# A note on strong summability of two-dimensional Walsh-Fourier series

G. Tephnadze  George Tephnadze, The University of Georgia, School of Science
and Technology, 77a Merab Kostava St, Tbilisi, 0128, Georgia.
<EMAIL_ADDRESS>
###### Abstract.

In this paper we investigate strong summability of the two-dimensional Walsh-
Fourier series obtained in Weisz [18] (see Theorem W) and prove sharpness of
this result.

The research was supported by Shota Rustaveli National Science Foundation
grant YS-18-043.

2010 Mathematics Subject Classification. 42C10.

Key words and phrases: Walsh system, two-dimensional Walsh-Fourier series,
Strong summability, martingale Hardy space.

Let $\mathbb{N}_{+}$ denote the set of positive integers,
$\mathbb{N}:=\mathbb{N}_{+}\cup\\{0\\}$. Denote by $Z_{2}$ the discrete cyclic
group of order 2, that is $Z_{2}=\\{0,1\\},$ where the group operation is the
modulo 2 addition and every subset is open. The Haar measure on $Z_{2}$ 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 $Z_{2}.$
The elements of $G$ are of the form $x=\left(x_{0},x_{1},...,x_{k},...\right)$
with $x_{k}\in\\{0,1\\}\left(k\in\mathbb{N}\right).$ The group operation on
$G$ is the coordinate-wise addition, the measure (denote by$\,\,\mu$) and the
topology are the product measure and topology of $Z_{2}=\\{0,1\\}$. The
compact Abelian group $G$ is called the Walsh group. A base for the
neighborhoods of $G$ can be given in the following way:

$\displaystyle I_{0}\left(x\right)$ $\displaystyle:$
$\displaystyle=G,\,\,\,I_{n}\left(x\right):=\,I_{n}\left(x_{0},...,x_{n-1}\right)$
$\displaystyle:$ $\displaystyle=\left\\{y\in
G:\,y=\left(x_{0},...,x_{n-1},y_{n},y_{n+1},...\right)\right\\},\,\left(x\in
G,n\in\mathbb{N}\right).$

These sets are called the dyadic intervals. Let
$0=\left(0:i\in\mathbb{N}\right)\in G$ denote the null element of
$G,\,\,\,I_{n}:=I_{n}\left(0\right)\,\left(n\in\mathbb{N}\right).$ Set
$e_{n}:=\left(0,...,0,1,0,...\right)\in G$ the $n\,$th coordinate of which is
1 and the rest are zeros $\left(n\in\mathbb{N}\right).$ Let
$\overline{I}_{n}:=G\backslash I_{n}.$

If $n\in\mathbb{N}$, then $n=\sum\limits_{i=0}^{\infty}n_{i}2^{i},$ where
$n_{i}\in\\{0,1\\}\,\,\left(i\in\mathbb{N}\right)$, i.e. $n$ is expressed in
the number system of base 2. Denote
$\left|n\right|:=\max\\{j\in\mathbb{N}:n_{j}\neq 0\\}$, that is,
$2^{\left|n\right|}\leq n<2^{\left|n\right|+1}.$

It is easy to show that for every odd number $n_{0}=1$ and we can write
$n=1+\sum_{i=1}^{\left|n\right|}n_{j}2^{i}$, where
$n_{j}\in\left\\{0,1\right\\},$ $~{}(j\in\mathbb{N}_{+})$.

For $k\in\mathbb{N}$ and $x\in G$ let as denote by

$r_{k}\left(x\right):=\left(-1\right)^{x_{k}}\,\,\,\,\,\,\left(x\in
G,k\in\mathbb{N}\right)$

the $k$-th Rademacher function.

