Ivane Javakhishvili Tbilisi State University

George Tephnadze
Faculty of Exact and Natural Sciences
Department of Mathematics

On the Partial Sums and Marcinkiewicz and Fejér Means
on the One- and Two-dimensional One-parameter Martingale Hardy Spaces

Georgian PhD Thesis

Tbilisi 2016

Abstract

Unlike the classical theory of Fourier series which deals with decomposition
of a function into continuous waves, the Walsh functions are “rectangular
waves”. Such waves have already been used frequently in the theory of signal
transmission, codic theory, cryptography, filtering, image enhancement and
digital signal processing.

The problems we have studied in this PhD thesis are central to Mathematical
Analysis. They involve techniques which have been developed a great deal
during the last three decades.

In this PhD thesis we are dealing with convergence and summability of partial
sums, Fejér and Marcinkiewicz means with respect to one- and two-dimensional
Walsh-Fourier series on the martingale Hardy spaces.

This thesis is focus to achieve the following main results:

• To find estimation of convergence and divergence of the subsequences of
partial sums of the one-dimensional Walsh-Fourier series on the martingale
Hardy spaces $H_{p}(G)$, when $0<p\leq 1$.

• To find necessary and sufficient conditions in terms of modulus of
continuity of martingale Hardy spaces, for which subsequences of partial sums
of the one-dimensional Walsh-Fourier series convergence in $H_{p}(G)$ norm,
when $0<p\leq 1$.

• To find estimation of convergence and divergence of the subsequences of
Fejér means of the one-dimensional Walsh-Fourier series on the martingale
Hardy spaces $H_{p}(G)$, when $0<p\leq 1/2$.

• To find necessary and sufficient conditions in terms of modulus of
continuity of martingale Hardy spaces, for which subsequences of Fejér means
of the one-dimensional Walsh-Fourier series converge in $H_{p}(G)$ norm, when
$0<p\leq 1/2$.

• To prove strong convergence of one-dimensional Fejér means with respect to
Walsh system on the martingale Hardy spaces $H_{p}(G)$, when $0<p\leq 1/2$.

• To prove strong convergence of diagonal partial sums with respect to the
two-dimensional Walsh-Fourier series on the martingale Hardy spaces
$H_{p}(G^{2})$, when $0<p<1$.

• To prove strong convergence of Marcinkiewicz means with respect to the two-
dimensional Walsh-Fourier series in $H_{2/3}(G^{2})$ norm.

• To find necessary and sufficient conditions in terms of modulus of
continuity of Hardy spaces, for which Marcinkiewicz means of the two-
dimensional Walsh-Fourier series converge in $H_{2/3}(G^{2})$ norm.

## Contents

toc

Key words: Walsh group, Walsh system, $L_{p}$ space, weak-$L_{p}$ space,
modulus of continuity, Walsh-Fourier coefficients, Walsh-Fourier series,
partial sums, Lebesgue constants, Fejér means, Marcinkiewicz means, dyadic
martingale, the one-dimensional Hardy space, the two-dimensional Hardy space,
maximal operator, strong convergence.

Preface

The classical theory of Fourier series deals with decomposition of a function
into sinusoidal waves. Unlike these continuous waves the Vilenkin (Walsh)
functions are rectangular waves. Such waves have already been used frequently
in the theory of signal transmission, multiplexing, filtering, image
enhancement, codic theory, digital signal processing and pattern recognition.
The development of the theory of Vilenkin-Fourier series has been strongly
influenced by the classical theory of trigonometric series. Because of this it
is inevitable to compare results of Vilenkin series to those on trigonometric
series. There are many similarities between these theories, but there exist
differences also. Much of these can be explained by modern abstract harmonic
analysis, which studies orthonormal systems from the point of view of the
structure of a topological group.

The problems studied in this PhD thesis are very important in Mathematical
Analysis and its applications. In particular, we consider convergence and
summability of partial sums, Fejér and Marcinkiewicz means with respect to the
one- and two-dimensional Walsh-Fourier series in the martingale Hardy spaces.
According to the problems considered in this PhD thesis, widely are used
methods of real analysis combined with methods of abstract and non-linear
harmonic analysis together with theory of approximation. Other research
methods include theory of function spaces. They involve techniques which have
been developed a great deal during the last three decades.

This PhD thesis consists the following chapters:

• Preliminaries

• Partial sums with respect to the one-dimensional Walsh-Fourier series on the
martingale Hardy spaces

• Fejér means means with respect to the one-dimensional Walsh-Fourier series
on the martingale Hardy spaces

• Convergence and summability of partial sums with respect to the two-
dimensional Walsh-Fourier series on the martingale Hardy spaces

In Chapter 1 we first present some classical results and well-known facts,
which are very important in the theory of Fourier analysis and is also very
crucial for the further investigation of problems considered in this thesis.
Moreover, there are presented results which are proved in the next chapters of
this thesis and is emphasized the actuality of them, there is also shown some
connections with known results.

In Chapter 2 we first define Walsh group and system, which are important to
develop theory of harmonic analysis on locally compact Abelian groups. We
consider some expressions and estimations of Lebesgue constants and Dirichlet
kernels, give basic definitions and notation of the theory of martingale Hardy
spaces and fundamental theorems which are very important to prove main results
of this thesis. We also construct martingales which we use to prove sharpness
of our positive results. Next, we find rate of convergence and divergence of
the subsequences of partial sums with respect to the one-dimensional Walsh-
Fourier series on the martingale Hardy spaces for $0<p\leq 1$. Finally, we
apply these results to find necessary and sufficient conditions for the
modulus of continuity which provide norm convergence of subsequences of the
partial sums in the martingale Hardy spaces, for $0<p\leq 1$.

In Chapter 3 we investigate Fejér means with respect to the one-dimensional
Walsh-Fourier series. First, we consider some expressions and estimations of
Fejér kernels and find rate of convergence and divergence of the subsequences
of Fejér means in the martingale Hardy spaces for $0<p\leq 1/2$. After that,
we apply these results to find necessary and sufficient conditions for the
modulus of continuity, which provide convergence of subsequences of Fejér
means in the martingale Hardy spaces. Finally, we prove some new strong
convergence theorems of Fejér means, for $0<p\leq 1/2$. We also prove
sharpness of all our main results in this Chapter.

In Chapter 4 we investigate basic definitions and notation of partial sums and
Marcinkiewicz means with respect to the two-dimensional Walsh-Fourier series.
First, we consider some expressions and estimations of Marcinkiewicz kernels,
give basic definitions and notation of the theory of the two-dimensional
martingale Hardy spaces and fundamental theorems which are very important to
prove main results of this thesis. Next, we present and prove strong
convergence results of diagonal partial sums with respect to the two-
dimensional Walsh-Fourier series in the martingale Hardy spaces for $0<p<1$.
Moreover, we consider strong convergence results of Marcinkiewicz means with
respect to the two-dimensional Walsh-Fourier series for $p=2/3$ and find
necessary and sufficient conditions for the modulus of continuity, which
provide convergence in $H_{2/3}(G^{2})$ norm of Marcinkiewicz means.

This PhD thesis is written as a monograph based on the following publications:

[1] G. Tephnadze, Strong convergence of two-dimensional Walsh-Fourier series,
Ukr. Math. J., 65, (6), (2013), 822-834.

[2] G. Tephnadze, Strong convergence theorems of Walsh-Fejér means, Acta Math.
Hung., 142 (1) (2014), 244–259.

[3] K. Nagy and G. Tephnadze, Approximation by Walsh-Marcinkiewicz means on
the Hardy space $H_{2/3}$, Kyoto J. Math., 54 (3), (2014), 641-652.

[4] G. Tephnadze, On the partial sums of Walsh-Fourier series, Colloquium
Mathematicum, 141, 2 (2015), 227-242.

[5] G. Tephnadze, On the convergence of Fejér means of Walsh-Fourier series in
the space $H_{p}$, J. Contemp. Math. Anal., 51, 2 (2016), 51-63.

[6] K. Nagy and G. Tephnadze, Strong convergence theorem for Walsh-Marcin-
kiewicz means, Math. Inequal. Appl., 19, 1 (2016), 185–195.

##  1 Preliminaries

It is well-known that (for details see e.g. [30] and [47]) for every $p>1$
there exists an absolute constant $c_{p}$, depending only on $p$, such that

$\left\|S_{n}f\right\|_{p}\leq c_{p}\left\|f\right\|_{p},\text{ \ when \
}p>1\text{ \ and \ }f\in H_{1}({G}).$

Moreover, Watari [76] (see also Gosselin [31] and Young [85]) proved that
there exists an absolute constant $c$ such that, for $n=1,2,...,$

$\displaystyle\lambda\mu\left(|S_{n}f|>\lambda\right)$ $\displaystyle\leq$
$\displaystyle c\left\|f\right\|_{1},\ \ \ f\in L_{1}(G_{m}),\ \ \lambda>0.$

On the other hand, it is also well-known that (for details see e.g. [1] and
[47]) Walsh system is not Schauder basis in $L_{1}({G})$ space. Moreover,
there exists function $f\in H_{1}({G})$, such that partial sums with respect
to Walsh system are not uniforml5y bounded in $L_{1}({G})$.

By applying Lebesgue constants

$L(n):=\left\|D_{n}\right\|_{1}$

we easily obtain that (for details see e.g. [2] and [47]) subsequences of
partial sums $S_{n_{k}}f$ with respect to Walsh system converge to $f$ in
$L_{1}$ norm if and only if

$\sup_{k\in\mathbb{N}}L(n_{k})\leq c<\infty.$ (1.1)

Since $n$-th Lebesgue constant with respect to Walsh system, where

$n=\sum_{j=0}^{\infty}n_{j}2^{j},(n_{j}\in Z_{2})$

can be estimated by variation of natural number

$V\left(n\right)=n_{0}+\overset{\infty}{\underset{k=1}{\sum}}\left|n_{k}-n_{k-1}\right|$

and it is also well known that (for details see e.g. [12] and [47]) the
following two-sided estimate is true

$\frac{1}{8}V\left(n\right)\leq L(n)\leq V\left(n\right)$

to obtain convergence of subsequences of partial sums $S_{n_{k}}f$ with
respect to Walsh system of $f\in L_{1}$ in $f\in L_{1}$-norm. Condition (1.1)
can be replaced by

$\sup_{k\in\mathbb{N}}V(n_{k})\leq c<\infty$

It follows that (for details see e.g. [47] and [78]) subsequence of partial
sums $S_{2^{n}}$ are bounded from $H_{p}(G)$ to $H_{p}(G)$ for every $p>0$,
from which we obtain that

$\left\|S_{2^{n}}f-f\right\|_{H_{p}(G)}\rightarrow 0,\text{ \ as \
}n\rightarrow\infty,$ (1.2)

On the other hand, (see e.g. [61]) there exist a martingale $f\in H_{p}(G)$
$\left(0<p<1\right),$ such that

$\underset{n\in\mathbb{N}}{\sup}\left\|S_{2^{n}+1}f\right\|_{weak-
L_{p}(G)}=\infty.$

The main reason of divergence of subsequence $S_{{2^{n}}+1}f$ of partial sums
it that (for details see [62]) Fourier coefficients of $f\in H_{p}(G)$ are not
uniformly bounded when $0<p<1$.

When $0<p<1$ in [74] was investigated boundedness of subsequences of partial
sums with respect to Walsh system from $H_{p}(G)$ to $H_{p}(G)$. In
particular, the following result is true:

Theorem T1. Let $0<p<1$ and $f\in H_{p}(G)$. Then there exists a absolute
constant $c_{p},$ depending only on $p$, such that

$\left\|S_{m_{k}}f\right\|_{H_{p}(G)}\leq c_{p}\left\|f\right\|_{H_{p}(G)}$

if and only if the following condition holds

$\underset{k\in\mathbb{N}}{\sup}d\left(m_{k}\right)<c<\infty,$ (1.3)

where

$d\left(m_{k}\right):=\left|m_{k}\right|-\left\langle m_{k}\right\rangle.$

In particular, Theorem T1 immediately follows:

Theorem T2. Let $p>0$ and $f\in H_{p}(G)$. Then there exists a absolute
constant $c_{p},$ depending only on $p$, such that

$\left\|S_{2^{n}}f\right\|_{H_{p}(G)}\leq c_{p}\left\|f\right\|_{H_{p}(G)}$

and

$\left\|S_{2^{n}+2^{n-1}}f\right\|_{H_{p}(G)}\leq
c_{p}\left\|f\right\|_{H_{p}(G)}.$

On the other hand, we have the following result:

Theorem T3. Let $p>0$. Then there exists a martingale $f\in H_{p}(G)$, such
that

$\sup_{n\in\mathbb{N}}\left\|S_{2^{n}+1}f\right\|_{{}_{H_{p}(G)}}=\infty.$

Taking into account these results it is interesting to find behaviour of rate
of divergence of subsequences of partials sums with respect to Walsh system of
martingale $f\in H_{p}(G)$ in the martingale Hardy spaces $H_{p}(G)$.

In the second chapter of this thesis (see also [63]) we investigate ebove
mentioned problem. For $0<p<1$ we have the following result:

Theorem 2.10. Let $f\in H_{p}(G)$. Then there exists a absolute constant
$c_{p},$ depending only on $p$, such that the following inequality is true

$\text{ }\left\|S_{n}f\right\|_{H_{p}(G)}\leq
c_{p}2^{d\left(n\right)\left(1/p-1\right)}\left\|f\right\|_{H_{p}(G)}.$ (1.4)

On the other hand, if $0<p<1,$ $\left\\{m_{k}:\text{ }k\geq 0\right\\}$ be
increasing subsequence of natural numbers, such that

$\sup_{k\in\mathbb{N}}d\left(m_{k}\right)=\infty$ (1.5)

and $\Phi:\mathbb{N}_{+}\rightarrow[1,\infty)$ be non-decreasing function
satisfying the condition

$\overline{\underset{k\rightarrow\infty}{\lim}}\frac{2^{d\left(m_{k}\right)\left(1/p-1\right)}}{\Phi\left(m_{k}\right)}=\infty,$

then there exists a martingale $f\in H_{p}(G),$ such that

$\underset{k\in\mathbb{N}}{\sup}\left\|\frac{S_{m_{k}}f}{\Phi\left(m_{k}\right)}\right\|_{weak-
L_{p}(G)}=\infty.$

Theorem 2.11 easily follows the following corollary:

Corollary 2.11. Let $0<p<1$ and $f\in H_{p}(G)$. Then there exists a absolute
constant $c_{p},$ depending only on $p$, such that

$\text{ }\left\|S_{n}f\right\|_{H_{p}(G)}\leq
c_{p}\left(n\mu\left\\{\text{supp}\left(D_{n}\right)\right\\}\right)^{1/p-1}\left\|f\right\|_{H_{p}(G)}.$

On the other hand, if $0<p<1$ and $\left\\{m_{k}:\text{ }k\geq 0\right\\}$ be
increasing sequence of natural numbers, such that

$\sup_{k\in\mathbb{N}}m_{k}\mu\left\\{\text{supp}\left(D_{m_{k}}\right)\right\\}=\infty$

and $\Phi:\mathbb{N}_{+}\rightarrow[1,\infty)$ be non-decreasing function
satisfying the condition

$\overline{\underset{k\rightarrow\infty}{\lim}}\frac{\left(m_{k}\mu\left\\{\text{supp}\left(D_{m_{k}}\right)\right\\}\right)^{1/p-1}}{\Phi\left(m_{k}\right)}=\infty,$

then there exists a martingale $f\in H_{p}(G),$ such that

$\underset{k\in\mathbb{N}}{\sup}\left\|\frac{S_{m_{k}}f}{\Phi\left(m_{k}\right)}\right\|_{weak-
L_{p}(G)}=\infty.$

In particular, we also get the proofs of Theorem T1 and Theorem T2.

In the second chapter of this thesis we also investigate case $p=1$. In this
case the following result is true:

Theorem 2.12. Let $n\in\mathbb{N}_{+}$ and $f\in H_{1}(G).$ Then there exists
a absolute constant $c,$ such that

$\left\|S_{n}f\right\|_{H_{1}(G)}\leq
cV\left(n\right)\left\|f\right\|_{H_{1}(G)}.$ (1.6)

Moreover, if $\left\\{m_{k}:\text{ }k\geq 0\right\\}$ be increasing sequence
of natural numbers $\mathbb{N}_{+},$ such that

$\sup_{k\in\mathbb{N}}V\left(m_{k}\right)=\infty$

and $\Phi:\mathbb{N}_{+}\rightarrow[1,\infty)$ be non-decreasing function
satisfying the condition

$\overline{\underset{k\rightarrow\infty}{\lim}}\frac{V\left(m_{k}\right)}{\Phi\left(m_{k}\right)}=\infty.$

Then there exists a martingale $f\in H_{1}(G),$ such that

$\underset{{k\in\mathbb{N}}}{\sup}\left\|\frac{S_{m_{k}}f}{\Phi\left(m_{k}\right)}\right\|_{1}=\infty.$

When $0<p<1$ in [74] was proved boundedness of maximal operators of
subsequences of partial sums from $H_{p}(G)$ to $L_{p}(G)$. In particular, the
following is true:

Theorem T4. Let $0<p<1$ and $f\in H_{p}(G)$. Then the maximal operator

$\underset{k\in\mathbb{N}}{\sup}|S_{m_{k}}f|$

is bounded from $H_{p}(G)$ to $L_{p}(G)$, if and only if condition (1.3) is
fulfilled.

In the special cases we obtain that the following is true:

Theorem T5. Let $p>0$ and $f\in H_{p}(G)$. Then there exists an absolute
constant $c_{p},$ depending only on $p$, such that

$\left\|\sup_{n\in\mathbb{N}}|S_{2^{n}}f|\right\|_{p}\leq
c_{p}\left\|f\right\|_{H_{p}(G)}$ (1.7)

and

$\left\|\sup_{n\in\mathbb{N}}|S_{2^{n}+2^{n-1}}f|\right\|_{p}\leq
c_{p}\left\|f\right\|_{H_{p}(G)}.$

On the other hand we have the following result:

Theorem T6. Let $p>0$. Then there exists a martingale $f\in H_{p}(G)$, such
that

$\left\|\sup_{n\in\mathbb{N}}|S_{2^{n}+1}f|\right\|_{p}=\infty.$

Above mentioned condition (1.3) is sufficient condition for the case $p=1$
also, but there exist subsequences which do not satisfy this condition, but
maximal operators of these subsequences of partial sums with respect to Walsh
system are not bounded from $H_{1}(G)$ to $L_{1}(G)$.

Such necessary and sufficient conditions which provides boundedness of maximal
operators of subsequences of partial sums with respect to Walsh system from
$H_{1}(G)$ to $L_{1}(G)$ is open problem.

In [62] and [74] was investigated boundedness of weighted maximal operators
from $H_{p}(G)$ to $L_{p}(G)$, when $0<p\leq 1$:

Theorem T7. Let $0<p\leq 1$. Then weighted maximal operator

$\overset{\sim}{S}_{p}^{\ast}f:=\underset{n\in\mathbb{N_{+}}}{\sup}\frac{\left|S_{n}f\right|}{\left(n+1\right)^{1/p-1}\log^{[p]}\left(n+1\right)}$

is bounded from $H_{p}(G)$ to $L_{p}(G)$, where $[p]$ denotes integer part of
$p$.

Moreover, for any non-decreasing function
$\varphi:\mathbb{N}_{+}\rightarrow[1,$ $\infty)$ satisfying the condition

$\overline{\lim_{n\rightarrow\infty}}\frac{\left(n+1\right)^{1/p-1}\log^{[p]}\left(n+1\right)}{\varphi\left(n+1\right)}=+\infty,$

there exists a martingale $f\in H_{p}(G)$ $(0<p\leq 1),$ such that

$\sup_{n\in\mathbb{N}}\left\|\frac{S_{n}f}{\varphi\left(n\right)}\right\|_{p}=\infty.$

According to negative result for weighted maximal operator of partial sums of
Walsh-Fourier series we immediately get the following result:

Theorem S1. There exists a martingale $f\in H_{p}(G)$, $(0<p\leq 1)$, such
that

$\underset{n\in\mathbb{N}}{\sup}\left\|S_{n}f\right\|_{p}=\infty.$

On the other hand, boundedness of weighted maximal operators immediately
follows the following estimation:

Theorem S2. Let $0<p\leq 1$. Then there exists a absolute constant $c_{p}$,
depending only on $p$, such that

$\left\|S_{n}f\right\|_{p}\leq
c_{p}{(n+1)^{1/p-1}\log^{[p]}(n+1)}\left\|f\right\|_{H_{p}(G)},\text{ \ for \
}0<p\leq 1,$

where $[p]$ denotes integer part of $p$.

By applying this inequality (see [60]) we find necessary and sufficient
conditions for martingale $f\in H_{p}(G)$ for which partial sums with respect
to Walsh system of martingale $f\in H_{p}(G)$ converge in $H_{p}(G)$ norm.

Theorem T8. Let $0<p\leq 1$, $[p]$ denotes integer part of $p$, $f\in
H_{p}(G)$ and

$\omega_{H_{p}(G)}\left(\frac{1}{2^{N}},f\right)=o\left(\frac{1}{2^{N(1/p-1)}N^{[p]}}\right),\text{
\ as \ }N\rightarrow\infty.$

Then

$\left\|S_{n}f-f\right\|_{p}\rightarrow 0,\text{ as }n\rightarrow\infty.$

Moreover, there exists a martingale $f\in H_{p}(G)$, where $0<p<1$, such that

$\omega_{H_{p}(G)}\left(\frac{1}{2^{N}},f\right)=O\left(\frac{1}{2^{N(1/p-1)}N^{[p]}}\right),\text{
\ as \ }N\rightarrow\infty$

and

$\left\|S_{n}f-f\right\|_{weak-L_{p}(G)}\nrightarrow 0,\,\,\,\text{ \ \ as \ \
}n\rightarrow\infty.$

By taking these results into account, it is interesting to find necessary and
sufficient conditions for modulus of continuity, such that subsequences of
partial sums with respect to Walsh system of martingale $f\in H_{p}(G)$
converge in $H_{p}(G)$ norm.

