

q0

≤ 2

. The proof of the theorem is complete.

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Department of Numerical Analysis L. E¨ otv¨ os University

M´ uzeum krt. 6-8

H-1088 Budapest, Hungary E-mail: weisz@ludens.elte.hu

Received 26 September 1995