Mathematical Notes, Vol. 56, No. 3-4, 1994

S T R O N G M E A N S A N D T H E O S C I L L A T I O N OF M U L T I P L E F O U R I E R - W A L S H S E R I E S

V . A . R o d i n

1. Using the me thod by Bochkarev, Kheladze proved [1] an analog of the Kolmogorov theorem on everywhere-divergent Fourier-Walsh-Paley series. Based on this result, it seems natura l to use the appa- ratus of strong means to represent functions integrable on the group I-Ik Zk(2) = G.

In [2] Shipp established the /z-s t rong summabil i ty of the Walsh-Paley series for f E L(G) a.e. on G. Since the techniques of the proof in [2] is complex, it is quite difficult to carry over this proof to multiple series. In this paper we establish the strong summabili ty of multiple Fourier-Walsh series by other methods as a consequence of one general s tatement.

2. In [3, 4] for simple Fourier-Walsh series the so-called BMO-property was established that makes possible to s t rengthen the results by Shipp on strong summability. The case of mult iple series is much more complex. In this paper we obtain a certain analog of the BMO-property of mult iple Fourier-Walsh series which cannot be used directly for proving the strong summability. This remark becomes clear after referring to results by Fefferman [5]. The obtained gap between the cases can be removed by considering a special functional related to the phenomenon scale.

The extreme points of this scale are occupied by the strong summabil i ty with an arbi t rary exponent and by the special oscillation of a sequence of rectangular partial sums. The author has obtained analogous statements for tr igonometrical series earlier [6].

3. For convenience, in this paper we consider the particular case N = 2. Using an analogous scheme, one can formulate and prove s ta tements for an arbitrary dimension N. Moreover, it seems na tura l to require that f E L(ln + L)N-I(G). With certain corrections, the statements of this paper can be established for multiple series relative to the Price system with a system of generating sequences bounded in the collection.

4. We denote by Wk a Walsh-Paley system. If

rk(x) = signsin(2k+lzrx), k = O, 1 , . . . ,

is a Rademacher function, and if the number n > 1 is represented as

n = 2 "~ + 2 "2 + �9 - �9 + 2 ~~ , P l > /12 > " ' " > Vs > O,

then

wo(x) = 1 , wk(x) =r~,(x)r~2(x) . . . r , , (x ).

Let Sk(f, x) be a partial sum of the Fourier series relative to the system wk. The strong summabil i ty (/~ < oo) of the series is connected with majorizing operators

(1)

2 n --1

nEZ+ k=O

l / / t

(2)

It is known [3, p. 1467] that the BMO-property is directly connected with majorizing operators

(jW1)2 n - 1 ( j + l ) 2 n - 1

p,jEZ+ kmj2P k=j2P (3)

The Higher Military School, Ministry for Internal Affairs of Russia. Translated from Matematicheskie Zametki, Vol. 56, No. 3, pp. 102-117, September, 1994. Original article submitted June 23, 1993.

948 0001-4346/94/5634-0948512.50 �9 1995 Plenmn Publishing Corporation

In [3, 4] it is shown that operators (2) and (3) have weak type (1.1). In this paper we formulate analogs of these statements for double Fourier-Wa.lsh series of a function from L in + L.

Let S , ( f , x ) (n = nl ,n2) be a rectangular partial sum of the Fourier series relative to the system {wk(xl)wi(x2)}, z = (Xl,X2) �9 G, where w~ are defined in (1), and let f be a function integrable on G. By equalities (2), (3), we define the following operators:

Rl ( f , x,#) = sup (2 -(n1+"2) 2n1--1 2n2--1

kl=O k2=O 2 nl --1

s u p 2 - n "

j~z+,nez?~ k~=o

11# ISk,,k~ ( f , x)l" , (4)

(j-l-l)2 n2 --1

E k2=j2'*2

Ski,k2 (f, x)