The Walsh-Paley system is defined as the sequence of Walsh-Paley functions:

$w_{n}\left(x\right):=\prod\limits_{k=0}^{\infty}\left(r_{k}\left(x\right)\right)^{n_{k}}=r_{\left|n\right|}\left(x\right)\left(-1\right)^{\sum\limits_{k=0}^{\left|n\right|-1}n_{k}x_{k}}\,\,\,\,\,\,\left(x\in
G,n\in\mathbb{N}_{+}\right).$

The Walsh-Dirichlet kernel is defined by

$D_{n}\left(x\right)=\sum\limits_{k=0}^{n-1}w_{k}\left(x\right).$

Recall that (see $\left[8,\text{ p. 7}\right]$)

(1) $D_{2^{n}}\left(x\right)=\left\\{\begin{array}[]{c}2^{n},\text{
\qquad}x\in I_{n}\\\
0,\,\,\,\text{\qquad}x\in\overline{I}_{n}\end{array}\right.,$

Let $n\in\mathbb{N}$ and $n=\sum\limits_{i=0}^{\infty}n_{i}2^{i}.$ Then

(2)
$D_{n}\left(x\right)=w_{n}\left(x\right)\sum\limits_{j=0}^{\infty}n_{j}w_{2^{j}}\left(x\right)D_{2^{j}}\left(x\right).$

Set $G^{2}:=G\times G.$ The rectangular partial sums $S_{M,N}$ of the
2-dimensional Walsh-Fourier series of a function $f\in
L_{2}\left(G^{2}\right)$ are defined as follows:

$S_{M,N}f\left(x,y\right):=\sum\limits_{i=0}^{M-1}\sum\limits_{j=0}^{N-1}\widehat{f}\left(i,j\right)w_{i}\left(x\right)w_{j}\left(y\right),$

where the number

$\widehat{f}\left(i,j\right)=\int\limits_{G^{2}}f\left(x,y\right)w_{i}\left(x\right)w_{j}\left(y\right)d\mu\left(x,y\right)$

is said to be the $\left(i,j\right)-$th Walsh-Fourier coefficient of the
function $f.$

The norms (or quasi-norms) of the spaces $L_{p}(G^{2})$ and $weak-
L_{p}\left(G^{2}\right)$ are respectively defined by

$\left\|f\right\|_{p}:=\left(\int_{G^{2}}\left|f\right|^{p}d\mu\right)^{1/p},\qquad\left\|f\right\|_{weak-
L_{p}(G^{2})}:=\underset{\lambda>0}{\sup}\lambda\mu\left(f>\lambda\right)^{1/p}\qquad\left(0<p<\infty\right).$

The $\sigma-$algebra generated by the dyadic 2-dimensional
$I_{n}\left(x\right)\times I_{n}\left(y\right)$ square of measure
$2^{-n}\times 2^{-n}$ will be denoted by
$\digamma_{n}\left(n\in\mathbb{N}\right).$ Denote by $f=\left(f_{n},\text{
}n\in\mathbb{N}\right)$ the one-parameter martingale with respect to
$\digamma_{n}\left(n\in\mathbb{N}\right),$ (for details see e.g. Weisz [16]
and [17]). The maximal function of a martingale $f$ is defined by

$f^{\ast}=\sup_{n\in\mathbb{N}}\left|f_{n}\right|.$

Let $f\in L_{1}(G^{2})$. Then the dyadic maximal function is given by

$f^{*}\left(x,y\right)=\sup\limits_{n\in\mathbb{N}}\frac{1}{\mu\left(I_{n}(x)\times
I_{n}(y)\right)}\left|\int\limits_{I_{n}(x)\times
I_{n}(y)}f\left(s,t\right)d\mu\left(s,t\right)\right|,\,\,\left(x,y\right)\in
G^{2}.$

The dyadic Hardy martingale space $H_{p}(G^{2})$ $\left(0<p<\infty\right)$
consists of all functions $f$ for which

$\left\|f\right\|_{H_{p}(G^{2})}:=\left\|f^{*}\right\|_{p}<\infty.$

If $f\in L_{1}(G^{2}),$ then it is easy to show that the sequence
$\left(S_{2^{n},2^{n}}f:n\in\mathbb{N}\right)$ is a martingale. If
$f=\left(f_{n},n\in\mathbb{N}\right)$ is a martingale, then the Walsh-Fourier
coefficients must be defined in a slightly different manner:

$\widehat{f}\left(i,j\right):=\lim_{k\rightarrow\infty}\int_{G}f_{k}\left(x,y\right)w_{i}\left(x\right)w_{j}\left(y\right)d\mu\left(x,y\right).$

The Walsh-Fourier coefficients of $f\in L_{1}\left(G^{2}\right)$ are the same
as those of the martingale $\left(S_{2^{n},2^{n}}f:n\in\mathbb{N}\right)$
obtained from $f.$

A bounded measurable function $a$ is a p-atom, if there exists a dyadic
2-dimensional cube $I\times I\mathbf{,}$ such that

$\int_{I\times I}ad\mu=0,\text{ \ \ }\left\|a\right\|_{\infty}\leq\mu(I\times
I)^{-1/p},\text{ \ \ \ supp}\left(a\right)\subset I\times I.$

The dyadic Hardy martingale spaces $H_{p}\left(G^{2}\right)$ for $0<p\leq 1$
have atomic characterizations (for details see e.g. Weisz [16] and [17]).

######  Lemma 1.

A martingale $f=\left(f_{n}:n\in\mathbb{N}\right)$ is in
$H_{p}\left(G^{2}\right)\left(0<p\leq 1\right)$ if and only if there exist a
sequence $\left(a_{k},k\in\mathbb{N}\right)$ of p-atoms and a sequence
$\left(\mu_{k},k\in\mathbb{N}\right)$ of a real numbers such that

(3) $\qquad\sum_{k=0}^{\infty}\mu_{k}S_{2^{n},2^{n}}a_{k}=f_{n}$

and

$\qquad\sum_{k=0}^{\infty}\left|\mu_{k}\right|^{p}<\infty.$

Moreover,

$\left\|f\right\|_{H_{p}}\backsim\inf\left(\sum_{k=0}^{\infty}\left|\mu_{k}\right|^{p}\right)^{1/p},$

where the infimum is taken over all decomposition of $f$ of the form (3).

It is known $\left[7,\text{ p. 125}\right]$ that the Walsh-Paley system is not
a Schauder basis in $L_{1}\left(G^{2}\right)$. Moreover, (for details see [8])
there exists a function in the dyadic martingale Hardy space
$H_{p}\left(G^{2}\right)$ $(0<p\leq 1)$ for which the partial sums are not
bounded in $L_{p}\left(G^{2}\right).$ However, Weisz [18] proved the following
estimation:

Theorem W. Let $\alpha\geq 0$, $0<p\leq 1$ and $f\in H_{p}\left(G^{2}\right).$
Then there exists an absolute constant $c_{p},$ depending only on $p,$ such
that

$\underset{n,m\geq 2}{\sup}\left(\frac{1}{\log n\log
m}\right)^{\left[p\right]}\underset{2^{-\alpha}\leq k/l\leq 2^{\alpha},\text{
}\left(k,l\right)\leq\left(n,m\right)}{\sum}\frac{\left\|S_{k,l}f\right\|_{p}^{p}}{\left(kl\right)^{2-p}}\leq
c_{p}\left\|f\right\|_{H_{p}}^{p},$

where $\left[p\right]$ denotes the integer part of $p.$

In the case when $\alpha=0$ and $p=1$ from Theorem W we obtain that the
following is true:

Theorem W1. Let $f\in H_{1}\left(G^{2}\right).$ Then there exists an absolute
constant $c,$ such that

(4)
$\frac{1}{\log^{2}n}\underset{k=0}{\sum^{n}}\frac{\left\|S_{k,k}f\right\|_{1}}{k^{2}}\leq
c\left\|f\right\|_{H_{1}\left(G^{2}\right)}.$

When $\alpha=0$ and $0<p<1$ from Theorem W follows the following result:

Theorem W2. Let $0<p<1$ and $f\in H_{p}\left(G^{2}\right).$ Then there exists
an absolute constant $c_{p},$ depending only on $p,$ such that

(5)
$\underset{k=0}{\sum^{\infty}}\frac{\left\|S_{k,k}f\right\|_{p}^{p}}{k^{4-2p}}\leq
c_{p}\left\|f\right\|_{H_{p}\left(G^{2}\right)}^{p}.$

Goginava and Gogoladze [3] generalized the estimate (4) (for details see [5]
and [11]) and proved that for any $f\in H_{1}\left(G^{2}\right)$, there exists
an absolute constant $c$, such that

(6)
$\sum\limits_{n=1}^{\infty}\frac{\left\|S_{n,n}f\right\|_{1}}{n\log^{2}n}\leq
c\left\|f\right\|_{H_{1}\left(G^{2}\right)}.$

The estimate (5) was generalized in [11] and it was proved that for any
$0<p<1$ and $f\in H_{p}\left(G^{2}\right)$, there exists an absolute constant
$c_{p}$, depending only on $p,$ such that

(7)
$\sum\limits_{n=1}^{\infty}\frac{\left\|S_{n,n}f\right\|_{p}^{p}}{n^{3-2p}}\leq
c_{p}\left\|f\right\|_{H_{p}\left(G^{2}\right)}^{p}.$

In [10] and [11] it was proved that the following is true:

Theorem T1. Let $0<p\leq 1$ and $\Phi:\mathbb{N}\rightarrow[1,$ $\infty)$ be
any nondecreasing function, satisfying the condition
$\underset{n\rightarrow\infty}{\lim}\Phi\left(n\right)=+\infty.$ Then there
exists a martingale $f\in H_{p}\left(G^{2}\right)$ such that

$\underset{n=1}{\overset{\infty}{\sum}}\frac{\left\|S_{n,n}f\right\|_{weak-
L_{p}}^{p}\Phi\left(n\right)}{n^{3-2p}\log^{2[p]}n}=\infty,$

where $\left[p\right]$ denotes the integer part of $p.$

Analogical problems for the one-dimensional case were proved in [1], [2], [9],
[12], [13], [15] and for the two-dimensional case in [5], [6], [7].

In view of Theorem T1 we can conclude that the sequence
$\left\\{1/(n^{3-2p}\log^{2[p]}n)\right\\}_{n=1}^{\infty}$ in inequalities (6)
and (7) can not be improved, which gives sharpness for $\alpha=0$.

In this paper we consider the analogous problem for $\alpha>0$ and prove the
sharpness of the sequence in Theorem W when $\alpha>0$:

######  Theorem 1.

Let $0<p<1,$ $\alpha>0$ and $\Phi:\mathbb{N}^{2}\rightarrow[1,$ $\infty)$
satisfies the conditions

(8) $\Phi\left(m_{1},n_{1}\right)\geq\Phi\left(m_{2},n_{2}\right)\text{ when
}m_{1}\geq m_{2}\text{ \ and }n_{1}\geq n_{2}$

and

(9) $\underset{m,n\rightarrow\infty}{\lim}\Phi\left(m,n\right)=+\infty.$

Then there exists a martingale $f\in H_{p}\left(G^{2}\right)$ such that

$\underset{n,m\in\mathbb{N}_{+}}{\sup}\underset{2^{-\alpha}\leq k/l\leq
2^{\alpha},\text{
}\left(k,l\right)\leq\left(n,m\right)}{\sum}\frac{\left\|S_{k,l}f\right\|_{p}^{p}\Phi\left(k,l\right)}{\left(kl\right)^{2-p}}=\infty.$

Proof. Under the condition (9) there exists an increasing sequence of positive
integers $\left\\{\alpha_{k}:\text{ }k\geq 0\right\\}$ such that
$\alpha_{0}\geq 2,$ $\alpha_{k}+[\alpha]+1<\alpha_{k+1}$ and