In the second chapter of this thesis (see also [63]) we investigate this
problem. By combining inequalities (1.4) and (1.6) we get the following
theorem:

Theorem 2.16. Let $2^{k}<n\leq 2^{k+1}.$ Then there exists an absolute
constant $c_{p},$ depending only on $p$, such that

$\left\|S_{n}f-f\right\|_{H_{p}(G)}\leq
c_{p}2^{d\left(n\right)\left(1/p-1\right)}\omega_{H_{p}(G)}\left(\frac{1}{2^{k}},f\right),\text{
\ \ \ }\left(0<p<1\right)$ (1.8)

and

$\left\|S_{n}f-f\right\|_{H_{1}(G)}\leq
c_{1}V\left(n\right)\omega_{H_{1}(G)}\left(\frac{1}{2^{k}},f\right)$ (1.9)

By applying inequality (1.8) the following result is proved in the second
chapter:

Theorem 2.17. Let $0<p<1,$ $f\in H_{p}(G)$ and $\\{m_{k}:k\geq 0\\}$ be
increasing sequence of natural number satisfying the condition

$\omega_{H_{p}(G)}\left(\frac{1}{2^{\left|m_{k}\right|}},f\right)=o\left(\frac{1}{2^{d\left(m_{k}\right)\left(1/p-1\right)}}\right)\text{
as \ }k\rightarrow\infty.$

Then

$\left\|S_{m_{k}}f-f\right\|_{H_{p}(G)}\rightarrow 0\text{ \ as \
}k\rightarrow\infty.$ (1.10)

On the other hand, if $\\{m_{k}:k\geq 0\\}$ be increasing sequence of natural
numbers satisfying the condition (1.5), then there exists a martingale $f\in
H_{p}(G)$ and subsequence $\\{\alpha_{k}:k\geq 0\\}\subset\\{m_{k}:k\geq
0\\},$ for which

$\omega_{H_{p}(G)}\left(\frac{1}{2^{\left|\alpha_{k}\right|}},f\right)=O\left(\frac{1}{2^{d\left(\alpha_{k}\right)\left(1/p-1\right)}}\right)\text{
as \ }k\rightarrow\infty\text{\ }$

and

$\limsup\limits_{k\rightarrow\infty}\left\|S_{\alpha_{k}}f-f\right\|_{weak-
L_{p}(G)}>c_{p}>0\,\,\,\text{as\thinspace\thinspace\thinspace}k\rightarrow\infty,$
(1.11)

where $c_{p}$ is an absolute constant depending only on $p$.

According to this theorem we immediately get that the following result is
true:

Corollary 2.18. Let $0<p<1,$ $f\in H_{p}(G)$ and $\\{m_{k}:k\geq 0\\}$ be
increasing sequence of natural number, satisfying the condition

$\omega_{H_{p}(G)}\left(\frac{1}{2^{\left|m_{k}\right|}},f\right)=o\left(\frac{1}{\left(m_{k}\mu\left(\text{supp}D_{m_{k}}\right)\right)^{1/p-1}}\right),\text{
\ as \ }k\rightarrow\infty.$

Then (1.10) holds.

On the other hand, if $\\{m_{k}:k\geq 0\\}$ be increasing sequence of natural
number, satisfying the condition

$\overline{\underset{k\rightarrow\infty}{\lim}}\frac{\left(m_{k}\mu\left\\{\text{supp}\left(D_{m_{k}}\right)\right\\}\right)^{1/p-1}}{\Phi\left(m_{k}\right)}=\infty,$

then there exists a martingale $f\in H_{p}(G)$ and subsequence
$\\{\alpha_{k}:k\geq 0\\}\subset\\{m_{k}:k\geq 0\\}$ such that

$\omega_{H_{p}(G)}\left(\frac{1}{2^{\left|\alpha_{k}\right|}},f\right)=O\left(\frac{1}{\left(\alpha_{k}\mu\left(\text{supp}D_{\alpha_{k}}\right)\right)^{1/p-1}}\right),\text{\
\ as \ }k\rightarrow\infty$

and (1.11) holds.

By applying (1.9) we prove that the following is true:

Theorem 2.19. Let $f\in H_{1}(G)$ and $\\{m_{k}:k\geq 0\\}$ be increasing
sequence of natural number, satisfying the condition

$\omega_{H_{1}(G)}\left(\frac{1}{2^{\left|m_{k}\right|}},f\right)=o\left(\frac{1}{V\left(m_{k}\right)}\right)\text{
\ as \ }k\rightarrow\infty.$

Then

$\left\|S_{m_{k}}f-f\right\|_{H_{1}(G)}\rightarrow 0\text{ \ as \
}k\rightarrow\infty.$

Moreover, if $\\{m_{k}:k\geq 0\\}$ be increasing sequence of natural number,
satisfying the condition (1.5), then there exists a martingale $f\in H_{1}(G)$
and subsequence $\\{\alpha_{k}:k\geq 0\\}\subset\\{m_{k}:k\geq 0\\}$ for which

$\omega_{H_{1}(G)}\left(\frac{1}{2^{\left|\alpha_{k}\right|}},f\right)=O\left(\frac{1}{V\left(\alpha_{k}\right)}\right)\text{
\ as \ }k\rightarrow\infty$

and

$\limsup\limits_{k\rightarrow\infty}\left\|S_{\alpha_{k}}f-f\right\|_{1}>c>0\text{
\ \ as \ \ }k\rightarrow\infty,$

where $c$ is an absolute constant.

By applying Theorem 2.17 and Theorem 2.19 we immediately get proof of Theorem
T8.

Weisz [79] consider convergence in norm of Fejér means of the one-dimensional
Walsh-Fourier and proved the following:

Theorem We1. Let $p>1/2$ and $f\in H_{p}(G)$. Then there exists a absolute
constant $c_{p}$, depending only on $p$, such that

$\left\|\sigma_{k}f\right\|_{H_{p}(G)}\leq c_{p}\left\|f\right\|_{H_{p}(G)}.$

Weisz (for details see e.g. [78]) also consider boundedness of subsequences of
Fejér means $\sigma_{2^{n}}$ of the one-dimensional Walsh-Fourier series from
$H_{p}(G)$ to $H_{p}(G)$ when $p>0$:

Theorem We2. Let $p>0$ and $f\in H_{p}(G)$. Then

$\left\|\sigma_{2^{k}}f-f\right\|_{H_{p}(G)}\rightarrow 0,\text{\ \ as \ \
}k\rightarrow\infty.$ (1.12)

On the other hand, in [56] was proved the following result:

Theorem T9. There exists a martingale $f\in H_{p}(G)$ $\left(0<p\leq
1/2\right)$ such that

$\underset{n\in\mathbb{N}}{\sup}\left\|\sigma_{2^{n}+1}f\right\|_{H_{p}(G)}=\infty.$

Goginava [27] (see also [44]) proved that the following result is true:

Theorem Gog1. Let $0<p\leq 1.$ Then the sequence of operators
$\left|\sigma_{2^{n}}f\right|$ are not bounded from $H_{p}(G)$ to $H_{p}(G)$.

When $0<p<1/2$ then in [45] was proved bondedness of subsequences of Fejér
means of the one-dimensional Walsh-Fourier from $H_{p}(G)$ to $H_{p}(G)$. In
particular, the following is true:

Theorem T10. Let $0<p<1/2$ and $f\in H_{p}(G)$. Then there exists a absolute
constant $c_{p}$, depending only on $p$, such that

$\left\|\sigma_{m_{k}}f\right\|_{H_{p}(G)}\leq
c_{p}\left\|f\right\|_{H_{p}(G)}$

estimation holds if and only if the condition (1.3) is fulfilled.

Theorem T10 immediately follows theorem of Weisz (see Theorem We2) and and
also interesting results:

Theorem T11. Let $p>0$ and $f\in H_{p}(G)$. Then there exists an absolute
constant $c_{p},$ depending only on $p$, such that

$\left\|\sigma_{2^{n}}f\right\|_{H_{p}(G)}\leq
c_{p}\left\|f\right\|_{H_{p}(G)}$

and

$\left\|\sigma_{2^{n}+2^{n-1}}f\right\|_{H_{p}(G)}\leq
c_{p}\left\|f\right\|_{H_{p}(G)}.$

On the other hand, we have the following result:

Theorem T12. Let $p>0$. Then there exists a martingale $f\in H_{p}(G)$, such
that

$\sup_{n\in\mathbb{N}}\left\|\sigma_{2^{n}+1}f\right\|_{{}_{H_{p}(G)}}=\infty.$

According to above mentioned results it is interesting to find rate of
divergence of subsequences $\sigma_{n_{k}}f$ of Fejér means of the one-
dimensional Walsh-Fourier series in the Hardy spaces $H_{p}(G)$.

In the third chapter of this thesis (see also [64]) we find rate of divergence
of subsequences of Fejér means of the one-dimensional Walsh-Fourier series on
the martingale Hardy spaces $H_{p}(G)$, when $0<p\leq 1/2$.

First, we consider case $p=1/2$:

Theorem 3.28. Let $n\in\mathbb{N}_{+}$ and $f\in H_{1/2}(G).$ Then there
exists an absolute constant $c,$ such that

$\left\|\sigma_{n}f\right\|_{H_{1/2}(G)}\leq
c{{V}^{2}}\left(n\right)\left\|f\right\|_{H_{1/2}(G)}.$ (1.13)

Moreover, if $\left\\{m_{k}:\text{ }k\geq 0\right\\}$ be increasing secuence
of natural numbers, such that

$\sup_{k\in\mathbb{N}}V\left(m_{k}\right)=\infty$

and $\Phi:\mathbb{N}_{+}\rightarrow[1,\infty]$ be non-decreasing function
satisfying the conditions

$\overline{\underset{k\rightarrow\infty}{\lim}}\frac{{V^{2}}\left(m_{k}\right)}{\Phi\left(m_{k}\right)}=\infty.$

then there exists a martingale $f\in H_{1/2}(G),$ such that

$\underset{{k\in\mathbb{N}}}{\sup}\left\|\frac{\sigma_{m_{k}}f}{\Phi\left(m_{k}\right)}\right\|_{1/2}=\infty.$

There was also considered case $0<p<1/2$ and was proved that the following is
true:

Theorem 3.29. Let $0<p<1/2$ and $f\in H_{p}(G)$. Then there exists an absolute
constant $c_{p},$ depending only on $p$ such that

$\text{ }\left\|\sigma_{n}f\right\|_{H_{p}(G)}\leq
c_{p}2^{d\left(n\right)\left(1/p-2\right)}\left\|f\right\|_{H_{p}(G)}.$ (1.14)

On the other hand, if $0<p<1/2,$ $\left\\{m_{k}:\text{ }k\geq 0\right\\}$ be
increasing sequence of natural numbers satisfying the condition (1.5) and
$\Phi:\mathbb{N}_{+}\rightarrow[1,\infty)$ be non-decreasing function such
that

$\overline{\underset{k\rightarrow\infty}{\lim}}\frac{2^{d\left(m_{k}\right)\left(1/p-2\right)}}{\Phi\left(m_{k}\right)}=\infty,$

then there exists a martingale $f\in H_{p}(G),$ such that

$\underset{k\in\mathbb{N}}{\sup}\left\|\frac{\sigma_{m_{k}}f}{\Phi\left(m_{k}\right)}\right\|_{weak-
L_{p}(G)}=\infty.$

From these results also follows proof of Theorem We2.

In 1975 Schipp [46] (see also [86]) proved that the maximal operator of Fejér
means $\sigma^{*}$ is of type weak-(1,1):

$\mu\left(\sigma^{*}f>\lambda\right)\leq\frac{c}{\lambda}\left\|f\right\|_{1},\text{
\qquad}\left(\lambda>0\right).$

By using Marcinkiewicz interpolation theorem it follows that $\sigma^{*}$ is
of strong type-$(p,p)$, when $p>1:$

$\left\|\sigma^{*}f\right\|_{p}\leq c\left\|f\right\|_{p},\text{
\qquad}\left(p>1\right).$

The boundedness does not hold for $p=1,$ but Fujji [15] (see also [84]) proved
that maximal operator of Fejér means is bounded from $H_{1}(G)$ to $L_{1}(G)$.
Weisz in [80] generalized result of Fujii and proved that maximal operator of
Fejér means is bounded from $H_{p}(G)$ to $L_{p}(G)$, when $p>1/2.$ Simon [48]
construct the counterexample, which shows that boundedness does not hold when
$0<p<1/2$. Goginava [21] (see also [9] and [10]) generalized this result for
$0<p\leq 1/2$ and proved that the following is true:

Theorem Gog2. There exists a martingale $f\in H_{p}(G)$ $\left(0<p\leq
1/2\right)$ such that

$\underset{n\in\mathbb{N}}{\sup}\left\|\sigma_{n}f\right\|_{p}=\infty.$

Weisz [81] (see also Goginava [23]) proved that the following is true:

Theorem We3. Let $f\in H_{1/2}(G)$. Then there exists an absolute constant
$c,$ such that

$\left\|\sigma^{*}f\right\|_{weak-L_{1/2}(G)}\leq
c\left\|f\right\|_{H_{1/2}(G)}.$

In [45] was considered boundedness of maximal operators of subsequences of
Fejér means of the one-dimensional Walsh-Fourier series from $H_{p}(G)$ to
$L_{p}(G)$ for $0<p<1/2$. In particular, the following is true:

Theorem T13. Let $0<p<1/2$ and $f\in H_{p}(G)$. Then the maximal operator

$\overset{\sim}{\sigma}^{*}f:=\underset{k\in\mathbb{N}}{\sup}|\sigma_{m_{k}}f|$

is bounded from $H_{p}(G)$ to $L_{p}(G)$ if and only if when condition (1.3)
is fulfilled.

As consequences the following results are true:

Theorem T14. Let $p>0$ and $f\in H_{p}(G)$. Then there exists an absolute
constant $c_{p}$ depending only on $p,$ such that

$\left\|\sup_{n\in\mathbb{N}}|\sigma_{2^{n}}f|\right\|_{p}\leq
c_{p}\left\|f\right\|_{H_{p}(G)}$ (1.15)

and

$\left\|\sup_{n\in\mathbb{N}}|\sigma_{2^{n}+2^{n-1}}f|\right\|_{p}\leq
c_{p}\left\|f\right\|_{H_{p}(G)}.$

On the other hand, we have the following negative result:

Theorem T15. Let $0<p<1/2.$ Then there exists a martingale $f\in H_{p}(G)$,
such that

$\left\|\sup_{n\in\mathbb{N}}|\sigma_{2^{n}+1}f|\right\|_{p}=\infty.$

above mentioned condition is sufficient for the case $p=1/2$ also, but there
exists subsequences, which do not satisfy condition (1.3), but maximal
operator of subsequences of Fejér means of the one-dimensional Walsh-Fourier
series are bounded from $H_{1/2}(G)$ to $L_{1/2}(G).$

However, it is open problem to find necessary and sufficient conditions on the
indexes, which provide boundedness of maximal operator of subsequences of
Fejér means of the one-dimensional Walsh-Fourier series from $H_{1/2}(G)$ to
$L_{1/2}(G)$.

In [22] and [56] (see also [43], [58], [29] and [55]) is proved that the
following is true:

Theorem GT1. Let $0<p\leq 1/2$ and $f\in H_{p}(G)$. Then the maximal operator

$\overset{\sim}{\sigma}_{p}^{\ast}f:=\underset{n\in\mathbb{N}}{\sup}\frac{\left|\sigma_{n}f\right|}{\left(n+1\right)^{1/p-2}\log^{2[1/2+p]}\left(n+1\right)}$

is bounded from $H_{p}(G)$ to $L_{p}(G)$.

Moreover, for any nondecreasing function
$\varphi:\mathbb{N}_{+}\rightarrow[1,$ $\infty)$ satisfying the condition

$\overline{\lim_{n\rightarrow\infty}}\frac{\left(n+1\right)^{1/p-2}\log^{2[1/2+p]}\left(n+1\right)}{\varphi\left(n\right)}=+\infty,$

there exists a martingale $f\in H_{p}(G)$, $(0<p<1/2)$ such that

$\sup_{n\in\mathbb{N}}\left\|\frac{\sigma_{n}f}{\varphi\left(n\right)}\right\|_{p}=\infty.$

From the divergence of weighted maximal operators we immediately get that
there exists a martingale $f\in H_{p}(G)$ $(0<p\leq 1/2)$, such that

$\underset{n\in\mathbb{N}}{\sup}\left\|\sigma_{n}f\right\|_{p}=\infty.$

and from the boundedness results of weighted maximal operators we immediately
get that for any $f\in H_{p}(G)$ there exists an absolute constant $c_{p}$,
such that the following inequality holds true:

$\left\|\sigma_{n}f\right\|_{p}\leq
c_{p}{n^{1/p-2}\log^{2[1/2+p]}\left(n+1\right)}\left\|f\right\|_{H_{p}(G)},\text{
\ as \ }0<p\leq 1/2.$ (1.16)

By applying inequality (1.16) in [60] was found necessary and sufficient
conditions for modulus of continuity of martingale $f\in H_{p}(G)$, for which
Fejér means of the one-dimensional Walsh-Fourier series converge in $H_{p}(G)$
norm.

Theorem T16. Let $0<p\leq 1/2$, $f\in H_{p}(G)$ and

$\omega_{H_{p}(G)}\left(\frac{1}{2^{N}},f\right)=o\left(\frac{1}{2^{N(1/p-2)}N^{2[1/2+p]}}\right),\text{\
as \ }N\rightarrow\infty.$

Then

$\left\|\sigma_{n}f-f\right\|_{p}\rightarrow 0,\text{ as }n\rightarrow\infty.$

Moreover, there exists a martingale $f\in H_{p}(G)$, for which

$\omega_{H_{1/2}(G)}\left(\frac{1}{2^{N}},f\right)=O\left(\frac{1}{2^{N(1/p-2)}N^{2[1/2+p]}}\right),\text{
\ as \ }N\rightarrow\infty$

and

$\left\|\sigma_{n}f-f\right\|_{p}\nrightarrow 0,\,\,\,\text{as\ \
}n\rightarrow\infty.$

According above mentioned results, it is interesting to find necessary and
sufficient conditions for the modulus of continuity, for which subsequences
$\sigma_{n_{k}}f$ of Fejér means of the one-dimensional Walsh-Fourier series
converge in $H_{p}(G)$ norm.

In the third chapter of this thesis we find necessary and sufficient
conditions for the modulus of continuity, for which subsequences
$\sigma_{n_{k}}f$ of Fejér means of the one-dimensional Walsh-Fourier series
converge in $H_{p}(G)$ norm (see also [64]).

By applying inequality (1.13) for the case $p=1/2$ the following necessary and
sufficient conditions are found:

Theorem 3.33. Let $f\in H_{1/2}(G)$ and $\\{m_{k}:k\geq 0\\}$ be increasing
sequence of natural numbers, such that

$\omega_{H_{1/2}(G)}\left(\frac{1}{2^{\left|m_{k}\right|}},f\right)=o\left(\frac{1}{{V^{2}}\left(m_{k}\right)}\right)\text{
\ as \ }k\rightarrow\infty.$

Then

$\left\|\sigma_{m_{k}}f-f\right\|_{H_{1/2}(G)}\rightarrow 0\text{ \ as \
}k\rightarrow\infty.$

Moreover, if $\\{m_{k}:k\geq 0\\}$ be increasing sequence of natural numbers,
such that (1.5) holds true, then there exists a martingale $f\in H_{1/2}(G)$
and subsequence $\\{\alpha_{k}:k\geq 0\\}\subset\\{m_{k}:k\geq 0\\}$ such that

$\omega_{H_{1/2}(G)}\left(\frac{1}{2^{\left|\alpha_{k}\right|}},f\right)=O\left(\frac{1}{{V^{2}}\left(\alpha_{k}\right)}\right)\text{
\ as\ }k\rightarrow\infty$

and

$\limsup\limits_{k\rightarrow\infty}\left\|\sigma_{\alpha_{k}}f-f\right\|_{1/2}>c>0\,\,\,\text{as\thinspace\thinspace\thinspace}k\rightarrow\infty,$

wher $c$ is an absolute constant.

By applying inequality (1.14) we also investigate case $0<p<1/2$. In the third
chapter of this thesis we prove that the following is true:

Theorem 3.34. Let $0<p<1/2,$ $f\in H_{p}(G)$ and $\\{m_{k}:k\geq 0\\}$ be
increasing sequence of natural numbers, such that

$\omega_{H_{p}(G)}\left(\frac{1}{2^{\left|m_{k}\right|}},f\right)=o\left(\frac{1}{2^{d\left(m_{k}\right)\left(1/p-2\right)}}\right),\text{
as \ }k\rightarrow\infty.$

Then

$\left\|\sigma_{m_{k}}f-f\right\|_{H_{p}(G)}\rightarrow 0,\text{ \ as \
}k\rightarrow\infty.$

On the other hand, if $\\{m_{k}:k\geq 0\\}$ be increasing sequence of natural
numbers satisfying the condition (1.5), then there exists a martingale $f\in
H_{p}(G)$ and subsequence $\\{\alpha_{k}:k\geq 0\\}\subset\\{m_{k}:k\geq
0\\},$ for which

$\omega_{H_{p}(G)}\left(\frac{1}{2^{\left|\alpha_{k}\right|}},f\right)=O\left(\frac{1}{2^{d\left(\alpha_{k}\right)\left(1/p-2\right)}}\right),\text{
as \ }k\rightarrow\infty$

and

$\limsup\limits_{k\rightarrow\infty}\left\|\sigma_{\alpha_{k}}f-f\right\|_{weak-
L_{p}(G)}>c_{p}>0,\text{\quad as\quad}k\rightarrow\infty,$

where $c_{p}$ is constant depending only on $p$.

