Almost everywhere convergence of Cesàro means of two variable Walsh – Fourier series with varying parameteres
UDC 517.5 We prove that the maximal operator of some $(C , \beta_{n})$ means of cubical partial sums of two variable Walsh – Fourier series of integrable functions is of weak type $(L_1,L_1)$. Moreover, the $ (C , \beta_{n})$-means $\sigma_{2^n}^{\beta_{n}} f$ of the function $ f \in L_{1} $ converg...
Gespeichert in:
| Datum: | 2021 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2021
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/196 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860506982272729088 |
|---|---|
| author | Abu Joudeh , A. A. Gát, G. Abu Joudeh, A. A. Gát, G. Abu Joudeh, A. A. Gát, G. |
| author_facet | Abu Joudeh , A. A. Gát, G. Abu Joudeh, A. A. Gát, G. Abu Joudeh, A. A. Gát, G. |
| author_sort | Abu Joudeh , A. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:48:21Z |
| description | UDC 517.5
We prove that the maximal operator of some $(C , \beta_{n})$ means of cubical partial sums of two variable Walsh – Fourier series of integrable functions is of weak type $(L_1,L_1)$. Moreover, the $ (C , \beta_{n})$-means $\sigma_{2^n}^{\beta_{n}} f$ of the function $ f \in L_{1} $ converge a.e. to $f$ for $ f \in L_{1}(I^2) $, where $I$ is the Walsh group for some sequences $1> \beta_n\searrow 0$. |
| doi_str_mv | 10.37863/umzh.v73i3.196 |
| first_indexed | 2026-03-24T02:02:04Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v73i3.196
UDC 517.5
A. A. Abu Joudeh, G. Gát (Inst. Math., Univ. Debrecen, Hungary)
ALMOST EVERYWHERE CONVERGENCE OF CESÀRO MEANS OF TWO
VARIABLE WALSH – FOURIER SERIES WITH VARYING PARAMETERS*
ЗБIЖНIСТЬ МАЙЖЕ СКРIЗЬ СЕРЕДНIХ ЧЕЗАРО ДЛЯ РЯДIВ
УОЛША – ФУР’Є ДВОХ ЗМIННИХ ЗI ЗМIННИМИ ПАРАМЕТРАМИ
We prove that the maximal operator of some (C, \beta n) means of cubical partial sums of two variable Walsh – Fourier series
of integrable functions is of weak type (L1, L1). Moreover, the (C, \beta n)-means \sigma \beta n
2n f of the function f \in L1 converge
a.e. to f for f \in L1(I
2), where I is the Walsh group for some sequences 1 > \beta n \searrow 0.
Доведено, що максимальний оператор вiд деяких середнiх (C, \beta n) кубiчних часткових сум рядiв Уолша – Фур’є
двох змiнних для iнтегровних функцiй має слабкий тип (L1, L1). Бiльш того, (C, \beta n)-середнi \sigma \beta n
2n f для функцiї
f \in L1 збiгаються мaйже скрiзь до f для f \in L1(I
2), де I — група Уолша для деяких послiдовностей 1 > \beta n \searrow 0.
1. Introduction and main results. In 1939, for the two-dimensional trigonometric Fourier partial
sums Sj,jf Marcinkiewicz [9] proved that for all f \in L \mathrm{l}\mathrm{o}\mathrm{g}L([0, 2\pi ]2) the relation
\sigma 1
nf =
1
n+ 1
n\sum
j=0
Sj,jf \rightarrow f
holds a.e. as n \rightarrow \infty . Zhizhiashvili [12] improved this result and showed that for f \in L([0, 2\pi ]2)
the (C,\alpha )-means
\sigma \alpha
nf =
1
A\alpha
n
n\sum
j=0
A\alpha - 1
n - jSj,jf
converge to f a.e. for any \alpha > 0. Dyachenko [4] proved this result for dimensions greater than
2. In papers [8, 11] by Goginava and Weisz one can find that the (C, 1)-means \sigma 1
nf of the double
Walsh – Fourier series of a function f \in L1([0, 1]
2) converges to f a.e. Recently, Gát [5] proved this
result with respect to two-dimensional Vilenkin systems. The d-dimensional Walsh – Fourier case is
discussed in [7].
For the one dimensional trigonometric system it can be found in Zygmund [13, p. 94] that the
Cesàro means or (C,\alpha )(\alpha > 0) means \sigma \alpha
nf of the Fourier series of a function f \in L1([ - \pi , \pi ])
converge a.e. to f as n \rightarrow \infty . Moreover, it is known that the maximal operator of the (C,\alpha )-means
\sigma \alpha
\ast := \mathrm{s}\mathrm{u}\mathrm{p}n\in \BbbN | \sigma \alpha
n | is of weak type (L1, L1), i.e.,
\mathrm{s}\mathrm{u}\mathrm{p}
\gamma >0
\gamma \lambda (\sigma \alpha
\ast f > \gamma ) \leq C\| f\| 1, f \in L1([ - \pi , \pi ]).
This result can be found implicitly in Zygmund [13, p. 154 – 156].
* This paper was supported by the European Union, co-financed by the European Social Fund (project EFOP-3.6.2-16-
2017-00015).
c\bigcirc A. A. ABU JOUDEH, G. GÁT, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3 291
292 A. A. ABU JOUDEH, G. GÁT
In 2007 Akhobadze [1] (see also [2]) introduced the notion of Cesàro means of Fourier series with
variable parameters for one-dimensional functions. In paper [3], we proved the almost everywhere
convergence of the the Cesàro (C,\alpha n)-means of integrable functions \sigma \alpha n
n f \rightarrow f, where \BbbN \supset \BbbN \alpha ,K \ni
\ni n \rightarrow \infty for f \in L1(I), where I is the Walsh group for every sequence \alpha = (\alpha n), 0 < \alpha n < 1.
The main aim of this paper is to investigate to two-dimensional version of this issue.
We follow the standard notions of dyadic analysis introduced by the mathematicians F. Schipp,
P. Simon, W. R. Wade (see, e.g., [10]) and others. Denote by \BbbN := \{ 0, 1, . . .\} , \BbbP := \BbbN \setminus \{ 0\} , the set of
natural numbers, the set of positive integers and I := [0, 1) the unit interval. Denote by \lambda (B) = | B|
the Lebesgue measure of the set B(B \subset I). Denote by Lp(I) the usual Lebesgue spaces and \| .\| p
the corresponding norms (1 \leq p \leq \infty ). Set
\scrJ :=
\biggl\{ \biggl[
p
2n
,
p+ 1
2n
\biggr\}
: p, n
\biggr\}
the set of dyadic intervals and for given x \in I set the definition of the nth (n \in \BbbN ) Walsh – Paley
function at point x \in I as
\omega n(x) :=
\infty \prod
j=0
( - 1)xjnj ,
where \BbbN \ni n =
\sum \infty
n=0
nj2
j , nj \in \{ 0, 1\} , j \in \BbbN , and x =
\sum \infty
j=0
xj2
- (j+1), xj \in \{ 0, 1\} , j \in \BbbN .
Remark that if x is a dyadic rational, that is, x \in
\bigl\{
p/2n : p, n \in \BbbN
\bigr\}
, then choose the expansion
terminates in zeros.
For any x, y \in I and k, n \in \BbbN define the so-called dyadic or logical addition as
x+ y :=
\infty \sum
n=0
| xn - yn| 2 - (n+1), k \oplus n :=
\infty \sum
n=0
| ki - ni| 2i.
By the definition of \omega n we have
\omega k\oplus n = \omega k\omega n, | \omega n| = 1,
and also the almost everywhere equality
\omega n(x+ y) = \omega n(x)\omega n(y).
