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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

1S2

generated by the quadratic variation is BMO

2

and that of the Hardy space H

1s2

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

S12

containing adapted function sequences. The inequality

(1)

 X

n=0

|E

Fn−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

1sq

is BMO

q0

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

1Sq

is BMO

q0

, 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

Spq

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

spq

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]

(2)

duality result with respect to h

S1q

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

Fn−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

Spq

and h

spq

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

n

) of adapted functions for which

kXk

hSqp

:= kS

q

(X)k

p

< ∞ and kXk

hsq

p

:= ks

q

(X)k

p

< ∞,

respectively. Note that h

Spq

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

Lp(lq)

:= h E  X

n=0

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

bmoq

=

sup

n∈N

 E

Fn

X

k=n+1

|X

k

|

q



1/q

< ∞ and

kXk

bmo q

=

sup

n∈N

 E

Fn

X

k=n

|X

k

|

q



1/q

< ∞,

respectively. Furthermore, let bmo

= bmo

= h

S

.

(3)

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

S12

is bmo

2

. It will be proved that the dual of h

S1q

is bmo

q0

and the one of h

s1q

is bmo

q0

(1 ≤ q < ∞, 1/q + 1/q

0

= 1).

3. Duality results. We shall embed the spaces h

Spq

and h

spq

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

pSq

and H

psq

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

n

) are defined with the help of the norms

kf k

HpSq

:=

 \

Ω 1

\

0

S

q

(f )

p

dP dλ



1/p

and kf k

Hsq

p

:=

 \

Ω 1

\

0

s

q

(f )

p

dP dλ



1/p

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

kf k

BMOq

= sup

n∈N

 E

An

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

.

(4)

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

Spq

in H

pSq

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

spq

to H

psq

(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

Spq

is h

Sp0q0

, 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

S1q

is bmo

q0

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

q0

consider the functional

l

Y

(X) := E

 X

n=0

X

n

Y

n



(X ∈ h

Sqq

).

Notice that h

Sqq

is dense in h

S1q

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

1Sq

is BMO

q0

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

H1Sq

kf

Y

k

BMO q0

= CkXk

hSq1

kY k

bmo q0

, which yields that l

Y

is bounded on h

S1q

.

Conversely, if l is in the dual of h

S1q

then it is also in the dual of h

Sqq

. Consequently, there exists Y ∈ h

Sq0q0

such that

(3) l(X) = E

 X

n=0

X

n

Y

n



(X ∈ h

Sqq

).

(5)

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

1Sq

. There- fore there exists g ∈ BMO

q0

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

Sqq

) 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

|

q0



1/q0

≤ sup

n∈N

 E

An

X

k=n

|d

k

g|

q0



1/q0

= kgk

BMO q0

and the theorem is proved.

The following theorem can be proved similarly.

Theorem 2. The dual of h

spq

is h

spq00

, 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

s1q

is bmo

q0

provided that 1 ≤ q < ∞ and 1/q + 1/q

0

= 1.

It is interesting to note that the duals of bmo

q0

and bmo

q0

are not h

s1q

and h

S1q

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

q0

and bmo

q0

can be defined, having duals h

s1q

and h

S1q

, 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

q0

is h

s1q

and

the dual of vmo

q0

is h

S1q

whenever 1 < q

0

< ∞ and 1/q + 1/q

0

= 1.

(6)

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

Fn−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

psq

and H

pSq

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

spq

and h

Spq

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

Fn−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

q0

) (1 ≤ q < ∞, 1/q + 1/q

0

= 1) we have

E  X

n=0

|E

Fn−1

X

n

|

q



1/q

= sup

Y ∈L(lq0) kY kL∞(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

hSq1

k(E

Fn−1

Y

n

, n ∈ N)k

bmo

q0

.

(7)

The inequality kY k

L(lq0)

≤ 1 implies E

Fn

X

k=n

|E

Fk−1

Y

k

|

q0

≤ |E

Fn−1

Y

n

|

q0

+ E

Fn

X

k=n+1

|Y

k

|

q0

≤ 2, which shows that

k(E

Fn−1

Y

n

, n ∈ N)k

bmo

q0

≤ 2

1/q0

. The proof of the theorem is complete.

REFERENCES

[1] N. A s m a r and S. M o n t g o m e r y - S m i t h, Littlewood–Paley theory on solenoids, Colloq. Math. 65 (1993), 69–82.

[2] D. L. B u r k h o l d e r, Distribution function inequalities for martingales, Ann.

Probab. 1 (1973), 19–42.

[3] C. D e l l a c h e r i e and P.-A. M e y e r, Probabilities and Potential B , North-Holland Math. Stud. 72, North-Holland, 1982.

[4] A. M. G a r s i a, Martingale Inequalities, Seminar Notes on Recent Progress, Math.

Lecture Notes Ser., Benjamin, New York, 1973.

[5] C. H e r z, Bounded mean oscillation and regulated martingales, Trans. Amer. Math.

Soc. 193 (1974), 199–215.

[6] —, H

p

-spaces of martingales, 0 < p ≤ 1, Z. Wahrsch. Verw. Gebiete 28 (1974), 189–205.

[7] D. L e p i n g l e, Quelques in´ egalit´ es concernant les martingales, Studia Math. 59 (1976), 63–83.

[8] —, Une in´ egalite de martingales, in: S´ eminaire de Probabilit´ es XII, Lecture Notes in Math. 649, Springer, Berlin, 1978, 134–137.

[9] E. M. S t e i n, Topics in Harmonic Analysis, Princeton Univ. Press, 1970.

[10] F. W e i s z, Duality results and inequalities with respect to Hardy spaces containing function sequences, J. Theor. Probab. 9 (1996), 301–316.

[11] —, Martingale Hardy Spaces and their Applications in Fourier-Analysis, Lecture Notes in Math. 1568, Springer, Berlin, 1994.

[12] —, Martingale operators and Hardy spaces generated by them, Studia Math. 114 (1995), 39–70.

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

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