Strong Convergence of Two-Dimensional Walsh–Fourier Series
We prove that certain means of quadratic partial sums of the two-dimensional Walsh–Fourier series are uniformly bounded operators acting from the Hardy space H p to the space L p for 0
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| author | Tephnadze, G. Тефнадзе, Г. |
| author_facet | Tephnadze, G. Тефнадзе, Г. |
| author_sort | Tephnadze, G. |
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| datestamp_date | 2020-03-18T19:16:10Z |
| description | We prove that certain means of quadratic partial sums of the two-dimensional Walsh–Fourier series are uniformly bounded operators acting from the Hardy space H p to the space L p for 0 |
| first_indexed | 2026-03-24T02:23:58Z |
| format | Article |
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UDC 517.5
G. Tephnadze (Tbilisi State Univ., Georgia)
STRONG CONVERGENCE
OF TWO-DIMENSIONAL WALSH – FOURIER SERIES
СИЛЬНА ЗБIЖНIСТЬ ДВОВИМIРНИХ РЯДIВ УОЛША – ФУР’Є
We prove that certain means of quadratic partial sums of the two-dimensional Walsh – Fourier series are uniformly bounded
operators acting from the Hardy space Hp to the space Lp for 0 < p < 1.
Доведено, що певнi середнi квадратичних часткових сум двовимiрних рядiв Уолша – Фур’є є рiвномiрно обмеженими
операторами, що дiють iз простору Хардi Hp у простiр Lp при 0 < p < 1.
1. Introduction. It is known [7, p. 125] that the Walsh – Paley system is not a Schauder basis in
L1(G). Moreover (see [8]), there exists a function in the dyadic Hardy space H1(G), the partial sums
of which are not bounded in L1(G). However, in Simon [9] the following strong convergence result
was obtained for all f ∈ H1:
lim
n→∞
1
log n
n∑
k=1
‖Skf − f‖1
k
= 0,
where Skf denotes the k th partial sum of the Walsh – Fourier series of f (for the trigonometric
analogue see Smith [11], for the Vilenkin system see Gát [1]).
Simon [10] proved that there is an absolute constant cp, depends only p, such that
∞∑
k=1
‖Skf‖pp
k2−p
≤ cp ‖f‖pHp , (1)
for all f ∈ Hp, where 0 < p < 1.
The author [13] proved that sequence
{
1/k2−p
}∞
k=1
in inequality (1) is important.
For the two-dimensional Walsh – Fourier series Weisz [16] generalized the result of Simon and
proved that if α ≥ 0 and f ∈ Hp(G×G), then
sup
n,m≥2
(
1
log n logm
)[p] ∑
2−α≤k/l≤2α,(k,l)≤(n,m)
‖Sk,lf‖pp
(kl)2−p
≤ c ‖f‖pHp ,
where 0 < p < 1 and [p] denotes the integer part of p.
Goginava and Gogoladze [5] proved that the following result is true:
Theorem G. Let f ∈ H1(G×G). Then there exists absolute constant c, such that
∞∑
n=1
‖Sn,nf‖1
n log2 n
≤ c ‖f‖H1
.
For two-dimensional trigonometric system analogical theorem was proved in [6].
Convergence of quadratical partial sums of two-dimensional Walsh – Fourier series was inves-
tigated in details by Weisz [15], Goginava [4], Gát, Goginava, Nagy [2], Gát, Goginava, Tke-
buchava [3].
c© G. TEPHNADZE, 2013
822 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
STRONG CONVERGENCE OF TWO-DIMENSIONAL WALSH – FOURIER SERIES 823
The main aim of this paper is to prove (see Theorem 1) that
∞∑
n=1
‖Sn,nf‖pp
n3−2p
≤ cp ‖f‖pHp , (2)
for all f ∈ Hp(G×G), where 0 < p < 1. We also proved that sequence
{
1/n3−2p
}∞
n=1
in inequality
(2) is important (see Theorem 2).