Denote by
\^f(n) :=
\int
I
f\omega nd\lambda , Dn :=
n - 1\sum
k=0
\omega k, K1
n :=
1
n+ 1
n\sum
k=0
Dk
the Fourier coefficients, the Dirichlet and the Fejér or (C, 1) kernels, respectively. At some places,
if it does not cause confusion, we simple write Kn
\bigl(
instead of K1
n
\bigr)
. It is also known that the Fejér
or (C, 1)-means of f is
\sigma 1
nf(y) :=
1
n+ 1
n\sum
k=0
Skf(y) =
\int
I
f(x)K1
n(y + x)d\lambda (x) =
=
1
n+ 1
n\sum
k=0
\int
I
f(x)Dk(y + x)d\lambda (x), n \in \BbbN , y \in I.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
ALMOST EVERYWHERE CONVERGENCE OF CESÀRO MEANS OF TWO VARIABLE WALSH – FOURIER . . . 293
Now, for the two variable case we have for x =
\bigl(
x1, x2
\bigr)
, y =
\bigl(
y1, y2
\bigr)
\in I2, n = (n1, n2) \in \BbbN 2
the two-dimensional Fourier coefficients
\^f(n1, n2) :=
\int
I\times I
f(x1, x2)\omega n1(x
1)\omega n2(x
2)d\lambda (x1, x2),
the rectangular partial sums of the two-dimensional Fourier series
Sn1,n2f(y
1, y2) :=
n1 - 1\sum
k1=0
n2 - 1\sum
k2=0
\^f(k1, k2)\omega k1(y
1)\omega k2(y
2)
and the rectangular Dirichlet kernels
Dn1,n2(z) := Dn1(z
1)Dn2(z
2) =
nk - 1\sum
k1=0
nk - 1\sum
k2=0
\omega k1(z
1)\omega k2(z
2), z = (z1, z2) \in I2.
We have the nth Marcinkiewicz mean and kernel
\sigma 1
nf(y) :=
1
n+ 1
n\sum
k=0
Sj,jf(y), K1
n(z) =
1
n+ 1
n\sum
j=0
Dj,j(z)
and so we get
\sigma 1
nf(y
1, y2) =
\int
I\times I
f(x1, x2)K1
n(y
1 + x1, y2 + x2)d\lambda (x1, x2).
Denote by K\alpha a
n the kernel of the summability method (C,\alpha a)-Marcinkiewicz and call it the (C,\alpha a)
kernel or the Cesàro – Marcinkiewicz kernel for \alpha a \in \BbbR \setminus \{ - 1, - 2, . . . \}
K\alpha a
n (x1, x2) =
1
A\alpha a
n
n\sum
k=0
A\alpha a - 1
n - k Dj,j(x1, x2),
where
A\alpha a
k =
(\alpha a + 1)(\alpha a + 2) . . . (\alpha a + k)
k!
.
The (C,\alpha n) Cesàro – Marcinkiewicz means of integrable function f for two variables are
\sigma \alpha n
n f(y1, y2) =
1
A\alpha n
n
n\sum
k=0
A\alpha n - 1
n - k Sk,kf(y
1, y2) =
\int
I\times I
f(x1, x2)K\alpha n
n (y1 + x1, y2 + x2)d\lambda (x1, x2),
\sigma \alpha n
n f(y1, y2) =
1
A\alpha n
n
n\sum
k=0
\int
I\times I
A\alpha n - 1
n - k f(x1, x2)Dk(y
1 + x1)Dk(y
2 + x2)d\lambda (x1, x2).
Over all of this paper we suppose that monotone decreasing sequences (\alpha n) and (\beta n) satisfy
\beta n = \alpha 2n ,
\alpha N
A\alpha N
N
\mathrm{l}\mathrm{o}\mathrm{g}\delta
\biggl(
1 +
N
n
\biggr)
\leq C
\alpha n
A\alpha n
n
, N \geq n, n,N \in \BbbP , (1.1)
for some \delta > 1 and for some positive constant C. We remark that from condition (1.1) it follows
that sequence
\Bigl( \alpha n
A\alpha n
n
\Bigr)
is quasimonotone decreasing. That is, for some C > 0 we have
\alpha N
A\alpha N
N
\leq C
\alpha n
A\alpha n
n
, N \geq n, n,N \in \BbbP .
The main aim of this paper is to prove the following theorem.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
294 A. A. ABU JOUDEH, G. GÁT
Theorem 1.1. Suppose that monotone decreasing sequence 1 > \beta n > 0 satisfies the condition
A\beta n
2n
\beta n
\beta N
A\beta N
2N
(N + 1 - n)\delta \leq C for every \BbbN \ni N \geq n \geq 1 and for some \delta > 1. Let f \in L1(I
2). Then
we have the almost everywhere convergence
\sigma \beta n
2n f \rightarrow f.
Remark 1.1. In the proof of Theorem 1.1 we define the sequence (\alpha n) in a way that \alpha 2k = \beta k,
and, for any 2k \leq n < 2k+1, let \alpha n = \alpha 2k = \beta k. Then the sequence (\alpha n) satisfies that it is
decreasing and
A\alpha n
n
\alpha n
\alpha N
A\alpha N
N
\mathrm{l}\mathrm{o}\mathrm{g}\delta
\biggl(
1 +
N
n
\biggr)
\leq C for every \BbbN \ni N \geq n \geq 1. That is, condition (1.1) is
fulfilled.
We give two examples for sequences (\beta n) like above. Example 1: \beta k = \alpha 2k = \alpha n = \alpha \in (0, 1)
for every natural number k, n.
Example 2: Let \alpha n = 1/n. Then it is not difficult to have that A\alpha n
n \rightarrow 1 and it should be fulfilled
for sequence (\alpha n) that CN/n \geq \mathrm{l}\mathrm{o}\mathrm{g}\delta (1 +N/n) for some \delta > 1 and it trivially holds.
Introduce the following notations: for a, n, j \in \BbbN , let n(j) :=
\sum j - 1
i=0
ni2
i, that is, n(0) = 0,
n(1) = n0 and, for 2B \leq n < 2B+1, let | n| := B, n = n(B+1). Moreover, introduce the following
functions and operators for n \in \BbbN and 1 > \alpha a > 0, a \in \BbbN , where (x1, x2), (y1, y2) \in I2 :
T\alpha a
n (x1, x2) :=
1
A\alpha a
n
2B - 1\sum
j=0
A\alpha a - 1
n - j Dj,j(x
1, x2),
\=T\alpha a
n (x1, x2) := D2B (x
1)
1
A\alpha a
n
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
2B - 1\sum
j=0
A\alpha a - 1
n(B)+jDj(x
2)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| , \=\=T\alpha a
n (x1, x2) := \=T\alpha a
n (x2, x1),
\~T\alpha a
n (x1, x2) :=
1
A\alpha a
n
D2B ,2B (x
1, x2)
2B - 1\sum
j=0
A\alpha a - 1
n(B)+j+
+
1 - \alpha a
A\alpha a
n
2B - 1\sum
j=0
A\alpha a - 1
n(B)+j
j + 1
n(B) + j + 1
\bigm| \bigm| K1
j (x
1, x2)
\bigm| \bigm| +A\alpha a - 1
n 2B
\bigm| \bigm| K1
2B - 1(x
1, x2)
\bigm| \bigm| ,
t\alpha a
n f(y1, y2) :=
\int
I\times I
f(x1, x2)T\alpha a
n (y1 + x1, y2 + x2)d\lambda (x1, x2),
\~t\alpha a
n f(y1, y2) :=
\int
I\times I
f(x1, x2) \~T\alpha a
n (y1 + x1, y2 + x2)d\lambda (x1, x2).
Now we need several lemmas in the next section.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
ALMOST EVERYWHERE CONVERGENCE OF CESÀRO MEANS OF TWO VARIABLE WALSH – FOURIER . . . 295
2. Proofs.
Lemma 2.1. Let 1 > \alpha a > 0, f \in L1(I \times I) such that
\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p} f \subset Ik(u
1)\times Ik(u
2),
\int
Ik(u1)\times Ik(u2)
fd\lambda = 0
for some dyadic rectangle (u1, u2) \in I2. Then we have\int
Ik(u1)\times Ik(u2)
\mathrm{s}\mathrm{u}\mathrm{p}
n,a\in \BbbN
| \~t\alpha a
n f | d\lambda \leq C\| f\| 1. (2.1)
We also prove that
| T\alpha a
n (x1, x2)| \leq \~T\alpha a
n (x1, x2) + \=T\alpha a
n (x1, x2) + \=\=T\alpha a
n (x1, x2). (2.2)
Proof. First, we start with the proof of the inequality | T\alpha a
n | \leq \~T\alpha a
n + \=T\alpha a
n + \=\=T\alpha a
n .