R'2(f , z ,# ) =

R3( f , x ,#) =

(j-l-l)2 n2 --1 /z) 1/#,

- 2 - n 2 E Ski,k2 I k2=j2"2

2 n2 --1 (j-{- 1)2 nl --1

sup (2-n2 E 2 -nl E Sk"k2(f' x) j EZ+ ,nEZ~ r k2=0 kl =j2 n 1

(j+1)2 nx --1 --2--n2 E Sk l , k2 ( f , x ) I'~) 1lId

kl=j2nl

(j+1)2 n1-1 (j2+1)2"2-1

s u p ( 2-(nl+n2, E E I Skl 'k ' ( f ' x) nEZ~., kx=j12nl k2=j22n2

(jl -I-1)2 "I --1 ( j2-{- 1) 2 "2 --1

Z sol,k2(f,x)-2-" sk,,o2(/,x) 01=j12nl 02=j22n2

(j1-t-1)2 nl --1 (j2+1)2"2--1

"q- 2--(nl+n2) E E Sol,o2(f, z) ) 1 / # ,

01=ja2nx 02=j22n2

(5)

(6)

where n = (nl,n2), j = ( j l , j2) . The operator R1 is connected with a strong mean with exponent #. The operators R2 and R~ have

symmetric formulas and are connected with the mean oscillation with respect to one of the numerical indices and averaging with respect to another index of the sequence Ski,k2. The operator R3 is directly connected with the classes of functions considered by Fefferman [5].

T h e o r e m 1. Each of the operators (4)-(6) has a weak type (Lln + L, 1). When proving this theorem, we distinguish the metric properties of functions of the space L ln + L,

related to operators (4)-(6). In particular, we distinguish the properties corresponding to the pointwise strong summability of Fourier-Walsh series.

Put x = (xl ,x2), n = (hi ,n2) �9 Z~_, and Inl = nl "n2. Using the standard reasoning applied to the operator R1, from Theorem 1 we obtain the following

statement. T h e o r e m 2. Let Sk(f , x) be a rectangular partial sum of the double Fourier-Walsh-Paley series of a

function f �9 L in + L. Then for any # < oo we have

n 1

li~rn~176 -~1 ~ ISk(f,x) -- f ( x ) l " = 0 (7)

949

for a.a. z E G. In this we characterize points of G at which relation (7) holds. 5. In this section we give some auxiliary lemmas. For numbers given in the binary number system, let us define the summation operation -i- as follows: if

x,y �9 [0, c~) and +oo + ~

�9 = y = 0 (y)2 k - - - - - ~ k = - c x 3

then we put -t-OO

k~----- (x)

10~(=) - 0~(y)12 - ~

We list the following properties of this operation ~-: (a) ~ q - ~ = 0 , n q - n = 0 ; (b) ~ , ( ~ - y ) = ~ , (~ )~ , (y ) ; (c) ~ , ~ ( ~ ) = ~,(~)~m(Y); (d) wj2.(x)wk(x) =wj2.+k(Z) for k=O, 1 , . . . , 2 n - l , j E 7/,}1_.

Let A~ = [k/2", (k + 1)/2"), k = 0 ,1 , . . . ,2" - 1, u E Z~_, be binary intervals. In studying the pointwise strong summability of a simple Walsh series, the following lemma has the central role.

L e m m a 1 (see [3]). If f �9 L([0, 1]) is a periodic function of period 1, then for any V > 1 we have

2~-1 2" / , x )7 lirn ~ ( ~ If(z~-t)- f(z)ldt = 0 k = 0

for a.a. x E [0, 1]. This lemma naturMly generates the majorizing operator

r ( / ) ( z ) = sup If(z4t)l de ~,EZ+

which is an operator of weak type (1,1) (see [7]). For a function of two variables, x = (xl, x2) E [0, 1] 2, periodic of period 1 with respect to each variable,

we consider coordinate analogs of F. In this case, we have

Fl(f,v)(Xl,X2) =

r2(f,v)(Zl,Z2) =

nl r2 _1.2nl/ s u p [ klan= 0 /~=~l ~ I f ( x l @ t l , x 2 ) l d t 1 , n~EZ+ i k~

n l

sup k2~_ ~ If(xl 'x2*t2)ldt2) 1/.y n 2 e Z + L _ \ k 2 n t- 1 ~

(s )

and, for the multiple analog,

r ( f , 7 ) (z ) = sup n=(,u ,n2)EZ?~

F 2n2 2nl n2/ L __ k2_~ 1 nl n2

If(x1-~t1,x25ct2)l) 7] 1/7

(9)

The following lemma makes clear the connection between the superposition of operators (8) and (9).