However, Simon in [49] and [51] (see also [13, 52]) consider strong
convergence theorems of the one-dimensional Walsh-Fouriere series and proved
the following:

Theorem Si1. Let $0<p\leq 1$ and $f\in H_{1}\left(G\right)$. Then there exists
an absolute constant $c_{p}$ depending only on $p,$ such that the following
inequality is true:

$\frac{1}{\log^{[p]}n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|S_{k}f\right\|_{H_{p}(G)}}{k^{2-p}}\leq
c_{p}\left\|f\right\|_{H_{p}(G)},$

Analigical result for trigonometric system was proved in [53], for unbounded
Walsh systems in [17].

In [57] was proved that the following is true:

Theorem T17. for any $0<p<1$ and non-decreasing function
$\varphi:\mathbb{N}_{+}\rightarrow[1,$ $\infty)$ satisfying the condition

$\overline{\lim_{n\rightarrow\infty}}\frac{{n^{2-p}}}{\varphi\left(n\right)}=+\infty,$

there exists a martingale $f\in H_{p}(G)$, such that

$\text{
}\underset{k=1}{\overset{\infty}{\sum}}\frac{\left\|S_{k}f\right\|_{weak-
L_{p}(G)}^{p}}{\varphi(k)}=\infty,\text{\qquad}(0<p<1).$

Theorem Si1 follows that if $f\in H_{1}(G)$ then the following equalities are
true:

$\underset{n\rightarrow\infty}{\lim}\frac{1}{\log
n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|S_{k}f-f\right\|_{1}}{k}=0$

and

$\underset{n\rightarrow\infty}{\lim}\frac{1}{\log
n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|S_{k}f\right\|_{1}}{k}=\left\|f\right\|_{H_{1}(G)}.$

When $0<p<1$ and $f\in H_{p}(G),$ then Theorem Si1 follows that there exists
an absolute constant $c_{p}$ depending only on $p,$ such that

$\frac{1}{n^{1/2-p/2}}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|S_{k}f\right\|_{H_{p}(G)}^{p}}{k^{3/2-p/2}}\leq
c_{p}\left\|f\right\|_{H_{p}(G)}^{p}.$

Moreover,

$\frac{1}{n^{1/2-p/2}}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|S_{k}f-f\right\|_{H_{p}(G)}^{p}}{k^{{}^{3/2-p}}}=0.$

It follows the following equality

$\frac{1}{n^{1/2-p/2}}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|S_{k}f\right\|_{H_{p}(G)}^{p}}{k^{3/2-p/2}}=\left\|f\right\|_{H_{p}(G)}^{p}.$

In the third chapter of this thesis we consider strong convergence results of
Fejér means of the one-dimensional Walsh-Fourier series. According to Theorem
We1 and Theorem Gog2 we only have to consider case $0<p\leq 1/2$ (for details
see [59], see also [4], [5], [7], [8], [12]):

Theorem 3.37. Let $0<p\leq 1/2$ and $f\in H_{p}(G)$. Then there exists an
absolute constant $c_{p}$ depending only on $p,$ such that

$\frac{1}{\log^{\left[1/2+p\right]}n}\overset{n}{\underset{m=1}{\sum}}\frac{\left\|\sigma_{m}f\right\|_{H_{p}(G)}^{p}}{m^{2-2p}}\leq
c_{p}\left\|f\right\|_{H_{p}(G)}^{p}.$

Moreover, let $0<p<1/2$ and $\Phi:\mathbb{N}_{+}\rightarrow[1,$ $\infty)$ be
non-decreasing, non-negative function, such that
$\Phi\left(n\right)\uparrow\infty$ and

$\overline{\underset{k\rightarrow\infty}{\lim}}\frac{k^{2-2p}}{\Phi\left({k}\right)}=\infty.$

Then there exists a martingale $f\in H_{p}(G),$ such that

$\underset{m=1}{\overset{\infty}{\sum}}\frac{\left\|\sigma_{m}f\right\|_{weak-
L_{p}(G)}^{p}}{\Phi\left(m\right)}=\infty.$

When $p=1/2$ is was also proved that the following is true:

Theorem 3.38. Let $f\in H_{1/2}(G).$ Then

$\underset{n\in\mathbf{\mathbb{N}}_{+}}{\sup}\underset{\left\|f\right\|_{H_{p}(G)}\leq
1}{\sup}\frac{1}{n}\underset{m=1}{\overset{n}{\sum}}\left\|\sigma_{m}f\right\|_{1/2}^{1/2}=\infty.$

Theorem 3.37 follows that if $f\in H_{1/2}(G)$ then the following equalities
are true:

$\lim_{n\rightarrow\infty}\frac{1}{\log
n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|\sigma_{k}f-f\right\|_{H_{1/2}(G)}^{1/2}}{k}=0$

and

$\lim_{n\rightarrow\infty}\frac{1}{\log
n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|\sigma_{k}f\right\|_{H_{1/2}(G)}^{1/2}}{k}=\left\|f\right\|_{H_{1/2}(G)}^{1/2}.$

When $0<p<1/2$ and $f\in H_{p}(G),$ then Theorem 3.37 follows that there
exists an absolute constant $c_{p}$ depending only on $p,$ such that

$\frac{1}{n^{1/2-p}}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|\sigma_{k}f\right\|_{H_{p}(G)}^{p}}{k^{3/2-p}}\leq
c_{p}\left\|f\right\|_{H_{p}(G)}^{p}.$

Moreover,

$\frac{1}{n^{1/2-p}}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|\sigma_{k}f-f\right\|_{H_{p}(G)}^{p}}{k^{{}^{3/2-p}}}=0.$

It follows that

$\frac{1}{n^{1/2-p}}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|\sigma_{k}f\right\|_{H_{p}(G)}^{p}}{k^{3/2-p}}=\left\|f\right\|_{H_{p}(G)}^{p}.$

For the two-dimensional case (for details see e.g. [47] and [78]) the
following is true:

Theorem S3. Let $p>0$ and $f\in H_{p}({G}^{2})$. Then

$\left\|S_{2^{n},2^{n}}f-f\right\|_{p}\rightarrow 0,\text{ \ as \
}n\rightarrow\infty.$ (1.17)

Moreover,

Theorem S4. Let $p>0$ and $f\in H_{p}({G}^{2})$. Then there exists an absolute
constant $c_{p}$ depending only on $p,$ such that

$\left\|\sup_{n\in\mathbb{N}}\left|S_{2^{n},2^{n}}f\right|\right\|_{p}\leq
c_{p}\left\|f\right\|_{H_{p}({G}^{2})},$ (1.18)

By applying Theorem S4 we can conclude that the following holds true (for
details see e.g. [47] and [78]):

Theorem S5. Let $p>0$ and $f\in H_{p}({G}^{2})$. Then there exists an absolute
constant $c_{p}$ depending only on $p,$ such that

$\left\|S_{2^{n},2^{n}}f\right\|_{H_{p}({G}^{2})}\leq
c_{p}\left\|f\right\|_{H_{p}({G}^{2})}.$ (1.19)

On the other hand, (see [61]) the following is true:

Theorem T18. Let $0<p\leq 1.$ Then there exists a martingale $f\in
H_{p}({G}^{2})$ such that

$\underset{n\in\mathbb{N}}{\sup}\left\|S_{n,n}f\right\|_{p}=\infty.$

However, for the two-dimensional case Weisz [77] proved the following:

Theorem We4. Let $\alpha\geq 0$ and $f\in H_{p}({G}^{2}).$ Then there exists
an absolute constant $c_{p}$ depending only on $p,$ such that

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

where $0<p\leq 1$ and $\left[p\right]$ denotes integer part of real number
$p$.

Moreover, sharpness of of the rates of weights are proved [72] was proved that
rate of the ${\left(kl\right)^{2-p}}$ is sharp.

Goginava and Gogoladze in [28] generalized this result in the case when
$\alpha=0$:

Theorem GG1. Let $f\in H_{1}({G}^{2})$. Then there exists an absolute constant
$c$, such that

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

In [65] was proved that rate of the weights
$\left\\{n\log^{2}n\right\\}_{n=1}^{\infty}$ is sharp. The following is true:

Theorem T19. Let $\Phi:\mathbb{N}\rightarrow[1,$ $\infty)$ be non-decreasing
function satisfying the condition
$\lim_{n\rightarrow\infty}\Phi\left(n\right)=+\infty.$ Then

$\underset{\left\|f\right\|_{H_{1}({G}^{2})}\leq
1}{\sup}\underset{n=1}{\overset{\infty}{\sum}}\frac{\left\|S_{n,n}f\right\|_{1}\Phi\left(n\right)}{n\log^{2}\left(n+1\right)}=\infty.$

Theorem GG1 follows that if $f\in H_{1}({G}^{2}),$ then

$\frac{1}{\log^{1/2}n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|S_{k,k}f\right\|_{H_{1}({G}^{2})}}{k\log^{3/2}k}\leq
c\left\|f\right\|_{H_{1}({G}^{2})},\text{ }$

Moreover,

$\lim_{n\rightarrow\infty}\frac{1}{\log^{1/2}n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|S_{k,k}f-f\right\|_{H_{1}({G}^{2})}^{2/3}}{k\log^{3/2}k}=0$

It follows the following equality

$\lim_{n\rightarrow\infty}\frac{1}{\log^{1/2}n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|S_{k,k}f\right\|_{H_{2/3}({G}^{2})}^{2/3}}{k{\log^{3/2}k}}=\left\|f\right\|_{H_{2/3}({G}^{2})}^{2/3}.$

In the fourth chapter (see also [66]) of this thesis we consider strong
convergence of Marcinkiewicz means with respect to the two-dimensional partial
sums of Walsh-Fourier series when $0<p<1$:

Theorem 4.49 Let $0<p<1$ and $f\in H_{p}({G}^{2}).$ Then there exists an
absolute constant $c_{p}$ depending only on $p,$ such that

$\sum\limits_{n=1}^{\infty}\frac{\left\|S_{n,n}f\right\|_{p}^{p}}{n^{3-2p}}\leq
c_{p}\left\|f\right\|_{H_{p}({G}^{2})}^{p}.$ (1.20)

Moreover, if $0<p<1$ and $\Phi:\mathbb{N}\rightarrow[1,$ $\infty)$ be non-
decreasing function satifying condition
$\underset{n\rightarrow\infty}{\lim}\Phi\left(n\right)=+\infty$, then there
exists martingale $f\in H_{p}({G}^{2})$ such that

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

Theorem 4.49 follows that, if $0<p<1$ and $f\in H_{p}({G}^{2}),$ then there
exists an absolute constant $c_{p}$ depending only on $p,$ such that

$\frac{1}{n^{1/2-p}}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|S_{k,k}f\right\|_{H_{p}({G}^{2})}^{p}}{k^{3/2-p}}\leq
c_{p}\left\|f\right\|_{H_{p}({G}^{2})}^{p},$

Moreover,

$\frac{1}{n^{1/2-p}}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|S_{k,k}f-f\right\|_{H_{p}({G}^{2})}^{p}}{k^{3/2-p}}=0.$

It follows the following equality

$\frac{1}{n^{1/2-p}}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|S_{k,k}f\right\|_{H_{p}(G)}^{p}}{k^{3/2-p}}=\left\|f\right\|_{H_{p}(G)}^{p}.$

Weisz (for details see e.g. [78]) consider Marcinkiewicz means with respect to
the two-dimensional partial sums of Walsh-Fourier series and proved the
following:

Theorem We5. Let $p>2/3$ and $f\in H_{p}({G}^{2})$. Then there exists an
absolute constant $c_{p}$ depending only on $p,$ such that

$\left\|\mathscr{M}_{n}f-f\right\|_{H_{p}({G}^{2})}\rightarrow 0,\text{ \ \
}n\rightarrow\infty.$

Goginava [26] proved that the following is true:

Theorem Gog2. Let $0<p\leq 2/3$. Then there exists a martingale $f\in
H_{p}({G}^{2})$, such that

$\underset{n\in\mathbb{N}}{\sup}\left\|\mathscr{M}_{n}f\right\|_{H_{p}({G}^{2})}=\infty.$

Goginava in [24] consider subsequence $\mathscr{M}_{2^{n}}$ of Marcinkiewicz
means with respect to the two-dimensional partial sums of Walsh-Fourier series
and proved that the following is true:

Theorem Gog3. Let $p>1/2$ and $f\in H_{p}({G}^{2})$. Then

$\left\|\mathscr{M}_{2^{k}}f-f\right\|_{H_{p}({G}^{2})}\rightarrow 0,\text{ \
as \ }k\rightarrow\infty.$ (1.21)

Moreover, there exists a martingale $f\in H_{p}({G}^{2}),$ $\text{ }(0<p\leq
1/2)$ such that

$\underset{n\in\mathbb{N}}{\sup}\left\|\mathscr{M}_{2^{n}}f\right\|_{H_{p}({G}^{2})}=\infty.$

In [37] was investigated strong convergence theorems of Marcinkiewicz means
with respect to the two-dimensional partial sums of Walsh-Fourier series when
$0<p<2/3$:

Theorem NT1. Let $0<p<2/3$ and $f\in H_{p}({G}^{2})$. Then there exists an
absolute constant $c_{p}$ depending only on $p,$ such that

$\sum_{m=1}^{\infty}\frac{\left\|\mathscr{M}_{m}f\right\|_{H_{p}({G}^{2})}^{p}}{m^{3-3p}}\leq
c_{p}\left\|f\right\|_{H_{p}({G}^{2})}^{p}.$

Moreover, if $0<p<2/3$ and $\Phi:\mathbb{N}_{+}\rightarrow[1,$ $\infty)$ be
non-decreasing function satisfying the condition
$\Phi\left(n\right)\uparrow\infty$ and

$\overline{\lim_{k\rightarrow\infty}}\frac{k^{3-3p}}{\Phi\left(k\right)}=\infty,$

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

${\sum_{m=1}^{\infty}}\frac{\left\|\mathscr{M}_{m}f\right\|_{weak-
L_{p}({G}^{2})}^{p}}{\Phi\left(m\right)}=\infty.$

Theorem NT1 follows that if $0<p<2/3$ and $f\in H_{p}({G}^{2}),$ then there
exists an absolute constant $c_{p}$ depending only on $p,$ such that

$\frac{1}{n^{1-3p/2}}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|\mathscr{M}_{k}f\right\|_{H_{p}({G}^{2})}^{p}}{k^{2-3p/2}}\leq
c_{p}\left\|f\right\|_{H_{p}({G}^{2})}^{p},$

Moreover,

$\frac{1}{n^{1-3p/2}}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|\mathscr{M}_{k}f-f\right\|_{H_{p}({G}^{2})}^{p}}{k^{2-3p/2}}=0,$

It follows the following equality:

$\frac{1}{n^{1-3p/2}}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|\mathscr{M}_{k}f\right\|_{H_{p}(G)}^{p}}{k^{2-3p/2}}=\left\|f\right\|_{H_{p}(G)}^{p}.$

In the fourth chapter (see [35]) we consider strong convergence results of
Marcinkiewicz means with respect to the two-dimensional partial sums of Walsh-
Fourier series, when $p=2/3$:

Theorem 4.50 Let $f\in H_{2/3}({G}^{2}).$ Then there exists an absolute
constant $c,$ such that

$\frac{1}{\log
n}\sum_{m=1}^{n}\frac{\left\|\mathscr{M}_{m}f\right\|_{H_{2/3}({G}^{2})}^{2/3}}{m}\leq
c\left\|f\right\|_{H_{2/3}({G}^{2})}^{2/3}.$

From these results we obtain that, if $f\in H_{2/3}({G}^{2})$ then

$\lim_{n\rightarrow\infty}\frac{1}{\log
n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|\mathscr{M}_{k}f-f\right\|_{H_{2/3}({G}^{2})}^{2/3}}{k}=0$

and

$\lim_{n\rightarrow\infty}\frac{1}{\log
n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|\mathscr{M}_{k}f\right\|_{H_{2/3}({G}^{2})}^{2/3}}{k}=\left\|f\right\|_{H_{2/3}({G}^{2})}^{2/3}.$

For the two-dimensional case Weisz [83] proved that the following is true:

Theorem We6. Let $p>2/3$ and $f\in H_{p}\left(G^{2}\right)$. Then the maximal
operator of Marcinkiewicz means with respect to the two-dimensional partial
sums of Walsh-Fourier series $\mathscr{M}^{\ast}f$ is bounded from
$H_{p}\left(G^{2}\right)$ to $L_{p}\left(G^{2}\right)$:

$\left\|\mathscr{M}^{\ast}f\right\|_{p}\leq
c_{p}\left\|f\right\|_{H_{p}(G^{2})},$

where $c_{p}$ is an absolute constant, depending only on $p$.

Goginava [25] also proved that the following is true:

Theorem Gog4. Let $f\in H_{2/3}\left(G^{2}\right)$. Then there exists an
absolute constant $c,$ such that

$\left\|\mathscr{M}^{\ast}f\right\|_{weak-L_{2/3}(G^{2})}\leq
c\left\|f\right\|_{H_{2/3}}.$

Moreover, there exists a martingale $f\in H_{p}({G}^{2}),$ $\text{ }(0<p\leq
2/3)$ such that

$\underset{n\in\mathbb{N}}{\sup}\left\|\mathscr{M}_{n}f\right\|_{p}=\infty.$

Goginava [26] also consider restricted maximal operator of Marcinkiewicz means
with respect to the two-dimensional partial sums of Walsh-Fourier series
$\sup_{n\in\mathbb{N}}\left|\mathscr{M}_{2^{n}}\right|$ and show that the
following is true:

Theorem Gog5. Let $p>1/2$ and $f\in H_{p}({G}^{2}).$ Then there exists an
absolute constant $c_{p}$ depending only on $p,$ such that

$\left\|\sup_{n\in\mathbb{N}}\left|\mathscr{M}_{2^{n}}f\right|\right\|_{p}\leq
c_{p}\left\|f\right\|_{H_{p}({G}^{2})}.$ (1.22)

Moreover, there exists a martingale $f\in H_{p}({G}^{2}),$ $\text{ }(0<p\leq
1/2)$ such that

$\underset{n\in\mathbb{N}}{\sup}\left\|\mathscr{M}_{2^{n}}f\right\|_{p}=\infty.$

In [34] and [37] we investigate boundedness of weighted maximal operators when
$0<p\leq 2/3$:

Theorem NT2. Let $0<p\leq 2/3.$ Then the maximal operator

$\widetilde{\mathscr{M}}^{\ast}:=\sup_{n\in\mathbb{N}}\frac{\left|\mathscr{M}_{n}\right|}{{(n+1)}^{2/p-3}\log^{3[1/3+p]/2}\left(n+1\right)}$

is bounded from $H_{p}(G^{2})$ to $L_{p}(G^{2})$.

Moreover, if $\varphi:\mathbb{N}\rightarrow[1,\infty)$ be non-decreasing
function satisfying the condition

${\lim_{n\rightarrow\infty}}\frac{n^{2/p-3}\log^{3[1/3+p]/2}\left(n\right)}{\varphi\left(n\right)}=+\infty,$

then

$\sup_{n\in\mathbb{N}}\left\|\frac{{\mathscr{M}}_{n}f}{\varphi\left(n\right)}\right\|_{p}=\infty.$

From Theorem NT2 we get that for $0<p\leq 1/2$ and $f\in H_{p}(G^{2})$, there
exists a absolute constant $c_{p},$ depending only on $p$, such that:

$\left\|\mathscr{M}_{n}f\right\|_{p}\leq
c_{p}{{(n+1)}^{2/p-3}\log^{3[1/3+p]/2}\left(n+1\right)}\left\|f\right\|_{H_{p}({G}^{2})}.$
(1.23)

By applying inequality (1.23) in [37] (see also [38]) we obtain necessary and
sufficient conditions for modulus of continuity of martingale $f\in
H_{p}(G^{2})$, for which Marcinkiewicz means with respect to the two-
dimensional partial sums of Walsh-Fourier series of $f\in H_{p}(G^{2})$
converge in $H_{p}(G^{2})$ norm.

