Every sequence converging to 0 weakly in contains an unconditional convergence sequence I

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Every sequence converging to 0 weakly in contains an unconditional convergence sequence I

JXNOS KOMLSS

H u n g a r i a n A c a d e m y of Sciences, Budapest, I-Iungary

T h e a i m o f t h i s p a p e r is t o p r o v e t h e a b o v e s t a t e m e n t , w h i c h is c l e a r l y e q u i v a l e n t t o t h e f o l l o w i n g :

THEOREM. For every sequence of measurable functions fn with f f2 ~= K (n 1, 2 . .

@)

there is a subsequence g~ and a square integrable function g such that the sequence h~ : g , - - g is an unconditional convergence sequence.

R e c a l l t h a t a s e q u e n c e hn is c a l l e d a c o n v e r g e n c e s e q u e n c e , i f t h e s e r i e s X cnhn is c o n v e r g e n t a l m o s t e v e r y w h e r e , w h e n e v e r t h e s e q u e n c e cn o f r e a l n u m b e r s s a t i s f i e s 27 c2n < m . T h e s e q u e n c e hn is c a l l e d a n u n c o n d i t i o n a l c o n v e r g e n c e s e q u e n c e , i f e v e r y r e a r r a n g e m e n t o f hn is a c o n v e r g e n c e s e q u e n c e . ( E . g . t h e s e q u e n c e r , ( o n [0, 1]) o f R a d e m a c h e r f u n c t i o n s is k n o w n t o b e a n u n c o n d i t i o n a l c o n v e r g e n c e s e q u e n c e ; w h i l e t h e s e q u e n c e % / 2 / ~ 9 cos (nx) (on [0, n]) is a c o n v e r g e n c e s e q u e n c e ( C a r l e s o n ) , b u t - - b e i n g a c o m p l e t e o r t h o n o r m a l s e q u e n c e - - i t is n o t a n u n c o n d i t i o n a l c o v e r g e n c e s e q u e n c e . )

1 Throughout the paper all functions are measurable functions on some measure space {X, cb ~,/~}. I t is clear t h a t it is sufficient to prove our Theorem in case of finite measure, thus we can t a k e /~(X) = 1.

As a rule, we do n o t indicate

the arguments of functions: writing ~ , f etc. instead of cf(x),f(x) etc., and /~(f > ~) instead of #({x;f(x) > 2}),

a n d the measure: writing jr ~, jr ~01~02 etc. instead of J x q~(x)tt(gx)' f x q;l(x)q~2(x)#(dx) etc.;

we also say }>almost everywhere}> instead of })/~-almost cverywhere}>.

c~n L~p ~" will s t a n d for weak convergence in Lp.

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42 J~_Nos KOMLbS

w O. The preliminaries

We summarize shortly the previous results in this direction.

a) Convergence sequence

The following theorem is a classical result of Menchov (see e.g. [1] p. 156 or [2]):

THEOREM A, Every orthonormal sequence contains a convergence sequence.

Rdvdsz proved t h a t orthogonality is not necessary, i.e.

T]~EO:~E]g B. [3]. For every L2-bounded sequence f , there is a subsequence gn and a square integrable g such that the sequence It, = g, - - g is a convergence sequence.

This theorem was independently proved also b y Gaposhkin ([4] p. 12), and a very simple proof was given b y Chatterji ([5] p. 243).

b) Unconditional convergence sequence The following theorem is due to 0.A. Ziza:

THEOREM C. [6]. I f the orthonormal sequence f , is pointwise bounded, i.e.

Ifn(x)l -<_ f(x)

where f ( x ) is f i n i t e a.e., then it contains an unconditional convergence sequence.

Here, obviously, the strong restriction is not that of the orthogonality, b u t the boundedness.

The problem of extending this result to arbitrary orthonormal sequences is proposed e.g. in Uljanov's survey on solved and unsolved problems in the theory of trigonometric and orthonormal series [7] p. 54.

For the proof first we established some maximal inequalities for strongly multi- plicative sequences, b u t I. Berkes remarked that the maximal inequalities of Billingsley would do the same.

w 1. Billingsley's theorem on 4-multiplieative sequences

In his book [8], Billingsley proves some very useful maximal inequalities; here we state one of them in the special case we are going to use.