2. Definitions and notations. Let P denote the set of positive integers, N := P ∪ {0}. Denote
by Z2 the discrete cyclic group of order 2, that is Z2 = {0, 1}, where the group operation is the
modulo 2 addition and every subset is open. The Haar measure on Z2 is given such that the measure
of a singleton is 1/2. Let G be the complete direct product of the countable infinite copies of the
compact groups Z2. The elements of G are of the form x = (x0, x1, . . . , xk, . . .) with xk ∈ {0, 1},
k ∈ N. The group operation on G is the coordinate-wise addition, the measure (denote by µ) and the
topology are the product measure and topology. The compact Abelian group G is called the Walsh
group. A base for the neighborhoods of G can be given in the following way:
I0 (x) := G,
In (x) := In (x0, . . . , xn−1) :=
:=
{
y ∈ G : y = (x0, . . . , xn−1, yn, yn+1, . . .)
}
, x ∈ G, n ∈ N.
These sets are called the dyadic intervals. Let 0 = (0: i ∈ N) ∈ G denote the null element of G,
In := In(0), n ∈ N. Set en := (0, . . . , 0, 1, 0, . . .) ∈ G the n th coordinate of which is 1 and the rest
are zeros (n ∈ N) . Let In := G\In.
If n ∈ N, then n =
∑∞
i=0
ni2
i, where ni ∈ {0, 1}, i ∈ N, i.e., n is expressed in the number
system of base 2. Denote |n| := max{j ∈ N :nj 6= 0}, that is, 2|n| ≤ n < 2|n|+1.
It is easy to show that for every odd number n0 = 1 and we can write n = 1 +
∑|n|
i=1
nj2
i,
where nj ∈ {0, 1} , j ∈ P.
For k ∈ N and x ∈ G let as denote by
rk (x) := (−1)xk , x ∈ G, k ∈ N,
the k th Rademacher function.
The Walsh – Paley system is defined as the sequence of Walsh – Paley functions:
wn (x) :=
∞∏
k=0
(rk (x))nk = r|n| (x) (−1)
∑|n|−1
k=0 nkxk , x ∈ G, n ∈ P.
The Walsh – Dirichlet kernel is defined by
Dn (x) =
n−1∑
k=0
wk (x) .
Recall that (see [8, p. 7])
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
824 G. TEPHNADZE
D2n (x) =
2n, x ∈ In,
0, x ∈ In.
(3)
Furthermore, the following representation holds for the Dn’s. Let n ∈ N and n =
∑∞
i=0
ni2
i.
Then
Dn (x) = wn (x)
∞∑
j=0
njw2j (x)D2j (x) . (4)
The rectangular partial sums of the 2-dimensional Walsh – Fourier series of function f ∈ L2(G×
×G) are defined as follows:
SM,Nf(x, y) :=
M−1∑
i=0
N−1∑
j=0
f̂ (i, j)wi (x)wj (y) ,
where the numbers
f̂ (i, j) =
∫
G×G
f(x, y)wi (x)wj (y) dµ(x, y)
is said to be the (i, j) th Walsh – Fourier coefficient of the function f.
Denote
S
(1)
M f(x, y) :=
∫
G
f (s, y)DM (x+ s) dµ (s)
and
S
(2)
N f(x, y) :=
∫
G
f (x, t)DN (y + t) dµ (t) .
The norm (or quasinorm) of the space Lp(G×G) is defined by
‖f‖p :=
∫
G×G
|f |p dµ
1/p , 0 < p <∞.
The space weak− Lp(G×G) consists of all measurable functions f for which
‖f‖weak−Lp(G×G) := sup
λ>0
λµ (f > λ)1/p < +∞.
The σ-algebra generated by the dyadic 2-dimensional In(x)×In(y) square of measure 2−n×2−n
will be denoted by zn,n, n ∈ N. Denote by f = (fn,n, n ∈ N) one-parameter martingale with
respect to zn,n, n ∈ N.
The expectation operator and the conditional expectation operator relative to the zn,n, n ∈ N,
are denoted by E and En,n, respectively.
The maximal function of a martingale f is defined by
f∗ = sup
n∈N
|fn,n| .