(10)
$\sum_{k=0}^{\infty}{\Phi^{-p/4}\left(2^{\alpha_{k}},2^{\alpha_{k}}\right)}<\infty.$

Let

$f_{n}=\sum_{\left\\{k;\text{
}\alpha_{k}+[\alpha]+1<n\right\\}}\lambda_{k}a_{k},$

where

$\lambda_{k}:=2^{2[\alpha]+2}{\Phi^{-1/4}\left(2^{\alpha_{k}},2^{\alpha_{k}}\right)}$

and

$a_{k}\left(x,y\right):=2^{\alpha_{k}\left(2/p-2\right)-2[\alpha]-2}\left(D_{2^{\alpha_{k}+[\alpha]+1}}\left(x\right)-D_{2^{\alpha_{k}}}\left(x\right)\right)\left(D_{2^{\alpha_{k}+[\alpha]+1}}\left(y\right)-D_{2^{\alpha_{k}}}\left(y\right)\right).$

Since

$S_{2^{n},2^{n}}a_{k}=\left\\{\begin{array}[]{l}a_{k}\text{,
\qquad}\alpha_{k}+[\alpha]+1<n,\\\ 0\text{, \qquad}\alpha_{k}+[\alpha]+1\geq
n,\end{array}\right.$

$\displaystyle\text{supp}(a_{k})=I_{\alpha_{k}}^{2},\text{ \qquad}\ \ \
\int_{I_{\alpha_{k}}^{2}}a_{k}d\mu=0,\text{ \qquad}\ \ \
\left\|a_{k}\right\|_{\infty}\leq
2^{2\alpha_{k}/p}=(\mu(\text{supp}a_{k}))^{-1/p}$

from Lemma 1 and (10) we obtain that $f\in H_{p}(G^{2}).$

It is obvious that

(11) $\displaystyle\widehat{f}(i,j)$ (14) $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{l}\frac{2^{\alpha_{k}\left(2/p-2\right)}}{\Phi^{1/4}\left(2^{\alpha_{k}},2^{\alpha_{k}}\right)},\text{
\ if \
}\left(i,j\right)\in\left\\{2^{\alpha_{k}},...,2^{\alpha_{k}+[\alpha]+1}-1\right\\}\times\left\\{2^{\alpha_{k}},...,2^{\alpha_{k}+[\alpha]+1}-1\right\\},\text{
}k\in\mathbb{N}\\\ 0,\text{ \ if \
}\left(i,j\right)\notin\bigcup\limits_{k=1}^{\infty}\left\\{2^{\alpha_{k}},...,2^{\alpha_{k}+[\alpha]+1}-1\right\\}\times\left\\{2^{\alpha_{k}},...,2^{\alpha_{k}+[\alpha]+1}-1\right\\}.\end{array}\right.$

Let $2^{\alpha_{k}}<m,n<2^{\alpha_{k}+[\alpha]+1}$. In view of (11) we can
conclude that

