ALMOST EVERYWHERE CONVERGENCE OF (C, α) - CiteSeerX

Georgian Mathematical Journal Volume 13 (2006), Number 3, 447–462

ALMOST EVERYWHERE CONVERGENCE OF (C, α)-MEANS OF QUADRATICAL PARTIAL SUMS OF DOUBLE VILENKIN–FOURIER SERIES ¨ ´ AND USHANGI GOGINAVA GYORGY GAT

Abstract. We prove that the maximal operator of the (C, α)-means of quadratical partial sums of double Vilenkin–Fourier series is of weak type (1,1). 1 Moreover, the (C, α)-means tα n f of a function f ∈ L converge a.e. to f as n → ∞. 2000 Mathematics Subject Classification: 42C10. Key words and phrases: Ces´aro means, Vilenkin system, a.e. convergence.

1. Introduction In 1939, for the two-dimensional trigonometric Fourier partial sums Sj,j f Marcinkiewicz [8] proved that for all f ∈ L log L([0, 2π]2 ) the relation n 1X t1n f = Sj,j f → f n j=0 holds a.e. as n → ∞. Zhizhiashvili [13] improved this result and showed that for f ∈ L([0, 2π]2 ) the (C, α) means n 1 X α−1 α tn f = α A Sj,j f An j=0 n−j converge to f a.e. for any α > 0. Dyachenko [3] proved this result for dimensions greater than 2. In papers [12, 6] by Weisz and Goginava one can find that the (C, 1) means t1n f of the double Walsh–Fourier series of a function f ∈ L([0, 1]2 ) converges to f a.e. Recently, G´at [4] proved this result with respect to two-dimensional Vilenkin systems. The d-dimensional Walsh–Fourier case is discussed in [7]. The aim of this paper is to generalize the result of Zhizhiashvili [13] concerning the (C, α)-means with respect to two-dimensional (bounded) Vilenkin systems. First, we give a brief introduction to the theory of Vilenkin systems. These orthonormal systems were introduced by N. Ya. Vilenkin in 1947 (see, e.g., [11, 1]) as follows. Let m := (mk , k ∈ N) (N := {0, 1, . . . }, P := N \ {0}) be a sequence of integers, each of them not less than 2. Let Zmk denote the discrete cyclic group of order mk . That is, Zmk can be represented by the set {0, 1, . . . , mk − 1}, with the group operation mod mk addition. Since the group is discrete, every subset c Heldermann Verlag www.heldermann.de ISSN 1072-947X / $8.00 / °

´ AND U. GOGINAVA G. GAT

448

is open. The normalized Haar measure µk on Zmk is defined by µk ({j}) := 1/mk (j ∈ {0, 1, . . . , mk − 1}). Let ∞

Gm := × Zmk . k=0

Then every x ∈ Gm can be represented by a sequence x = (xi , i ∈ N), where xi ∈ Zmi (i ∈ N). The group operation on Gm (denoted by +) is the coordinatewise addition (the inverse operation is denoted by −), the measure (denoted by µ), which is the normalized Haar measure, and the topology are respectively the product measure and the topology. Consequently, Gm is a compact Abelian group. If supn∈N mn < ∞, then we call Gm a bounded Vilenkin group. If the generating sequence m is not bounded, then Gm is said to be an unbounded Vilenkin group. In this paper we discuss bounded Vilenkin groups only. A Vilenkin group is metrizable in the following way: ∞ X |xi − yi | d(x, y) := (x, y ∈ Gm ). M i+1 i=0 The topology induced by this metric, the product topology, and the topology given by intervals defined below, are the same. A base for the neighborhoods of Gm can be given by the intervals I0 (x) := Gm ,

In (x) := {y = (yi , i ∈ N) ∈ Gm : yi = xi for i < n}

for x ∈ Gm , n ∈ P. Let 0 = (0, i ∈ N) ∈ Gm denote the null element of Gm and In (0) := In , I n = Gm \In . Furthermore, let Lp (Gm ) (1 ≤ p ≤ ∞) denote the usual Lebesgue spaces (k · kp are the corresponding norms) on Gm , n the σ-algebra generated by the sets In (x) (x ∈ Gm ), and En the conditional expectation operator with respect to n (n ∈ N). Let 1 ≤ p ≤ +∞ be real. We say that an operator T is of type (p, p) if there exists an absolute constant C > 0 such that kT f kp ≤ Ckf kp for all f ∈ Lp . T is said to be of weak type (1, 1) if there exist an absolute constant C > 0 such that kT f kweak-L1 ≤ Ckf k1 for all f ∈ L1 (Gm ), where kf kweak-L1 = supλ>0 λµ(|f | > λ). It is known that the operator which maps a function f on the maximal function f ∗ := sup |En f | is of weak type (1, 1), and of type (p, p) for all 1 < p ≤ ∞ (see, e.g., [2]). Let M0 := 1, Mn+1 := mn Mn (n ∈ N) be the so-called generalized powers. Then each natural number n can be uniquely expressed as ∞ X n= ni Mi (ni ∈ {0, 1, . . . , mi − 1}, i ∈ N), i=0