Let f ∈ L1(G×G). Then the dyadic maximal function is given by
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
STRONG CONVERGENCE OF TWO-DIMENSIONAL WALSH – FOURIER SERIES 825
f∗(x, y) = sup
n∈N
1
µ (In(x)× In(y))
∣∣∣∣∣∣∣
∫
In(x)×In(y)
f (s, t) dµ (s, t)
∣∣∣∣∣∣∣ , (x, y) ∈ G×G.
The dyadic Hardy space Hp(G×G) (0 < p <∞) consists of all functions for which
‖f‖Hp := ‖f∗‖p <∞.
If f ∈ L1(G×G), then it is easy to show that the sequence (S2n,2n(f) : n ∈ N) is a martingale.
If f = (fn,n, n ∈ N) is a martingale, then the Walsh – Fourier coefficients must be defined in a
slightly different manner:
f̂ (i, j) := lim
k→∞
∫
G
fk,k(x, y)wi (x)wj (y) dµ(x, y).
It is known [12] that that Fourier coefficients of f ∈ Hp(G×G) are not bounded when 0 < p < 1.
The Walsh – Fourier coefficients of f ∈ L1(G × G) are the same as those of the martingale
(S2n,2nf : n ∈ N) obtained from f.
A bounded measurable function a is a p-atom, if there exists a dyadic 2-dimensional cube I × I,
such that
a)
∫
I×I
adµ = 0,
b) ‖a‖∞ ≤ µ(I × I)−1/p,
c) supp(a) ⊂ I × I.
3. Formulation of main results.
Theorem 1. Let 0 < p < 1 and f ∈ Hp(G×G). Then
∞∑
n=1
‖Sn,nf‖pp
n3−2p
≤ cp ‖f‖pHp .
Theorem 2. Let 0 < p < 1 and Φ: N→ [1, ∞) is any nondecreasing function, satisfying the
condition limn→∞Φ(n) = +∞. Then there exists a martingale f ∈ Hp(G×G) such that
∞∑
n=1
‖Sn,nf‖pweak−Lp Φ(n)
n3−2p
=∞.
4. Auxiliary propositions.
Lemma 1 [14]. A martingale f ∈ Lp(G×G) is in Hp(G×G), 0 < p ≤ 1, if and only if there
exist a sequence (ak, k ∈ N) of p-atoms and a sequence (µk, k ∈ N) of a real numbers such that
∞∑
k=0
µkEn,nak = fn,n (5)
and
∞∑
k=0
|µk|p <∞.
Moreover, ‖f‖Hp v inf
(∑∞
k=0
|µk|p
)1/p
, where the infimum is taken over all decomposition
of f of the form (5).
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
826 G. TEPHNADZE
5. Proof of the theorems. Proof of Theorem 1. If we apply Lemma 1 we only have to prove
that
∞∑
n=1
‖Sn,na‖pp
n3−2p
≤ cp <∞, (6)
for every p atom a.
Let a be an arbitrary p-atom with support IN (z′) × IN (z′′) and µ (IN ) = µ (IN ) = 2−N . We
can suppose that z′ = z′′ = 0.