Theorem NT3. Let $1/2<p<2/3,$ $f\in H_{p}\left(G^{2}\right)\ $ and

$\omega_{H_{p}({G}^{2})}\left(\frac{1}{2^{k}},f\right)=o\left(\frac{1}{2^{k\left(2/p-3\right)}}\right),\text{
as \ }k\rightarrow\infty.$

Then

$\left\|\mathscr{M}_{n}f-f\right\|_{H_{p}({G}^{2})}\rightarrow 0,\text{ as
}n\rightarrow\infty.$

Moreover, if $0<p<2/3,$ then there exists a martingale $f\in H_{p}(G^{2}),$
such that

$\omega_{H_{p}({G}^{2})}\left(\frac{1}{2^{k}},f\right)=O\left(\frac{1}{2^{k\left(2/p-3\right)}}\right),\text{
as \ }k\rightarrow\infty$

and

$\left\|\mathscr{M}_{n}f-f\right\|_{weak-L_{p}({G}^{2})}\nrightarrow 0,\text{
\ as \ }n\rightarrow\infty.$

If we apply again (1.23) and improve method which was investigated in [37] in
the fourth chapter (see also [36]) we obtain that the following is true:

Theorem 4.52. Let $f\in H_{2/3}(G^{2})$ and

$\omega_{H_{2/3}({G}^{2})}\left(\frac{1}{2^{k}},f\right)=o\left(\frac{1}{{k}^{3/2}}\right),\text{
as \ }k\rightarrow\infty.$

Then

$\left\|\mathscr{M}_{n}f-f\right\|_{H_{2/3}({G}^{2})}\rightarrow 0,\text{ as
}n\rightarrow\infty.$

On the other hand, there exists a martingale $f\in H_{2/3}(G^{2}),$ such that

$\omega_{H_{2/3}({G}^{2})}\left(\frac{1}{2^{k}},f\right)=O\left(\frac{1}{{k}^{3/2}}\right),\text{
as \ }k\rightarrow\infty$

and

$\left\|\mathscr{M}_{n}f-f\right\|_{2/3}\nrightarrow 0,\text{ \ as \
}n\rightarrow\infty.$

##  2 Partial sums with respect to the one-dimensional Walsh-Fourier series on
the martingale Hardy spaces

###  2.1 Basic notations

Denote by $\mathbb{N}_{+}$ the set of the positive integers and by
$\mathbb{N}:=\mathbb{N}_{+}\cup\\{0\\}$ the set of non-negative integers.
Denote by $Z_{2}$ the additive group of integers modulo-$2,$ which contains
only two elements $Z_{2}:=\\{0,1\\},$ group operation is modulo-$2$ sum and
all sets are open.

Define the group $G$ as the complete direct product of the groups $Z_{2}$ with
the product of the discrete topologies $Z_{2}$. The direct product $\mu$ of
the measures $\mu_{n}\left(\\{j\\}\right):=1/2,\ (j\in Z_{2})$ is the Haar
measure on $G$ with $\mu\left(G\right)=1.$

The elements of $G$ are represented by sequences

$x:=(x_{0},x_{1},...,x_{j},...)\qquad\left(x_{k}=0,1\right).$

It is easy to give a base for the neighbourhood of $G$

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

Set $I_{n}:=I_{n}\left(0\right)$ for any $n\in\mathbb{N}$ and
$\overline{I_{n}}:=G$ $\backslash$ $I_{n}$.

It is evident that

$\overline{I_{M}}=\left(\overset{M-2}{\underset{k=0}{\bigcup}}\overset{M-1}{\underset{l=k+1}{\bigcup}}I_{l+1}\left(e_{k}+e_{l}\right)\right)\bigcup\left(\underset{k=0}{\bigcup\limits^{M-1}}I_{M}\left(e_{k}\right)\right)=\underset{k=0}{\bigcup\limits^{M-1}}I_{k}\backslash
I_{k+1}.$ (2.1)

If $n\in\mathbb{N},$ then it can be uniquely expressed as

$n=\sum_{k=0}^{\infty}n_{j}2^{j}$

where $n_{j}\in Z_{2}$ $~{}(j\in\mathbb{N})$ and only a finite number of
$n_{j}$s differ from zero.

Set

$\left\langle n\right\rangle:=\min\\{j\in\mathbb{N},n_{j}\neq 0\\}\text{ \ \
and \ \ \ }\left|n\right|:=\max\\{j\in\mathbb{N},n_{j}\neq 0\\},$

It is evident that $2^{\left|n\right|}\leq n\leq 2^{\left|n\right|+1}.$

Let

$d\left(n\right):=\left|n\right|-\left\langle n\right\rangle,\text{ \ for any
\ }n\in\mathbb{N}.$

Denote by $V(n)$ variation of natural number $n\in\mathbb{N}$

$V\left(n\right)=n_{0}+\overset{\infty}{\underset{k=1}{\sum}}\left|n_{k}-n_{k-1}\right|.$

Define $k$-th Rademacher functions by

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

By using Rademacher functions we define Walsh system
$w:=(w_{n}:n\in\mathbb{N})$ $G$ as:

$w_{n}(x):=\underset{k=0}{\overset{\infty}{\prod}}r_{k}^{n_{k}}\left(x\right)=r_{\left|n\right|}\left(x\right)\left(-1\right)^{\underset{k=0}{\overset{\left|n\right|-1}{\sum}}n_{k}x_{k}}\text{\qquad}\left(n\in\mathbb{N}\right).$

The norm (quasi-norm) of space $L_{p}(G),\ \left(0<p<\infty\right)$ is defined
as

$\left\|f\right\|_{p}^{p}:=\left(\int_{G}\left|f(x)\right|^{p}d\mu(x)\right).$

and norm (quasi-norm) of space $weak-L_{p}(G)$ is defined by

$\left\|f\right\|_{weak-
L_{p}(G)}^{p}:=\underset{\lambda>0}{\sup}\lambda^{p}\mu\left(x\in{G}:|f|>\lambda\right)<+\infty.$

Walsh system is orthonormal and complete in $L_{2}\left(G\right)$ (see [47]).

For any $f\in L_{1}\left(G\right)$ the numbers

$\displaystyle\widehat{f}\left(n\right):=\int\limits_{G}f(x)w_{n}(x)d\mu({x})$

are called $n$-th Walsh-Fourier coefficient of $f$.

$n$-th partial sum is denoted by

$S_{n}(f;{x}):=\sum\limits_{i=0}^{n-1}\widehat{f}\left(i\right)w_{i}\left(x\right).$

Dirichlet kernels are defined by

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

We also define the following maximal operators

$\displaystyle{S}^{\ast}f$ $\displaystyle=$
$\displaystyle\sup_{n\in\mathbb{N}}\left|S_{{n}}f\right|$
$\displaystyle\widetilde{S}_{\\#}^{\ast}f$ $\displaystyle=$
$\displaystyle\sup_{n\in\mathbb{N}}\left|S_{2^{n}}f\right|$

The $\sigma$-algebra generated by the intervals $I_{n}(x)$ with measure
$2^{-n}$ is denoted by $\digamma_{n}\left(n\in\mathbb{N}\right).$ Conditional
exponential operator with respect to $\digamma_{n}\left(n\in\mathbb{N}\right)$
is denoted by $E_{n}$ and it is given by

$\displaystyle E_{n}f(x)$ $\displaystyle=$ $\displaystyle
S_{2^{n}}f\left(x\right)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{2^{n}-1}\widehat{f}\left(k\right)w_{k}(x)$
$\displaystyle=$
$\displaystyle\frac{1}{\left|I_{n}\left(x\right)\right|}\int_{I_{n}\left(x\right)}f(x)d\mu(x),$

where $\left|I_{n}\left(x\right)\right|=2^{-n}$ denotes length of set
$I_{n}\left(x\right)$.

Sequence $f=\left(f_{n},\text{ }n\in\mathbb{N}\right)$ of functions $f_{n}\in
L_{1}\left(G\right)$ is called dyadic martingale (for details see [39], [47])
if

$\left(i\right)$ $f_{n}$ is measurable with respect to $\sigma-$ algebras
$\digamma_{n}$ for any $n\in\mathbb{N}$,

$\left(ii\right)$ $E_{n}f_{m}=f_{n}$ for any $n\leq m$.

The maximal function of a martingale $f$ is defined by

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

In the case $f\in L_{1}\left(G\right),$ the maximal functions are also be
given by:

$f^{\ast}\left(x\right)=\sup\limits_{n\in\mathbb{N}}\frac{1}{\mu\left(I_{n}\left(x\right)\right)}\left|\int\limits_{I_{n}\left(x\right)}f\left(u\right)d\mu\left(u\right)\right|.$

For $0<p<\infty$ the Hardy martingale space $H_{p}\left(G\right)$ consists of
all martingales, for which

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

A bounded measurable function $a$ is said to be a $p$-atom if there exists an
dyadic interval $I$, such that

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

It is easy to show that for martingale $f=(f_{n},n\in\mathbb{N})$ and for any
${k}\in{\mathbb{N}}$ there exists a limit

$\widehat{f}\left(k\right):=\lim_{{n}\rightarrow\infty}\int_{{G}}f_{{{n}}}\left({x}\right)w_{{k}}\left({x}\right)d\mu\left({x}\right)$

and it is called $k$-th Walsh-Fourier coefficients of $f$.

If $f_{0}\in L_{1}\left(G\right)$ and $f:=(E_{n}f_{0}:n\in\mathbb{N})$ is
regular martingale then

$\displaystyle\widehat{f}\left(k\right)$ $\displaystyle=$
$\displaystyle\int_{{G}}f\left({x}\right)w_{k}(x)d\mu\left({x}\right)=\widehat{f_{0}}\left(k\right),\
\ k\in\mathbb{N}.$

The modulus of continuity in $H_{p}(G)$ space is defined by

$\omega_{H_{p}\left(G\right)}\left(\frac{1}{2^{n}},f\right):=\left\|f-S_{2^{n}}f\right\|_{H_{p}\left(G\right)}.$

It is important to describe how can be understood difference $f-S_{2^{n}}f$,
where $f$ be martingale $S_{2^{n}}f$ is a function:

###### Remark 2.1

Let $0<p\leq 1.$ Since

$S_{2^{n}}f=f^{\left(n\right)}\in L_{1}(G),\text{ \ where \
}f=\left(f^{\left(n\right)}:n\in\mathbb{N}\right)\in H_{p}(G)$

and

$\displaystyle\left(S_{2^{k}}f^{\left(n\right)}:k\in\mathbb{N}\right)$
$\displaystyle=$ $\displaystyle\left(S_{2^{k}}S_{2^{n}},k\in\mathbb{N}\right)$
$\displaystyle=$
$\displaystyle\left(S_{2^{0}}f,\ldots,S_{2^{n-1}}f,S_{2^{n}}f,S_{2^{n}}f,\ldots\right)$
$\displaystyle=$
$\displaystyle\left(f^{\left(0\right)},\ldots,f^{\left(n-1\right)},f^{\left(n\right)},f^{\left(n\right)},\ldots\right).$

Under the difference $f-S_{2^{n}}f$ we mean the following martingale:

$f:=(\left(f-S_{2^{n}}f\right)^{\left(k\right)},\ \ k\in\mathbb{N})$

where

$\left(f-S_{2^{n}}f\right)^{\left(k\right)}=\left\\{\begin{array}[]{ll}0,&k=0,\ldots,n,\\\
f^{\left(k\right)}-f^{\left(n\right)},&k\geq n+1,\end{array}\right.$

Consequently, the norm $\left\|f-S_{2^{n}}f\right\|_{H_{p}\left(G\right)}$ is
understood as $H_{p}$-norm of

$f-S_{2^{n}}f=(\left(f-S_{2^{n}}f\right)^{\left(k\right)},k\in\mathbb{N})$

.

Watari [75] showed that there are strong connections between

$\omega_{p}\left(\frac{1}{2^{n}},f\right),\text{ \
}E_{2^{n}}\left(L_{p},f\right)\text{ \ \ and\ \ \
}\left\|f-S_{2^{n}}f\right\|_{p},\ \ p\geq 1,\text{ \ }n\in\mathbb{N}.$

In particular,

$\frac{1}{2}\omega_{p}\left(\frac{1}{2^{n}},f\right)\leq\left\|f-S_{2^{n}}f\right\|_{p}\leq\omega_{p}\left(\frac{1}{2^{n}},f\right)$

and

$\frac{1}{2}\left\|f-S_{2^{n}}f\right\|_{p}\leq
E_{2^{n}}\left(L_{p},f\right)\leq\left\|f-S_{2^{n}}f\right\|_{p}.$

###  2.2 Auxiliary lemmas

First we present and prove equalities and estimations of Dirichlet kernel and
Lebesgue constants with respect to the one-dimensional Walsh-Fourier systems
(see Lemmas 2.2-3.26).

First equality of the following Lemma is proved in [47] and second identity is
proved in [18]:

###### Lemma 2.2

Let $j,n\in\mathbb{N}$. Then

$D_{j+2^{n}}=D_{2^{n}}+w_{2^{n}}D_{j},\text{ \ when \ }j\leq 2^{n},$

and

$D_{2^{n}-j}=D_{2^{n}}-\psi_{2^{n}-1}{D_{j}},\text{ \ when \ }j<2^{n}.$

The following estimation of Dirichlet kernel with respect to the one-
dimensional Walsh-Fourier systems is proved in [47]:

###### Lemma 2.3

Let $n\in\mathbb{N}$. Then

$D_{2^{n}}\left(x\right)=\left\\{\begin{array}[]{l}2^{n},\text{ \ if \ }x\in
I_{n},\\\ 0,\text{ \ if \ }x\notin I_{n},\end{array}\right.$

and

$D_{n}=w_{n}\overset{\infty}{\underset{k=0}{\sum}}n_{k}r_{k}D_{2^{k}}=w_{n}\overset{\infty}{\underset{k=0}{\sum}}n_{k}\left(D_{2^{k+1}}-D_{2^{k}}\right),\text{
\ for \ }n=\overset{\infty}{\underset{i=0}{\sum}}n_{i}2^{i}.$

The following two-sided estimations of Lebesgue constants with respect to the
one-dimensional Walsh-Fourier systems is proved in [47] and second equality is
proved in [14]:

###### Lemma 2.4

Let $n\in\mathbb{N}.$ Then

$\frac{1}{8}V\left(n\right)\leq\left\|D_{n}\right\|_{1}\leq V\left(n\right)$

and

$\frac{1}{n\log
n}\underset{k=1}{\overset{n}{\sum}}V\left(k\right)=\frac{1}{4\log
2}+o\left(1\right).$

Hardy martingale space $H_{p}\left(G\right)$ for any $0<p\leq 1$ can be
characterize by simple functions which are called $p$-atoms. The following is
true (for details see [50], [78] and [82]):

###### Lemma 2.5

A martingale $f=\left(f_{n},\text{ }n\in\mathbb{N}\right)$ belongs to
$H_{p}(G)\left(0<p\leq 1\right)$ if and only if there exists a sequence of
$p$-atoms of $\left(a_{k},k\in\mathbb{N}\right)$ and sequence of real numbers
$\left(\mu_{k},\text{ }k\in\mathbb{N}\right)$ such that for all
$n\in\mathbb{N}$,

$\qquad\sum_{k=0}^{\infty}\mu_{k}S_{2^{n}}a_{k}=f_{n}$ (2.2)

and

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

Moreover,

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

where the infimum is taken over all decomposition of $f$ of the form (2.2).

The next five Examples of martingales will be used many times to prove
sharpness of our main results. Such counterexamples first appear in the papers
of Goginava [24] (see also [23]). Such constructions of martingales are also
used in the papers [3], [4], [11], [32], [33], [36], [40], [41], [54], [59],
[63], [64], [66], [67], [68], [72], [73], [71], [73]. For the one-dimensional
case we use martingales which were used in [74]. So, we leave out the details
of proof.

###### Example 2.6

Let $0<p\leq 1,$ $\left\\{\lambda_{k}:k\in\mathbb{N}\right\\}$ be sequence of
real numbers

$\sum_{k=0}^{\infty}\left|\lambda_{k}\right|^{p}\leq c_{p}<\infty$ (2.3)

and $\left\\{a_{k}:k\in\mathbb{N}\right\\}$ be sequence of $p$-atoms, given by

$a_{k}(x):={2^{\left|\alpha_{k}\right|(1/p-1)}}\left(D_{2^{\left|\alpha_{k}\right|+1}}(x)-D_{2^{{\left|\alpha_{k}\right|}}}(x)\right),$

where $\left|\alpha_{k}\right|:=\max$ $\\{j\in\mathbb{N}:$
$\left(\alpha_{k}\right)_{j}\neq 0\\}$ and $\left(\alpha_{k}\right)_{j}$
denotes $j$-th binary coefficients of real number of
$\alpha_{k}\in\mathbb{N}_{+}$. Then $\,f=\left(f_{n}:\text{
}n\in\mathbb{N}\right),$ where

$f_{n}(x):=\sum_{\left\\{k:\text{
}\left|\alpha_{k}\right|<n\right\\}}\lambda_{k}a_{k}(x)$

is martingale, which belongs to $H_{p}(G)$ for any $0<p\leq 1$.

It is easy to show that

$\widehat{f}(j)$ (2.4)
$=\left\\{\begin{array}[]{ll}{\lambda_{k}2^{(1/p-1)\left|\alpha_{k}\right|}},&j\in\left\\{2^{\left|\alpha_{k}\right|},...,2^{\left|\alpha_{k}\right|+1}-1\right\\},\text{
}k\in\mathbb{N}_{+},\\\
0,&\text{\thinspace}j\notin\bigcup\limits_{k=1}^{\infty}\left\\{2^{\left|\alpha_{k}\right|},...,2^{\left|\alpha_{k}\right|+1}-1\right\\}.\end{array}\right.$

Let $2^{\left|\alpha_{l-1}\right|+1}\leq j\leq 2^{\left|\alpha_{l}\right|},$
$l\in\mathbb{N}_{+}.$ Then

$\displaystyle S_{j}f$ $\displaystyle=$ $\displaystyle
S_{2^{\left|\alpha_{l-1}\right|+1}}$ $\displaystyle=$
$\displaystyle\sum_{\eta=0}^{l-1}{\lambda_{\eta}2^{{}_{\left|\alpha_{\eta}\right|}(1/p-1)}}\left(D_{2^{\left|\alpha_{\eta}\right|+1}}-D_{2^{\left|\alpha_{\eta}\right|}}\right).$

Let $2^{\left|\alpha_{l}\right|}\leq j<2^{\left|\alpha_{l}\right|+1},$
$l\in\mathbb{N}_{+}.$ Then

$\displaystyle S_{j}f$ $\displaystyle=$ $\displaystyle
S_{2^{\left|\alpha_{l}\right|}}+{\lambda_{l}2^{(1/p-1)\left|\alpha_{l}\right|}w_{2^{\left|\alpha_{l}\right|}}D_{j-2^{{}_{\left|\alpha_{l}\right|}}}}$
$\displaystyle=$
$\displaystyle\sum_{\eta=0}^{l-1}{\lambda_{\eta}2^{(1/p-1)\left|\alpha_{\eta}\right|}}\left(D_{2^{{\left|\alpha_{\eta}\right|+1}}}-D_{2^{{\left|\alpha_{\eta}\right|}}}\right)$
$\displaystyle+$
$\displaystyle{\lambda_{l}2^{(1/p-1)\left|\alpha_{l}\right|}w_{2^{\left|\alpha_{l}\right|}}D_{j-2^{\left|\alpha_{l}\right|}}}.$

Moreover, for the modulus of continuity for $0<p\leq 1$ we have the following
estimation:

$\omega_{H_{p}}\left(\frac{1}{2^{n}},f\right)=O\left(\sum_{\left\\{k:\text{
}\left|\alpha_{k}\right|\geq
n\right\\}}^{\infty}\left|\lambda_{k}\right|^{p}\right)^{1/p},\text{ \ \ as \
\ }n\rightarrow\infty.$ (2.7)

By applying Lemma 2.5 we easily obtain that the following is true (see [82]):

###### Lemma 2.7

Let $0<p\leq 1$ and $T$ be $\sigma$-sub-linear operator, such that, for any
$p$-atom $a$,

$\int\limits_{G}\left|Ta({x})\right|^{p}d\mu({x})\leq c_{p}<\infty.$

Then

$\left\|Tf\right\|_{p}\leq c_{p}\left\|f\right\|_{H_{p}(G)}.$ (2.8)

In addition, if $T$ is bounded from $L_{\infty}(G)$ to $L_{\infty}(G)$ then to
prove (2.8) it is suffices to show that

$\int\limits_{\overset{-}{I}}\left|Ta({x})\right|^{p}d\mu({x})\leq
c_{p}<\infty,$

for every $p$-atom $a$, where $I$ denotes support of the atom $a$.

In the concrete cases the norm of Hardy martingale spaces can be calculated by
simpler formulas (for details see [50], [78] and [79]):

###### Lemma 2.8

If $g\in L_{1}\left(G\right)$ and $f:=(E_{n}g:n\in\mathbb{N})$ be regular
martingale, then $H_{p}\left(G\right)\text{ \ for \ }0<p\leq 1$ norm can be
calculated by

$\left\|f\right\|_{H_{p}(G)}=\left\|\sup\limits_{n\in\mathbb{N}}|S_{2^{n}}g|\right\|_{p}.$

The following lemmas are proved in [59], [63], [64].

###### Lemma 2.9

Let $0<p\leq 1$, $2^{k}\leq n<2^{k+1}$ and $S_{n}f$ be $n$-th partial sum with
respect to the one-dimensional Walsh-Fourier series, where $f\in H_{p}(G)$.
Then for any fixed $n\in\mathbb{N}$,

$\displaystyle\left\|S_{n}f\right\|_{H_{p}(G)}^{p}$ $\displaystyle\leq$
$\displaystyle\left\|\sup_{0\leq l\leq
k}\left|S_{2^{l}}f\right|\right\|_{p}^{p}+\left\|S_{n}f\right\|_{p}^{p}$
$\displaystyle\leq$
$\displaystyle\left\|\widetilde{S}_{\\#}^{\ast}f\right\|_{p}^{p}+\left\|S_{n}f\right\|_{p}^{p}.$

Proof: Let consider the following martingales

$f_{\\#}:=\left(S_{2^{k}}S_{n}f,\text{ }k\in\mathbb{N}_{+}\right)$
$=\left(S_{2^{0}},S_{2^{k}}f,\text{ }S_{n}f,...,S_{n}f,...\right),$

Hence, Lemma 2.8 immediately follows that

$\displaystyle\left\|S_{n}f\right\|_{H_{p}(G)}^{p}$ $\displaystyle\leq$
$\displaystyle\left\|\sup_{0\leq l\leq
k}\left|S_{2^{l}}f\right|\right\|_{p}^{p}+\left\|S_{n}f\right\|_{p}^{p}$
$\displaystyle\leq$
$\displaystyle\left\|\widetilde{S}_{\\#}^{\ast}f\right\|_{p}^{p}+\left\|S_{n}f\right\|_{p}^{p}.$

Lemma is proved.