T~EO~EM D. [8] p. 87--89. I f for all integers

~n satisfies the inequalities

1 ~_ a ~ b ~ A ~= B the sequence

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E V E I ~ Y S E Q U E N C E CO:NVEI~GI:NG TO 0 W E A K L Y IbT L 2 C O N T A I N S A:N U N C O N D I T I O : N A L 43

f k~__~ K1 ~b 2

f ( v~)2( ~ ~,)~ < K~( ~

^~ ^B ⁼ ^bc~)( y~ c~), ^B (2)

k=a I=A k=a I=A

( K 1 and K s are absolute constants, ck are real numbers), then for all 1 <~ a <~ b ,and 2 > 0

b b

2

k=a k~a

#( m a x I ~v~l >~ ~.) ~< 106K2 ~t4 ~- 4K1 ~2 (3)

a ~ t < b k=a

As a n i m m e d i a t e consequence of T h e o r e m D, one has t h e following corollary ,of i n d e p e n d e n t interestS):

COROLLARY 1. I f the sequence ~vn satisfies the following four conditions for all different indices k, l, m, n:

f q~ ~ g (4)

f

~k~, =< ^{2 2} ^{K 2}

(5)

~v1~(v,~v ~ = 0 (6)

f ~kq~lq~mq~n ~- O,

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then it is an unconditional convergence sequence.

I n d e e d , (5), (6) a n d (7) i m p l y (2) for t h e sequence Fn ~ Cnq:n w i t h Ks = K 2.

Using (4) we get

f 1/fl~

k=a a<=k< l ~ b k=a _ k < l ~ b

b b b

s 2 3 K ~ c~, 2

k=a k=a k=a

i.e. (1) holds w i t h K 1 = 3K. Thus, ~v k satisfies (3), t h a t is k n o w n to be sufficient in order t h a t t h e series X cn~v~ converge almost everywhere, i f o n l y ~c2n <

(see e.g. [ l l ] pp. 1--4).

3) Corollary 1 is similar to the Theorem in [9] (see also Theorem C in [10]); but it does not require the existence of the fourth moments.

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44 aXNos ~o~L6s

Now Fn is clearly an unconditional convergence sequence, since conditions (4), (5), (6) and (7) are invariant under rearrangements of qn.

Corollary 1 can be slightly sharpened as follows:

COROLLARY 2. I f the sequence q~n satisfies the following two conditions:

where the summation runs over all indices k, l, m, n such that at least three

of

them are different, then it is an unconditional convergence sequence.

Conditions (A) and (B) are invariant under rearrangements of ~. thus it is sufficient to check t h a t t h e y imply t h a t the sequence ~ = c,q~,, satisfies conditions (1) and (2).

Using the inequality

~k~_a 2 ( a < k < b )

I c k l < c k - -

one easily gets t h a t (2) holds with K 2 = K 2 - ~ 4 C , where C = ~ v l ~ , ~ ,

and using the inequality

b b /

<=

y. c~ ~ + 2 ~c,~,) ~

we get t h a t (1) holds with K1 = K § 2 ~ r -~ 6C.

The proof of our Theorem will be based on C3rollary 2.

w 2. Proof of the Theorem

Two sequences H,, and q}n are said to be equivalent (this will be indicated by Hn ~ ~b.), if for an arbitrary sequence cn of real numbers the two sets

{x; ~ cnH.(x) is convergent}

n ~ l

and

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E V E R Y S E Q U E N C E C O N V E R G I N G TO 0 W E A K L Y I N L 2 C O N T A I N S A N U N C O N D I T I O N A L 4 5

{x; ~ c~q~(x) is c o n v e r g e n t }

differ in o n l y a set o f zero measure, a n d this holds also for a n a r b i t r a r y r e a r r a n g e m e n t Hp(k), q~p(k) (19(1), 19(2) . . . . is a p e r m u t a t i o n o f t h e n u m b e r s 1, 2, . . .). I t is clear t h a t if Hn ~ Cn a n d n~ < n2 < . . . are positive integers, t h e n t h e subsequences N o w t h e basic step before a p p l y i n g Corollary 2, can be f o r m u l a t e d as follows:

PROPOSITION. F o r every sequence H~ w i t h H~ ~ 0 one can construct a sequence q)~, which is equivalent to a subsequence B~ of H , , and has the following p r o - 19erties :

(i) r (ii)

with 0<_ ~ < 1 a.e.