where only a finite number of ni ’s differs from zero. For 1 ≤ n ∈ N we denote by |n| := max {k ∈ N : M k ≤ n} the order of a natural number n. In other words, M|n| ≤ n < M|n|+1 . The generalized Rademacher functions are defined as ¶ µ √ xn (x ∈ Gm , n ∈ N, ı := −1). rn (x) := exp 2πı mn

ALMOST EVERYWHERE CONVERGENCE

449

The nth Vilenkin function is ψn :=

∞ Y

n

rj j

(n ∈ N).

j=0

The system ψ := (ψn : n ∈ N) is called a Vilenkin system. Each ψn is a character of Gm , and all the characters of Gm are of this form. Define the m-adic addition as ∞ X k ⊕ n := (kj + nj ( mod mj ))Mj (k, n ∈ N). j=0

Then ψk⊕n = ψk ψn , ψn (x + y) = ψn (x)ψn (y), ψn (−x) = ψ¯n (x), |ψn | = 1 (k, n ∈ N, x, y ∈ Gm ). Set Aαn := (1+α)...(n+α) for any n ∈ N, α ∈ R. It is known that Aαn ∼ nα . Define n! the Fourier coefficients, the partial sums of the Fourier series, the Dirichlet kernels, the (C, α) means, kernels, and the Fej´er means and kernels with respect to the Vilenkin system ψ as follows: Z fˆ(n) :=

f ψ¯n dµ, Gm

Sn f :=

n−1 X

fˆ(k)ψk ,

k=0

Dn :=

n−1 X

ψk ,

k=0

σnα f =

n 1 X α−1 A Sk f, Aαn k=0 n−k

Knα :=

n 1 X α−1 A Dk , Aαn k=0 n−k

σn f := σn1 f, Kn := Kn1 (f ∈ L1 (Gm ). It is well-known that Z Sn f (y) = f (x)Dn (y − x)dµ(x) (n ∈ N, y ∈ Gm , f ∈ L1 (Gm )). Gm

It is also well-known [1] that ( Mn if x ∈ In := In (0), DMn (x) = 0 if x ∈ / In , Z SMn f (x) = Mn f dµ = En f (x) (f ∈ L1 (Gm ), n ∈ N). In (x)

(1)

´ AND U. GOGINAVA G. GAT

450

Next, we introduce some notation for the theory of two-dimensional Vilenkin systems. Let m ˜ be a sequence like m. The relation between the sequences (m ˜ n) ˜ and (Mn ) is the same as between the sequences (mn ) and (Mn ). The group Gm × Gm˜ is called a two-dimensional Vilenkin group. The normalized Haar measure is denoted by µ as in the one-dimensional case. We also suppose that m = m, ˜ that is, Gm × Gm˜ = G2m . The two-dimensional Fourier coefficients, the rectangular partial sums of the Fourier series, the Dirichlet kernels, the (C, α) means, the kernels of the (C, α) means with respect to a two-dimensional Vilenkin system are defined as follows: Z ˆ f (x, y)ψ¯n1 (y)ψ¯n2 (y)dµ(x, y), f (n1 , n2 ) := G2m

Sn1 ,n2 f (u, v) :=

nX 2 −1 1 −1 n X

fˆ(k1 , k2 )ψk1 (u)ψk2 (v),

k1 =0 k2 =0

Dn1 ,n2 (x, y) := Dn1 (x)Dn2 (y), n 1 X α−1 α tn f := α A Sj,j f, An j=0 n−j Tnα

n 1 X α−1 A Dj,j . := α An j=0 n−j

It is also well-known that tαn f (u, v)

Z f (x, y)Tnα (u − x, v − y)dµ(x, y).

= G2m

For the two-dimensional variable (x, y) ∈ G2m we use the notation ψn1 (x, y) = ψn (x),

Dn1 (x, y) = Dn (x),

Knα,1 (x, y) = Knα (x),

ψn2 (x, y) = ψn (y),

Dn2 (x, y) = Dn (y),

Knα,2 (x, y) = Knα (y)

for any α > 0 and n ∈ N. Set the maximal operator tα∗ f := sup |tαn f | for any f ∈ L1 (G2m ) and α > 0. n∈N

2. Main Results Theorem 1. Let f ∈ L1 (G2m ) and α > 0. Then ktα∗ f kweak-L1 ≤ C kf k1 . Corollary 1. Let f ∈ L1 (G2m ) and α > 0. Then tαn (f ) → f a.e. as n → ∞.