Let (x, y) ∈ IN × IN . In this case D2i (x+ s) 1IN (s) = 0 and D2i (y + t) 1IN (t) = 0 for
i ≥ N . Recall that w2j (x+ t) = w2j (x) for t ∈ IN and j < N . Consequently, from (4) we obtain
Sn,na(x, y) =
=
∫
G×G
a (s, t)Dn (x+ s)Dn (y + t) dµ (s, t) =
=
∫
IN×IN
a (s, t)Dn (x+ s)Dn (y + t) dµ (s, t) =
=
∫
IN×IN
a (s, t)wn (x+ s+ y + t)
N−1∑
i=0
niw2i (x+ s)D2i (x+ s)×
×
N−1∑
j=0
njw2j (y + t)D2j (y + t) dµ (s, t) =
= wn (x)
N−1∑
i=0
niw2i (x)D2i (x)wn (y)
N−1∑
j=0
njw2j (y)D2j (y)×
×
∫
IN×IN
a (s, t)wn (s+ t) dµ (s, t) =
= wn (x+ y)
N−1∑
i=0
niw2i (x)D2i (x)
N−1∑
j=0
njw2j (y)D2j (y)×
×
∫
IN
∫
IN
a (t+ τ, t) dµ (t)
wn (τ) dµ (τ) =
= wn (x+ y)
N−1∑
i=0
niw2i (x)D2i (x)
N−1∑
j=0
njw2j (y)D2j (y)
∫
IN
Φ (τ)wn (τ) dµ (τ) =
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
STRONG CONVERGENCE OF TWO-DIMENSIONAL WALSH – FOURIER SERIES 827
= wn (x+ y)
N−1∑
i=0
niw2i (x)D2i (x)
N−1∑
j=0
njw2j (y)D2j (y) Φ̂(n),
where
Φ (τ) =
∫
IN
a (t+ τ, t) dµ (t) .
Let x ∈ Is\Is+1. Using (3) we get
N−1∑
i=0
D2i (x) ≤ c2s.
Since
−
IN =
N−1⋃
s=0
Is\Is+1
we obtain ∫
IN
(
N−1∑
i=0
D2i (x)
)p
dµ (x) ≤ cp
N−1∑
s=0
∫
Is\Is+1
2psdµ (x) ≤
≤ cp
∞∑
s=0
2(p−1)s < cp <∞, 0 < p < 1, (7)
applying (7) we can write
∞∑
n=1
1
n3−2p
∫
IN×IN
|Sn,na(x, y)|p dµ(x, y) ≤
≤
∞∑
n=1
∣∣∣Φ̂ (n)
∣∣∣p
n3−2p
∫
IN
(
N−1∑
i=0
D2i (x)
)p
dµ (x)
2
≤
≤ cp
∞∑
n=1
∣∣∣Φ̂ (n)
∣∣∣p
n3−2p
.
Let n < 2N . Since wn (τ) = 1, for τ ∈ IN we have
Φ̂(n) =
∫
IN
Φ (τ)wn (τ) dµ (τ) =
=
∫
IN
∫
IN
a (t+ τ, t) dµ (t)
wn (τ) dµ (τ) =
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
828 G. TEPHNADZE
=
∫
IN×IN
a (s, t) dµ (s, t) = 0.
Hence, we can suppose that n ≥ 2N . By Hölder inequality we obtain
∞∑
n=1
∣∣∣Φ̂(n)
∣∣∣p
n3−2p
≤
( ∞∑
n=2N
∣∣∣Φ̂ (n)
∣∣∣2)p/2( ∞∑
n=2N
1
n(3−2p)·(2/(2−p))
)(2−p)/2
≤
≤
(
1
2N(2(3−2p)/(2−p)−1)
)(2−p)/2
∫
G
|Φ (τ)|2 dµ (τ)
p/2
≤
≤ cp
2N(4−3p)/2
∫
IN
∣∣∣∣∣∣∣
∫
IN
a (t+ τ, t) dµ (t)
∣∣∣∣∣∣∣
2
dµ (τ)
p/2
≤
≤ cp
2N(4−3p)/2 ‖a‖
p
∞
1
2Np/2
1
2Np
≤
≤ cp
2N(4−3p)/2 22N
1
23pN/2
< cp <∞. (8)
Let (x, y) ∈ IN × IN . Then we have
Sn,na(x, y) = wn (x)
N−1∑
j=0
njw2j (x)D2j (x)×
×
∫
G×G
a (s, t)wn (s)Dn (y + t) dµ (s, t) =
= wn (x)
N−1∑
j=0
njw2j (x)D2j (x)
∫
G
S(2)
n a (s, y)wn (s) dµ (s) =
= wn (x)
N−1∑
j=0
njw2j (x)D2j (x) Ŝ(2)
n a (n, y) .