950

L e m m a 2 . If T > 1, then

(r f ) (z) <_ (r2Flf)(z),

Proof . Let us rewrite the double sum in (9) as

2n2( 2~ ~ 2 n l ( r ? nl )

_ L

x E [0,1].

'7 If(x~-i-h ), (x2-i-t2)l dq dr2 .

For 7 > 1, applying the Minkovskii inequality [8, p. 178] to the internM sum, we obtain

2",-1 ( 2m ~ kx dt2)~} l/'r

< L o : . \ (k, + 1) I,(=,-i-,,), (:= "

From (i i), (12) it follows that

2"2-1 2. 2 2 z [ 2 ~ i f (x14t l ) , (x24t2) idt 1 ( r f p ( x ) < ~ k 2 + l k2 k(kl+Z) k,

k2=O n2 kl=O nl

< 2"~-1 [ ~ Lk2+ 1-2~ ~2 (Flf)(xlz2-i-t2)dt2 _< {(rlr2f)(~)} ~. k2=O n2

l/-y

(10)

(11)

(12)

dr2] ~

If(xl,x2+t2)ldt2 = (M2d f)(x),

If (zl -i-h, x2-i-t2 )l dtl dt2 = ( Md f)( x ).

In the sequel we omit the index d for convenience. By means of the standard arguments these inequalities and Lemma 3 enables us to obtain the following

lemma.

951

(rlf)(z) > sup 2"' [2--1 nlEZ+ JO

(r2f)(z) > sup 2 '~2 i 2-"2 n~EZ+ dO

2--nl 2--n2 ( r f ) (z )> sup2" '+"2[ [

nEZ~ JO JO

[f(x~4q,x2)[ dtl = (Md f)(x),

The lemma is proved.

Each of the operators (8), considered as an operator of a function of one variable with the other variable fixed, has weak type (1,1). Since [fl -< 1, we have (r'if)7(x) < 7/ (7 - 1). Therefore, Fi are operators of strong type (o%o0). In this case, using the standard arguments (see [8, p. 168] and [8, pp. 178, 462]) we obtain the following lemma.

L e m m a 3. I f 7 > 1, then the operator (Ff)(x) has weak type (L ln+L , 1).

In conclusion, we note that operators (8)-(9) majorize the known maximal Hardy operators with respect to the binary net (see [9, p. 117]).

Now we have

We divide the integral in (18) into integrals over the intervals A i = ( i /2- '~, , ( i + 1) /2 -n , ) , i = r/1

1 , 2 , . . . ,2 '~' - 1. Thus, we have

I 2 n l - - I 1 n

- h i k l ~ 0 - 1 ~"~Q ( f , :r,)Tn, ( t l ) dta.

The polynomial under the integral sign takes a constant value in the interval A i �9 this value is denoted n 1 , by Tnl (Ai). Using the H51der inequality, we obtain the estimate of this integral in the form of the product ( 3 , - I + #-1 = i)

_ i~1 \ i fain ~t'If'x) dtl) ) _ ~ (2~l[Tn'I n,)[) Now the problem consists of estimating the right-hand multiplier,

2 n - - 1

i = 1

2 n - - 1 (') = ~ ~ u2" lT,,(t)l "dr /----1

n--1 n--1 n--1 2 - i

=E E i"2"r163 IT.(t)I"dt_<2"E(2-')"]. i----0 j.=2 "~n-; ;" / : 0 - i - 1

]T.(t)[" dr. (19)

Applying (15) for x E (2 - j -1 , 2-J), we get

2 n --I 2n - - I j - - I

v----O u=O " i : 0

2 j --I 2 n - j - - I 2 n - - I

: Z b' E O['v+i2JOJv-l-i2J -- 2j E ~J(b')O~vLQv(t)= E l + E 2 " v----I i : 0 v : O