Recall that B = | n| . Then, by equality D2B - j = D2B - \omega 2B - 1Dj and n(B) =
\sum B - 1
j=0
nj2
j , n(B)+
+ 2B = n,
A\alpha a
n T\alpha a
n (x) =
2B - 1\sum
j=0
A\alpha a - 1
2B+n(B) - j
Dj,j(x) =
2B - 1\sum
j=0
A\alpha a - 1
n(B)+jD2B - j,2B - j(x) =
= D2B (x
1)D2B (x
2)
2B - 1\sum
j=0
A\alpha a - 1
n(B)+j - \omega 2B - 1(x
1)D2B (x
2)
2B - 1\sum
j=0
A\alpha a - 1
n(B)+jDj(x
1) -
- \omega 2B - 1(x
2)D2B (x
1)
2B - 1\sum
j=0
A\alpha a - 1
n(B)+jDj(x
2)+
+\omega 2B - 1(x
1)\omega 2B - 1(x
2)
2B - 1\sum
j=0
A\alpha a - 1
n(B)+jDj,j(x
1, x2) =: (1) - (2) - (3) + (4).
So, by the help of the Abel transform, we get
| (4)| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
2B - 1\sum
j=0
A\alpha a - 1
n(B)+jDj,j(x
1, x2)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
2B - 1\sum
j=0
(A\alpha a - 1
n(B)+j - A\alpha a - 1
n(B)+j+1)
j\sum
i=0
Di,i(x
1, x2) +A\alpha a - 1
n(B)+2B
2B - 1\sum
i=0
Di,i(x
1, x2)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| (1 - \alpha a)
2B - 1\sum
j=0
A\alpha a - 1
n(B)+j
j + 1
n(B) + j + 1
K1
j (x
1, x2) +A\alpha a - 1
n 2BK1
2B - 1(x
1, x2)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq (1 - \alpha a)
2k - 1\sum
j=0
A\alpha a - 1
n(B)+j
j + 1
n(B) + j + 1
\bigm| \bigm| K1
j (x
1, x2)
\bigm| \bigm| +
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
296 A. A. ABU JOUDEH, G. GÁT
+(1 - \alpha a)
2B - 1\sum
j=2k
A\alpha a - 1
n(B)+j
j + 1
n(B) + j + 1
\bigm| \bigm| K1
j (x
1, x2)
\bigm| \bigm| +A\alpha a - 1
n 2B
\bigm| \bigm| K1
2B - 1(x
1, x2)
\bigm| \bigm| =:
=: I + II + III.
By the above written, we have
A\alpha a
n
\bigm| \bigm| T\alpha a
n (x1, x2)
\bigm| \bigm| \leq D2B ,2B (x
1, x2)
2B - 1\sum
j=0
A\alpha a - 1
n(B)+j +D2B (x
1)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
2B - 1\sum
j=0
A\alpha a - 1
n(B)+jDj(x
2)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| +
+D2B (x
2)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
2B - 1\sum
j=0
A\alpha a - 1
n(B)+jDj(x
1)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| +
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
2B - 1\sum
j=0
A\alpha a - 1
n(B)+jDj,j(x
1, x2)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| .
Thus,
\bigm| \bigm| T\alpha a
n (x1, x2)
\bigm| \bigm| \leq \~T\alpha a
n (x1, x2) +D2B (x
1)
1
A\alpha a
n
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
2B - 1\sum
j=0
A\alpha a - 1
n(B)+jDj(x
2)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| +
+D2B (x
2)
1
A\alpha a
n
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
2B - 1\sum
j=0
A\alpha a - 1
n(B)+jDj(x
1)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| = \~T\alpha a
n (x1, x2) + \=T\alpha a
n (x1, x2) + \=\=T\alpha a
n (x1, x2).
For n < 2k and (x1, x2) \in Ik(u
1) \times Ik(u
2), we have that \~T\alpha a
n (y + x) depends (with respect to x)
only on coordinates x10, . . . , x
1
k - 1, x
2
0, . . . , x
2
k - 1, thus \~T\alpha a
n (y + x) = \~T\alpha a
n (y + u) and, consequently,\int
Ik(u1)\times Ik(u2)
f(x1, x2) \~T\alpha a
n (y1 + x1, y2 + x2)d\lambda (x1, x2) =
= \~T\alpha a
n (y1 + u1, y2 + u2)
\int
Ik(u1)\times Ik(u2)
f(x1, x2)d\lambda (x1, x2) = 0.
Observe that
Ik(u1)\times Ik(u2) = Ik(u1)\times Ik(u2) \cup Ik(u
1)\times Ik(u2) \cup Ik(u1)\times Ik(u
2).
Since for any j < 2k we obtain that the kernel K1
j (y + x) depends (with respect to x) only on
coordinates x10, . . . , x
1
k - 1, x
2
0, . . . , x
2
k - 1, then\int
Ik(u1)\times Ik(u2)
f(x)| K1
j (y + x)| d\lambda (x) = | K1
j (y + u)|
\int
Ik(u1)\times Ik(u2)
f(x)d\lambda (x) = 0
gives
\int
Ik(u1)\times Ik(u2)
f(x)I(y + x)d\lambda (x) = 0. On the other hand,
II = (1 - \alpha a)
2B - 1\sum
j=2k
A\alpha a - 1
n(B)+j
j + 1
n(B) + j + 1
| K1
j (y
1 + x1, y2 + x2)| \leq
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
ALMOST EVERYWHERE CONVERGENCE OF CESÀRO MEANS OF TWO VARIABLE WALSH – FOURIER . . . 297
\leq \mathrm{s}\mathrm{u}\mathrm{p}
j\geq 2k
| K1
j (x
1, x2)| (1 - \alpha a)
n\sum
j=0
A\alpha a - 1
j = A\alpha a
n (1 - \alpha a) \mathrm{s}\mathrm{u}\mathrm{p}
j\geq 2k
| K1
j (x
1, x2)| .
This, by Lemma 3 in [5], gives\int
Ik\times Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k,a\in \BbbN
1
A\alpha a
n
IId\lambda \leq
\int
Ik\times Ik
\mathrm{s}\mathrm{u}\mathrm{p}
j\geq 2k
| K1
j (x
1, x2)| d\lambda \leq C.
The situation with III is similar. So, just as in the case of II we apply Lemma 3 in [5]:\int
Ik\times Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k,a\in \BbbN
1
A\alpha a
n
IIId\lambda \leq
\int
Ik\times Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k
| K1
2| n| - 1
| d\lambda \leq C.
Therefore, substituting z1 = (x1 + y1), z2 = (x2 + y2), where z \in Ik \times Ik and, consequently,
D2B ,2B (z
1, z2) = 0, we obtain \int
Ik\times Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k,a\in \BbbN
\~t\alpha a
n fd\lambda =
=
\int
Ik\times Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k,a\in \BbbN
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
Ik\times Ik
f(x1, x2) \~T\alpha a
n (y1 + x1, y2 + x2)d\lambda (x1, x2)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| d\lambda (y1, y2) \leq
\leq
\int
Ik\times Ik
\int
Ik\times Ik
| f(x1, x2)| \mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k,a\in \BbbN
1
A\alpha a
n
[II(y1 + x1, y2 + x2)+
+III(y1 + x1, y2 + x2)]d\lambda (x1, x2)d\lambda (y1, y2) =
=
\int
Ik\times Ik
| f(x1, x2)|
\int
Ik\times Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k,a\in \BbbN
1
A\alpha a
n
II(z1, z2) + III(z1, z2)d\lambda (z1, z2)d\lambda (x1, x2) \leq
\leq C
\int
Ik\times Ik
| f(x1, x2)| d\lambda (x1, x2).
This gives \int
Ik\times Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n,a\in \BbbN
\bigm| \bigm| \~t\alpha a
n f
\bigm| \bigm| d\lambda \leq C\| f\| 1.
Lemma 2.1 is proved.
Now, we just proved the lemma which means that maximal operator \mathrm{s}\mathrm{u}\mathrm{p}n,a | \~t\alpha a
n | is quasilocal.
The following lemma shows that the one-dimensional operator which maps f \in L1(I) to
\mathrm{s}\mathrm{u}\mathrm{p}
n
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f \ast 1
A\alpha n
n
n\sum
j=0
A\alpha n - 1
j | Kj |
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
is quasilocal. This lemma is interesting itself if one investigates Cesàro means with variable para-
meters and in the proof we introduce methods which will also be used later.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
298 A. A. ABU JOUDEH, G. GÁT
Lemma 2.2. Let (\alpha n) be a monotone decreasing sequence and
\Bigl( \alpha n
n\alpha n
\Bigr)
be a quasidecreasing
sequences with 1 > \alpha n > 0, n \in \BbbN . Then\int
Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k
1
A\alpha n
n
n\sum
j=0
A\alpha n - 1
j | Kj | \leq C.