ALMOST EVERYWHERE CONVERGENCE

451

3. Auxiliary results Lemma 1. Let 0 ≤ j < ns Ms and 0 ≤ ns < ms . Then Dns Ms −j = Dns Ms − ψns Ms −1 Dj . Proof. It is clear that nsX Ms −1

Dns Ms = Dns Ms −j +

ψk = Dns Ms −j +

k=ns Ms −j

j−1 X

ψns Ms −k−1 .

k=0

Consequently, ψns Ms −k−1 (x) = ψ(ns −1)Ms +(ms−1 −1)Ms−1 +···+(m0 −1)M0 −k (x) = ψ(ns −ks −1)Ms +(ms−1 −ks−1 −1)Ms−1 +...(m0 −k0 −1)M0 (x) = ψ(n −1)M +(m −1)M +...(m −1)M (x)ψ¯k (x) s

s

s−1

s−1

0

0

= ψns Ms −1 (x)ψ¯k (x). Lemma 1 is proved.

¤

Lemma 2. Let α ∈ (0, 1) and n := n(A) = nA MA + · · · + n0 M0 . Then in the one-dimensional case   Mp −1 A X i  X X c (α) |Knα | ≤ α Mpα−1 |Kj | + Miα |KMi −1 | + Miα DMi .  p=1  n i=0

j=Mp−1

Proof. It is evident that n X

Aα−1 n−j Dj =

nX A MA

j=1

n X

Aα−1 n−j Dj +

j=1

Aα−1 n−j Dj = I + II.

(2)

j=nA MA +1

Since [5] for r ∈ {0, . . . , mA − 1} ! Ã r−1 X q r ψMA DMA + ψM Dj , Dj+rMA = A q=0

then for I we write ! Ã r−1 nX MA A −1 X X q I= ψMA DMA Aα−1 n−j−rMA r=0 j=1

+

ÃM −1 nX A −1 A X r=0

j=0

q=0

!

Aα−1 D (nA −r−1)MA +n(A−1) +j MA −j

r ψM = I1 + I2 . A

It is evident that |I1 | ≤ c (α) MAα DMA . Using Lemma 1, for I2 we obtain ¯) ¯ −1 ( nX A A −1 ¯M ¯ X ¯ ¯ α−1 α A(nA −r−1)MA +n(A−1) +j Dj ¯ . |I2 | ≤ c (α) MA DMA + ¯ ¯ ¯ r=0

j=1

(3)

(4)

´ AND U. GOGINAVA G. GAT

452

Since Dj+nA MA = DnA MA + ψnA MA Dj , we write

¯ ¯ª © |II| ≤ c (α) MAα DMA + n(A−1) ¯Knα(A−1) ¯ .

Combining (2)–(5), we obtain ( n |Knα | ≤ c (α) MAα DMA

(5)

¯ ¯ ¯ ¯

¯ ¯ ¯ + Aα−1 D ¯ (nA −r−1)MA +n(A−1) +j j ¯ r=0 j=1 ) ¯ ¯ + n(A−1) ¯K α(A−1) ¯ . nX A −1 ¯M A −1 X

n

Iterating this inequality, we obtain ¯ −1 ¯) ( A A n i −1 ¯M i ¯ X X X X ¯ ¯ α−1 n |Knα | ≤ c (α) Miα DMi + A D ¯ ¯ . (ni −r−1)Mi +n(i−1) +j j ¯ ¯ i=0 r=0

i=0

j=1

Applying Abel’s transformation, we write M i −1 X

D Aα−1 (ni −r−1)Mi +n(i−1) +j j

=−

j=1

M i −2 X

Aα−2 jK j (ni −r−1)Mi +n(i−1) +j

j=1

+Aα−1 (Mi − 1) KMi −1 , (ni −r)Mi +n(i−1) −1 consequently, n |Knα | ≤ c (α)

 A X 

Miα DMi +

A X i X

Mp −1

Mpα−1

i=0 p=1

i=0

X

j=Mp−1

|Kj | +

A X

Miα |KMi −1 |

i=0

Lemma 2 is proved.