###  2.3 Boundedness of subsequences of partial sums with respect to the one-
dimensional Walsh-Fourier series on the martingale Hardy spaces

In this section we consider boundedness of subsequences of partial sums with
respect to the one-dimensional Walsh-Fourier series in the martingale Hardy
spaces (for details see [63]).

###### Theorem 2.10

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

$\text{ }\left\|S_{n}f\right\|_{H_{p}(G)}\leq
c_{p}2^{d\left(n\right)\left(1/p-1\right)}\left\|f\right\|_{H_{p}(G)}.$

b) Let $0<p<1,$ $\left\\{m_{k}:\text{ }k\in\mathbb{N}_{+}\right\\}$ be non-
negative, increasing sequence of natural numbers such that

$\sup_{k\in\mathbb{N}}d\left(m_{k}\right)=\infty$ (2.9)

and $\Phi:\mathbb{N}_{+}\rightarrow[1,\infty)$ be non-decreasing function
satisfying the condition

$\overline{\underset{k\rightarrow\infty}{\lim}}\frac{2^{d\left(m_{k}\right)\left(1/p-1\right)}}{\Phi\left(m_{k}\right)}=\infty.$
(2.10)

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

$\underset{k\in\mathbb{N}}{\sup}\left\|\frac{S_{m_{k}}f}{\Phi\left(m_{k}\right)}\right\|_{weak-
L_{p}(G)}=\infty.$

Proof: Suppose that

$\left\|2^{\left(1-1/p\right)d\left(n\right)}S_{n}f\right\|_{p}\leq
c_{p}\left\|f\right\|_{H_{p}(G)}.$ (2.11)

By combining Lemma 2.9 and inequalities (1.7) and (2.11), since
$2^{\left(1-1/p\right)d\left(n\right)}\leq c_{p}$ we obtain that

$\displaystyle\left\|2^{\left(1-1/p\right)d\left(n\right)}S_{n}f\right\|_{H_{p}(G)}^{p}$
$\displaystyle\leq$
$\displaystyle\left\|2^{\left(1-1/p\right)d\left(n\right)}S_{n}f\right\|_{p}^{p}+2^{\left(1-1/p\right)d\left(n\right)}\left\|\widetilde{S}_{\\#}^{\ast}f\right\|_{p}^{p}$
$\displaystyle\leq$ $\displaystyle
c_{p}\left\|f\right\|_{H_{p}(G)}^{p}+c_{p}\left\|\widetilde{S}_{\\#}^{\ast}f\right\|_{p}^{p}$
$\displaystyle\leq$ $\displaystyle c_{p}\left\|f\right\|_{H_{p}(G)}^{p}.$

By combining Lemma 2.7 and (2.3) it is suficies to show that

$\int\limits_{G}\left|2^{\left(1-1/p\right)d\left(n\right)}S_{n}a\right|^{p}d\mu\leq
c_{p}<\infty,$ (2.13)

for every $p$-atom $a$, with support $I$, such that
$\mu\left(I\right)=2^{-M}$.

Without loss the generality we may assume that $p$-atom $a$ has support
$I=I_{M}.$ Then it is easy to see that $S_{n}a=0,$ where $2^{M}$ $\geq n$. So,
we may assume that $2^{M}<n$. Since $\left\|a\right\|_{\infty}\leq 2^{M/p}$ we
can conclude that

$\displaystyle\left|2^{\left(1-1/p\right)d\left(n\right)}S_{n}a\left(x\right)\right|$
$\displaystyle\leq$ $\displaystyle
2^{\left(1-1/p\right)d\left(n\right)}\left\|a\right\|_{\infty}\int_{I_{M}}\left|D_{n}\left(x+t\right)\right|d\mu\left(t\right)$
$\displaystyle\leq$ $\displaystyle
2^{M/p}2^{\left(1-1/p\right)d\left(n\right)}\int_{I_{M}}\left|D_{n}\left(x+t\right)\right|d\mu\left(t\right).$

Let $x\in I_{M}$. Since $V\left(n\right)\leq 2d\left(n\right),$ by using first
estimations of Lemma 2.4 we can conclude that

$\displaystyle\left|2^{\left(1-1/p\right)d\left(n\right)}S_{n}a\right|$
$\displaystyle\leq$ $\displaystyle
2^{M/p}2^{\left(1-1/p\right)d\left(n\right)}V\left(n\right)$
$\displaystyle\leq$ $\displaystyle
2^{M/p}d\left(n\right)2^{\left(1-1/p\right)d\left(n\right)}$

and

$\displaystyle\int_{I_{M}}\left|2^{\left(1-1/p\right)d\left(n\right)}S_{n}a\right|^{p}d\mu$
$\displaystyle\leq$ $\displaystyle
d\left(n\right)2^{\left(1-1/p\right)d\left(n\right)}<c_{p}<\infty.$

Let $t\in I_{M}$ and $x\in I_{s}\backslash I_{s+1},\,$ where $0\leq s\leq
M-1<\left\langle n\right\rangle$ or $0\leq s<\left\langle n\right\rangle\leq
M-1.$ Then $x+t$ $\in I_{s}\backslash I_{s+1}$ and if we use both equality of
Lemma 2.3 we get that $D_{n}\left(x+t\right)=0$ and it follows that

$\left|2^{\left(1-1/p\right)d\left(n\right)}S_{n}a\left(x\right)\right|=0.$
(2.16)

Let $x\in I_{s}\backslash I_{s+1},$ $\,\left\langle n\right\rangle\leq s\leq
M-1.$ Then $x+t\in I_{s}\backslash I_{s+1},$ where $t\in I_{M}$. Then by using
again both equality of Lemma 2.3 we have that

$\left|D_{n}\left(x+t\right)\right|\leq\sum_{j=0}^{s}n_{j}2^{j}\leq c2^{s}.$

If we apply again (2.3) we can conclude that

$\displaystyle\left|2^{\left(1-1/p\right)d\left(n\right)}S_{n}a\left(x\right)\right|$
$\displaystyle\leq$ $\displaystyle
2^{\left(1-1/p\right)d\left(n\right)}2^{M/p}\frac{2^{s}}{2^{M}}$
$\displaystyle\leq$ $\displaystyle
2^{\left<n\right>\left(1/p-1\right)}2^{M(1/p-1)}\frac{2^{s}}{2^{\left|n\right|\left(1/p-1\right)}}$
$\displaystyle\leq$ $\displaystyle 2^{\left\langle
n\right\rangle\left(1/p-1\right)}2^{s}.$

By identity (2.1) and inequalities (2.16) and (2.3) we find that

$\displaystyle\int_{\overline{I_{M}}}\left|2^{\left(1-1/p\right)d\left(n\right)}S_{n}a\left(x\right)\right|^{p}d\mu\left(x\right)$
$\displaystyle=$ $\displaystyle\overset{M-1}{\underset{s=\left\langle
n\right\rangle}{\sum}}\int_{I_{s}\backslash I_{s+1}}\left|2^{\left\langle
n\right\rangle\left(1/p-1\right)}2^{s}\right|^{p}d\mu\left(x\right)$
$\displaystyle\leq$ $\displaystyle c\overset{M-1}{\underset{s=\left\langle
n\right\rangle}{\sum}}\frac{2^{\left\langle
n\right\rangle\left(1-p\right)}}{2^{s\left(1-p\right)}}\leq c_{p}<\infty.$

Now, we prove part b) of Theorem 2.10. By using condition (2.10) there exists
sequence of natural numbers $\left\\{\alpha_{k}:\text{
}k\in\mathbb{N}_{+}\right\\}\subset\left\\{m_{k}:\text{
}k\in\mathbb{N}_{+}\right\\},$ such that

$\sum_{\eta=0}^{\infty}\frac{\Phi^{p/2}\left(\alpha_{\eta}\right)}{2^{d\left(\alpha_{\eta}\right)\left(1-p\right)/2}}<\infty,$
(2.18)

Let $f=\left(f_{n},n\in\mathbb{N}_{+}\right)\in H_{p}(G)$ be a martingale from
the Example 2.6, where

$\lambda_{k}=\frac{\Phi^{1/2}\left(\alpha_{k}\right)}{2^{d\left(\alpha_{k}\right)\left(1/p-1\right)/2}}.$
(2.19)

Then, if we use (2.18) we obtain that condition (2.3) is fulfilled and it
follows that $f=\left(f_{n},n\in\mathbb{N}_{+}\right)\in H_{p}(G).$

If we apply (2.4) when $\lambda_{k}$ are given by the formula (2.19) then we
get that

$\widehat{f}(j)$ (2.20)
$=\left\\{\begin{array}[]{c}\Phi^{1/2}\left(\alpha_{k}\right)2^{\left(\left|\alpha_{k}\right|+\left\langle\alpha_{k}\right\rangle\right)\left(1/p-1\right)/2},\text{
\ if
\thinspace\thinspace}j\in\left\\{2^{{}_{\left|\alpha_{k}\right|}},...,2^{{}_{\left|\alpha_{k}\right|+1}}-1\right\\},\text{
}k\in\mathbb{N}_{+}\\\ 0,\text{ \ if \
}j\notin\bigcup\limits_{k=0}^{\infty}\left\\{2^{{}_{\left|\alpha_{k}\right|}},...,2^{{}_{\left|\alpha_{k}\right|+1}}-1\right\\}.\text{
}\end{array}\right.$

In the view of (2.6) when $\lambda_{k}$ are given by (2.19) we get that

$\displaystyle\frac{S_{\alpha_{k}}f}{\Phi\left(\alpha_{k}\right)}$
$\displaystyle=$
$\displaystyle\frac{1}{\Phi\left(\alpha_{k}\right)}\sum_{\eta=0}^{k-1}\Phi^{1/2}\left(\alpha_{\eta}\right)2^{\left(\left|\alpha_{\eta}\right|+\left\langle\alpha_{\eta}\right\rangle\right)\left(1/p-1\right)/2}\left(D_{2^{\left|\alpha_{\eta}\right|+1}}-D_{2^{\left|\alpha_{\eta}\right|}}\right)$
$\displaystyle+$
$\displaystyle\frac{2^{\left(\left|\alpha_{k}\right|+\left\langle\alpha_{k}\right\rangle\right)\left(1/p-1\right)/2}w_{2^{\left|\alpha_{k}\right|}}D_{\alpha_{k}-2^{\left|\alpha_{k}\right|}}}{\Phi^{1/2}\left(\alpha_{k}\right)}:=I+II.$

by using (2.18) for $I$ we have that

$\displaystyle\left\|I\right\|_{weak-L_{p}(G)}^{p}$ $\displaystyle\leq$
$\displaystyle\frac{1}{\Phi^{p}\left(\alpha_{k}\right)}\sum_{\eta=0}^{k-1}\frac{\Phi^{p/2}\left(\alpha_{\eta}\right)}{2^{d\left(\alpha_{\eta}\right)\left(1-p\right)/2}}\left\|2^{\left|\alpha_{\eta}\right|\left(1/p-1\right)}\left(D_{2^{\left|\alpha_{\eta}\right|+1}}-D_{2^{\left|\alpha_{\eta}\right|}}\right)\right\|_{weak-
L_{p}(G)}^{p}$ $\displaystyle\leq$
$\displaystyle\frac{1}{\Phi^{p}\left(\alpha_{k}\right)}\sum_{\eta=0}^{\infty}\frac{\Phi^{p/2}\left(\alpha_{\eta}\right)}{2^{d\left(\alpha_{\eta}\right)\left(1-p\right)/2}}\leq
c<\infty.$

Let $x\in I_{\left\langle\alpha_{k}\right\rangle}\backslash
I_{\left\langle\alpha_{k}\right\rangle+1}.$ Since
$\left|\alpha_{k}\right|\neq\left\langle\alpha_{k}\right\rangle$ and
$\left\langle\alpha_{k}-2^{\left|\alpha_{k}\right|}\right\rangle=\left\langle\alpha_{k}\right\rangle.$

By using both inequalities of Lemma 2.3 we get that

$\displaystyle\left|D_{\alpha_{k}-2^{\left|\alpha_{k}\right|}}(x)\right|$
$\displaystyle=$
$\displaystyle\left|\left(D_{2^{\left\langle\alpha_{k}\right\rangle+1}}(x)-D_{2^{\left\langle\alpha_{k}\right\rangle}}(x)\right)+\overset{\left|\alpha_{k}\right|-1}{\underset{j=\left\langle\alpha_{k}\right\rangle+1}{\sum}}\left(\alpha_{k}\right)_{j}\left(D_{2^{i+1}}(x)-D_{2^{i}}(x)\right)\right|$
$\displaystyle=$
$\displaystyle\left|-D_{2^{\left\langle\alpha_{k}\right\rangle}}(x)\right|=2^{\left\langle\alpha_{k}\right\rangle}$

and

$\displaystyle\left|II\right|$ $\displaystyle=$
$\displaystyle\frac{2^{\left(\left|\alpha_{k}\right|+\left\langle\alpha_{k}\right\rangle\right)\left(1/p-1\right)/2}}{\Phi^{1/2}\left(\alpha_{k}\right)}\left|D_{\alpha_{k}-2^{\left|\alpha_{k}\right|}}\left(x\right)\right|$
$\displaystyle=$
$\displaystyle\frac{2^{\left|\alpha_{k}\right|\left(1/p-1\right)/2}2^{\left\langle\alpha_{k}\right\rangle\left(1/p+1\right)/2}}{\Phi^{1/2}\left(\alpha_{k}\right)}.$

By combining (2.3) and (2.3) we get that

$\displaystyle\left\|\frac{S_{\alpha_{k}}f}{\Phi\left(\alpha_{k}\right)}\right\|_{weak-
L_{p}(G)}^{p}$ $\displaystyle\geq$ $\displaystyle\left\|II\right\|_{weak-
L_{p}(G)}^{p}-\left\|I\right\|_{weak-L_{p}(G)}^{p}$ $\displaystyle\geq$
$\displaystyle\frac{2^{\left(\left|\alpha_{k}\right|\right)\left(1/p-1\right)/2}2^{\left\langle\alpha_{k}\right\rangle\left(1/p+1\right)/2}}{\Phi^{1/2}\left(\alpha_{k}\right)}\mu\left\\{x\in
G:\text{
}\left|II\right|\geq\frac{2^{\left(\left|\alpha_{k}\right|\right)\left(1/p-1\right)/2}2^{\left\langle\alpha_{k}\right\rangle\left(1/p+1\right)/2}}{\Phi^{1/2}\left(\alpha_{k}\right)}\right\\}^{1/p}$
$\displaystyle\geq$
$\displaystyle\frac{2^{\left(\left|\alpha_{k}\right|\right)\left(1/p-1\right)/2}2^{\left\langle\alpha_{k}\right\rangle\left(1/p+1\right)/2}}{\Phi^{1/2}\left(\alpha_{k}\right)}\left(\mu\left\\{I_{\left\langle\alpha_{k}\right\rangle}\backslash
I_{\left\langle\alpha_{k}\right\rangle+1}\right\\}\right)^{1/p}$
$\displaystyle\geq$ $\displaystyle
c\frac{2^{d\left(\alpha_{k}\right)\left(1/p-1\right)/2}}{\Phi^{1/2}\left(\alpha_{k}\right)}\rightarrow\infty,\
\text{ \ as \ }k\rightarrow\infty.$

The proof of Theorem 2.10 is complete.

###### Corollary 2.11

a) Let $n\in\mathbb{N}_{+}$, $0<p<1$ and $f\in H_{p}(G)$. Then there exists an
absolute constant $c_{p},$ depending only on $p$ such that

$\text{ }\left\|S_{n}f\right\|_{H_{p}(G)}\leq
c_{p}\left(n\mu\left\\{\text{supp}\left(D_{n}\right)\right\\}\right)^{1/p-1}\left\|f\right\|_{H_{p}(G)}.$

b) Let $0<p<1,$ $\left\\{m_{k}:\text{ }k\in\mathbb{N}_{+}\right\\}$ be
increasing sequence of natural numbers, such that

$\sup_{k\in\mathbb{N}}m_{k}\mu\left\\{\text{supp}\left(D_{m_{k}}\right)\right\\}=\infty$
(2.25)

and $\Phi:\mathbb{N}_{+}\rightarrow[1,\infty)$ be non-decreasing function
satisfying the condition

$\overline{\underset{k\rightarrow\infty}{\lim}}\frac{\left(m_{k}\mu\left\\{\text{supp}\left(D_{m_{k}}\right)\right\\}\right)^{1/p-1}}{\Phi\left(m_{k}\right)}=\infty.$
(2.26)

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

$\underset{k\in\mathbb{N}}{\sup}\left\|\frac{S_{m_{k}}f}{\Phi\left(m_{k}\right)}\right\|_{weak-
L_{p}(G)}=\infty.$

Proof: By applying both inequalities of Lemma 2.3 we get that

$I_{\left\langle n\right\rangle}\backslash I_{\left\langle
n\right\rangle+1}\subset\text{supp}\left\\{D_{n}\right\\}\subset
I_{\left\langle n\right\rangle}\text{ \ and \ }2^{-\left\langle
n\right\rangle-1}\leq\mu\left\\{\text{supp}\left(D_{n}\right)\right\\}\leq
2^{-\left\langle n\right\rangle}.$

Hence,

$\frac{2^{d\left(n\right)\left(1/p-1\right)}}{4}\leq\left(n\mu\left\\{\text{supp}\left(D_{n}\right)\right\\}\right)^{1/p-1}\leq
2^{d\left(n\right)\left(1/p-1\right)}.$

Corollary 2.11 is proved.

###### Theorem 2.12

a) Let $n\in\mathbb{N}_{+}$ and $f\in H_{1}(G).$ Then there exists an absolute
constant $c,$ such that

$\left\|S_{n}f\right\|_{H_{1}(G)}\leq
cV\left(n\right)\left\|f\right\|_{H_{1}(G)}.$

b) Let $\left\\{m_{k}:\text{ }k\in\mathbb{N}_{+}\right\\}$ be non-negative
increasing sequence of natural numbers such that

$\sup_{k\in\mathbb{N}}V\left(m_{k}\right)=\infty$ (2.27)

and $\Phi:\mathbb{N}_{+}\rightarrow[1,\infty)$ be non-decreasing function
satisfying the condition

$\overline{\underset{k\rightarrow\infty}{\lim}}\frac{V\left(m_{k}\right)}{\Phi\left(m_{k}\right)}=\infty.$
(2.28)

Then there exists a martingale $f\in H_{1}(G),$ such that

$\underset{{k\in\mathbb{N}}}{\sup}\left\|\frac{S_{m_{k}}f}{\Phi\left(m_{k}\right)}\right\|_{1}=\infty.$

Proof: Since

$\left\|\frac{S_{n}f}{V\left(n\right)}\right\|_{1}\leq\left\|f\right\|_{1}\leq\left\|f\right\|_{H_{1}(G)}$
(2.29)

by combining Lemmas 2.9 and (2.29) we can conclude that

$\displaystyle\left\|\frac{S_{n}f}{V\left(n\right)}\right\|_{H_{1}(G)}$
$\displaystyle\leq$
$\displaystyle\left\|\frac{S_{n}f}{V\left(n\right)}\right\|_{1}+\frac{1}{V\left(n\right)}\left\|\widetilde{S}_{\\#}^{\ast}f\right\|_{1}$
$\displaystyle\leq$ $\displaystyle
c\left\|f\right\|_{H_{1}(G)}+c\left\|\widetilde{S}_{\\#}^{\ast}f\right\|_{1}\leq
c\left\|f\right\|_{H_{1}(G)}.$

Now prove second part of Theorem 2.12. Let $\left\\{m_{k}:\text{
}k\in\mathbb{N}_{+}\right\\}$ be increasing sequence of natural numbers and
function $\ \Phi:\mathbb{N}_{+}\rightarrow[1,\infty)$ satisfies conditions
(2.27) and (2.28). Then there exists non-negative, increasing sequence
$\left\\{\alpha_{k}:k\in\mathbb{N}_{+}\right\\}\subset\left\\{m_{k}:k\in\mathbb{N}_{+}\right\\}$
such that

$\sum_{k=1}^{\infty}\frac{\Phi^{1/2}\left(\alpha_{k}\right)}{V^{1/2}\left(\alpha_{k}\right)}\leq\beta<\infty.$
(2.31)

Let $f=\left(f_{n},n\in\mathbb{N}_{+}\right)$ be martingale from Example 2.6,
where

$\text{\
}\lambda_{k}=\frac{\Phi^{1/2}\left(\alpha_{k}\right)}{V^{1/2}\left(\alpha_{k}\right)}.$
(2.32)

By applying condition (2.31) we can conclude that condition (2.3) is fulfilled
and it follows that $f=\left(f_{n},n\in\mathbb{N}_{+}\right)\in H_{1}(G).$

In the view of (2.4) when $\lambda_{k}$ are given by (2.32) we get that

$\widehat{f}(j)=\left\\{\begin{array}[]{ll}\frac{\Phi^{1/2}\left(\alpha_{k}\right)}{V^{1/2}\left(\alpha_{k}\right)},&\text{
\ if \
}j\in\left\\{2^{{}_{\left|\alpha_{k}\right|}},...,2^{{}_{\left|\alpha_{k}\right|+1}}-1\right\\},\text{
}k=0,1,...\\\ 0,&\text{ \ if \
}j\notin\bigcup\limits_{k=0}^{\infty}\left\\{2^{{}_{\left|\alpha_{k}\right|}},...,2^{{}_{\left|\alpha_{k}\right|+1}}-1\right\\}.\text{
}\end{array}\right.$ (2.33)

Analogously to (2.3) if we apply (2.6) when $\lambda_{k}$ are given by (2.32)
we get that

$\displaystyle S_{\alpha_{k}}f$ $\displaystyle=$
$\displaystyle\sum_{\eta=0}^{k-1}\frac{\Phi^{1/2}\left(\alpha_{\eta}\right)}{V^{1/2}\left(\alpha_{\eta}\right)}\left(D_{2^{\left|\alpha_{\eta}\right|+1}}-D_{2^{\left|\alpha_{\eta}\right|}}\right)$
$\displaystyle+$
$\displaystyle\frac{\Phi^{1/2}\left(\alpha_{k}\right)}{V^{1/2}\left(\alpha_{k}\right)}w_{2^{\left|\alpha_{k}\right|}}D_{\alpha_{k}-2^{\left|\alpha_{k}\right|}}.$

By applying first estimation of Lemma 2.4 and (2.31) we can conclude that

$\displaystyle\left\|\frac{S_{\alpha_{k}}f}{\Phi\left(\alpha_{k}\right)}\right\|_{1}$
$\displaystyle\geq$
$\displaystyle\frac{\Phi^{1/2}\left(\alpha_{k}\right)}{\Phi\left(\alpha_{k}\right)V^{1/2}\left(\alpha_{k}\right)}\left\|D_{\alpha_{k}-2^{\left|\alpha_{k}\right|}}\right\|_{1}$
$\displaystyle-$
$\displaystyle\frac{1}{\Phi\left(\alpha_{k}\right)}\sum_{\eta=0}^{k-1}\frac{\Phi^{1/2}\left(\alpha_{\eta}\right)}{V^{1/2}\left(\alpha_{\eta}\right)}\left\|D_{2^{\left|\alpha_{\eta}\right|+1}}-D_{2^{\left|\alpha_{\eta}\right|}}\right\|_{1}$
$\displaystyle\geq$
$\displaystyle\frac{V\left(\alpha_{k}-2^{\left|\alpha_{k}\right|}\right)\Phi^{1/2}\left(\alpha_{k}\right)}{8\Phi\left(\alpha_{k}\right)V^{1/2}\left(\alpha_{k}\right)}$
$\displaystyle-$
$\displaystyle\frac{1}{\Phi\left(\alpha_{k}\right)}\sum_{\eta=0}^{\infty}\frac{\Phi^{1/2}\left(\alpha_{\eta}\right)}{V^{1/2}\left(\alpha_{\eta}\right)}$
$\displaystyle\geq$
$\displaystyle\frac{cV^{1/2}\left(\alpha_{k}\right)}{\Phi^{1/2}\left(\alpha_{k}\right)}\rightarrow\infty,\text{
\ as \ }k\rightarrow\infty.$

Theorem 2.12 is proved.