(iii) ]q)~l <= K~ a.e.,

where K~ is some sequence of 19ositive numbers.

T h e P r o p o s i t i o n will be p r o v e d in w 3. A f t e r h a v i n g this proposition, we can c o m p l e t e t h e p r o o f of t h e t h e o r e m as follows: Since f~ is L2-bounded, it contains a s u b s e q u e n e e f~' such t h a t f~' L-~ g w i t h some s q u a r e i n t e g r a b l e g. T a k i n g H n = fs - - g we h a v e H~ L-~ 0. A p p l y i n g t h e P r o p o s i t i o n , we get a s u b s e q u e n e e /4n of

H~, a n d a sequence ~n w i t h q~n ~ / T n a n d q~ satisfies (i), ( i i ) a n d (iii).

N o w we c o n s t r u c t a s u b s e q u e n c e ~vn of q5 satisfying (A) a n d (B). D e f i n e t h e sequence nk o f positive integers b y i n d u c t i o n as follows:

Since 0 ~ q ~ < 1,

andf

^{qSq5 ~} ^{< 1 ,}

one cart choose nl so large t h a t for all n ~ nl f q~2 n < 1 a n d J q ~ n r 2 < 1.

Assume t h a t nl < n2 < . . . < nk h a v e a l r e a d y b e e n d e f i n e d m such a w a y t h a t we h~ve

f 2 2

l < j < i = < / c

f

~2n~q~n.~ n. < 2~ l < _ _ l < j < i < - - k

J t - - - -

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46 JXNOS KO~LdS

f

q~n ~b2.~bn,j < 2~

I

1 ~ 1 < j < i ~ k

f

q~ q~,.q)zj < 2 ~

1

1 < l < j < i <= k

f

~ n ~ n t ~ ^. < 2~ l < j < i _

f

t~)nm~)nl~)nj~-)ni < ~ 1 <= m < l < j < i <= k,

a n d t r y to define t h e n u m b e r nk+~ > nk in such a w a y t h a t the above six conditions hold if k is replaced b y k ~ - 1, i.e. also for i = k - ~ ].

Since On1 . . . Cnk are b o u n d e d (by (iii)), for n - + oc

t h u s

nj n nj < 1 ,

j = l , . . . , k

f q)2 q)2 nj nk+x < 1, j = 1 , . . . , k

if nk+~ is chosen sufficiently large. Since q~n ~ 0, t h e second, third, f i f t h a n d sixth conditions will be satisfied for i = k + 1, if nk+~ is chosen large enough.

Now for l < = l < j < k

1

f q52 ,

~- 2 j -~- ~nl~)nj( n - ~2)

a n d since t h e second t e r m in t h e r i g h t - h a n d side of this i n e q u a l i t y t e n d s to zero as n--~ oo, t h e f o u r t h condition also holds, if nk+l is chosen sufficiently large.

Choosing % = ~bnk, we o b t a i n e d a sequence % satisfying (A) a n d (B) w i t h K = 1, t h u s % is a n u n c o n d i t i o n a l convergence sequence, a n d (since /Tn ~ ~bn) so is /Ink, t h a t will be t h e sequence hk in our Theorem. Qu.e.d.

w 3. Proof of the Proposition

W e will use a l e m m a of G a p o s h k i n [4], p. 14 (which is also c o n t a i n e d i m p l i e i t e l y in [12], however it is n o t explicitely s t a t e d there):

LEMMA. I f an is an Ll-bounded sequence:

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then it contains a subsequence ft. such that ft. can be written as

where riO)fiT) ~- O, n = l, 2 , . . . , fl!~,) is weakly convergent riO) ~ fl, a n d

~(~) # o) < oo.

n = l

W e will also use the following simple remark: if y , ~ 7 and ~v is a b o u n d e d function, then

R .

~ y . ~ y (p is arbitary, 1 < p < oo).