Using (7) we get
∞∑
n=1
1
n3−2p
∫
IN×IN
|Sn,na(x, y)|p dµ (x, y) ≤
≤
∞∑
n=1
1
n3−2p
∫
IN×IN
N−1∑
j=0
D2j (x)
∣∣∣Ŝ(2)
n a (n, y)
∣∣∣
p
dµ(x, y) ≤
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
STRONG CONVERGENCE OF TWO-DIMENSIONAL WALSH – FOURIER SERIES 829
≤
∞∑
n=1
1
n3−2p
∫
IN
(
N−1∑
i=0
D2i (x)
)p
dµ (x) ·
∫
IN
∣∣∣Ŝ(2)
n a (n, y)
∣∣∣p dµ (y) ≤
≤
∞∑
n=1
1
n3−2p
∫
IN
∣∣∣Ŝ(2)
n a (n, y)
∣∣∣p dµ (y) .
Let n < 2N . Then by the definition of the atom we have
Ŝ(2)
n a (n, y) =
∫
G
∫
G
a (s, t)Dn (y + t) dµ (t)
wn (s) dµ (s) =
= Dn (y)
∫
IN×IN
a (s, t) dµ (s, t) = 0.
Therefore, we can suppose that n ≥ 2N . Hence
∞∑
n=1
1
n3−2p
∫
IN×IN
|Sn,na(x, y)|p dµ(x, y) ≤
≤
∞∑
n=2N
1
n3−2p
∫
IN
∣∣∣Ŝ(2)
n a (n, y)
∣∣∣p dµ (y) .
Since ∥∥∥S(2)
n a (n, y)
∥∥∥
2
≤ c ‖a‖2
from Hölder inequality we can write
∫
IN
∣∣∣Ŝ(2)
n a (n, y)
∣∣∣p dµ (y) ≤ cp
2N(1−p)
∫
IN
∣∣∣Ŝ(2)
n a (n, y)
∣∣∣ dµ (y)
p
=
=
cp
2N(1−p)
∫
IN
∣∣∣∣∣∣∣
∫
IN
S(2)
n a (s, y)wn (s) dµ (s)
∣∣∣∣∣∣∣ dµ (y)
p
=
=
cp
2N(1−p)
∫
IN
∣∣∣∣∣∣∣
∫
IN
∫
IN
a (s, t)Dn (y + t) dµ (t)
wn (s) dµ (s)
∣∣∣∣∣∣∣ dµ (y)
p
≤
≤ cp
2N(1−p)
∫
IN
∫
IN
∣∣∣∣∣∣∣
∫
IN
a (s, t)Dn (y + t) dµ (t)
∣∣∣∣∣∣∣ dµ (y)
dµ (s)
p
≤
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
830 G. TEPHNADZE
≤ cp
2N(1−p)
1
2N/2
∫
IN
∫
IN
∣∣∣∣∣∣∣
∫
IN
a (s, t)Dn (y + t) dµ (t)
∣∣∣∣∣∣∣
2
dµ (y)
1/2
dµ (s)
p
≤
≤ cp
2N(1−p)
1
2N/2
∫
IN
∫
IN
|a (s, t)|2 dµ (t)
1/2
dµ (s)
p
≤
≤ cp
2N(1−p)
(
‖a‖∞
2N/2
1
2N
1
2N/2
)p
≤ cp
2N(1−p)
(
22N/p
22N
)p
≤ cp2N(1−p).
Consequently,
∞∑
n=1
1
n3−2p
∫
IN×IN
|Sn,na(x, y)| dµ(x, y) ≤
≤ cp
∞∑
n=2N
1
n3−2p
2N(1−p) ≤ cp
2N(1−p) ≤ cp <∞. (9)
Analogously, we can prove that
∞∑
n=1
1
n3−2p
∫
IN×IN
|Sn,na(x, y)|p dµ (x, y) ≤ cp <∞. (10)
Let (x, y) ∈ IN × IN . Then by the definition of the atom we can write∫
IN×IN
|Sn,na(x, y)|p dµ(x, y) ≤
≤ 1
2N(2−p)
∫
IN×IN
|Sn,na(x, y)|2 dµ(x, y)
p/2
≤
≤ 1
2N(2−p)
∫
IN×IN
|a(x, y)|2 dµ(x, y)
p/2
≤
≤
‖a‖p∞
2N(2−p)
1
2Np
≤ cp
1
2N(2−p) 22N
1
2Np
≤ cp <∞.