###### Corollary 2.13

Let $n\in\mathbb{N}$, $0<p\leq 1$ and $f\in H_{p}(G)$. Then there exists an
absolute constant $c_{p},$ depending only on $p,$ such that

$\left\|S_{2^{n}}f\right\|_{H_{p}(G)}\leq c_{p}\left\|f\right\|_{H_{p}(G)}.$
(2.34)

Proof: To prove Theorem 2.13 we only have to show that

$\left|2^{n}\right|=n,\text{ \ \ }\left\langle 2^{n}\right\rangle=n-1\text{ \
და \ }d\left(2^{n}\right)=0.$

By applying first part of Theorem 2.10 we immediately get that (2.34) for any
$0<p\leq 1$ and proof of Corollary 2.13 is proved.

###### Corollary 2.14

Let $n\in\mathbb{N}$, $0<p\leq 1$ and $f\in H_{p}(G)$. Then there exists an
absolute constant $c_{p},$ depending only on $p$ such that

$\left\|S_{2^{n}+2^{n-1}}f\right\|_{H_{p}(G)}\leq
c_{p}\left\|f\right\|_{H_{p}(G)}.$ (2.35)

Proof: Since

$\left|2^{n}+2^{n-1}\right|=n,\left\langle
2^{n}+2^{n-1}\right\rangle=n-1\text{ \ and \ }d\left(2^{n}+2^{n-1}\right)=1$

by first part of Theorem 2.10 we get that (2.35) holds, for any $0<p\leq 1$
and proof of Corollary 2.14 is complete.

###### Corollary 2.15

Let $n\in\mathbb{N}$ and $0<p<1.$ Then there exists a martingale $f\in
H_{p}(G),$ such that

$\underset{n\in\mathbb{N}}{\sup}\left\|S_{2^{n}+1}f\right\|_{weak-
L_{p}(G)}=\infty.$ (2.36)

On the other hand, there exists an absolute constant $c$, such that

$\left\|S_{2^{n}+1}f\right\|_{H_{1}(G)}\leq c\left\|f\right\|_{H_{1}(G)}.$
(2.37)

Proof: Since

$\left|2^{n}+1\right|=n,\left\langle 2^{n}+1\right\rangle=0\text{ \ და \
}d\left(2^{n}+1\right)=n.$ (2.38)

by applying second part of Theorem 2.10 we get that there exists a martingale
$f=\left(f_{n},n\in\mathbb{N}_{+}\right)\in H_{p}(G),$ for $0<p<1,$ such that
(2.36) holds.

On the other hand, proof of (2.37) follows simple observation that

$V(2^{n}+1)=4<\infty.$

Corollary 2.15 is proved.

###  2.4 Modulus of continuity and convergence in norm of subsequences of
partial sums with respect to the one-dimensional Walsh-Fourier series on the
martingale Hardy spaces

In this section we apply Theorem 2.10 and Theorem 2.12 to find necessary and
sufficient conditions for modulus of continuity, for which subsequences of
partial sums with respect to the one-dimensional Walsh-Fourier series are
bounded in the martingale Hardy spaces.

First, we prove the following estimation:

###### Theorem 2.16

Let $n\in\mathbb{N}_{+}$ and $2^{k}<n\leq 2^{k+1}.$ Then there exists an
absolute constant $c_{p},$ depending only on $p$ such that

$\left\|S_{n}f-f\right\|_{H_{p}(G)}\leq
c_{p}2^{d\left(n\right)\left(1/p-1\right)}\omega_{H_{p}(G)}\left(\frac{1}{2^{k}},f\right),\text{
\ \ \ }(f\in H_{p}(G))\text{ \ \ \ }\left(0<p<1\right)$ (2.39)

and

$\left\|S_{n}f-f\right\|_{H_{1}(G)}\leq
c_{1}V\left(n\right)\omega_{H_{1}(G)}\left(\frac{1}{2^{k}},f\right),\text{ \ \
\ }(f\in H_{1}(G)).$ (2.40)

Proof: Let $0<p<1$ and $2^{k}<n\leq 2^{k+1}.$ By applying first part of
Theorem 2.10 we get that

$\displaystyle\left\|S_{n}f-f\right\|_{H_{p}(G)}^{p}$ $\displaystyle\leq$
$\displaystyle
c_{p}\left\|S_{n}f-S_{2^{k}}f\right\|_{H_{p}(G)}^{p}+c_{p}\left\|S_{2^{k}}f-f\right\|_{H_{p}(G)}^{p}$
$\displaystyle=$ $\displaystyle
c_{p}\left\|S_{n}\left(S_{2^{k}}f-f\right)\right\|_{H_{p}(G)}^{p}+c_{p}\left\|S_{2^{k}}f-f\right\|_{H_{p}(G)}^{p}$
$\displaystyle\leq$ $\displaystyle
c_{p}\left(1+2^{d\left(n\right)\left(1-p\right)}\right)\omega_{H_{p}(G)}^{p}\left(\frac{1}{2^{k}},f\right)$
$\displaystyle\leq$ $\displaystyle
c_{p}2^{d\left(n\right)\left(1-p\right)}\omega_{H_{p}(G)}^{p}\left(\frac{1}{2^{k}},f\right).$

The proof of (2.40) is analogical to (2.39). Analogously to (2.39) we can also
prove estimation (2.40). So, we leave out the details.

Theorem 2.16 is proved.

###### Theorem 2.17

a) Let $k\in\mathbb{N}_{+}$, $0<p<1,$ $f\in H_{p}(G)$ and
$\\{m_{k}:k\in\mathbb{N}_{+}\\}$ be increasing sequence of natural numbers,
such that

$\omega_{H_{p}(G)}\left(\frac{1}{2^{\left|m_{k}\right|}},f\right)=o\left(\frac{1}{2^{d\left(m_{k}\right)\left(1/p-1\right)}}\right)\text{
\ as \ }k\rightarrow\infty.$ (2.42)

Then

$\left\|S_{m_{k}}f-f\right\|_{H_{p}(G)}\rightarrow 0\text{ \ as \
}k\rightarrow\infty.$ (2.43)

b) Let $\\{m_{k}:k\in\mathbb{N}_{+}\\}$ be increasing sequence of natural
numbers,such that condition (2.9) is fulfilled. Then there exists a martingale
$f\in H_{p}(G)$ and increasing sequence of natural numbers
$\\{\alpha_{k}:k\in\mathbb{N}_{+}\\}\subset\\{m_{k}:k\in\mathbb{N}_{+}\\},$
such that

$\omega_{H_{p}(G)}\left(\frac{1}{2^{\left|\alpha_{k}\right|}},f\right)=O\left(\frac{1}{2^{d\left(\alpha_{k}\right)\left(1/p-1\right)}}\right)\text{
\ as \ }k\rightarrow\infty\text{\ }$

and

$\limsup\limits_{k\rightarrow\infty}\left\|S_{\alpha_{k}}f-f\right\|_{weak-
L_{p}(G)}>c_{p}>0,\text{ \ as \ }k\rightarrow\infty,$ (2.44)

where $c_{p}$ is an absolute constant depending only on $p$.

Proof: Let $0<p<1$, $f\in H_{p}(G)$ and $\\{m_{k}:k\in\mathbb{N}_{+}\\}$ be
increasing sequence of natural numbers, such that condition (2.42) is
fulfilled. By combining Theorem 2.16 and estimation (2.39) we get that (2.43)
holds true.

Now, prove second part of Theorem 2.17. In the view of (2.9) we simply get
that there exists sequence
$\\{\alpha_{k}:k\in\mathbb{N}_{+}\\}\subset\\{m_{k}:k\in\mathbb{N}_{+}\\},$
such that

$\ 2^{d\left(\alpha_{k}\right)}\uparrow\infty,\,\,\,\,\text{ \ as\ }\ \
k\rightarrow\infty\text{, \ \ \
}2^{2\left(1/p-1\right)d\left(\alpha_{k}\right)}\leq
2^{\left(1/p-1\right)d\left(\alpha_{k+1}\right)}.$ (2.45)

Let $f=\left(f_{n},n\in\mathbb{N}\right)$ be a martingale from Example 2.6,
such that

$\lambda_{i}={2^{-(1/p-1)d\left(\alpha_{i}\right)}}.$ (2.46)

By applying (2.45) we obtain that condition (2.3) is fulfilled and it follows
that $\ f\in H_{p}(G).$

By applying (2.4), whene $\lambda_{k}$ are given by (2.46), then

$\widehat{f}(j)=\left\\{\begin{array}[]{ll}2^{\left(1/p-1\right)\left\langle\alpha_{k}\right\rangle},&\text{
\ if \
}j\in\left\\{2^{{}_{\left|\alpha_{k}\right|}},...,2^{{}_{\left|\alpha_{k}\right|+1}}-1\right\\},\text{
}k\in\mathbb{N}_{+},\\\ 0,&\text{\ if \
}j\notin\bigcup\limits_{k=0}^{\infty}\left\\{2^{{}_{\left|\alpha_{k}\right|}},...,2^{{}_{\left|\alpha_{k}\right|+1}}-1\right\\}.\text{
}\end{array}\right.$ (2.47)

By combining (2.45) and (2.7) we have that

$\displaystyle\omega_{H_{p}(G)}(\frac{1}{2^{\left|\alpha_{k}\right|}},f)$
$\displaystyle\leq$
$\displaystyle\sum\limits_{i=k}^{\infty}\frac{1}{2^{\left(1/p-1\right)d\left(\alpha_{i}\right)}}$
$\displaystyle=$ $\displaystyle
O\left(\frac{1}{2^{\left(1/p-1\right)d\left(\alpha_{k}\right)}}\right),\text{
\ as \ }k\rightarrow\infty.$

By using (2.3) we get that

$\left|D_{\alpha_{k}-2^{\left\langle\alpha_{k}\right\rangle}}\right|\geq
2^{\left\langle\alpha_{k}\right\rangle},\text{ \ \ \ \ where \ \ \
}I_{\left\langle\alpha_{k}\right\rangle}\backslash
I_{\left\langle\alpha_{k}\right\rangle+1}.$

In the view of (2.6) we can conclude that

$S_{\alpha_{k}}f=S_{2^{|\alpha_{k}|}}f+2^{\left(1/p-1\right)\left\langle\alpha_{k}\right\rangle}{w_{2^{|\alpha_{k}|}}D_{{\alpha_{k}}-2^{|\alpha_{k}|}}},$

Since

$\displaystyle\|D_{\alpha_{k}}\|_{weak-L_{p}(G)}$ $\displaystyle\geq$
$\displaystyle 2^{\left\langle\alpha_{k}\right\rangle}\mu\left\\{x\in
I_{\left\langle\alpha_{k}\right\rangle}\backslash
I_{\left\langle\alpha_{k}\right\rangle+1}:\text{
}\left|D_{\alpha_{k}}\right|\geq
2^{\left\langle\alpha_{k}\right\rangle}\right\\}^{1/p}$ $\displaystyle\geq$
$\displaystyle
2^{\left\langle\alpha_{k}\right\rangle}\left(\mu\left\\{I_{\left\langle\alpha_{k}\right\rangle}\backslash
I_{\left\langle\alpha_{k}\right\rangle+1}\right\\}\right)^{1/p}\geq
2^{\left\langle\alpha_{k}\right\rangle\left(1-1/p\right)},$

if we apply (1.2) (see also Theorem T2) we obtain that

$\displaystyle\|f-S_{\alpha_{k}}f\|_{weak-L_{p}(G)}^{p}$ $\displaystyle\geq$
$\displaystyle
2^{\left(1-p\right)\left\langle\alpha_{k}\right\rangle}\|{w_{2^{|\alpha_{k}|}}D_{{\alpha_{k}}-2^{|\alpha_{k}|}}}\|_{weak-
L_{p}(G)}^{p}$ $\displaystyle-$
$\displaystyle\|f-S_{2^{|\alpha_{k}|}}f\|_{weak-L_{p}(G)}^{p}$
$\displaystyle\geq$ $\displaystyle c-o(1)>c>0,\text{\ \ \ as \ \
}k\rightarrow\infty.$

Proof of Theorem 2.17 is complete.

###### Corollary 2.18

a) Let $0<p<1,$ $f\in H_{p}(G)$ and $\\{m_{k}:k\in\mathbb{N}_{+}\\}$ be
increasing sequence of natural numbers, such that

$\omega_{H_{p}(G)}\left(\frac{1}{2^{\left|m_{k}\right|}},f\right)=o\left(\frac{1}{\left(m_{k}\mu\left(\text{supp}D_{m_{k}}\right)\right)^{1/p-1}}\right)\text{
\ as\ }k\rightarrow\infty.$ (2.49)

Then (2.43) holds.

b) Let $\\{m_{k}:k\in\mathbb{N}_{+}\\}$ be increasing sequence of natural
numbers, such that

$\sup_{k\in\mathbb{N}_{+}}m_{k}\mu\left\\{\text{supp}\left(D_{m_{k}}\right)\right\\}=\infty.$
(2.50)

Then there exist a martingale $f\in H_{p}(G)$ and sequence
$\\{\alpha_{k}:k\in\mathbb{N}_{+}\\}\subset\\{m_{k}:k\in\mathbb{N}_{+}\\},$
such that

$\omega_{H_{p}(G)}\left(\frac{1}{2^{\left|\alpha_{k}\right|}},f\right)=O\left(\frac{1}{\left(\alpha_{k}\mu\left(\text{supp}D_{\alpha_{k}}\right)\right)^{1/p-1}}\right)\text{\
\ as \ }k\rightarrow\infty$

and (2.44) holds.

###### Theorem 2.19

a) Let $f\in H_{1}(G)$ and $\\{m_{k}:k\in\mathbb{N}_{+}\\}$ be increasing
sequence of natural numbers, such that

$\omega_{H_{1}(G)}\left(\frac{1}{2^{\left|m_{k}\right|}},f\right)=o\left(\frac{1}{V\left(m_{k}\right)}\right)\text{
\ as \ }k\rightarrow\infty.$ (2.51)

Then

$\left\|S_{m_{k}}f-f\right\|_{H_{1}(G)}\rightarrow 0\text{ \ as \
}k\rightarrow\infty.$ (2.52)

b) Let $\\{m_{k}:k\in\mathbb{N}_{+}\\}$ be increasing sequence of natural
numbers, such that condition (2.27) is fulfilled. Then there exists a
martingale $f\in H_{1}(G)$ and increasing sequence of natural numbers
$\\{\alpha_{k}:k\in\mathbb{N}_{+}\\}\subset\\{m_{k}:k\in\mathbb{N}_{+}\\}$
such that

$\omega_{H_{1}(G)}\left(\frac{1}{2^{\left|\alpha_{k}\right|}},f\right)=O\left(\frac{1}{V\left(\alpha_{k}\right)}\right)\text{
\ as \ }k\rightarrow\infty$

and

$\limsup\limits_{k\rightarrow\infty}\left\|S_{\alpha_{k}}f-f\right\|_{1}>c>0\,\,\,\text{\
as \ }k\rightarrow\infty,$ (2.53)

where $c$ is an absolute constant.

Proof: Let $f\in H_{1}(G)$ and $\\{m_{k}:k\in\mathbb{N}_{+}\\}$ be increasing
sequence of natural numbers, such that (2.51). By applying Theorem 2.16 we get
that condition (2.52) is fulfilled.

Now, we prove second part of Theorem 2.19. By applying (2.27) we conclude that
there exists sequence
$\\{\alpha_{k}:k\in\mathbb{N}_{+}\\}\subset\\{m_{k}:k\in\mathbb{N}_{+}\\}$,
such that

$V(\alpha_{k})\uparrow\infty,\text{ \ \ as \ \ }k\rightarrow\infty\text{ \ \
and \ \ }V^{2}(\alpha_{k})\leq V(\alpha_{k+1})\text{ \ \ }k\in\mathbb{N}_{+}.$
(2.54)

Let $f=\left(f_{n},n\in\mathbb{N}_{+}\right)$ be a martingale from the Example
2.6, where

$\lambda_{k}=\frac{1}{V(\alpha_{k})}.$

By applying (2.54) we conclude that (2.3) is fulfilled and we conclude that
$f=\left(f_{n},n\in\mathbb{N}_{+}\right)\in H_{1}(G).$

In the view of (2.4) we have that

$\widehat{f}(j)=\left\\{\begin{array}[]{ll}\frac{1}{V(\alpha_{k})},&\text{ თუ
\thinspace\thinspace}j\in\left\\{2^{{}_{\left|\alpha_{k}\right|}},...,2^{{}_{\left|\alpha_{k}\right|+1}}-1\right\\},\text{
}k=0,1,...\\\ 0,&\text{\ თუ
\thinspace\thinspace\thinspace}j\notin\bigcup\limits_{k=0}^{\infty}\left\\{2^{{}_{\left|\alpha_{k}\right|}},...,2^{{}_{\left|\alpha_{k}\right|+1}}-1\right\\}.\text{
}\end{array}\right.$ (2.55)

According to (2.7) we get that

$\displaystyle w_{H_{1}(G)}(1/2^{n},f)$ $\displaystyle=$
$\displaystyle\left\|f-S_{2^{n}}f\right\|_{H_{1}(G)}\leq\sum\limits_{i=n+1}^{\infty}\frac{1}{V(\alpha_{i})}$
$\displaystyle=$ $\displaystyle O\left(\frac{1}{V(\alpha_{n})}\right),\text{\
\ as \ \ }n\rightarrow\infty.$

By applying (2.6) we can conclude that

$S_{\alpha_{k}}f=S_{2^{|\alpha_{k}|}}f+\frac{w_{2^{|\alpha_{k}|}}D_{{\alpha_{k}}-2^{|\alpha_{k}|}}}{V(\alpha_{k})},$

If we use (1.2) and Theorem T2 we get that

$\displaystyle\|f-S_{\alpha_{k}}f\|_{1}$ $\displaystyle\geq$
$\displaystyle\|\frac{w_{2^{|\alpha_{k}|}}D_{{\alpha_{k}}-2^{|\alpha_{k}|}}}{V(\alpha_{k})}\|_{1}-\|f-S_{2^{|\alpha_{k}|}}f\|_{1}$
$\displaystyle\geq$
$\displaystyle\frac{V(\alpha_{k}-2^{|\alpha_{k}|})}{8V(\alpha_{k})}-o(1)>c>0,\text{\
\ as \ \ }k\rightarrow\infty.$

The proof of Theorem 2.19 is proved.