  

. ¤

Lemma 3. Let A ≥ k. Then Z sup |KMn | ≤ c

n≥A

Mk . MA

Ik

Proof. Since [9] |KMA (x)| ≤ cMs

m s −1 X

1In (0)+es xs (x) , x ∈ Is \Is+1 , s = 0, . . . , n − 1,

xs =1

where 1E is the characteristic function of a set E and es := (0, . . . , 0, 1, 0, . . . ) ∈ G, the s-th coordinate of which is 1 and the rest are zeros, we obtain Z ∞ Z ∞ X k−1 Z X X sup |KMn | ≤ |KMn | = |KMn | n≥A

Ik

n=A

Ik

n=A s=0

Is+1 \Is

ALMOST EVERYWHERE CONVERGENCE

≤c

∞ X k−1 X n=A s=0

Ms

Z

m s −1 X xs =1

1In (0)+es xs ≤ c

Is+1 \Is

453

∞ k−1 X 1 X Mk Ms ≤ c . M M n s=0 n n=A

Lemma 3 is proved.

¤

Lemma 4. Let A ≥ k. Then Z Mk (A − k + 1) sup |Kn | ≤ c . MA n≥MA Ik

Proof. Since A X ¯ Mj ¯¯ |Kn | ≤ c KMj ¯ MA j=0

by Lemma 3 and the fact that [1]

for MA ≤ n < MA+1 ,

Z

sup n≥1

|Kn | < ∞,

(6)

Gm

we obtain Z Z ∞ X v X ¯ ¯ Mj ¯KM ¯ sup |Kn | ≤ j Mv n≥MA v=A j=0 Ik

Ik

Z Z ∞ X v ∞ X k X ¯ ¯ X ¯ ¯ M Mj j ¯KM ¯ + ¯KM ¯ = j j Mv Mv v=A j=k+1 v=A j=0 ( ≤c

Ik

∞

Mk X Mk (v − k + 1) + MA v=A Mv

Ik

) ≤c

Mk (A − k + 1) . MA

Lemma 4 is proved.

¤

Lemma 5. Let α ∈ (0, 1) and A ≥ k. Then Z A−k+1 sup |Knα | ≤ c (α) . (MA /Mk )α n≥MA Ik

Proof. From Lemma 2 we get Z Z Mp −1 N i 1 X X α−1 X ¯¯ ¯¯ α sup |Kn | ≤ c (α) sup α Mp Kj n≥MA N ≥A MN i=0 p=1 j=M Ik

Ik

p−1

Z

N 1 X α Mi |KMi −1 | α N ≥A MN i=0

sup

+ c (α) Ik

´ AND U. GOGINAVA G. GAT

454

Z + c (α)

N 1 X α Mi DMi = I + II + III. α N ≥A MN i=0

sup

Ik

(7)

From (1), (6) and Lemma 4 we obtain Z ∞ k X Mkα 1 X α ≤ c (α) D III ≤ c (α) , M Mi i α α M M N A i=0 N =A

(8)

Ik

Z N ∞ X 1 X α Mi |KMi −1 | II ≤ c (α) α M N i=0 N =A Ik

Z ∞ k X 1 X α = c (α) Mi |KMi −1 | α M N i=0 N =A Ik

Z N ∞ X 1 X α Mi + c (α) |KMi −1 | α M N N =A i=k+1 Ik

∞ N X Mkα 1 X Mkα α (i − k) Mk M ≤ c (α) α + c (α) ≤ c (α) α , MA MNα i=k+1 i Mi MA N =A

Z I ≤ c (α) Ik

Mp −1 k i 1 X X α−1 X sup α Mp |Kj | N ≥A MN i=0 p=1 j=M p−1

Z + c (α) Ik

Z + c (α) Ik

Mp −1 k N 1 X X α−1 X Mp |Kj | sup α N ≥A MN j=M i=k+1 p=1 p−1

Mp −1 i N X 1 X X α−1 Mp |Kj | sup α N ≥A MN j=M i=k+1 p=k+1 p−1

= I1 + I2 + I3 ; Mα I1 ≤ c (α) kα , MA Mα I2 ≤ c (α) kα (A − k + 1) , MA

(10) (11) (12)

Z ∞ N i X 1 X X α I3 ≤ c (α) M MNα i=k+1 p=k+1 p N =A

sup |Kl |

l≥Mp−1

Ik

≤ c (α)

(9)

∞ X

1 MNα N =A

N X

i X

i=k+1 p=k+1

Mpα

p−k (Mp /Mk )

ALMOST EVERYWHERE CONVERGENCE ∞ X N −k+1 A−k+1 . ≤ c (α) α ≤ c (α) (MN /Mk ) (MA /Mk )α N =A

(13)

Combining (7)–(13), we complete the proof of Lemma 5. Lemma 6. Let α ∈ (0, 1) and n = nA MA + · · · + n0 M0 . Then 10 X

|Tnα | ≤ c (α)