It follows that
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
STRONG CONVERGENCE OF TWO-DIMENSIONAL WALSH – FOURIER SERIES 831
∞∑
n=1
1
n3−2p
∫
IN×IN
|Sn,na(x, y)| dµ(x, y) ≤ cp
∞∑
n=1
1
n3−2p
≤ cp <∞. (11)
Combining (6) – (11) we complete the proof of Theorem 1.
Proof of Theorem 2. Let 0 < p < 1 and Φ(n) is any nondecreasing, nonnegative function,
satisfying condition
lim
n→∞
Φ(n) =∞.
For this function Φ(n), there exists an increasing sequence of the positive integers {αk : k ≥ 0} such
that:
α0 ≥ 2
and
∞∑
k=0
1
Φp/4 (2αk)
<∞. (12)
Let
fA,A(x, y) =
∑
{k;αk<A}
λkak,
where
λk =
1
Φ1/4 (2αk)
and
ak(x, y) = 2αk(2/p−2) (D2αk+1 (x)−D2αk (x)) (D2αk+1 (y)−D2αk (y)) .
It is easy to show that the martingale f = (f1,1, f2,2, . . . , fA,A, . . .) ∈ Hp.
Indeed, since
S2Aak(x, y) =
ak(x, y), αk < A,
0, αk ≥ A,
(13)
supp(ak) = Iαk ,∫
Iαk
akdµ = 0
and
‖ak‖∞ ≤ 2αk(2/p−2)22αk ≤ 22αk/p = (supp ak)
−1/p
from Lemma 1 and (12) we conclude that f ∈ Hp.
It is easy to show that
f̂(i, j) =
2αk(2/p−2)
Φ1/4 (2αk)
, if (i, j) ∈
{
2αk , . . . , 2αk+1 − 1
}
×
×
{
2αk , . . . , 2αk+1 − 1
}
, k = 0, 1, 2 . . . ,
0, if (i, j) /∈
∞⋃
k=1
{
2αk , . . . , 2αk+1 − 1
}
×
{
2αk , . . . , 2αk+1 − 1
}
.
(14)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
832 G. TEPHNADZE
Let 2αk < n < 2αk+1. From (14) we have
Sn,nf(x, y) =
2αk−1+1−1∑
i=0
2αk−1+1−1∑
j=0
f̂(i, j)wi (x)wj (y) +
+
n−1∑
i=2αk
n−1∑
j=2αk
f̂(i, j)wi (x)wj (y) =
=
k−1∑
η=0
2αη+1−1∑
i=2αη
2αη+1−1∑
j=2αη
f̂(i, j)wi (x)wj (y) +
+
n−1∑
i=2αk
n−1∑
j=2αk
f̂(i, j)wi (x)wj (y) =
=
k−1∑
η=0
2αη+1−1∑
i=2αη
2αη+1−1∑
j=2αη
2αη(2/p−2)
Φ1/4 (2αη)
wi (x)wj (y) +
+
n−1∑
i=2αk
n−1∑
j=2αk
2αk(2/p−2)
Φ1/4 (2αk)
wi (x)wj (y) =
=
k−1∑
η=0
2αη(2/p−2)
Φ1/4 (2αη)
(D2αη+1 (x)−D2αη (x)) (D2αη+1 (y)−D2αη (y)) +
+
2αk(2/p−2)
Φ1/4 (2αk)
(Dn (x)−D2αk (x)) (Dn (y)−D2αk (y)) =
= I + II. (15)
Let (x, y) ∈ (G\I1)× (G\I1) and n is odd number. Since n− 2αk is odd number too and
Dn+2αk (x) = D2αk (x) + w2αk (x)Dn (x) , when n < 2αk ,
from (3) and (4) we can write
|II| = 2αk(2/p−2)
Φ1/4 (2αk)
|w2αk (x)Dn−2αk (x)w2αk (y)Dn−2αk (y)| =
=
2αk(2/p−2)
Φ1/4 (2αk)
|w2αk (x)wn−2αk (x)D1 (x)w2αk (y)wn−2αk (y)D1 (y)| =
=
2αk(2/p−2)
Φ1/4 (2αk)
. (16)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
STRONG CONVERGENCE OF TWO-DIMENSIONAL WALSH – FOURIER SERIES 833
Applying (3) and condition αn ≥ 2 (n ∈ N) for I we have
I =
k−1∑
η=0
2αk(2/p−2)
Φ1/4 (2αη)
(D2αη+1 (x)−D2αη (x))(D2αη+1 (y)−D2αη (y)) = 0. (17)
Hence
‖Sn,nf(x, y)‖weak−Lp ≥
≥ 2αk(2/p−2)
2Φ1/4 (2αk)
(
µ
{
(x, y) ∈ (G\I1)× (G\I1) : |Sn,nf(x, y)| ≥ 2αk(2/p−2)