Proof. Recall that Kn denotes the one-dimensional Fejér kernel, that is, Kn = K1
n. By [6]
we get \int
Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k
1
A\alpha n
n
n\sum
j=2k
A\alpha n - 1
j | Kj(x)| dx \leq
\int
Ik
\mathrm{s}\mathrm{u}\mathrm{p}
j\geq 2k
| Kj(x)| \mathrm{s}\mathrm{u}\mathrm{p}
n
1
A\alpha n
n
n\sum
l=2k
A\alpha n - 1
l dx \leq
\leq
\int
Ik
\mathrm{s}\mathrm{u}\mathrm{p}
j\geq 2k
| Kj(x)| dx \leq C.
On the other hand, if j < 2k, by \=Ik =
\bigcup k - 1
a=0
(Ia\setminus Ia+1) , we have
\int
Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k
1
A\alpha n
n
2k - 1\sum
j=0
A\alpha n - 1
j | Kj | \leq
\leq
k - 1\sum
a=0
\int
Ia\setminus Ia+1
\mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k
1
A\alpha n
n
2k - 1\sum
j=2a
A\alpha n - 1
j | Kj | +
k - 1\sum
a=0
\int
Ia\setminus Ia+1
\mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k
1
A\alpha n
n
2a - 1\sum
j=0
A\alpha n - 1
j | Kj | =:
=: I + II.
For I we obtain
I \leq
k - 1\sum
a=0
\int
Ia\setminus Ia+1
\mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k
1
A\alpha n
n
k - 1\sum
b=a
2b+1 - 1\sum
j=2b
A\alpha n - 1
j | Kj | \leq
\leq
k - 1\sum
a=0
k - 1\sum
b=a
\int
Ia\setminus Ia+1
\mathrm{s}\mathrm{u}\mathrm{p}
j\geq 2b
| Kj | \mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k
1
A\alpha n
n
2b+1 - 1\sum
l=2b
A\alpha n - 1
l ,
where
\mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k
1
A\alpha n
n
2b+1 - 1\sum
l=2b
A\alpha n - 1
l \leq \mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k
A\alpha n
2b+1 - 1
- A\alpha n
2b - 1
A\alpha n
n
=
= \mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k
A\alpha n
2b - 1
A\alpha n
n
\biggl[
(2b + \alpha n) . . . (2
b+1 - 1 + \alpha n)
2b(2b + 1) . . . (2b+1 - 1)
- 1
\biggr]
=
= \mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k
A\alpha n
2b - 1
A\alpha n
n
\biggl[ \Bigl(
1 +
\alpha n
2b
\Bigr) \biggl(
1 +
\alpha n
2b + 1
\biggr)
. . .
\biggl(
1 +
\alpha n
2b + 2b - 1
\biggr)
- 1
\biggr]
\leq
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
ALMOST EVERYWHERE CONVERGENCE OF CESÀRO MEANS OF TWO VARIABLE WALSH – FOURIER . . . 299
\leq \mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k
A\alpha n
2b
A\alpha n
n
\biggl[ \Bigl(
1 +
\alpha n
2b
\Bigr) 2b
- 1
\biggr]
\leq C \mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k
A\alpha n
2b
A\alpha n
n
\alpha n \leq C \mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k
\biggl(
2b
n
\biggr) \alpha n
\alpha n \leq
\leq C \mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k
\Bigl(
2b
\Bigr) \alpha
2k
\Bigl( \alpha n
n\alpha n
\Bigr)
\leq C
\Bigl(
2b
\Bigr) \alpha
2k
\biggl(
\alpha 2k
(2k)
\alpha
2k
\biggr)
,
where the inequality
A\alpha n
2b
A\alpha n
n
\leq C
\biggl(
2b
n
\biggr) \alpha n
is given from [3] (Lemma 2.4). Besides, since (\alpha n) is
a monotone decreasing sequences, then (2b)
\alpha n \leq (2b)
\alpha
2k . Sequence
\Bigl( \alpha n
n\alpha n
\Bigr)
is quasidecreasing.
Moreover,
\Bigl(
1 +
\alpha n
2b
\Bigr) 2b
- 1 \leq C\alpha n for any 0 < \alpha n < 1, b \in \BbbN .
Thus, by (2.3) [8],
I \leq C
k - 1\sum
a=0
k - 1\sum
b=a
2a
2b
(b - a)\alpha 2k
\biggl(
2b
2k
\biggr) \alpha
2k
= C
k - 1\sum
b=0
b\sum
a=0
2a
2b
(b - a)\alpha 2k
\biggl(
2b
2k
\biggr) \alpha
2k
\leq
\leq C
k - 1\sum
b=0
\alpha 2k
\biggl(
2b
2k
\biggr) \alpha
2k
\leq C\alpha 2k
\infty \sum
l=0
1
2l\alpha k
\leq C\alpha 2k
1
1 - 2\alpha 2k
\leq C.
We have to discuss II in the case when j < 2a and, thus, | Kj(x)| \leq j. Besides, A\alpha n - 1
j j = \alpha nA
\alpha n
j - 1
and we get
2a - 1\sum
j=0
A\alpha n - 1
j | Kj(x)| \leq \alpha n
2a - 1\sum
j=0
A\alpha n
j \leq \alpha nA
\alpha n+1
2a = \alpha nA
\alpha n
2a+1
\biggl(
2a + 1
\alpha n + 1
\biggr)
.
Besides, by [3] (Lemma 2.4) and by the fact that the sequence (\alpha n/n
\alpha n) is quasidecreasing, we have
\mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k
\alpha nA
\alpha n
2a+1
A\alpha n
n
2a + 1
\alpha n + 1
\leq C2a \mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k
\alpha n
\biggl(
2a + 1
n
\biggr) \alpha n
\leq C2a\alpha 2k
\biggl(
2a
2k
\biggr) \alpha
2k
.
Then
II \leq C
k - 1\sum
a=0
1
2a
2a\alpha 2k
\biggl(
2a
2k
\biggr) \alpha
2k
\leq C \mathrm{s}\mathrm{u}\mathrm{p}
k
\alpha 2k
\infty \sum
l=0
1
2l\alpha 2k
\leq C.
Lemma 2.2 is proved.
Next we prove the following lemma.
Lemma 2.3. Suppose that for the monotone decreasing sequence (\alpha n) the condition (1.1) is
fulfilled. Let a : I \setminus \{ 0\} \mapsto - \rightarrow \BbbN be defined as a(x) = a for x \in (Ia\setminus Ia+1) . Then the inequality
\int
Ik\times Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k
1
A\alpha n
n
| n| \sum
s=k
2a(x
2)\sum
j=0
A\alpha n - 1
j
\bigm| \bigm| Kj(x
2)
\bigm| \bigm| D2s(x
1)d(x1, x2) \leq C
holds.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
300 A. A. ABU JOUDEH, G. GÁT
Proof. Since
\int
Ik\times Ik
=
\sum k - 1
a=0
\int
Ik\times (Ia\setminus Ia+1)
, we have to check the values of the integrand on
Ik \times (Ia\setminus Ia+1). That is, x2 \in Ia\setminus Ia+1. Thus,
\bigm| \bigm| Kj(x
2)
\bigm| \bigm| \leq Cj gives
A\alpha n - 1
j .j =
\alpha n . . . (\alpha n + j - 1)
j!
j = \alpha n
(1 + \alpha n) . . . (j - 1 + \alpha n)
(j - 1)!
= \alpha nA
\alpha n
j - 1.
Hence it follows that
2a\sum
j=0
A\alpha n - 1
j
\bigm| \bigm| Kj(x
2)
\bigm| \bigm| \leq C
2a\sum
j=1
\alpha nA
\alpha n
j - 1 = C\alpha nA
\alpha n+1
2a - 1 =
= C\alpha n
(2 + \alpha n) . . . (2
a + \alpha n)
(2a - 1)!
= C\alpha n
\biggl(
2a
1 + \alpha n
\biggr)
A\alpha n
2a \leq C\alpha n2
aA\alpha n
2a ,
that is, we have to investigate
k - 1\sum
a=0
\int
Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k
\alpha n
A\alpha n
n
A\alpha n
2a
| n| \sum
s=k
D2s(x
1)d(x1).