Bi ,

i=1

where

Mr+1 −1 A+1 s−1 1 X X α−1 X ¯¯ 1 ¯¯ M Tj , B1 = α n s=0 r=0 r j=M r

B2 = 1 B3 = α n

A+1 X

1 nα

A X

s=0

¯ ¯ Msα ¯Tn1s Ms ¯ ,

s=0

|Dn1 s Ms |

s−1 X

Mr+1 −1

X ¯ 1,2 ¯ ¯K ¯ ,

Mrα−1

r=0

j

j=Mr

A ¯ ¯ 1 X 1 ¯, B4 = α |Dns Ms |Msα ¯Kn1,2 M s s n s=0 Mr+1 −1 A+1 s−1 X X ¯ 1,1 ¯ 1 X 2 α−1 ¯K ¯ , |Dns Ms | Mr B5 = α j n s=0 r=0 j=M r

B6 =

1 nα

A X

¯ ¯ ¯, |Dn2 s Ms |Msα ¯Kn1,1 M s s

s=0

A 1 X α B7 = α M |Dns Ms ,ns Ms |, n s=0 s A ¯ ¯ 1 X 2 ¯ B8 = α |Dns Ms |Aαn(s−1) ¯Knα,1 (s−1) , n s=1 A ¯ ¯ 1 X 1 ¯ B9 = α |Dns Ms |Aαn(s−1) ¯Knα,2 (s−1) , n s=1

B10 ≤ c (α) . Proof. It is evident that Aαn Tnα

=

nX A MA

Aα−1 n−j Dj,j

+

j=1

=

nAX MA −1 j=0

Aα−1 D + n(A−1) +j nA MA −j,nA MA −j

n X

Aα−1 n−j Dj,j

j=nA MA +1 (A−1) nX

j=1

455

Aα−1 D . n(A−1) −j j+nA MA ,j+nA MA

¤

´ AND U. GOGINAVA G. GAT

456

Since (see Lemma 1) 2

DnA MA −j,nA MA −j = DnA MA ,nA MA − Dn1 A MA ψn2 A MA −1 Dj 1

−Dn2 A MA ψn1 A MA −1 Dj + ψn1 A MA −1 ψn2 A MA −1 Dj,j and Dj+nA MA ,j+nA MA = DnA MA ,nA MA + Dn1 A MA ψn2 A MA Dj2 +Dn2 A MA ψn1 A MA Dj1 + ψn1 A MA ψn2 A MA Dj,j , we get Aαn Tnα = Aαn(A) −1 DnA MA ,nA MA ! Ãn M −1 AX A 2 D Dn1 A MA ψn2 A MA −1 − Aα−1 n(A−1) +j j Ãn − Ãn +

j=0 A MA −1

X

! 1 Aα−1 D n(A−1) +j j

j=0 MA −1 AX

Dn2 A MA ψn1 A MA −1

! D Aα−1 n(A−1) +j j,j

ψn1 A MA −1 ψn2 A MA −1

j=0 α,1 1 α 2 2 1 +Aαn(A−1) Knα,2 (A−1) DnA MA ψnA MA + An(A−1) Kn(A−1) DnA MA ψnA MA +ψn1 A MA ψn2 A MA Aαn(A−1) Tnα(A−1) .

Consequently,

¯n M −1 ¯ A ¯ AX ¯ ¯ 2¯ ¯ 1 ¯ ¯Dn M ¯ Aαn |Tnα | ≤ Aαn(A) −1 |DnA MA ,nA MA | + ¯ D Aα−1 ¯ (A−1) j A A n +j ¯ ¯ j=0 ¯ ¯n M −1 ¯n M −1 ¯ A A ¯ ¯ AX ¯ AX ¯ 1 ¯ ¯ ¯ ¯ α−1 2 D + D +¯ D A Aα−1 ¯ ¯ ¯ j,j (A−1) +j (A−1) +j j nA MA n n ¯ ¯ ¯ ¯ j=0 j=0 ¯ ¯¯ 1 ¯ ¯ ¯¯ 2 ¯ ¯ ¯ ¯ ¯Dn M ¯ + Aα(A−1) ¯K α,1 ¯ ¯Dn M ¯ + Aα(A−1) ¯T α(A−1) ¯ . +Aαn(A−1) ¯Knα,2 (A−1) n n n A A A A n(A−1) Iterating this inequality, we obtain ¯ ¯n M −1 ( A s ¯ sX ¯¯ X 2 ¯ ¯ 1 ¯¯ ¯ Aα−1 D Aαn |Tnα | ≤ Aαn(s) −1 |Dns Ms ,ns Ms | + ¯ (s−1) +j j ¯ Dns Ms n ¯ ¯ s=1 j=0 ¯ ¯ ¯ ¯ Ms −1 Ms −1 ¯nsX ¯ ¯nsX ¯¯ ¯ 1¯ ¯ ¯ ¯ α−1 α−1 2 ¯ ¯ An(s−1) +j Dj,j ¯ An(s−1) +j Dj ¯ Dns Ms + ¯ +¯ ¯ ¯ ¯ ¯ j=0 j=0 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ª ¯ ¯Dn1 M ¯ + Aα(s−1) ¯K α,1 ¯ ¯Dn2 M ¯ + Aα(0) ¯T α(0) ¯ . (14) +Aαn(s−1) ¯Knα,2 (s−1) (s−1) n n n s s s s n Applying Abel’s transformation, we write ¯ ¯n M −1 s ¯ ¯ sX ¯ ¯ α−1 An(s−1) +j Dj,j ¯ ¯ ¯ ¯ j=0