$\displaystyle S_{m,n}f\left(x,y\right)$ $\displaystyle=$
$\displaystyle\sum_{\eta=0}^{k-1}\sum_{i=2^{\alpha_{\eta}}}^{2^{\alpha_{\eta}+1}-1}\sum_{j=2^{\alpha_{\eta}}}^{2^{\alpha_{\eta}+1}-1}\widehat{f}(i,j)w_{i}\left(x\right)w_{j}\left(y\right)+\sum_{i=2^{\alpha_{k}}}^{m-1}\sum_{j=2^{\alpha_{k}}}^{n-1}\widehat{f}(i,j)w_{i}\left(x\right)w_{j}\left(y\right)$
$\displaystyle=$
$\displaystyle\sum_{\eta=0}^{k-1}\sum_{i=2^{\alpha_{\eta}}}^{2^{\alpha_{\eta}+1}-1}\sum_{j=2^{\alpha_{\eta}}}^{2^{\alpha_{\eta}+1}-1}\frac{2^{\alpha_{\eta}\left(2/p-2\right)}w_{i}\left(x\right)w_{j}\left(y\right)}{\Phi^{1/4}\left(2^{\alpha_{\eta}},2^{\alpha_{\eta}}\right)}+\sum_{i=2^{\alpha_{k}}}^{m-1}\sum_{j=2^{\alpha_{k}}}^{n-1}\frac{2^{\alpha_{k}\left(2/p-2\right)}w_{i}\left(x\right)w_{j}\left(y\right)}{\Phi^{1/4}\left(2^{\alpha_{k}},2^{\alpha_{k}}\right)}$
$\displaystyle=$
$\displaystyle\sum_{\eta=0}^{k-1}\frac{2^{\alpha_{\eta}\left(2/p-2\right)}}{\Phi^{1/4}\left(2^{\alpha_{\eta}},2^{\alpha_{\eta}}\right)}\left(D_{2^{\alpha_{\eta}+1}}\left(x\right)-D_{2^{\alpha_{\eta}}}\left(x\right)\right)\left(D_{2^{\alpha_{\eta}+1}}\left(y\right)-D_{2^{\alpha_{\eta}}}\left(y\right)\right)$
$\displaystyle+\frac{2^{\alpha_{k}\left(2/p-2\right)}}{\Phi^{1/4}\left(2^{\alpha_{k}},2^{\alpha_{k}}\right)}\left(D_{m}\left(x\right)-D_{2^{\alpha_{k}}}\left(x\right)\right)\left(D_{n}\left(y\right)-D_{2^{\alpha_{k}}}\left(y\right)\right)$
$\displaystyle:=$ $\displaystyle I+II.$

Let $\left(x,y\right)\in\left(G\backslash I_{1}\right)\times\left(G\backslash
I_{1}\right),$ $n$ and $m$ are odd numbers, such that
$2^{\alpha_{k}}<m,n<2^{\alpha_{k}+[\alpha]+1}$. Since $\alpha_{k}\geq 2$
$\left(k\in\mathbb{N}\right),$ if we combine (1) and (2) it follows that

$D_{2^{\alpha_{k}}}\left(x\right)=D_{2^{\alpha_{k}}}\left(y\right)=0$

and

(16)
$\displaystyle\left|II\right|=\frac{2^{\alpha_{k}\left(2/p-2\right)}}{\Phi^{1/4}\left(2^{\alpha_{k}},2^{\alpha_{k}}\right)}\left|w_{m}\left(x\right)w_{1}\left(x\right)D_{1}\left(x\right)w_{n}\left(y\right)D_{1}\left(y\right)\right|=\frac{2^{\alpha_{k}\left(2/p-2\right)}}{\Phi^{1/4}\left(2^{\alpha_{k}},2^{\alpha_{k}}\right)}.$

By applying (1) and the condition $\alpha_{n}\geq 2$
$\left(n\in\mathbb{N}\right)$ for $I$ we have that

(17)
$I=\sum_{\eta=0}^{k-1}\frac{2^{\alpha_{\eta}\left(2/p-2\right)}}{\Phi^{1/4}\left(2^{\alpha_{\eta}}\right)}\left(D_{2^{\alpha_{\eta}+1}}\left(x\right)-D_{2^{\alpha_{\eta}}}\left(x\right)\right)\left(D_{2^{\alpha_{\eta}+1}}\left(y\right)-D_{2^{\alpha_{\eta}}}\left(y\right)\right)=0.$

By combining (16) and (17), for $2^{\alpha_{k}}<m,n<2^{\alpha_{k}+[\alpha]+1}$
we get that