(20)

Applying the Riesz theorem to the system of f u n c t i o n s 2(J+l)/2Wv+i2J+l , V > O, i > O, orthonormal in the interval (2 - j - l , 2 - i ) , we conclude that

f2 j --1 1 _< 2"-'2 - j ( .t .l

x v ' ~ l

2 . / - - I

v~---I

2 j --I

--< C l ( # ) [ 2 - - J ( E / ] [ ~ v 0

+[ J2-J-1 [ v = l i = 0

2 j --1

~ 1

. ] + 2 j ( ' a - 1 / 2 ) .

2 n - ' / - - 1 /L

,~.+i2Jw~+i2J (t)] dt

2 n - j - - I

i = l

(21)

Similarly, for the second sum we obtain the estimate

953

We continue to estimate the right-hand side of (19) by means of (21), (22). Then we have

) ~IT.(Zx'~)I #.l n - -1 2 ̀7 --1

-< c.(,.).." v_(..-:).+' (~ .. ,o.0 j=o " - n - - 1 2 " / + 1 - - 1 k #.l

_.c.(,.),-z v. k-. j = 0 k=2./ v = l

2'* --1

_< ~,(.)2" ~ ~'"-'1<:<,.1" --- ~(.)" 2". t , ,~ l

(23)

The last inequality in (23) is valid for # ~_ 2 taking account that ~ la.l~ _< 1.

Combining the estimates (16)-(23), we see that the right-hand side of inequality (17) does not exceed the superposition of the operators F1M2. In a similar way, the mean value of I21 is estimated by the superposition of the operators P2M1. Let k = (kl,k2), n = (nl,n2). If we take into account the duality argument for dimension two, we obtain

kl~__O ~ '-~22]#) 1/# : i 2 nl - - I 2 "2 --I

su. v. v. J, s ~(..,,....,,.I Z Io<.,1" < l .= 71=1 i 2 : 1 " l

2 "1 --1 2 "2-1

• ~ ~ a~l,.2D,'.l(t')Dm:(t2)dt. m l . ~ . O 1-rl. 2 ~--0

(24)

The polynomial ~ amDmlDm2 o n the rectangle A~I x A~2 takes a constant value denoted by T(A~). Applying the H61der inequality to the right-hand side of this equality, we see that it does not exceed (see (9)) the product

2 n 1 - 1 2 n 2 - - 1 �9 �9

,~ ~.~: .....+ ~ .~,~,s'p [ ~ ~ , . ~ ~/~,/'] ~ ' ~ . , - _ i 2=1

(25)

For the binary expansions

o o

rnv= ~k(rn~ , )2 k, O k = 0 , 1 , u = l , 2 , k = 0

the products of the Dirichlet kernels on the rectangle ( 2 - j - 1 , 2 -j) x (2 -i-1 , 2-i), as in (20), are represented in the form

j - -1 i--1

(26)

Let us remove the brackets and substitute (26) in the polynomial T. Then, for

t = ( t l , t 2 ) E Aj,i = ( 2 - J - - 1 , 2 - j ) X ( 2 - - i - - 1 , 2 - - i ) ,

954

we obtain the representation of the polynomial as the sum of the following four functions:

2 nl --i 2 n2-1 j--I i--1

ffl(t) = E E Otml'm2Wml(tl)OJm2(t2)E~:k(ml)2k E s mi=0 m2=0 k=0 k=0 2 h i - 1 2 n2-1 j--1

if2('/;) = E E Otmi'm2Wmi(tl)Wm2(t2)E Ek(ml)2kCi(rrz2)2i, ml=0 m2=0 k=0 2 nl --I 2 n2 --I i--I

ff3(t) = E E ~ml'm20"lml('tl)O')m2(l~2)E Ek(m2)2kEj(i721)2j' ml=0 m2=0 k=0 2 nl --12 n2 --1

a 4 ( t ) = E E C~ml'm2Wml(tl)Wm2(t2)~j(m)2J~i(m2)2i" ml=0 m2=0

(27)

If ml < 2 i - 1 and m2 < 2 i -- 1, then for t E hj,i we have

=_ I.