ALMOST EVERYWHERE CONVERGENCE

(n ≤ c (α)

s Ms −2

X ¡

n(s−1) + j

j=0

(

≤ c (α)

s MX r+1 −1 X ¡ r=0

≤ c (α)

¢α−2 ¯ 1 ¯ ¡ (s) ¢α−1 ¡ (s) ¢¯ j ¯Tj ¯ + n n − 1 ¯Tn1s Ms

) ¯ ¯ ¯ ¯ ¢ ¡ ¢ α−2 α n(s−1) + j j ¯Tj1 ¯ + n(s) ¯Tn1s Ms −1 ¯

j=Mr

(

s X

) X ¯ ¯ ¡ ¯ ¯ ¢ α ¯Tj1 ¯ + n(s) ¯Tn1 M −1 ¯ . s s

Mr+1 −1

Mrα−1

r=0

(15)

j=Mr

Analogously,

≤ c (α)

457

) ¯ ¯ −1

¯ ¯ Ms −1 ¯nsX ¯ l ¯ ¯ α−1 An(s−1) +j Dj ¯ ¯ ¯ ¯ ( s X

j=0

Mrα−1

r=0

) X ¯ 1,l ¯¯ ¡ ¯ ¯ ¢ α ¯Kj ¯ + n(s) ¯Tn1s Ms −1 ¯ , l = 1, 2.

Mr+1 −1 ¯

(16)

j=Mr

Combining (14)–(16) we complete the proof of Lemma 6.

¤

Corollary 2. Let α ∈ (0, 1) . Then Z sup |Tnα | < ∞. n≥1

G2m

Proof. Since [1], [4]

Z sup n≥1

and

¯ 1¯ ¯Tn ¯ < ∞

(17)

G2m

Z |Knα | < ∞, α > 0,

sup n≥1

Gm

from Lemma 6 we obtain the proof of Corollary 2.

¤

Lemma 7. Let α ∈ (0, 1) . Then Z sup |Tnα | ≤ c (α) < ∞. n≥Mk

Ik ×Ik

Proof. Since in [4] one can find Z

¯ ¯ sup ¯Tn1 ¯ ≤ c < ∞,

n≥Mk Ik ×Ik

by (17) we can write Z Z sup B1 ≤ c (α) n≥Mk

Ik ×Ik

Ik ×Ik

Mr+1 −1 A+1 s−1 1 X X α−1 X ¯¯ 1 ¯¯ sup α Mr Tj A≥k MA s=0 r=0 j=M r

´ AND U. GOGINAVA G. GAT

458

Z ≤ c (α) Ik ×Ik

Z

Mr+1 −1 k s−1 1 X X α−1 X ¯¯ 1 ¯¯ sup α Mr Tj A≥k MA s=0 r=0 j=M r

Mr+1 −1 A+1 k−1 1 X X α−1 X ¯¯ 1 ¯¯ Tj sup α Mr A≥k MA j=M s=k+1 r=0

+c (α)

r

Ik ×Ik

Z

Mr+1 −1 A+1 s−1 1 X X α−1 X ¯¯ 1 ¯¯ sup α Mr Tj A≥k MA j=M s=k+1 r=k

+c (α)

r

Ik ×Ik

Mr+1 −1 Z k s−1 1 X X α−1 X M ≤ c (α) α Mk s=0 r=0 r j=M r

Ik ×Ik

Mr+1 −1 Z k−1 1 X α−1 X +c (α) α M Mk r=0 r j=M r

Z

¯ 1¯ ¯Tj ¯ ¯ 1¯ ¯Tj ¯

Ik ×Ik

A+1 s−1 ¯ ¯ 1 X X α Mr sup ¯Tl1 ¯ ≤ c (α) < ∞. α l≥Mk A≥k MA s=k+1 r=k

sup

+c (α) Ik ×Ik

Analogously, we obtain

(18)

Z sup B2 < ∞.