Recall that
\int
Ia\setminus Ia+1
2a \leq 1, A\alpha n
2a \leq A
\alpha
2k
2a , since \alpha n \searrow and n \geq 2k. Also recall that
\alpha n
A\alpha n
n
\leq C
\mathrm{l}\mathrm{o}\mathrm{g}\delta
\Bigl(
1 +
n
2k
\Bigr) \alpha 2k
A
\alpha
2k
2k
,
which gives
\alpha n
A\alpha n
n
A\alpha n
2a \leq C\alpha 2k
A
\alpha
2k
2a
A
\alpha
2k
2k
1
\mathrm{l}\mathrm{o}\mathrm{g}\delta
\Bigl(
1 +
n
2k
\Bigr) .
That is, we have to investigate
k - 1\sum
a=0
\alpha 2k
A
\alpha
2k
2a
A
\alpha
2k
2k
\int
Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k
1
\mathrm{l}\mathrm{o}\mathrm{g}\delta
\Bigl(
1 +
n
2k
\Bigr) | n| \sum
s=k
D2s(x
1)d(x1).
Check the integral above
\int
Ik
=
\sum \infty
t=k
\int
It\setminus It+1
and the integral on It \setminus It+1 can be estimated by
\int
It\setminus It+1
\mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2k
C
(1 + | n| - k)\delta
min(t,| n| )\sum
s=k
2sd(x1) \leq C
(t+ 1 - k)\delta
and henceforth, by \delta > 1,
\sum \infty
t=k
1
(1 + t - k)\delta
\leq C. We have, by Lemma 2.4 in [3], that
k - 1\sum
a=0
\alpha 2k
A
\alpha
2k
2a
A
\alpha
2k
2k
\leq 2
k - 1\sum
a=0
\alpha 2k
\biggl(
2a + 1
2k
\biggr) \alpha
2k
\leq C
k - 1\sum
a=0
\alpha 2k
\biggl(
2a
2k
\biggr) \alpha
2k
\leq
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
ALMOST EVERYWHERE CONVERGENCE OF CESÀRO MEANS OF TWO VARIABLE WALSH – FOURIER . . . 301
\leq C\alpha 2k
\infty \sum
j=0
\biggl(
1
2\alpha 2k
\biggr) j
=
C\alpha 2k
1 -
\bigl(
1
2
\bigr) \alpha
2k
\leq C.
Lemma 2.3 is proved.
Let (\alpha n) be a monotone decreasing sequences such that 0 < \alpha n < 1 with property (1.1). That
is, for some \delta > 1, C > 0,
A\alpha n
n
\alpha n
\alpha N
A\alpha N
N
\mathrm{l}\mathrm{o}\mathrm{g}\delta
\biggl(
1 +
N
n
\biggr)
\leq C
for every \BbbN \ni N \geq n \geq 1.
We prove the following lemma.
Lemma 2.4.
k - 1\sum
a=0
\int
Ik\times (Ia\setminus Ia+1)
\mathrm{s}\mathrm{u}\mathrm{p}
n>2k
1
A\alpha n
n
| n| \sum
s=k
k - 1\sum
b=a
2b+1\sum
j=2b+1
A\alpha n - 1
j
\bigm| \bigm| Kj(x
2)
\bigm| \bigm| D2s(x
1)d(x1, x2) \leq C.
Proof. By the result of Goginava [8], that is, by\int
Ia\setminus Ia+1
\mathrm{s}\mathrm{u}\mathrm{p}
n\geq 2b
\bigm| \bigm| Kj(x
2)
\bigm| \bigm| d(x2) \leq C
\biggl(
b - a
2b - a
\biggr)
, (2.3)
we have to investigate
\bfB \bfone :=
\sum
a<k
\int
Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n>2k
1
A\alpha n
n
| n| \sum
s=k
k - 1\sum
b=a
b - a
2b - a
2b+1\sum
j=2b+1
A\alpha n - 1
j D2s(x
1)d(x1).
So, we have
2b+1\sum
j=2b+1
A\alpha n - 1
j = A\alpha n
2b+1 - A\alpha n
2b
= A\alpha n
2b
\biggl[
(2b + 1 + \alpha n) . . . (2
b+1 + \alpha n)
(2b + 1) . . . (2b+1)
- 1
\biggr]
=
= A\alpha n
2b
\biggl[ \biggl(
1 +
\alpha n
2b + 1
\biggr)
. . .
\Bigl(
1 +
\alpha n
2b+1
\Bigr)
- 1
\biggr]
\leq A\alpha n
2b
\biggl[
0
\Bigl(
1 +
\alpha n
2b
\Bigr) 2b
- 1
\biggr]
\leq C\alpha nA
\alpha n
2b
.
On the other hand, by
\int
Ik
=
\sum \infty
t=k
\int
It\setminus It+1
it follows that
\int
Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n>2k
1
(| n| + 1 - k)\delta
| n| \sum
s=k
D2s(x
1)d(x1) =
\infty \sum
t=k
\int
It\setminus It+1
\mathrm{s}\mathrm{u}\mathrm{p}
n>2k
1
(| n| + 1 - k)\delta
min(t,| n| )\sum
s=k+1
2s \leq
\leq
\infty \sum
t=k
\left( \int
It\setminus It+1
\mathrm{s}\mathrm{u}\mathrm{p}
t\geq | n| >k
1
(| n| + 1 - k)\delta
2| n| +
\int
It\setminus It+1
\mathrm{s}\mathrm{u}\mathrm{p}
| n| >t
1
(| n| + 1 - k)\delta
2t
\right) =:
=:
\infty \sum
t=k
(\bfB \bftwo ,\bfone +\bfB \bftwo ,\bftwo ) .
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
302 A. A. ABU JOUDEH, G. GÁT
Now we have
\infty \sum
t=k
(\bfB \bftwo ,\bftwo ) \leq
\infty \sum
t=k
1
(t+ 1 - k)\delta
\leq C,
\infty \sum
t=k
(\bfB \bftwo ,\bfone ) \leq
\infty \sum
t=k
\mathrm{s}\mathrm{u}\mathrm{p}
t\geq | n| >k
2| n| +1 - t
(| n| - k)\delta
\leq
\infty \sum
t=k+1
1
(t - k)\delta
\leq C.
That is, for \bfB \bfone we get
\bfB \bfone \leq C
\sum
a<k
\mathrm{s}\mathrm{u}\mathrm{p}
n>2k
1
A\alpha n
n
k - 1\sum
b=a
\alpha nA
\alpha n
2b
b - a
2b - a
\mathrm{l}\mathrm{o}\mathrm{g}\delta
\Bigl(
1 +
n
2k
\Bigr)
\times
\times
\infty \sum
t=k
\int
It\setminus It+1
\mathrm{s}\mathrm{u}\mathrm{p}
n>2k
1
(| n| + 1 - k)\delta
min(t,| n| )\sum
s=k
D2s(x
1)dx1 \leq
\leq C
\sum
a<k
\mathrm{s}\mathrm{u}\mathrm{p}
n>2k
\alpha n
A\alpha n
n
\mathrm{l}\mathrm{o}\mathrm{g}\delta
\Bigl(
1 +
n
2k
\Bigr) k - 1\sum
b=a
A\alpha n
2b
b - a
2b - a
\leq
\leq C
\sum
a<k
k - 1\sum
b=a
A
\alpha
2k
2b
b - a
2b - a
\mathrm{s}\mathrm{u}\mathrm{p}
n>2k
\alpha n
A\alpha n
n
\mathrm{l}\mathrm{o}\mathrm{g}\delta
\Bigl(
1 +
n
2k
\Bigr)
=:
=: \bfB \bfthree .
Recall that A\alpha n
2b
\leq A
\alpha
2k
2b
. Since n > 2k and (\alpha n) is a monotone decreasing sequence, by the
properties of (\alpha n) we have
\alpha n
A\alpha n
n
\mathrm{l}\mathrm{o}\mathrm{g}\delta
\Bigl(
1 +
n
2k
\Bigr)
\leq C
\alpha 2k
A
\alpha
2k
2k
, and then by Lemma 2.4 for the Cesàro
numbers in [3]
\bfB \bfthree \leq C
\alpha 2k
A
\alpha
2k
2k
\sum
a<k
k - 1\sum
b=a
A
\alpha
2k
2b
b - a
2b - a
= C
\alpha 2k
A
\alpha
2k
2k
k - 1\sum
b=0
A
\alpha
2k
2b
b\sum
a=0
b - a
2b - a
\leq
\leq C
\alpha 2k
A
\alpha
2k
2k
k - 1\sum
b=0
A
\alpha
2k
2b
\leq C
k - 1\sum
b=0
\alpha 2k
\biggl(
2b + 1
2k
\biggr) \alpha
2k
\leq C
k - 1\sum
b=0
\alpha 2k
\biggl(
2b
2k
\biggr) \alpha
2k
\leq C,
again just as at the end of the proof of Lemma 2.3.