Theorem 3.34 follows the following corollaries which are [61]:

###### Corollary 2.20

a) Let $0<p<1,$ $f\in H_{p}(G)$ and

$\omega_{H_{p}(G)}\left(1/2^{k},f\right)=o\left(1/2^{k(1/p-1)}\right),\text{
as }k\rightarrow\infty.$

Then

$\left\|S_{k}f-f\right\|_{H_{p}(G)}\rightarrow 0,\text{ \ as \
}k\rightarrow\infty.$

b) There exists a martingale $f\in H_{p}(G)$ $\left(0<p<1\right),$ such that

$\omega_{H_{p}(G)}\left(1/2^{k},f\right)=O\left(1/2^{k\left(1/p-1\right)}\right),\text{
\ as \ }k\rightarrow\infty$

and

$\left\|S_{k}f-f\right\|_{weak-L_{p}(G)}\nrightarrow 0,\text{ \ as \
}k\rightarrow\infty.$

###### Corollary 2.21

a) Let $f\in H_{1}(G)$ and

$\omega_{H_{1}(G)}\left(1/2^{k},f\right)=o\left(\frac{1}{k}\right),\text{ as \
}k\rightarrow\infty.$

Then

$\left\|S_{k}f-f\right\|_{H_{1}(G)}\rightarrow 0,\text{\ as \
}k\rightarrow\infty.$

b) There exists a martingale $f\in H_{1}(G),$ such that

$\omega_{H_{1}(G)}\left(1/2^{k},f\right)=O\left(\frac{1}{k}\right),\text{ \ as
\ }k\rightarrow\infty$

and

$\left\|S_{k}f-f\right\|_{1}\nrightarrow 0,\text{ \ as \ }k\rightarrow\infty.$

##  3 Fejér means with respect to the one-dimensional Walsh-Fourier series on
the martingale Hardy spaces

###  3.1 Basic notations

Fot the one-dimensional case Fejér means with respect to the one-dimensional
Walsh-Fourier series $\sigma_{n}$ is defined by:

$\displaystyle\qquad\sigma_{n}f(x)$ $\displaystyle:$
$\displaystyle=\frac{1}{n}\sum_{k=1}^{n}S_{k}f(x)\text{\qquad\ \ \
}\left(\text{ }n\in\mathbb{N}_{+}\right).$

The following equality is true (for details see [2] and [47]):

$\sigma_{n}f\left(x\right)=\frac{1}{n}\overset{n-1}{\underset{k=0}{\sum}}\left(D_{k}\ast
f\right)\left(x\right)$ $=\left(f\ast
K_{n}\right)\left(x\right)=\int_{G_{m}}f\left(t\right)K_{n}\left(x-t\right)d\mu\left(t\right).$

where

$\displaystyle K_{n}(x)$ $\displaystyle:$
$\displaystyle=\frac{1}{n}\overset{n}{\underset{k=1}{\sum}}D_{k}(x)\text{
\qquad\thinspace}\left(\text{ }n\in\mathbb{N}_{+}\text{ }\right)$

In the literature $K_{n}$ is called $n$-th Fejér kernel.

We also define the following maximal operators

$\displaystyle{\sigma}^{\ast}f$ $\displaystyle=$
$\displaystyle\sup_{n\in\mathbb{N}}\left|\sigma_{{n}}f\right|$
$\displaystyle\widetilde{\sigma}_{\\#}^{\ast}f$ $\displaystyle=$
$\displaystyle\sup_{n\in\mathbb{N}}\left|\sigma_{2^{n}}f\right|.$

.

For any natural number $n\in\mathbb{N}$ we also need the following expression

$n=\sum_{i=1}^{s}2^{n_{i}},\text{ \qquad\thinspace}n_{1}<n_{2}<...<n_{s}.$

Set

$n^{\left(i\right)}:=2^{n_{1}}+...+2^{n_{i-1}},\text{ \ }i=2,...,s$

and

$\mathbb{A}_{0,2}:=\left\\{n\in\mathbb{N}:\text{
}n=2^{0}+2^{2}+\sum_{i=3}^{s_{n}}2^{n_{i}}\right\\}.$

Then, for any natural number $n\in\mathbb{N}$ there exists numbers

$0\leq l_{1}\leq m_{1}\leq l_{2}-2<l_{2}\leq m_{2}\leq...\leq
l_{s}-2<l_{s}\leq m_{s}$

such that it can be written as

$n=\sum_{i=1}^{s}\sum_{k=l_{i}}^{m_{i}}2^{k},$

where $s$ is depending on $n$.

It is evident that

$s\leq V\left(n\right)\leq 2s+1.$

###  3.2 Auxiliary lemmas

The following equality and estimation of Fejér kernels with respect to the
one-dimensional Walsh-Fourier series is proved in [47]:

###### Lemma 3.22

Let $n\in\mathbb{N}$ and $n=\sum_{i=1}^{s}2^{n_{i}},$ $n_{1}<n_{2}<...<n_{s}$.
Then

$nK_{n}=\sum_{r=1}^{s}\left(\underset{j=r+1}{\overset{s}{\prod}}w_{2^{n_{j}}}\right)2^{n_{r}}K_{2^{n_{r}}}+\sum_{t=2}^{s}\left(\underset{j=t+1}{\overset{s}{\prod}}w_{2^{n_{j}}}\right)n^{\left(t\right)}D_{2^{n_{t}}},$

and

$\sup_{n\in\mathbb{N}}\int_{G}\left|K_{n}\left(x\right)\right|d\mu\left(x\right)\leq
c<\infty,$

where $c$ is an absolute constant.

The following equality is proved in [47] (see also [16]):

###### Lemma 3.23

Let $n>t$ and $t,n\in\mathbb{N}$. Then we have the following expression for
$2^{n}$-th Fejér kernels with respect to the one-dimensional Walsh-Fourier
series:

$K_{2^{n}}\left(x\right)=\left\\{\begin{array}[]{c}\text{ }2^{t-1},\text{\ if
\ }x\in I_{n}\left(e_{t}\right),\\\ \frac{2^{n}+1}{2},\text{ \ if \ }x\in
I_{n},\text{\ }\\\ 0,\text{ \ otherwise.\ }\end{array}\right.$

The following estimation is proved by Goginava [22]:

###### Lemma 3.24

Let $x\in I_{l+1}\left(e_{k}+e_{l}\right),\ \ k=0,...,M-2,\ \ l=0,...,M-1.$
Then

$\int_{I_{M}}\left|K_{n}\left(x+t\right)\right|d\mu\left(t\right)\leq\frac{c2^{l+k}}{n2^{M}},\text{
\ where \ }n>2^{M}.$

Let $x\in I_{M}\left(e_{k}\right),m=0,...,M-1.$ Then

$\int_{I_{M}}\left|K_{n}\left(x+t\right)\right|d\mu\left(t\right)\leq\frac{c2^{k}}{2^{M}},\text{
\ \ for \ \ }n>2^{M},$

where $c$ is an absolute constant.

The following estimations of Fejér kernels with respect to the one-dimensional
Walsh-Fourier series is proved in [64]:

###### Lemma 3.25

Let

$n=\sum_{i=1}^{r}\sum_{k=l_{i}}^{m_{i}}2^{k},$

where

$m_{1}\geq l_{1}>l_{1}-2\geq m_{2}\geq l_{2}>l_{2}-2>...>m_{s}\geq l_{s}\geq
0.$

Then

$\left|nK_{n}\right|\leq
c\sum_{A=1}^{r}\left(2^{l_{A}}\left|K_{2^{l_{A}}}\right|+2^{m_{A}}\left|K_{2^{m_{A}}}\right|+2^{l_{A}}\sum_{k=l_{A}}^{m_{A}}D_{2^{k}}\right)+cV\left(n\right),$

where $c$ is an absolute constant.

Proof: Let

$n=\sum_{i=1}^{r}2^{n_{i}},n_{1}>n_{2}>...>n_{r}\geq 0.$

By using Lemma 3.22 for $n$-th Fejér kernels we can conclude that

$\displaystyle nK_{n}$ $\displaystyle=$
$\displaystyle\sum_{A=1}^{r}\left(\underset{j=1}{\overset{A-1}{\prod}}w_{2^{n_{j}}}\right)\left(\left(2^{n_{A}}K_{2^{n_{A}}}+\left(2^{n_{A}}-1\right)D_{2^{n_{A}}}\right)\right)$
$\displaystyle-$
$\displaystyle\sum_{A=1}^{r}\left(\underset{j=1}{\overset{A-1}{\prod}}w_{2^{n_{j}}}\right)\left(2^{n_{A}}-1-n^{\left(A\right)}\right)D_{2^{n_{A}}}=I_{1}-I_{2}.$

For $I_{1}$ we have the following equality

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\sum_{v=1}^{r}\left(\underset{j=1}{\overset{v-1}{\prod}}\underset{i=l_{j}}{\overset{m_{j}}{\prod}}w_{2^{i}}\right)\left(\sum_{k=l_{v}}^{m_{v}}\left(\underset{j=k+1}{\overset{m_{v}}{\prod}}w_{2^{j}}\right)\left(2^{k}K_{2^{k}}-\left(2^{k}-1\right)D_{2^{k}}\right)\right)$
$\displaystyle=$
$\displaystyle\sum_{v=1}^{r}\left(\underset{j=1}{\overset{v-1}{\prod}}\underset{i=l_{j}}{\overset{m_{j}}{\prod}}w_{2^{i}}\right)\left(\sum_{k=0}^{m_{v}}-\sum_{k=0}^{l_{v}-1}\right)\left(\underset{j=k+1}{\overset{m_{v}}{\prod}}w_{2^{j}}\right)\left(2^{k}K_{2^{k}}-\left(2^{k}-1\right)D_{2^{k}}\right)$
$\displaystyle=$
$\displaystyle\sum_{v=1}^{r}\left(\underset{j=1}{\overset{v-1}{\prod}}\underset{i=l_{j}}{\overset{m_{j}}{\prod}}w_{2^{i}}\right)\left(\sum_{k=0}^{m_{v}}\left(\underset{j=k+1}{\overset{m_{v}}{\prod}}w_{2^{j}}\right)\left(2^{k}K_{2^{k}}-\left(2^{k}-1\right)D_{2^{k}}\right)\right)$
$\displaystyle-$
$\displaystyle\sum_{v=1}^{r}\left(\underset{j=1}{\overset{v}{\prod}}\underset{i=l_{j}}{\overset{m_{j}}{\prod}}w_{2^{i}}\right)\left(\sum_{k=0}^{l_{v}-1}\left(\underset{j=k+1}{\overset{l_{v}-1}{\prod}}w_{2^{j}}\right)\left(2^{k}K_{2^{k}}-\left(2^{k}-1\right)D_{2^{k}}\right)\right).$

Since

$2^{n}-1=\sum_{k=0}^{n-1}2^{k}$

and

$\left(2^{n}-1\right)K_{2^{n}-1}=\sum_{k=0}^{n-1}\left(\prod_{j=k+1}^{n-1}w_{2^{j}}\right)\left(2^{k}K_{2^{k}}-\left(2^{k}-1\right)D_{2^{k}}\right),$

we obtain that

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\sum_{v=1}^{r}\left(\underset{j=1}{\overset{v-1}{\prod}}\underset{i=l_{j}}{\overset{m_{j}}{\prod}}w_{2^{i}}\right)\left(2^{m_{v}+1}-1\right)K_{2^{m_{v}+1}-1}$
$\displaystyle-$
$\displaystyle\sum_{v=1}^{r}\left(\underset{j=1}{\overset{v}{\prod}}\underset{i=l_{j}}{\overset{m_{j}}{\prod}}w_{2^{i}}\right)\left(2^{l_{v}}-1\right)K_{2^{l_{v}}-1}.$

If we apply estimations

$\left|K_{2^{n}}\right|\leq c\left|K_{2^{n-1}}\right|$

and

$\left|K_{2^{n}-1}\right|\leq c\left|K_{2^{n}}\right|+c\ $

we get that

$\left|I_{1}\right|\leq
c\sum_{v=1}^{r}\left(2^{l_{v}}\left|K_{2^{l_{v}}}\right|+2^{m_{v}}\left|K_{2^{m_{v}}}\right|+cr\right).$
(3.1)

Let $l_{j}<n_{A}\leq m_{j},$ where $j=1,...,s.$ Then

$n^{\left(A\right)}\geq\sum_{v=l_{j}}^{n_{A}-1}2^{v}\geq 2^{n_{A}}-2^{l_{j}}$

and

$2^{n_{A}}-1-n^{\left(A\right)}\leq 2^{l_{j}}.$

If $l_{j}=n_{A}$ where $j=1,...,s,$ then

$n^{\left(A\right)}\leq 2^{m_{j-1}+1}<2^{l_{j}}.$

By using these estimations we can conclude that

$\left|I_{2}\right|\leq
c\sum_{v=1}^{r}2^{l_{v}}\sum_{k=l_{v}}^{m_{v}}D_{2^{k}}.$ (3.2)

By combining (3.1)-(3.2) we get the proof of Lemma 3.25.

The following estimations of Fejér kernels with respect to the one-dimensional
Walsh-Fourier series is proved in [64]:

###### Lemma 3.26

Let

$n=\sum_{i=1}^{s}\sum_{k=l_{i}}^{m_{i}}2^{k},$

where

$0\leq l_{1}\leq m_{1}\leq l_{2}-2<l_{2}\leq m_{2}\leq...\leq
l_{s}-2<l_{s}\leq m_{s}.$

Then

$n\left|K_{n}\left(x\right)\right|\geq\frac{2^{2l_{i}}}{16},\text{ \ \ for \ \
}x\in I_{l_{i}+1}\left(e_{l_{i}-1}+e_{l_{i}}\right).$

Proof: If we apply Lemma 3.22 for
$n=\sum_{i=1}^{s}\sum_{k=l_{i}}^{m_{i}}2^{k}$ we can write that

$\displaystyle nK_{n}$ $\displaystyle=$
$\displaystyle\sum_{r=1}^{s}\sum_{k=l_{r}}^{m_{r}}\left(\underset{j=r+1}{\overset{s}{\prod}}\underset{q=l_{j}}{\overset{m_{j}}{\prod}}w_{2^{q}}\underset{j=k+1}{\overset{m_{r}}{\prod}}w_{2^{j}}\right)2^{k}K_{2^{k}}$
$\displaystyle+$
$\displaystyle\sum_{r=1}^{s}\sum_{k=l_{r}}^{m_{r}}\left(\underset{j=r+1}{\overset{s}{\prod}}\underset{q=l_{j}}{\overset{m_{j}}{\prod}}w_{2^{q}}\underset{j=k+1}{\overset{m_{r}}{\prod}}w_{2^{j}}\right)\left(\sum_{t=1}^{r-1}\sum_{q=l_{t}}^{m_{t}}2^{q}+\sum_{q=l_{r}}^{k-1}2^{q}\right)D_{2^{k}}.$

Let $x\in I_{l_{i}+1}\left(e_{l_{i}-1}+e_{l_{i}}\right).$ Then

$\displaystyle
n\left|K_{n}\right|\geq\left|2^{l_{i}}K_{2^{l_{i}}}\right|-\sum_{r=1}^{i-1}\sum_{k=l_{r}}^{m_{r}}\left|2^{k}K_{2^{k}}\right|-\sum_{r=1}^{i-1}\sum_{k=l_{r}}^{m_{r}}\left|2^{k}D_{2^{k}}\right|=I-II-
III.$

Lemma 3.23 follows that

$I=\left|2^{l_{i}}K_{2^{l_{i}}}\left(x\right)\right|=\frac{2^{2l_{i}}}{4}.$
(3.3)

Since $m_{i-1}\leq l_{i}-2$, we easily obtain that the following estimation is
true:

$\displaystyle II$ $\displaystyle\leq$
$\displaystyle\sum_{n=0}^{l_{i}-2}\left|2^{n}K_{2^{n}}\left(x\right)\right|$
$\displaystyle\leq$
$\displaystyle\sum_{n=0}^{l_{i}-2}2^{n}\frac{\left(2^{n}+1\right)}{2}$
$\displaystyle\leq$
$\displaystyle\frac{2^{2l_{i}}}{24}+\frac{2^{l_{i}}}{4}-\frac{2}{3}.$

For $III$ we get that

$III\leq\sum_{k=0}^{l_{i}-2}\left|2^{k}D_{2^{k}}\left(x\right)\right|\leq\sum_{k=0}^{l_{i}-2}4^{k}=\frac{2^{2l_{i}}}{12}-\frac{1}{3}.$
(3.5)

By combining (3.3-3.5) we can conclude that

$n\left|K_{n}\left(x\right)\right|\geq I-II-
III\geq\frac{2^{2l_{i}}}{8}-\frac{2^{l_{i}}}{4}+1.$ (3.6)

Suppose that $l_{i}\geq 2$. Then

$n\left|K_{n}\left(x\right)\right|\geq\frac{2^{2l_{i}}}{8}-\frac{2^{2l_{i}}}{16}\geq\frac{2^{2l_{i}}}{16}.$

If $l_{i}=0$ or $l_{i}=1$, then by applying (3.6) we get that

$n\left|K_{n}\left(x\right)\right|\geq\frac{7}{8}\geq\frac{2^{2l_{i}}}{16},$

Lemma is proved.

The following estimations of Fejér kernels with respect to the one-dimensional
Walsh-Fourier series is proved in [64] (see also [74]):

###### Lemma 3.27

Let $0<p\leq 1$, $2^{k}\leq n<2^{k+1}$ and $\sigma_{n}f$ be Fejér means with
respect to the one-dimensional Walsh-Fourier series, where $f\in H_{p}(G)$.
Then, for any fixed $n\in\mathbb{N}$,

$\displaystyle\left\|\sigma_{n}f\right\|_{H_{p}(G)}$
$\displaystyle\leq\left\|\sup_{0\leq l\leq
k}\left|\sigma_{2^{l}}f\right|\right\|_{p}+\left\|\sup_{0\leq l\leq
k}\left|S_{2^{l}}f\right|\right\|_{p}+\left\|\sigma_{n}f\right\|_{p}$
$\displaystyle\leq\left\|\widetilde{\sigma}_{\\#}^{\ast}f\right\|_{p}+\left\|\widetilde{S}_{\\#}^{\ast}f\right\|_{p}+\left\|\sigma_{n}f\right\|_{p}.$

Proof: Let consider the following martingale

$\displaystyle f_{\\#}$ $\displaystyle=$
$\displaystyle\left(S_{2^{k}}\sigma_{n}f,\text{ }k\in\mathbb{N}\right)$
$\displaystyle=$
$\displaystyle\left(\frac{2^{0}\sigma_{2^{0}}}{n}+\frac{(n-2^{0})S_{2^{0}}f}{n},...,\frac{2^{k}\sigma_{2^{k}}f}{n}+\frac{(n-2^{k})S_{2^{k}}f}{n},\sigma_{n}f,...,\sigma_{n}f,...\right).$

By using Lemma 2.8 we immediately get

$\displaystyle\left\|\sigma_{n}f\right\|_{H_{p}(G^{2})}^{p}$
$\displaystyle\leq\left\|\sup_{0\leq l\leq
k}\left|\sigma_{2^{l}}f\right|\right\|_{p}^{p}+\left\|\sup_{0\leq l\leq
k}\left|S_{2^{l}}f\right|\right\|_{p}^{p}+\left\|S_{n}f\right\|_{p}^{p}$
$\displaystyle\leq\left\|\widetilde{\sigma}_{\\#}^{\ast}f\right\|_{p}^{p}+\left\|\widetilde{S}_{\\#}^{\ast}f\right\|_{p}^{p}+\left\|\sigma_{n}f\right\|_{p}^{p}.$

Lemma is proved.

###  3.3 Boundedness of subsequences of Fejér means with respect to the one-
dimensional Walsh-Fourier series on the martingale Hardy spaces

In this section we study boundedness of subsequences of Fejér means with
respect to the one-dimensional Walsh-Fourier series in the martingale Hardy
spaces (For details see [64]).

First, we consider case $p=1/2$. The following estimation is true:

###### Theorem 3.28

a) Let $f\in H_{1/2}(G).$ Then there exists an absolute constant $c,$ such
that

$\left\|\sigma_{n}f\right\|_{H_{1/2}(G)}\leq
cV^{2}\left(n\right)\left\|f\right\|_{H_{1/2}(G)}.$

b) Let $\left\\{n_{k}:k\in\mathbb{N}_{+}\right\\}$ be increasing sequence of
natural numbers, such that
$\sup_{k\in\mathbb{N}_{+}}V\left(n_{k}\right)=\infty$ and
$\Phi:\mathbb{N}_{+}\rightarrow[1,\infty)$ be non-decreasing function
satisfying the conditions $\Phi\left(n\right)\uparrow\infty$ and

$\overline{\underset{k\rightarrow\infty}{\lim}}\frac{V^{2}\left(n_{k}\right)}{\Phi\left(n_{k}\right)}=\infty.$
(3.7)

Then there exists a martingale $f\in H_{1/2}(G),$ such that

$\underset{k\in\mathbb{N}}{\sup}\left\|\frac{\sigma_{n_{k}}f}{\Phi\left(n_{k}\right)}\right\|_{1/2}=\infty.$

Proof: Suppose that

$\left\|\frac{\sigma_{n}f}{V^{2}\left(n\right)}\right\|_{1/2}\leq
c\left\|f\right\|_{H_{1/2}(G)}.$ (3.8)

By combining estimations (1.7), (1.15) and Lemma 3.27 we can conclude that

$\displaystyle\left\|\frac{\sigma_{n}f}{V^{2}\left(n\right)}\right\|_{H_{1/2}(G)}^{1/2}$
$\displaystyle\leq$
$\displaystyle\left\|\frac{\sigma_{n}f}{V^{2}\left(n\right)}\right\|_{1/2}^{1/2}+\frac{1}{V^{2}\left(n\right)}\left\|\sigma_{\\#}^{\ast}f\right\|_{1/2}^{1/2}+\frac{1}{V^{2}\left(n\right)}\left\|\widetilde{S}_{\\#}^{\ast}\right\|_{1/2}^{1/2}$
$\displaystyle\leq$
$\displaystyle\left\|\frac{\widetilde{\sigma}_{n}f}{V^{2}\left(n\right)}\right\|_{1/2}^{1/2}+\left\|\widetilde{\sigma}_{\\#}^{\ast}f\right\|_{1/2}^{1/2}+\left\|\widetilde{S}_{\\#}^{\ast}f\right\|_{1/2}^{1/2}\leq
c\left\|f\right\|_{H_{1/2}(G)}^{1/2}.$

By combining Lemma 2.7 and (3.3), Theorem 3.28 will be proved if we show that

$\int_{\overline{I_{M}}}\left(\frac{\left|\sigma_{n}a\right|}{V^{2}\left(n\right)}\right)^{1/2}d\mu\leq
c<\infty,$

for any $1/2$-atom $a$.