(19)

n≥Mk Ik ×Ik

We have Z

Z sup B3 =

sup B3 +

n≥Mk Ik ×Ik

Z

It is evident that Z Z sup B3 ≤ c (α) n≥Mk

I k ×Ik

I k ×Ik

Z ≤ c (α) I k ×Ik

sup B3 +

n≥Mk I k ×I k

Z

I k ×Ik

A+1 1 X sup α A≥k MA s=0

k 1 X sup α A≥k MA s=0

Ãm −1 s X

Ãm −1 s X

(20)

n≥Mk Ik ×I k

|Dn1 s Ms |

ns =0

|Dn1 s Ms |

ns =0

! s−1 X

Mr+1 −1

Mrα−1

r=0

! s−1 X r=0

≤ c (α) < ∞. Analogously,

sup B3 .

n≥Mk

X ¯ 1,2 ¯ ¯K ¯ j

j=Mr

Mr+1 −1

Mrα−1

X ¯ 1,2 ¯ ¯K ¯ j

j=Mr

(21)

Z sup B3 < ∞.

n≥Mk I k ×I k

(22)

ALMOST EVERYWHERE CONVERGENCE

We write Z Z sup B3 ≤ c (α) n≥Mk

Ik ×I k

Ik ×I k

Z ≤ c (α) Ik ×I k

Z +c (α) Ik ×I k

Z +c (α) Ik ×I k

Ãm −1 s X

A+1 1 X sup α A≥k MA s=0

k 1 X sup α A≥k MA s=0

Ãm −1 s X

A+1 1 X sup α A≥k MA s=k+1 A+1 1 X sup α A≥k MA s=k+1

|Dn1 s Ms |

! s−1 X

ns =0

Mr+1 −1

Mrα−1

r=0

|Dn1 s Ms |

! s−1 X

ns =0

Ãm −1 s X

459

j=Mr

Mr+1 −1

X ¯ 1,2 ¯ ¯K ¯ j

Mrα−1

r=0

|Dn1 s Ms |

! k−1 X

ns =0

r=0

Ãm −1 s X

! s−1 X

|Dn1 s Ms |

ns =0

X ¯ 1,2 ¯ ¯K ¯ j

j=Mr Mr+1 −1

X ¯ 1,2 ¯ ¯K ¯ j

Mrα−1

j=Mr Mr+1 −1

X ¯ 1,2 ¯ ¯K ¯ j

Mrα−1

j=Mr

r=k

=: M + N + R.

(23)

It is evident that M ≤ c (α) < ∞, N ≤ c (α)

∞ X Mα

(24) A+1 m s −1 Z X X

k MAα s=k+1 n =0 A=k s I

|Dn1 s Ms |

k

≤ c (α)

∞ X

A−k+1 ≤ c (α) < ∞. (MA /Mk )α A=k

(25)

Using Lemma 4 and (1), for R we get Z s−1 ∞ A+1 ms −1 X X 1 X X α M R ≤ c (α) MAα s=k+1 n =0 r=k r A=k s

¯ ¯ |Dn1 s Ms | sup ¯Kl1,2 ¯ l≥Mr

Ik ×I k

A+1 s−1 ∞ X 1 X X αr − k + 1 M ≤ c (α) MAα s=k+1 r=k r (Mr /Mk ) A=k

≤ c (α) ≤ c (α)

∞ A+1 s−1 X Mkα X X r − k + 1 MAα s=k+1 r=k (Mr /Mk )1−α A=k ∞ X Mα

k MAα A=k

(A − k + 1) ≤ c (α) < ∞.

(26)

Combining (23)–(26), we obtain Z sup B3 < ∞.

n≥Mk Ik ×I k

(27)

´ AND U. GOGINAVA G. GAT

460

After substituting (21), (22) and (27) into (20), we have Z sup B3 < ∞.

(28)

n≥Mk

Ik ×Ik

Analogously, we obtain Z sup Bj < ∞,

j = 4, 5, 6, 7.