For the sequence of these functions with indices ml >__ 2J and m2 > 2 i, taking into account their orthogo- nality on the rectangle Aj,i, by the Riesz theorem we obtain

2./_1 2 i_ i

S, Io,(t)l" at <_ c{2-(J+i'( E E I'r........l"",m,)" j,i m l = l m2=l

2 - i 2 -/--I 21--1

m l = l m2=l

__2-./ 2/--1 2 ./--I

+ 2--i J2_j_l ml=lm2 E r o l e

2 n2 --i

v=0

2 nl --J --I

E Otmx+v2J,m2LOml+v2J(tl ) "dtl m2=l v=l

2/--1 2hi-J--1 2i_1 2 n z - J _ l

-'1"-2--~ (i"t'J)[ E 'r#27 E E r'r'/'7 E I ~rn '"1- vl 2"/,m2"t- v22J I 3t] pt / "l' } m l = l v i= l rn2=l v2=l

2 i --1 2 j --i

~ C{2--(J+i'(E m2=1 m l = l

2 h i - 1 2 ./--I

m2=1 m l = l

2 nl --I 2 i-I (E

m2=1 rn i= l

mliC~mx,m2i)'r) t~l'r

+ 2 ~(i+j) ). (2s)

The integrals of the symmetric sums ~2 and ~3 are estimated in the same way:

2 n I - 1 2 n2-1 j--1 it J

f A 10"2(/~)1/1 dt = fA E E Otml'm2OJmx(tl)OJm2(t2)EEk(VI2l)2k~i(~'lZ2)2il d~ j,i j,i ml=0 m21~.1 k=0

( 2 ; - - I 2 j - I )/.t

< c 2 - j+ i (" - l ) E E m lc r,,,, 21 ml=0 m2=l

955

2 n2-I 2 j -1 -{-2J(#--1)-I-i~( E ( E IOtm"m'l)7) I~/'Y

m2=O ml=0 2 n l - I 2 i-1

+ 2i(t~-l'-I-J~( E ( E ]Otrn"mz07) 1~/7 ml=0 m2=O

The symmetry arguments and (28), (29) lead to the following estimate:

2;--1 2 j -1

L (10-21" "j- 10"31")dt ~ c{2-Jq-i(#-1) ( E E r/21[Ceml'rn2i)/~ j,i m2=0ml=l

2J_1 21_1

-J-2--i'J-J(/~--I)(E E rn210~m,,rn.I)/~ ml=0 rn2=l 2 n2 --1 2 j --I

-J-2i~'J-J(/~--l)( E (E ]OLm1'm21)'7)/~/'7 rn2=0 rnl=0 2 nl --1 2 i --1

~-2J~q-i(/~-l)( E ( E ]Olml'm20"l) t~/'Y rnl=0 m2=0

In an artalogous way, we have

2J--1 21--i

IA 10-41'u d, ~ c{2(/~-1)(i-I-J)( E E i'rnl'm20 # j,i ml=0 m2=0

-t- 2 (i+j)~ }.

+ 2~z(i+j)

2 "2-1 25-1 "J-2~(iq-J' "]-2i~q'J(tt--1)( E ( E IO~ml'm21)7) #/'~

m2=O ml=0 2 nl --1 21--1

ml=0 m2=0

.

(29)

(30)

(31)

Applying the triangle inequality, we obtain the estimate for the full sum

2 nl --I 2n2--I n1--1 2nl--3 n2--1 2n2--i f /]1/,i 2 /.t

E 1+ = E E E VI=I /12=I j=O lJl=2nl--J -I i=O lJ2=2n2 - I - I

X [ (vl'~1)/2n1 f (l'2"]-j)/2rt2 IT(t)l/~ dt

Jvl/2nl Ju2/2n2 nl--1 n2--1 2-J 2 -~ 4

3=0 i=0 - j -1 -i-1 k=l

In these estimates, the free constants depend only on p. Substituting estimates (29)-(31) in the right- hand side of inequality (32), we see that the problem is to study the behavior of the numerical sums

nl--I n2--1 2 i --I 2 j --I

E 1 = E E 2--(iq-J)( E E IC~ml,m21) #' j=0 i=0 m2=0 ml=0

-1 -1 -2 -1 2"2-1 25 (33)

j=0 i=0 m2=0 rnl=0

956

and also of the sum symmetric to ~-~2 under the conditions p > 2, II{am,,m=}llt-, < 1. The first sum in (33) is majorized by