F u r t h e r , if 17"1 < ]Y.], #(Y: # ?.) --> 0, then ~'~ ~ 7.

N o w applying the L e m m a (H2. is Ll-bounded), one can choose a s u b s e q u e n c e of H . such t h a t

where

and

g l)~2)= o, g ) ) < ~,

g(fl~) # o) <

~.

n ~ l

Define the sequence y . as follows

Since y~. = riO), we h a v e

(*) ~ < ~.

/7. if riO)~0

Y"~- 0 if r i O ) = O .

Since [y.[ ~ ]/7.I, /t(y. # / t . ) - + 0 and /~. ~ 0, we h a v e

(**) ~. ~ o.

Since

g(~. # / 7 . ) < oo,

t h u s y . ~--/1. (actually on almost all x, y= = / 7 . for n > nc = no(x)).

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48 JANOS K O ~ L Q S

D e f i n e

v " -

+ [3

Since O ~ 1/~/1 + f i ~ 1 a n d a n d 9 ] ~ { 3 / 1 + f i . N o w p u t

Since #(q)~ r ~.) ~

f 1+.1/2+,

^{t h u s}

a n d h e n c e

a n d

Since }r =~ 2",

O ~ < l / ( l + f i ) ~ l ,

~. if I~.] =<2"

0 otherwise.

(*) a n d (**) i m p l y r

~b~ -~

q)~ satisfies all conditions o f t h e P r o p o s i t i o n .

L e t us m e n t i o n t h a t t h e a b o v e t r a n s f o r m a t i o n (dividing b y ~ / 1 -4- fl) m a k e s it possible t o choose a s u b s e q u e n c e of ~bn w i t h

f

< K ( t h a t was done in t h e p r o o f of t h e T h e o r e m ) . B u t ( t h o u g h

f < 1)

it c a n n o t ensure t h e e x i s t e n c e o f h i g h e r m o m e n t s , e.g.

f

~ ~s k - ~ ~ e~n still h a p p e n . T h u s , t h e a b o v e used Billingsley's t h e o r e m c a n n o t be r e p l a c e d b y a n y t h e o r e m using h i g h e r m o m e n t s t h a n second.

References

1. ALExI2S, G., Convergence problems of orthogonal series. Publishing House of the Hungarian Academy of Sciences, Budapest, 1961.

2. ME~Oaov, D., Sur ]a convergence et 1~ sommation des s6ries de fonctions orthogonales.

Bull. Soc. Math. France 64 (1936), 147--170.

3. 1R.~v~sz, P.: On a problem of Steinhaus. Acta Math. Acad. Sci. Hungar 16 (1965), 310--318.

4. GAPOSHKI~, V. ~'., Lacunary series and independent functions. Uspehi Mat. N a u k 6/21 (1966), 3--82. (In Russian).

5. C~A2~E~aI, S. D., A general strong law. Invent. Math. 9 (1970), 235--245.

6. ZIZA, O. A.: On a property of orthonormal sequences (in Russian). Mat. Sbornik, 55 (100) (1962), 3--16.

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E V E R Y S E Q U E N C E C O N V E R S I N G TO 0 W E A K L Y I N L 2 C O N T A I N S A N U N C O N D I T I O N A L 49

7. ULJA~qOV, P. L., Solved a n unsolved problems in the theory of trigonometric a n d orthonormal series. Uspehi Mat. Naulc 19/1 (1964), 3--69.. (In Russian.)

8. BILLIN~ST,EY, P., Convergence of probability measures. J o h n Wiley a n d Sons Inc., 1968.

9. KoMn6s, J. & R~v]~sz, P., R e m a r k to a paper of Gaposhkin. Acta Sci. Math. Szeged 33 (1972), 237--241.

10. --~)-- On the series Xc~q;~. Studia Sci. Math. Hungar 7 (1972), 451--458.

11. G ~ s I s A , A., Topics in almost everywhere convergence. Chicago, 1970.

12. KOML6S, J., A generalization of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar.

18 (1967), 217--229.

Received December 7, 1972 J~nos Koml6s

Mathematical I n s t i t u t e

of the H u n g a r i a n Academy of Sciences Budapest V., Re~ltanoda u. 13--15 l~ungary