Lemma 2.4 is proved.
Corollary 2.1. Let 1 > \alpha a > 0 fulfill property (1.1). Then by Lemmas 2.3 and 2.4, as a direct
consequence, we have
\int
Ik\times Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n>2k
1
A\alpha n
n
| n| \sum
s=k
2k\sum
j=0
A\alpha n - 1
j
\bigm| \bigm| Kj(x
2)
\bigm| \bigm| D2s(x
1)d(x1, x2) \leq C.
Moreover, we prove the following lemma.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
ALMOST EVERYWHERE CONVERGENCE OF CESÀRO MEANS OF TWO VARIABLE WALSH – FOURIER . . . 303
Lemma 2.5. \int
Ik\times Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n>2k
1
A\alpha n
n
| n| \sum
s=k
2| n| \sum
j=2k+1
A\alpha n - 1
j
\bigm| \bigm| Kj(x
2)
\bigm| \bigm| D2s(x
1)d(x1, x2) \leq C,
where 1 > \alpha a > 0 is a decreasing sequence with property (1.1).
Proof. By the result of Goginava [8] (see (2.3)) we have
\int
I\setminus Ik
\mathrm{s}\mathrm{u}\mathrm{p}j\geq 2b
\bigm| \bigm| Kj(x
1)
\bigm| \bigm| d(x1) \leq
\leq C
b - k + 1
2b - k
for any b \geq k. That is the integral in Lemma 2.5 is bounded by
C
\int
Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n>2k
1
A\alpha n
n
| n| \sum
s=k
| n| - 1\sum
b=k
b - k + 1
2b - k
2b+1\sum
j=2b+1
A\alpha n - 1
j D2s(x
1)d(x1) =: \bfB \bffour .
As in the proof of Lemma 2.4 we have
\sum 2b+1
j=2b+1
A\alpha n - 1
j \leq C\alpha nA
\alpha n
2b
. In the proof of Lemma 2.4
we can find inequality
\int
Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n>2k
1
(| n| + 1 - k)\delta
| n| \sum
s=k
D2s(x
1)d(x1) \leq C
and henceforth
\bfB \bffour \leq
\int
Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n>2k
1
A\alpha n
n
| n| - 1\sum
b=k
b - k + 1
2b - k
\alpha nA
\alpha n
2b
(| n| + 1 - k)\delta
1
(| n| + 1 - k)\delta
| n| \sum
s=k
D2s(x
1)d(x1) \leq
\leq C \mathrm{s}\mathrm{u}\mathrm{p}
n>2k
1
A\alpha n
n
| n| \sum
b=k
b - k + 1
2b - k
\alpha nA
\alpha n
2b
\mathrm{l}\mathrm{o}\mathrm{g}\delta
\Bigl(
1 +
n
2k
\Bigr) \int
Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n>2k
1
(| n| + 1 - k)\delta
| n| \sum
s=k
D2s(x
1)d(x1) \leq
\leq C \mathrm{s}\mathrm{u}\mathrm{p}
n>2k
\alpha n
A\alpha n
n
| n| \sum
b=k
b - k + 1
2b - k
A\alpha n
2b
\mathrm{l}\mathrm{o}\mathrm{g}\delta
\Bigl(
1 +
n
2k
\Bigr)
=: \bfB \bffive .
So, by
\alpha n
A\alpha n
n
\mathrm{l}\mathrm{o}\mathrm{g}\delta
\Bigl(
1 +
n
2k
\Bigr)
\leq C
\alpha 2k
A
\alpha
2k
2k
, we have
\bfB \bffive \leq C
\alpha 2k
A
\alpha
2k
2k
\mathrm{s}\mathrm{u}\mathrm{p}
n>2k
| n| \sum
b=k
b - k + 1
2b - k
A\alpha n
2b
.
Since (\alpha n) is a monotone decreasing, then A\alpha n
2b
\leq A
\alpha
2k
2b
. Thus, by [3] (Lemma 2.4) (second inequa-
lity below)
\bfB \bffive \leq C
\alpha 2k
A
\alpha
2k
2k
\biggl[
A
\alpha
2k
2k
+
2
2
A
\alpha
2k
2k+1 +
3
22
A
\alpha
2k
2k+2 +
4
23
A
\alpha
2k
2k+3 + . . .
\biggr]
\leq
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
304 A. A. ABU JOUDEH, G. GÁT
\leq C\alpha 2k
\infty \sum
j=0
\biggl(
2k+j + 1
2k
\biggr) \alpha
2k j
2j
\leq C\alpha 2k
\infty \sum
j=0
j
2j(1 - \alpha
2k
)
\leq C
as it holds 0 < \alpha 2k \leq 1 - \alpha 2 < 1.
Lemma 2.5 is proved.
Corollary 2.1 and Lemma 2.5 give the following corollary.
Corollary 2.2. Let 0 < \alpha n < 1 be a monotone decreasing sequence and
\alpha N
A\alpha N
N
A\alpha n
n
\alpha n
\mathrm{l}\mathrm{o}\mathrm{g}\delta
\biggl(
1 +
N
n
\biggr)
\leq C
for every N \geq n \geq 1. Then
\int
Ik\times Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n>2k
1
A\alpha n
n
| n| \sum
s=k+1
2| n| \sum
j=0
A\alpha n - 1
j
\bigm| \bigm| Kj(x
2)
\bigm| \bigm| D2s(x
1)d(x1, x2) \leq C.
By the help of Corollary 2.2 and Lemma 2.1 we prove that operator
t\ast f(y) := \mathrm{s}\mathrm{u}\mathrm{p}
n
| t\ast ,\alpha n
n f(y)| := \mathrm{s}\mathrm{u}\mathrm{p}
n
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
I\times I
f(x)| T\alpha n
n (x+ y)| d\lambda (x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
is quasilocal.
Lemma 2.6. Suppose that sequence (\alpha n) fulfills the conditions of Corollary 2.2. Let f \in L1(I\times
\times I) such that supp f \subset Ik(u
1)\times Ik(u
2),
\int
Ik(u1)\times Ik(u2)
fd\lambda = 0 for some dyadic rectangle. Then
we have \int
Ik(u1)\times Ik(u2)
t\ast fd\lambda \leq C\| f\| 1.
Besides, operator t\ast is of strong type (L\infty , L\infty ).
Proof. Recall that for any m, n \leq 2k we have \^f(m,n) = 0, and then
t\ast f(y) := \mathrm{s}\mathrm{u}\mathrm{p}
n>2k
| t\ast ,\alpha n
n f(y)| .
The proof this lemma is based on Lemma 2.1. More precisely, on inequalities (2.1) and (2.2), that is,\int
Ik(u1)\times Ik(u2)
t\ast fd\lambda \leq
\int
Ik(u1)\times Ik(u2)
\mathrm{s}\mathrm{u}\mathrm{p}
n>2k
| \~t\alpha n
n f | d\lambda +
+
\int
Ik(u1)\times Ik(u2)
\mathrm{s}\mathrm{u}\mathrm{p}
n>2k
| \=t\alpha n
n f | d\lambda +
\int
Ik(u1)\times Ik(u2)
\mathrm{s}\mathrm{u}\mathrm{p}
n>2k
| \=\=t\alpha n
n f | d\lambda =: A1 +A2 +A3.
Lemma 2.1 means that A1 \leq C\| f\| 1. Since the difference between terms A2 and A3 is only the
interchange of variables therefore it is enough to discuss A2 only. By the theorem of Fubini and the
shift invariance of the Lebesgue measure, we have
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
ALMOST EVERYWHERE CONVERGENCE OF CESÀRO MEANS OF TWO VARIABLE WALSH – FOURIER . . . 305
A2 \leq
\int
Ik(u1)\times Ik(u2)
| f(x1, x2)|
\int
Ik\times Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n>2k
\=T\alpha n
n (z1, z2)d\lambda (z)d\lambda (x).
Therefore, if we could prove the inequality
\int
Ik\times Ik
\mathrm{s}\mathrm{u}\mathrm{p}
n>2k
\=T\alpha n
n (z1, z2)d\lambda (z) \leq C, then the proof of
Lemma 2.6 would be complete.