2Φ1/4 (2αk)
})1/p
≥
≥ 2αk(2/p−2)
2Φ1/4 (2αk)
|(G\I1)× (G\I1)| ≥
cp2
αk(2/p−2)
Φ1/4 (2αk)
. (18)
Using (18) we have
2αk+1−1∑
n=1
‖Sn,nf‖pweak−Lp Φ(n)
n3−2p
≥
≥
2αk+1−1∑
n=2αk+1
‖Sn,nf‖pweak−Lp Φ(n)
n3−2p
≥
≥ cpΦ (2αk)
2αk−1∑
n=2αk−1+1
‖S2n+1,2n+1f‖pweak−Lp
(2n+ 1)3−2p
≥
≥ cpΦ (2αk)
22αk(1−p)
Φ1/4 (2αk)
2αk−1∑
n=2αk−1+1
1
(2n+ 1)3−2p
≥
≥ cpΦ3/4 (2αk)→∞, when k →∞. (19)
Combining (12) – (19) we complete the proof of Theorem 2.
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Received 18.04.12,
after revision — 14.11.12
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
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| id | umjimathkievua-article-2467 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:23:58Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c9/bdd9f61743c889072a2a44145f489dc9.pdf |
| spelling | umjimathkievua-article-24672020-03-18T19:16:10Z Strong Convergence of Two-Dimensional Walsh–Fourier Series Сильна збіжність двовимірних рядів Уолша-Фур'є Tephnadze, G. Тефнадзе, Г. We prove that certain means of quadratic partial sums of the two-dimensional Walsh–Fourier series are uniformly bounded operators acting from the Hardy space H p to the space L p for 0 Доведено, що певш середш квадратичних часткових сум двовимiрних рядiв Уолша-Фур'е е рівномірно обмеженими операторами, що дiють із простору Хаpдi H p у проспр L p при 0 Institute of Mathematics, NAS of Ukraine 2013-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2467 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 6 (2013); 822–834 Український математичний журнал; Том 65 № 6 (2013); 822–834 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2467/1700 https://umj.imath.kiev.ua/index.php/umj/article/view/2467/1701 Copyright (c) 2013 Tephnadze G. |
| spellingShingle | Tephnadze, G. Тефнадзе, Г. Strong Convergence of Two-Dimensional Walsh–Fourier Series |
| title | Strong Convergence of Two-Dimensional Walsh–Fourier Series |
| title_alt | Сильна збіжність двовимірних рядів Уолша-Фур'є |
| title_full | Strong Convergence of Two-Dimensional Walsh–Fourier Series |
| title_fullStr | Strong Convergence of Two-Dimensional Walsh–Fourier Series |
| title_full_unstemmed | Strong Convergence of Two-Dimensional Walsh–Fourier Series |
| title_short | Strong Convergence of Two-Dimensional Walsh–Fourier Series |
| title_sort | strong convergence of two-dimensional walsh–fourier series |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2467 |
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