$\displaystyle\left\|S_{m,n}f\right\|_{weak-L_{p}}$ $\displaystyle\geq$
$\displaystyle\frac{2^{\alpha_{k}\left(2/p-2\right)}}{2\Phi^{1/4}\left(2^{\alpha_{k}},2^{\alpha_{k}}\right)}\left(\mu\left\\{\left(x,y\right)\in\left(G\backslash
I_{1}\right)\times\left(G\backslash
I_{1}\right):\left|S_{m,n}f\left(x,y\right)\right|\geq\frac{2^{\alpha_{k}\left(2/p-2\right)}}{2\Phi^{1/4}\left(2^{\alpha_{k}},2^{\alpha_{k}}\right)}\right\\}\right)^{1/p}$
$\displaystyle\geq$
$\displaystyle\frac{2^{\alpha_{k}\left(2/p-2\right)}}{2\Phi^{1/4}\left(2^{\alpha_{k}},2^{\alpha_{k}}\right)}\mu\left(\left(G\backslash
I_{1}\right)\times\left(G\backslash
I_{1}\right)\right)\geq\frac{c_{p}2^{\alpha_{k}\left(2/p-2\right)}}{\Phi^{1/4}\left(2^{\alpha_{k}},2^{\alpha_{k}}\right)}.$

According to (8), (9) and (A note on strong summability of two-dimensional
Walsh-Fourier series) we can conclude that

$\displaystyle\underset{s,l\in\mathbb{N}_{+}}{\sup}\underset{2^{-\alpha}\leq
m/n\leq 2^{\alpha},\text{
}\left(m,n\right)\leq\left(s,l\right)}{\sum}\frac{\left\|S_{m,n}f\right\|_{weak-
L_{p}}^{p}\Phi\left(m,n\right)}{\left(mn\right)^{2-p}}$ $\displaystyle\geq$
$\displaystyle\underset{2^{\alpha_{k}}<m,n\leq
2^{\alpha_{k}+\alpha}}{\sum}\frac{\left\|S_{m,n}f\right\|_{weak-
L_{p}}^{p}\Phi\left(m,n\right)}{\left(mn\right)^{2-p}}$ $\displaystyle\geq$
$\displaystyle\frac{c_{p}\Phi\left(2^{\alpha_{k}},2^{\alpha_{k}}\right)}{2^{\alpha_{k}\left(4-2p\right)}}\underset{2^{\alpha_{k}}<m,n\leq
2^{\alpha_{k}+\alpha}}{\sum}\left\|S_{m,n}f\right\|_{weak-L_{p}}^{p}$
$\displaystyle\geq$
$\displaystyle\frac{c_{p}\Phi\left(2^{\alpha_{k}},2^{\alpha_{k}}\right)}{2^{\alpha_{k}\left(4-2p\right)}}\underset{2^{\alpha_{k}-1}<m,n\leq
2^{\alpha_{k}-1+\alpha}}{\sum}\left\|S_{2m+1,2n+1}f\right\|_{weak-L_{p}}^{p}$
$\displaystyle\geq$
$\displaystyle\frac{c_{p}\Phi\left(2^{\alpha_{k}},2^{\alpha_{k}}\right)}{2^{\alpha_{k}\left(4-2p\right)}}\frac{2^{\alpha_{k}\left(2-2p\right)}}{\Phi^{1/4}\left(2^{\alpha_{k}},2^{\alpha_{k}}\right)}\underset{2^{\alpha_{k}-1}<m,n\leq
2^{\alpha_{k}-1+\alpha}}{\sum}1$ $\displaystyle\geq$
$\displaystyle\frac{c_{p}\Phi^{3/4}\left(2^{\alpha_{k}},2^{\alpha_{k}}\right)}{2^{2\alpha_{k}}}\left(2^{\alpha_{k}-1+\alpha}-2^{\alpha_{k}-1}-1\right)^{2}$
$\displaystyle\geq$ $\displaystyle
c_{p}\Phi^{3/4}\left(2^{\alpha_{k}},2^{\alpha_{k}}\right)\rightarrow\infty,\qquad\text{as}\text{\qquad}k\rightarrow\infty.$

By combining (10-A note on strong summability of two-dimensional Walsh-Fourier
series) we complete the proof of Theorem 1.

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