By the help of the Abel transform, we get
A\alpha n
n
\=T\alpha n
n (z1, z2) = D2B (z
1)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
2B - 1\sum
j=0
A\alpha a - 1
n(B)+jDj(z
2)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
= D2B (z
1)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
2B - 1\sum
j=0
(A\alpha a - 1
n(B)+j - A\alpha a - 1
n(B)+j+1)
j\sum
i=0
Di +A\alpha n - 1
n(B)+2B
2B - 1\sum
i=0
Di(z
2)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
= D2B (z
1)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| (1 - \alpha n)
2B - 1\sum
j=0
A\alpha a - 1
n(B)+j
j + 1
n(B) + j + 1
K1
j (z
2) +A\alpha n - 1
n 2BK1
2B - 1(z
2)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq D2B (z
1)
2B - 1\sum
j=0
A\alpha n - 1
j
\bigm| \bigm| K1
j (z
2)
\bigm| \bigm| +D2B (z
1)A\alpha n - 1
n 2B
\bigm| \bigm| K1
2B - 1(z
2)
\bigm| \bigm| . (2.4)
Use the facts that Ik \times Ik = \=Ik \times Ik \cup \=Ik \times \=Ik \cup Ik \times \=Ik and D2B (z
1) = 0 for n > 2k, that is,
B = | n| \geq k in the case of z1 \in \=Ik. Moreover, 2BA\alpha n - 1
n /A\alpha n
n \leq 1, then by Corollary 2.2 the proof
of the sublinearity of operator t\ast f is complete. On the other hand,
\| t\ast f\| \infty \leq \mathrm{s}\mathrm{u}\mathrm{p}
n
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
I\times I
\| f\| \infty | T\alpha n
n (x+ y)| d\lambda (x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq C\| f\| \infty
as it comes from (2.4) and the fact that the L1-norms of the Fejér kernels and also the Dirichlet
kernels with indices of the form 2m are uniformly bounded.
Lemma 2.6 is proved.
Now, we can prove the main tool in order to have Theorem 1.1 for operators
\sigma \beta
\ast f := \mathrm{s}\mathrm{u}\mathrm{p}
n\in \BbbN
\bigm| \bigm| \bigm| \sigma \beta n
2n f
\bigm| \bigm| \bigm| = \mathrm{s}\mathrm{u}\mathrm{p}
n\in \BbbN
\bigm| \bigm| \bigm| f \ast K\beta n
2n
\bigm| \bigm| \bigm|
and
\~\sigma \beta
\ast f := \mathrm{s}\mathrm{u}\mathrm{p}
n\in \BbbN
\bigm| \bigm| \bigm| \~\sigma \beta n
2n f
\bigm| \bigm| \bigm| = \mathrm{s}\mathrm{u}\mathrm{p}
n\in \BbbN
\bigm| \bigm| \bigm| f \ast | K\beta n
2n |
\bigm| \bigm| \bigm| .
Lemma 2.7. The operators \~\sigma \beta
\ast and \sigma \beta
\ast are of weak type (L1, L1).
Proof. First, we prove Lemma 2.7 for operator \~\sigma \beta
\ast . We apply the Calderon – Zygmund decom-
position lemma [10]. That is, let f \in L1(I
2) and \| f\| 1 < \eta . Then there is a decomposition
f = f0 +
\infty \sum
j=1
fj
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
306 A. A. ABU JOUDEH, G. GÁT
such that \| f0\| \infty \leq C\eta , \| f0\| 1 \leq C\| f\| 1 and Ij \times Ij = Ikj (u
j,1) \times Ikj (u
j,2) are disjoint dyadic
rectangles for which
\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p} fj \subset Ij \times Ij ,
\int
Ij\times Ij
fjd\lambda = 0, \lambda (F ) \leq C\| f1\|
\eta
, (uj,1, uj,2) \in I \times I, kj \in \BbbN , j \in \BbbP ,
where F = \cup \infty
j=1I
j \times Ij . By the \sigma -sublinearity of the maximal operator with an appropriate con-
stant C, we have
\lambda (\~\sigma \beta
\ast f > 2C\eta ) \leq \lambda (\~\sigma \beta
\ast f0 > C\eta ) + \lambda
\Biggl(
\~\sigma \beta
\ast
\Biggl( \infty \sum
i=1
fi
\Biggr)
> C\eta
\Biggr)
:= I + II.
Notice that
K\beta n
2n (x) = T\alpha 2n
2n (x) +
D2n(x
1)D2n(x
2)
A\alpha 2n
2n
and keep in mind that operator \mathrm{s}\mathrm{u}\mathrm{p}n | f \ast (D2n \times D2n)| is quasilocal and it is of weak type (L1, L1)
and it is also of type (Lp, Lp) for each 1 < p \leq \infty [10]. Since by Lemma 2.6 \| \~\sigma \alpha
\ast f0\| \infty \leq
\leq C\| f0\| \infty \leq C\eta , then we have I = 0. So,
\lambda
\Biggl(
\~\sigma \beta
\ast
\Biggl( \infty \sum
i=1
fi
\Biggr)
> C\eta
\Biggr)
\leq \lambda (F ) + \lambda
\Biggl(
\=F \cap
\Biggl\{
\~\sigma \beta
\ast
\Biggl( \infty \sum
i=1
fi
\Biggr)
> C\eta
\Biggr\} \Biggr)
\leq
\leq C\| f\| 1
\eta
+
C
\eta
\infty \sum
i=1
\int
Ij\times Ij
\~\sigma \beta
\ast fjd\lambda =:
C\| f\| 1
\eta
+
C
\eta
\infty \sum
i=1
IIIj ,
where
IIIj :=
\int
Ij\times Ij
\~\sigma \beta
\ast fjd\lambda =
=
\int
Ikj (u
j)\times Ikj (u
j)
\mathrm{s}\mathrm{u}\mathrm{p}
n\in \BbbN
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
Ikj (u
j)\times Ikj (u
j)
fj(x)
\bigm| \bigm| \bigm| K\beta n
2n (y + x)
\bigm| \bigm| \bigm| d\lambda (x1, x2)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| d\lambda (y
1, y2).
The forthcoming estimation of IIIj is given by the help Lemma 2.6:
IIIj \leq C\| fj\| 1.
That is, operator \~\sigma \beta
\ast is of weak type (L1, L1) and same holds for operator \sigma \beta
\ast .
Lemma 2.7 is proved.
Proof of Theorem 1.1. Let P \in P be a two-dimensional Walsh polynomial, that is, P (x) =
=
\sum 2k - 1
i,j=0
ci,j\omega i(x
1)\omega j(x
2). Then for all natural number m \geq 2k we have that Sm,mP \equiv P. Conse-
quently, the statement \sigma \beta n
2nP \rightarrow P holds everywhere. This follows from the fact that for any fixed j
it holds
A\beta n - 1
2n - j
A\beta n
2n
\rightarrow 0 since, for instance, for j = 1 we have
A\beta n - 1
2n - 1
A\beta n
2n
=
\beta n2
n
(2n - 1 + \beta n)(2n + \beta n)
\rightarrow 0.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
ALMOST EVERYWHERE CONVERGENCE OF CESÀRO MEANS OF TWO VARIABLE WALSH – FOURIER . . . 307
Now, let \eta , \epsilon > 0, f \in L1(I
2). Let P \in P be a two-dimensional Walsh polynomial such that
\| f - P\| 1 < \eta . Then
\lambda
\biggl(
\mathrm{l}\mathrm{i}\mathrm{m}
n\in \BbbN
\bigm| \bigm| \bigm| \sigma \beta n
2n f - f
\bigm| \bigm| \bigm| > \epsilon
\biggr)
\leq
\leq \lambda
\biggl(
\mathrm{l}\mathrm{i}\mathrm{m}
n\in \BbbN
\bigm| \bigm| \bigm| \sigma \beta n
2n (f - P )
\bigm| \bigm| \bigm| > \epsilon
3
\biggr)
+ \lambda
\biggl(
\mathrm{l}\mathrm{i}\mathrm{m}
n\in \BbbN
\bigm| \bigm| \bigm| \sigma \beta n
2nP - P
\bigm| \bigm| \bigm| > \epsilon
3
\biggr)
+ \lambda
\biggl(
\mathrm{l}\mathrm{i}\mathrm{m}
n\in \BbbN
\bigm| \bigm| \bigm| \sigma \beta n
2nP - f
\bigm| \bigm| \bigm| > \epsilon
3
\biggr)
\leq
\leq \lambda
\biggl(
\mathrm{l}\mathrm{i}\mathrm{m}
n\in \BbbN
\bigm| \bigm| \bigm| \sigma \beta n
2n (f - P )
\bigm| \bigm| \bigm| > \epsilon
3
\biggr)
+ 0 +
3
\epsilon
\| P - f\| 1 \leq C\| P - f\| 1
3
\epsilon
\leq C
\epsilon
\eta
because \sigma \beta
\ast is of weak type (L1, L1). This holds for all \eta > 0. That is, for an arbitrary \epsilon > 0 we
have
\lambda
\biggl(
\mathrm{l}\mathrm{i}\mathrm{m}
n\in \BbbN
\bigm| \bigm| \bigm| \sigma \beta n
2n f - f
\bigm| \bigm| \bigm| > \epsilon
\biggr)
= 0
and, consequently,
\lambda
\biggl(
\mathrm{l}\mathrm{i}\mathrm{m}
n\in \BbbN
\bigm| \bigm| \bigm| \sigma \beta n
2n f - f
\bigm| \bigm| \bigm| > 0
\biggr)
= 0.