2"1 2"2 I

E m2El\ 'm?rrt2 E E I ~vl'v2 (rrtlm2)tt-2" (34) ml=l = b'l~---0 v2=0

Applying the Hardy inequality to the double sum, we get

2 nz --I 2n2--i

E1 ~_C E E IOtml'rn21t~(ml'm2)#-2 ~- Cl[Ot[l~u ' "/--1 n t-/.t -1 = 1. (35) rnl~--0 m2=O

The second inequality in (35) is established following the standard scheme [8, vol. 2, p. 186]. The second sum on (33) and the sum symmetric to the former are estimated analogously. Note that the mean of (4) increases with respect to #; therefore, without loss of generality we can assume that # /7 > 1. Then we have

n i -1 2n2-1 2J-1 \'Y\/~/3' C( 2n2-1 2hi

E2 ~--C E 2--J( E ( E Olm"rn2 ) ) ~- \ E E (--'~1 j~---0 m2=0 ml=0 m2=0 ml=l 2n2--1 2 nl --1 "7 #/'7

___~ C( E E [O~ml'rn2[) ) =CH{Olrn1'm2}Hl~ " rn2=O rn i --.-~0

b'~0

Applying together all the equalities and inequalities (16)-(35'), we obtain the estimate

< c(Mf)(x) + (rlM24r2M1)(f,7)( ) +

(35')

Taking into account the strict subjection of M to F, and using Lemma 2, we have

Rl(f,x,#)<_C(FxF2+F2F1)(f,z,"/), /2-1 dl-"/-1 = 1, 7 > 1 .

(36)

(37)

Since Fk (k = 1, 2) are operators of weak type (1,1) and of strong type (b, b) for b > 1, from (37) we obtain, by the standard scheme, the conclusion of Theorem 1 for operator (4).

An analogous inequality for trigonometrical series was obtained in [7]. For 27r-periodic functions, oper- ators F we introduced by Gabisonia [10]. Inequality (37) shows that Gabisonia's operators play the same role in problems on the strong summation of multiple series as maximal Hardy operators in problems on the strong differentiation of multiple integrals (see [8, p. 461].

Before the introduction of operators Fk, attempts of confining ourselves to the operators Mk give neg- ative results. As is known, Hardy and Littlewood showed that the strong summability of the multiple trigonometrical series of an integrable function must not hold at a Lebesgue point.

By the standard arguments, we obtain the conclusion of Theorem 2 from Lemma 4. In the sequel we show that the operators R2, R'2, R3 are Mso estimated by the superposition of opera-

tors (8). First, let us consider R2. By (14) we get

2 n2 --1 Skl,j2"2+k2(f'x)--2-n2 E Sk,,j2,,2+v(f,x)

v=0

=folfo 1 f(x14t1,x25rt2)wj2.2(t2) [Dk2(t2) - K2.2(t2)] Dkl (tl) dt2 dt,

f ( xa +t l, x2 4 t2 )wj2.2 ( t2 )Dk2 ( t2 )Dkl ( tl ) dt

(3s) f (xl 4t l , x2 4t2)t.dj2-2 (t2)Du (t2)Dk, (tl) dtl dt2. 2 n2 --1 1 1

lJ-=0

957

Taking into account (38), we have

F R2 ~- sup |2 - m nEZ~_, ]- kx = 0 jEz+

< c sup [2 -(n~+n2)

n EZ~F, k

jEZ+

2 nl - - I 2 n2 - a

2 - n 2 Z S k l , j 2 n 2 + k 2 -- 2 - n 2

k2=0

2n2--1 /~] i /#

S k 1 ,IJ-l-j2n2

v--~O

2 nl --1 2 n2 --I I 1

Z/o/o k l = O k2=O

f(xl~-tl,x24t2)wj2.~(t2)Dk~(t2)Dk,(tl)dt u] 1/u (39)

