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

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(1)

BY

1S2

2

1s2

2

S12

n=0

Fn−1

n

q

1/qp

p

n=0

n

q

1/q

p

n

1sq

q0

1Sq

q0

0

q

q

Spq

spq



(2)

S1q

n

n

n=0

n

n=0

n

C

p

p

p

p

1/p

n

n

n

0

q

q

q

n=0

n

q

1/q

q

n=0

Fn−1

n

q

1/q

n∈N

n

Spq

spq

n

hSqp

q

p

hsq

p

q

p

Spq

p

q

n

Lp(lq)

n=0

n

q

p/q

1/p

q

q

n

bmoq

n∈N

Fn

k=n+1

k

q

1/q

bmo q

n∈N

Fn

k=n

k

q

1/q

S

(3)

S12

2

S1q

q0

s1q

q0

0

Spq

spq

k

0

k−1

−k

−k

k

k

k

k+1

k+1

k

k+1

k+1

k

n

n

n

n

n

n

n

An−1

n

q

n=0

n

q

1/q

q

n=0

An−1

n

q

1/q

k∈N

k

pSq

psq

n

HpSq

Ω 1

0

q

p

1/p

Hsq

p

Ω 1

0

q

p

1/p

BMOq

n∈N

An

k=n+1

k

q

1/q

BMO q

n∈N

An

k=n

k

q

1/q

(4)

S

X

n

n−1

Spq

pSq

n

X

n

n−1

n

An−1

n

X

Fn−1

n

Dn−1

n−1

q

X

q

An−1

n

X

q

Fn−1

n

q

q

X

q

spq

psq

q

q

q

q

Spq

Sp0q0

0

0

S1q

q0

0

q0

Y

n=0

n

n

Sqq

Sqq

S1q

1Sq

q0

0

Y

Ω 1

0

n=0

n

X

n

Y

X

H1Sq

Y

BMO q0

hSq1

bmo q0

Y

S1q

S1q

Sqq

Sq0q0

n=0

n

n

Sqq

(5)

1Sq

q0

X

Ω 1

0

n=0

n

n−1

n

Sqq

BMO q0

n

1

0

n−1

n

bmo

q0

n∈N

Fn

k=n

k

q0

1/q0

n∈N

An

k=n

k

q0

1/q0

BMO q0

spq

spq00

0

0

s1q

q0

0

q0

q0

s1q

S1q

q0

q0

s1q

S1q

q

q

q

q

n→∞

Fn

k=n+1

k

q

1/q

n→∞

Fn

k=n

k

q

1/q

n

q0

s1q

q0

S1q

0

0

(6)

n=0

Fn−1

n

q

1/qp

p

n=0

n

q

1/q

p

n=0

n

q

1/qp

p

n=0

Fn−1

n

q

1/q

p

n

1

Fn

Fn−1

n

n

n

psq

pSq

spq

Spq

n

n

n

n

n=0

Fn−1

n

q

1/q1

n=0

n

q

1/q1

1

q

q0

0

n=0

Fn−1

n

q

1/q

### = sup

Y ∈L(lq0) kY kL∞(lq0 )≤1

n=0

Fn−1

n

n

n=0

Fn−1

n

n

hSq1

Fn−1

n

bmo

q0

(7)

L(lq0)

Fn

k=n

Fk−1

k

q0

Fn−1

n

q0

Fn

k=n+1

k

q0

Fn−1

n

bmo

q0

1/q0

p

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

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