This finally gives \sigma \beta n
2n f - \rightarrow f a.e.
Theorem 1.1 is proved.
References
1. T. Akhobadze, On the convergence of generalized Cesàro means of trigonometric Fourier series. I, Acta Math.
Hungar., 115, № 1-2, 59 – 78 (2007).
2. T. Akhobadze, On the generalized Cesàro means of trigonometric Fourier series, Bull. TICMI, 18, № 1, 75 – 84
(2014).
3. A. Abu Joudeh, G. Gát, Almost everywhere convergence of Cesàro means with varying parameters of Walsh – Fourier
series, Miskolc Math. Notes, 19, № 1, 303 – 317 (2018).
4. M. I. Dyachenko, On (C,\alpha )-summability of multiple trigonometric Fourier series (in Russian), Soobshch. Akad. Nauk
Gruzin. SSR, 131, № 2, 261 – 263 (1988).
5. G. Gát, Convergence of Marcinkiewicz means of integrable functions with respect to two-dimensional Vilenkin systems,
Georg. Math. J., 11, № 3, 467 – 478 (2004).
6. G. Gát, On (C,1) summability for Vilenkin-like systems, Stud. Math., 144, № 2, 101 – 120 (2001).
7. U. Goginava, Marcinkiewicz – Fejér means of d-dimensional Walsh – Fourier series, J. Math. Anal. and Appl., 307,
№ 1, 206 – 218 (2005).
8. U. Goginava, Almost everywhere convergence of (C,\alpha )-means of cubical partial sums of d-dimensional Walsh –
Fourier series, J. Approxim. Theory, 141, № 1, 8 – 28 (2006).
9. J. Marcinkiewicz, Sur une nouvelle condition pour la convergence presque partout des séries de Fourier, Ann. Scuola
Norm. Super. Pisa Cl. Sci., 8, № 3-4, 239 – 240 (1939).
10. F. Schipp, W. R. Wade, P. Simon, J. Pál, Walsh series: an introduction to dyadic harmonic analysis, Adam Hilger,
Bristol, New York (1990).
11. F. Weisz, Convergence of double Walsh – Fourier series and Hardy spaces, Approxim. Theory and Appl., 17, № 2,
32 – 44 (2001).
12. L. V. Zhizhiashvili, A generalization of a theorem of Marcinkiewicz., Izv. Ross. Akad. Nauk. Ser. Mat., 32, № 5,
1112 – 1122 (1968).
13. A. Zygmund, Trigonometric series, Univ. Press, Cambridge (1959).
Received 11.07.18,
after revision — 17.10.19
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
|
| id | umjimathkievua-article-196 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:04Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/78/c633dd8d7514f33924b5cd1efaf53e78.pdf |
| spelling | umjimathkievua-article-1962025-03-31T08:48:21Z Almost everywhere convergence of Cesàro means of two variable Walsh – Fourier series with varying parameteres Almost everywhere convergence of Cesàro means of two variable Walsh – Fourier series with varying parameteres Almost everywhere convergence of Cesàro means of two variable Walsh – Fourier series with varying parameteres Abu Joudeh , A. A. Gát, G. Abu Joudeh, A. A. Gát, G. Abu Joudeh, A. A. Gát, G. Cesàro means with varying parameters two-dimensional Walsh-Fourier series Marcinkiewicz means Cesàro means with varying parameters two-dimensional Walsh-Fourier series Marcinkiewicz means UDC 517.5 We prove that the maximal operator of some $(C , \beta_{n})$ means of cubical partial sums of two variable Walsh – Fourier series of integrable functions is of weak type $(L_1,L_1)$. Moreover, the $ (C , \beta_{n})$-means $\sigma_{2^n}^{\beta_{n}} f$ of the function $ f \in L_{1} $ converge a.e. to $f$ for $ f \in L_{1}(I^2) $, where $I$ is the Walsh group for some sequences $1&gt; \beta_n\searrow 0$. УДК 517.5 Збiжнiсть майже скрізь середнiх Чезаро для рядiв Уолша–Фур’є вiд двох змiнних зi змiнними параметрами Доведено, що максимальний оператор вiд деяких середнiх $(C, \beta n)$ кубiчних часткових сум рядiв Уолша – Фур’є двох змiнних для iнтегровних функцiй має слабкий тип $(L_1,L_1)$. Бiльш того, $ (C , \beta_{n})$-середнi $\sigma_{2^n}^{\beta_{n}} f$ для функцiї $ f \in L_{1} $ збiгаються мaйже скрiзь до $f$ для $ f \in L_{1}(I^2) $, де $I$ — група Уолша для деяких послiдовностей $1&gt; \beta_n \searrow 0$. Institute of Mathematics, NAS of Ukraine 2021-03-11 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/196 10.37863/umzh.v73i3.196 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 3 (2021); 291 - 307 Український математичний журнал; Том 73 № 3 (2021); 291 - 307 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/196/8974 Copyright (c) 2021 ANAS AHMAD MOHAMMAD ABU JOUDEH, GYÖRGY Tamás GÁT |
| spellingShingle | Abu Joudeh , A. A. Gát, G. Abu Joudeh, A. A. Gát, G. Abu Joudeh, A. A. Gát, G. Almost everywhere convergence of Cesàro means of two variable Walsh – Fourier series with varying parameteres |
| title | Almost everywhere convergence of Cesàro means of two variable Walsh – Fourier series with varying parameteres |
| title_alt | Almost everywhere convergence of Cesàro means of two variable Walsh – Fourier series with varying parameteres Almost everywhere convergence of Cesàro means of two variable Walsh – Fourier series with varying parameteres |
| title_full | Almost everywhere convergence of Cesàro means of two variable Walsh – Fourier series with varying parameteres |
| title_fullStr | Almost everywhere convergence of Cesàro means of two variable Walsh – Fourier series with varying parameteres |
| title_full_unstemmed | Almost everywhere convergence of Cesàro means of two variable Walsh – Fourier series with varying parameteres |
| title_short | Almost everywhere convergence of Cesàro means of two variable Walsh – Fourier series with varying parameteres |
| title_sort | almost everywhere convergence of cesàro means of two variable walsh – fourier series with varying parameteres |
| topic_facet | Cesàro means with varying parameters two-dimensional Walsh-Fourier series Marcinkiewicz means Cesàro means with varying parameters two-dimensional Walsh-Fourier series Marcinkiewicz means |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/196 |
| work_keys_str_mv | AT abujoudehaa almosteverywhereconvergenceofcesaromeansoftwovariablewalshfourierserieswithvaryingparameteres AT gatg almosteverywhereconvergenceofcesaromeansoftwovariablewalshfourierserieswithvaryingparameteres AT abujoudehaa almosteverywhereconvergenceofcesaromeansoftwovariablewalshfourierserieswithvaryingparameteres AT gatg almosteverywhereconvergenceofcesaromeansoftwovariablewalshfourierserieswithvaryingparameteres AT abujoudehaa almosteverywhereconvergenceofcesaromeansoftwovariablewalshfourierserieswithvaryingparameteres AT gatg almosteverywhereconvergenceofcesaromeansoftwovariablewalshfourierserieswithvaryingparameteres |