(29)

n≥Mk Ik ×Ik

For B9 we write Z Z sup B9 =

Z sup B9 +

n≥Mk

Ik ×Ik

sup B9 +

n≥Mk

I k ×J k

Using Lemma 2, we have Z Z sup B9 ≤ c (α) n≥Mk

I k ×Ik

Z

I k ×Ik

I k ×Ik

A 1 X sup α A≥k MA s=1

k Z 1 X ≤ c (α) α Mk s=1

sup B9 .

n≥Mk

(30)

n≥Mk Ik ×I k

Ãm −1 s X

! |Dn1 s Ms |

ns =0

¯ ¯ sup Aαn ¯Knα,2 ¯

1≤n≤Ms

¯ ¯ sup Aαn ¯Knα,2 ¯

Gm

1≤n≤Ms

 Mp −1 Z k X s X i X X ¯ 1,2 ¯ 1 α−1 ¯K ¯ Mp ≤ c (α) α j Mk s=1 i=0  p=1 j=Mp −1G  m Z Z  ¯ 1,2 ¯ ¯K ¯ + Miα +Miα |D | ≤ c (α) < ∞. Mi Mi −1  Gm

(31)

Gm

Analogously, we obtain

Z sup B9 < ∞.

(32)

n≥Mk Ik ×I k

By Lemmas 2, 5 and (1) we have à A m −1 ! A Z Z s XX X ¯ ¯ 1 sup B9 ≤ c (α) sup α Maα sup ¯Kqα,2 ¯ |Dn1 s Ms | n≥Mk A≥k MA |q|=a a=0 s=k n =0 Ik ×I k

s

Ik ×I k

Z +c (α) Ik ×I k

k−1 ¯ ¯ 1 X 1 ¯ sup α |Dns Ms |Aαn(s−1) ¯Knα,2 (s−1) A≥k MA s=1

∞ k−1 Z X 1 X ≤ c (α) MAα a=0 A=k

Ik ×I k

A X

Ãm −1 s X

s=k

ns =0

!

¯ ¯ |Dn1 s Ms | Maα sup ¯Kqα,2 ¯ |q|=a

ALMOST EVERYWHERE CONVERGENCE

Ãm −1 A s X X

Z ∞ A X 1 X +c (α) MAα a=k A=k

Ik ×I k

Z +c (α) Ik ×I k

( ≤ c (α)

k−1 1 X Mkα s=1

∞ k−1 X A−k+1X A=k

MAα

Ãm −1 s X

a=0

|q|=a

! |Dn1 s Ms |

ns =0

Maα

¯ ¯ |Dn1 s Ms | Maα sup ¯Kqα,2 ¯

ns =0

s=k

+

¯ ¯ sup Aαl ¯Klα,2 ¯

1≤l≤Ms

∞ A X A−k+1X A=k

461

!

MAα

a=k

Maα

a−k+1 (Ma /Mk )α

( i )) k−1 s 1 XX X α + α M + Miα ≤ c (α) < ∞. Mk s=1 i=0 p=1 p By virtue of (30)–(33) we have Z sup B9 ≤ c (α) < ∞.

(33)

(34)

n≥Mk

Ik ×Ik

Analogously, we get Z sup B8 ≤ c (α) < ∞.

(35)

n≥Mk Ik ×Ik

Combining (18), (19), (28), (29), (34) and (35), by Lemma 6 we complete the proof of Lemma 7. ¤ Proof of Theorem 1. For α ≥ 1 Theorem 1 is proved in [4]. Hence it can be assumed that 0 < α < 1. As a consequence of Corollary 2 we have that the maximal operator tα∗ f := supn∈N |tαn f | is of type (∞, ∞). Since this sublinear operator is quasi-local (this is what Lemma 4 means), then by standard arguments (see, e.g., the book [10]) it follows that it is of weak type (1, 1). That is, the proof of Theorem 1 is complete. ¤ Proof of Corollary 1. The set of Vilenkin polynomials is dense in L1 (G2m ), so by the well-known density argument we have that tαn f → f a.e. for all integrable two-variable functions f . We remark that the Marcinkiewicz interpolation theorem (see, e.g., [10]) also gives that the maximal operator tα∗ is of type (p, p) for all 1 < p ≤ ∞. The proof of Corollary 1 is complete. ¤ Acknowledgement The first author is supported by the Hungarian National Foundation for Scientific Research (OTKA), grant no. M 36511/2001,T 048780, and by the Sz´echenyi fellowship of the Hungarian Ministry of Education Sz¨o 184/2003.

462

´ AND U. GOGINAVA G. GAT

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(Received 11.10.2005) Authors’ addresses: G. G´at Institute of Mathematics and Computer Science College of Ny´ıregyh´aza P.O. Box 166, Nyiregyh´aza, H-4400 Hungary E-mail: [email protected] U. Goginava Institute of Mechanics and Mathematics Faculty of Exact and Natural Sciences I. Javakhishvili Tbilisi State University 1, Chavchavadze Ave., Tbilisi 0128 Georgia E-mail: z [email protected]