The expression in the right-hand side differs from/~a only by the multiplier wj2n2 (t2). The module of this function is equal to unit and

Therefore, all the arguments we have given above are valid for both R2 and R~; moreover, we have

(n2 + R~2)(f,x,#) <_ C(rlr2 + r 2 r l )(/ , x, 7), # - a + ~ , - 1 = 1, ~ > 1. (40)

Let us study the operator R3. Since the kernel in the integral representation of the expression under the module sign in (16) is of the form

2 n2 --I 2 nl --I

y----0 Vl ~ 0

we see that the mean value of R3, as in (38)-(39), is estimated by the mean value of partial sums of f multiplied by wj12.1 (ta) "wj22.2 (t2). Therefore, R3 also holds for the following inequality:

R3(f,x,#) < c(rar + (41)

This completes the proof of Theorems 1, 2.

R e m a r k . Operators (4)-(6) give direct two-dimensional analogs of operators (2)-(3). In the gen- eral case, we consider a partial sum of a multiple series as a sequence indexed with the multi-index n = (na, n 2 , . . . , aN). Operators that constitute a scale of the form (4)-(6) are organized as follows : the arithmetic mean is taken over some coordinates of n, and the special oscillation is taken over the remaining coordinates.

In [6] we show that operator (3) is related to the functional space of the bounded mean oscillation. By means of the John-Nirenberg inequality for simple series, we established [3, 4] a considerable strengthening of the strong summability, connected with a change of a function of exponential type to the function �9 (u) = exp ] u ] - 1. On the other hand, in [6] we show that a multiple analog connected with the N- dimensional BMO does not hold. The function classes, connected with operators of the type (6), are well-known. These classes are described, for example, by Fefferman [5]; moreover, he showed that John- Nirenberg type inequalities do not hold for these classes. In this connection the author states the following hypothesis: an analog of the BMO-property of simple series for multiple series is a statement connected with the space dual to Ha (R x R); more precisely, with its "restriction" to the product [0, 1] • [ 0, 1]. By means of the embedding established for this space in [11], we arrive at the hypothesis on the ~l-s t rong summability a.e. for N-multiple series of functions of L(ln + L) g-1 for ~l(U) = exp(lu] I/N) - 1. The complex atomic structure of Ha (R • I~) does not yet enable the author to confirm this hypothesis.

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R E F E R E N C E S

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2. F. Schipp, "Bemerkung zur starken Summation der Walsh-Fourier-Reihe," Acta Math. Acad. Sci. Hung., 20, No. 3-4, 263-274 (1969).

3. V. A. Rodin, "Totik's conjecture for Chrestenson and Walsh systems," Anal. Math., 18, 295-305 (1992).

4. V. A. Rodin, "A BMO space and strong means of Fourier-Walsh series," Mat. Sb., 182, No. 10, 1463-1478 (1991).

5. R. Fefferman, "Bounded mean oscillation on the polydisk," Annals Math., 110, No. 2, 395-406 (1979).

6. V. A. Rodin, "The rectangular oscillation of the sequence of partial sums for multiple Fourier series and the absence of the BMO-property," Mat. Zametki, 52, No. 2, 152-154 (1992).

7. V. A. Rodin, "Pointwise strong summability of multiple Fourier series," Mat. Zametki, 50, No. 1, 148-150 (1991).

8. A. Zygmund, Trigonometrical Series [Russian translation], Vol. 2, Mir, Moscow (1965). 9. B. I. Golubov, A. V. Efimov, and V. A. Skvortsov, Walsh Series and Transformations [in Russian],

Nauka, Moscow (1991). 10. O. D. Gabsonia, "On strong summability points for Fourier series," Mat. Zametki, 14, No. 5,615-626

(1973). 11. A. Chang Sun-Yung and R. Fefferman, "A continuous version of H 1 with BMO on the bidisk,"

Annal. Math., 112, No. 1, 179-201 (1980).

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