Without loss the generality we may assume that $a$ is $1/2$-atom, with support
$I,$ for which $\mu\left(I\right)=2^{-M},$ $I=I_{M}.$ It is easy to check that
$\sigma_{n}\left(a\right)=0,$ when $n\leq 2^{M}.$ Therefore, we may assume
that $n>2^{M}.$ Set

$\displaystyle
II_{\alpha_{A}}^{1}\left(x\right):=2^{M}\int_{I_{M}}2^{\alpha_{A}}\left|K_{2^{\alpha_{A}}}\left(x+t\right)\right|d\mu\left(t\right),\text{\
}$ $\displaystyle
II_{l_{A}}^{2}\left(x\right)=2^{M}\int_{I_{M}}2^{l_{A}}\sum_{k=l_{A}}^{m_{A}}D_{2^{k}}\left(x+t\right)d\mu\left(t\right).$

Let $x\in I_{M}.$ Since $\sigma_{n}$ is bounded from $L_{\infty}(G)$ to
$L_{\infty}(G)$, for $n>2^{M}$ and $\left\|a\right\|_{\infty}\leq 2^{2M},$ by
using Lemma 3.24 we can conclude that

$\displaystyle\frac{\left|\sigma_{n}a\left(x\right)\right|}{V^{2}\left(n\right)}$
$\displaystyle\leq\frac{c}{V^{2}\left(n\right)}\int_{I_{M}}\left|a\left(x\right)\right|\left|K_{n}\left(x+t\right)\right|d\mu\left(t\right)$
$\displaystyle\leq\frac{c\left\|a\right\|_{\infty}}{V^{2}\left(n\right)}\int_{I_{M}}\left|K_{n}\left(x+t\right)\right|d\mu\left(t\right)$
$\displaystyle\leq\frac{c2^{2M}}{V^{2}\left(n\right)}\int_{I_{M}}\left|K_{n}\left(x+t\right)\right|d\mu\left(t\right)$
$\displaystyle\leq\frac{c2^{M}}{V^{2}\left(n\right)}\left\\{\sum_{A=1}^{s}\int_{I_{M}}2^{l_{A}}\left|K_{2^{l_{A}}}\left(x+t\right)\right|d\mu\left(t\right)+\int_{I_{M}}2^{m_{A}}\left|K_{2^{m_{A}}}\left(x+t\right)\right|d\mu\left(t\right)\right\\}$
$\displaystyle+\frac{c2^{M}}{V^{2}\left(n\right)}\sum_{A=1}^{s}\int_{I_{M}}2^{l_{A}}\sum_{k=l_{A}}^{m_{A}}D_{2^{k}}\left(x+t\right)d\mu\left(t\right)+\frac{c2^{M}}{V^{2}\left(n\right)}\int_{I_{M}}V\left(n\right)d\mu\left(t\right)$
$\displaystyle=\frac{c}{V^{2}\left(n\right)}\sum_{A=1}^{s}\left(II_{{}^{l_{A}}}^{1}\left(x\right)+II_{{}^{m_{A}}}^{1}\left(x\right)+II_{l_{A}}^{2}\left(x\right)\right)+c.$

Hence,

$\displaystyle\int_{\overline{I_{M}}}\left|\frac{\sigma_{n}a\left(x\right)}{V^{2}\left(n\right)}\right|^{1/2}d\mu\left(x\right)$
$\displaystyle\leq$
$\displaystyle\frac{c}{V\left(n\right)}\left(\sum_{A=1}^{s}\int_{\overline{I_{M}}}\left|II_{l_{A}}^{1}\left(x\right)\right|^{1/2}d\mu\left(x\right)\right.$
$\displaystyle\left.+\int_{\overline{I_{M}}}\left|II_{m_{A}}^{1}\left(x\right)\right|^{1/2}d\mu\left(x\right)+\int_{\overline{I_{M}}}\left|II_{l_{A}}^{2}\left(x\right)\right|^{1/2}d\mu\left(x\right)\right)+c.$

Since $s\leq 4V\left(n\right)$ we obtain that Theorem 3.28 will be proved if
we show that

$\int_{\overline{I_{M}}}\left|II_{\alpha_{A}}^{1}\left(x\right)\right|^{1/2}d\mu\left(x\right)\leq
c<\infty,\text{\
}\int_{\overline{I_{M}}}\left|II_{l_{A}}^{2}\left(x\right)\right|^{1/2}d\mu\left(x\right)\leq
c<\infty,\text{\ }$ (3.10)

where $\alpha_{A}=l_{A}$ or $\alpha_{A}=m_{A},$ $A=1,...,s$.

Let $t\in I_{M}$ and $x\in I_{l+1}\left(e_{k}+e_{l}\right),$ $0\leq
k<l<\alpha_{A}\leq M$ or $0\leq k<l\leq M\leq\alpha_{A}.$ Since $x+t\in
I_{l+1}\left(e_{k}+e_{l}\right),$ by applying Lemma 3.23 we can conclude that

$K_{2^{\alpha_{A}}}\left(x+t\right)=0\text{ \ \ and \ \
}II_{\alpha_{A}}^{1}\left(x\right)=0.$ (3.11)

Let $x\in I_{l+1}\left(e_{k}+e_{l}\right),$ $0\leq k<\alpha_{A}\leq l\leq M.$
Then $x+t\in I_{l+1}\left(e_{k}+e_{l}\right),$ where $t\in I_{M}$ and if we
apply again Lemma 3.23 we get that

$2^{\alpha_{A}}\left|K_{2^{\alpha_{A}}}\left(x+t\right)\right|\leq
2^{\alpha_{A}+k}\text{ \ \ and \ \ }II_{\alpha_{A}}^{1}\left(x\right)\leq
2^{\alpha_{A}+k}.$ (3.12)

Analogously to (3.12) for $0\leq\alpha_{A}\leq k<l\leq M$ we can prove that

$2^{\alpha_{A}}\left|K_{2^{\alpha_{A}}}\left(x+t\right)\right|\leq
2^{2\alpha_{A}},\text{ \ }II_{\alpha_{A}}^{1}\left(x\right)\leq
2^{2\alpha_{A}},\text{\ }t\in I_{M},\text{\ }x\in
I_{l+1}\left(e_{k}+e_{l}\right).$ (3.13)

Let $0\leq\alpha_{A}\leq M-1,$ where $A=1,...,s.$ According to (2.1) and
(3.11-3.13) we find that

$\displaystyle\int_{\overline{I_{M}}}\left|II_{\alpha_{A}}^{1}\left(x\right)\right|^{1/2}d\mu\left(x\right)$
$\displaystyle=$
$\displaystyle\overset{M-2}{\underset{k=0}{\sum}}\overset{M-1}{\underset{l=k+1}{\sum}}\int_{I_{l+1}\left(e_{k}+e_{l}\right)}\left|II_{\alpha_{A}}^{1}\left(x\right)\right|^{1/2}d\mu\left(x\right)$
$\displaystyle+$
$\displaystyle\overset{M-1}{\underset{k=0}{\sum}}\int_{I_{M}\left(e_{k}\right)}\left|II_{\alpha_{A}}^{1}\left(x\right)\right|^{1/2}d\mu\left(x\right)$
$\displaystyle\leq$ $\displaystyle
c\overset{\alpha_{A}-1}{\underset{k=0}{\sum}}\overset{M-1}{\underset{l=\alpha_{A}+1}{\sum}}\int_{I_{l+1}\left(e_{k}+e_{l}\right)}2^{\left(\alpha_{A}+k\right)/2}d\mu\left(x\right)$
$\displaystyle+$ $\displaystyle
c\overset{M-2}{\underset{k=\alpha_{A}}{\sum}}\overset{M-1}{\underset{l=k+1}{\sum}}\int_{I_{l+1}\left(e_{k}+e_{l}\right)}2^{\alpha_{A}}d\mu\left(x\right)$
$\displaystyle+$ $\displaystyle
c\overset{\alpha_{A}-1}{\underset{k=0}{\sum}}\int_{I_{M}\left(e_{k}\right)}2^{\left(\alpha_{A}+k\right)/2}d\mu\left(x\right)+c\overset{M-1}{\underset{k=\alpha_{A}}{\sum}}\int_{I_{M}\left(e_{k}\right)}2^{\alpha_{A}}d\mu\left(x\right)$
$\displaystyle\leq$ $\displaystyle
c\overset{\alpha_{A}-1}{\underset{k=0}{\sum}}\overset{M-1}{\underset{l=\alpha_{A}+1}{\sum}}\frac{2^{\left(\alpha_{A}+k\right)/2}}{2^{l}}+c\overset{M-2}{\underset{k=\alpha_{A}}{\sum}}\overset{M-1}{\underset{l=k+1}{\sum}}\frac{2^{\alpha_{A}}}{2^{l}}$
$\displaystyle+$ $\displaystyle
c\overset{\alpha_{A}-1}{\underset{k=0}{\sum}}\frac{2^{\left(\alpha_{A}+k\right)/2}}{2^{M}}+c\overset{M-1}{\underset{k=\alpha_{A}}{\sum}}\frac{2^{\alpha_{A}}}{2^{M}}\leq
c<\infty.$

Let $\alpha_{A}\geq M.$ Analogously to $II_{\alpha_{A}}^{1}\left(x\right)$ we
can prove (3.10), for $A=1,...,s.$

Now, prove boundedness of $II_{l_{A}}^{2}$. Let $t\in I_{M}$ and $x\in
I_{i}\backslash I_{i+1},$ $i\leq l_{A}-1.$ Since $x+t\in I_{i}\backslash
I_{i+1},$ if we apply first equality of Lemma 2.3 we get that

$II_{l_{A}}^{2}\left(x\right)=0.$ (3.14)

Let $x\in I_{i}\backslash I_{i+1},$ $l_{A}\leq i\leq m_{A}.$ Since $n\geq
2^{M}$ and $t\in I_{M},$ if we apply first equality of Lemma 2.3 we get that

$II_{l_{A}}^{2}\left(x\right)\leq
2^{M}\int_{I_{M}}2^{l_{A}}\sum_{k=l_{A}}^{i}D_{2^{k}}\left(x+t\right)d\mu\left(t\right)\leq
c2^{l_{A}+i}.$ (3.15)

Let $x\in I_{i}\backslash I_{i+1},$ $m_{A}<i\leq M-1.$ Then $x+t\in
I_{i}\backslash I_{i+1},$ for any $t\in I_{M}$ and by first equality of Lemma
2.3 we have that

$II_{l_{A}}^{2}\left(x\right)\leq c2^{M}\int_{I_{M}}2^{l_{A}+m_{A}}\leq
c2^{l_{A}+m_{A}}.$ (3.16)

Let $0\leq l_{A}\leq m_{A}\leq M.$ Then, in the view of (2.1) and (3.14-3.16)
we can conclude that

$\displaystyle\int_{\overline{I_{M}}}\left|II_{l_{A}}^{2}\left(x\right)\right|^{1/2}d\mu\left(x\right)$
$\displaystyle=$
$\displaystyle\left(\sum_{i=0}^{l_{A}-1}+\sum_{i=l_{A}}^{m_{A}}+\sum_{i=m_{A}+1}^{M-1}\right)\int_{I_{i}\backslash
I_{i+1}}\left|II_{l_{A}}^{2}\left(x\right)\right|^{1/2}d\mu\left(x\right)$
$\displaystyle\leq$ $\displaystyle
c\sum_{i=l_{A}}^{m_{A}}\int_{I_{i}\backslash
I_{i+1}}2^{\left(l_{A}+i\right)/2}d\mu\left(x\right)$ $\displaystyle+$
$\displaystyle c\sum_{i=m_{A}+1}^{M-1}\int_{I_{i}\backslash
I_{i+1}}2^{\left(l_{A}+m_{A}\right)/2}d\mu\left(x\right)$ $\displaystyle\leq$
$\displaystyle
c\sum_{i=l_{A}}^{m_{A}}2^{\left(l_{A}+i\right)/2}\frac{1}{2^{i}}$
$\displaystyle+$ $\displaystyle
c\sum_{i=m_{A}+1}^{M-1}2^{\left(l_{A}+m_{A}\right)/2}\frac{1}{2^{i}}\leq
c<\infty.$

Analogously, we can prove same estimations in the cases $0\leq l_{A}\leq
M<m_{A}$ and $M\leq l_{A}\leq m_{A}.$

Now, we prove part b) of Theorem 3.28. According to (3.7), there exists
increasing sequence $\left\\{\alpha_{k}:\text{
}k\in\mathbb{N}_{+}\right\\}\subset\left\\{n_{k}\text{
}:k\in\mathbb{N}_{+}\right\\}$ of natural numbers such that

$\sum_{k=1}^{\infty}\frac{\Phi^{1/4}\left(\alpha_{k}\right)}{V^{1/2}\left(\alpha_{k}\right)}\leq
c<\infty.$ (3.17)

Let $f=\left(f_{n},n\in\mathbb{N}_{+}\right)$ be martingale form Example 2.6,
where

$\lambda_{k}:=\Phi^{1/2}\left(\alpha_{k}\right)/V\left(\alpha_{k}\right).$

According to (3.17) we get that condition (2.3) is fulfilled and it follows
that $f=\left(f_{n},n\in\mathbb{N}_{+}\right)$.

By applying (2.4) we get that

$\widehat{f}(j)$ (3.18)
$=\left\\{\begin{array}[]{ll}2^{\left|\alpha_{k}\right|}\Phi^{1/2}\left(\alpha_{k}\right)/V\left(\alpha_{k}\right),&\text{\thinspace\thinspace}j\in\left\\{2^{\left|\alpha_{k}\right|},...,2^{{}_{\left|\alpha_{k}\right|+1}}-1\right\\},\text{
}k\in\mathbb{N}_{+}\\\
0\,,&\text{\thinspace}j\notin\bigcup\limits_{k=0}^{\infty}\left\\{2^{{}_{\left|\alpha_{k}\right|}},...,2^{{}_{\left|\alpha_{k}\right|+1}}-1\right\\}.\end{array}\right.$

Let $2^{\left|\alpha_{k}\right|}<j<\alpha_{k}.$ If we apply (2.6) we get that

$S_{j}f=S_{2^{\left|\alpha_{k}\right|}}f+\frac{w_{2^{\left|\alpha_{k}\right|}}D_{j-2^{\left|\alpha_{k}\right|}}\Phi^{1/2}\left(\alpha_{k}\right)}{V\left(\alpha_{k}\right)}$
(3.19)

Hence,

$\displaystyle\frac{\sigma_{{}_{\alpha_{k}}}f}{\Phi\left(\alpha_{k}\right)}$
$\displaystyle=$
$\displaystyle\frac{1}{\Phi\left(\alpha_{k}\right)\alpha_{k}}\sum_{j=1}^{2^{\left|\alpha_{k}\right|}}S_{j}f$
$\displaystyle+$
$\displaystyle\frac{1}{\Phi\left(\alpha_{k}\right)\alpha_{k}}\sum_{j=2^{\left|\alpha_{k}\right|}+1}^{\alpha_{k}}S_{j}f$
$\displaystyle=$
$\displaystyle\frac{\sigma_{{}_{2^{\left|\alpha_{k}\right|}}}f}{\Phi\left(\alpha_{k}\right)\alpha_{k}}$
$\displaystyle+$
$\displaystyle\frac{\left(\alpha_{k}-2^{\left|\alpha_{k}\right|}\right)S_{2^{\left|\alpha_{k}\right|}}f}{\Phi\left(\alpha_{k}\right)\alpha_{k}}$
$\displaystyle+$
$\displaystyle\frac{w_{2^{\left|\alpha_{k}\right|}}2^{\left|\alpha_{k}\right|}\Phi^{1/2}\left(\alpha_{k}\right)}{\Phi\left(\alpha_{k}\right)V\left(\alpha_{k}\right)\alpha_{k}}\sum_{j=2^{{}_{\left|\alpha_{k}\right|}}+1}^{\alpha_{k}}D_{j-2^{\left|\alpha_{k}\right|}}$
$\displaystyle=$ $\displaystyle III_{1}+III_{2}+III_{3}.$

For $III_{3}$ we can conclude that

$\displaystyle\left|III_{3}\right|$ $\displaystyle=$
$\displaystyle\frac{2^{\left|\alpha_{k}\right|}\Phi^{1/2}\left(\alpha_{k}\right)}{\Phi\left(\alpha_{k}\right)V\left(\alpha_{k}\right)\alpha_{k}}\left|\sum_{j=1}^{\alpha_{k}-2^{{}_{\left|\alpha_{k}\right|}}}D_{j}\right|$
$\displaystyle=$
$\displaystyle\frac{2^{\left|\alpha_{k}\right|}\Phi^{1/2}\left(\alpha_{k}\right)}{\Phi\left(\alpha_{k}\right)V\left(\alpha_{k}\right)\alpha_{k}}\left(\alpha_{k}-2^{{}_{\left|\alpha_{k}\right|}}\right)\left|K_{\alpha_{k}-2^{{}_{\left|\alpha_{k}\right|}}}\right|$
$\displaystyle\geq$
$\displaystyle\frac{c\left(\alpha_{k}-2^{{}_{\left|\alpha_{k}\right|}}\right)\left|K_{\alpha_{k}-2^{{}_{\left|\alpha_{k}\right|}}}\right|}{\Phi^{1/2}\left(\alpha_{k}\right)V\left(\alpha_{k}\right).}$

Let

$\alpha_{k}=\sum_{i=1}^{r_{k}}\sum_{k=l_{i}^{k}}^{m_{i}^{k}}2^{k},$

where

$m_{1}^{k}\geq l_{1}^{k}>l_{1}^{k}-2\geq m_{2}^{k}\geq
l_{2}^{k}>l_{2}^{k}-2\geq...\geq m_{s}^{k}\geq l_{s}^{k}\geq 0.$

Since (see theorems 2.10 and 3.28)

$\left\|III_{1}\right\|_{1/2}\leq c,\left\|III_{2}\right\|_{1/2}\leq c,$

and

$\mu\left\\{E_{l_{i}^{k}}\right\\}\geq 1/2^{l_{i}^{k}-1},$

By combining (3.3), (3.3) and Lemma 3.26 we get that

$\displaystyle\int_{G}\left|\sigma_{\alpha_{k}}f(x)/\Phi\left(\alpha_{k}\right)\right|^{1/2}d\mu\left(x\right)$
$\displaystyle\geq$
$\displaystyle\left\|III_{3}\right\|_{1/2}^{1/2}-\left\|III_{2}\right\|_{1/2}^{1/2}-\left\|III_{1}\right\|_{1/2}^{1/2}$
$\displaystyle\geq$ $\displaystyle c\text{
}\underset{i=2}{\overset{r_{k}-2}{\sum}}\int_{E_{l_{i}^{k}}}\left|2^{2l_{i}^{k}}/\left(\Phi^{1/2}\left(\alpha_{k}\right)V\left(\alpha_{k}\right)\right)\right|^{1/2}d\mu\left(x\right)-2c$
$\displaystyle\geq$ $\displaystyle
c\overset{r_{k}-2}{\underset{i=2}{\sum}}1/\left(V^{1/2}\left(\alpha_{k}\right)\Phi^{1/4}\left(\alpha_{k}\right)\right)-2c$
$\displaystyle\geq$ $\displaystyle
cr_{k}/\left(V^{1/2}\left(\alpha_{k}\right)\Phi^{1/4}\left(\alpha_{k}\right)\right)$
$\displaystyle\geq$ $\displaystyle
cV^{1/2}\left(\alpha_{k}\right)/\Phi^{1/4}\left(\alpha_{k}\right)\rightarrow\infty,\text{
as }k\rightarrow\infty.$

Theorem 3.28 is proved.

###### Theorem 3.29

a) Let $0<p<1/2,$ $f\in H_{p}(G).$ Then there exists an absolute constant
$c_{p}$ depending only on $p$ such that

$\text{ }\left\|\sigma_{n}f\right\|_{H_{p}(G)}\leq
c_{p}2^{d\left(n\right)\left(1/p-2\right)}\left\|f\right\|_{H_{p}(G)}.$

b) Let $0<p<1/2$ and $\Phi\left(n\right):\mathbb{N}_{+}\rightarrow[1,\infty)$
be non-decreasing function such that

$\sup_{k\in\mathbb{N}_{+}}d\left(n_{k}\right)=\infty,\text{ \ \
}\overline{\underset{k\rightarrow\infty}{\lim}}\frac{2^{d\left(n_{k}\right)\left(1/p-2\right)}}{\Phi\left(n_{k}\right)}=\infty.$
(3.22)

Then there exist a martingale $f\in H_{p}(G),$ such that

$\underset{k\in\mathbb{N}_{+}}{\sup}\left\|\frac{\sigma_{n_{k}}f}{\Phi\left(n_{k}\right)}\right\|_{weak-
L_{p}(G)}=\infty.$

Proof: Let $n\in\mathbb{N}.$ Analogously to (3.3) it is sufficient to prove
that

$\int\limits_{\overline{I}_{M}}\left(2^{d\left(n\right)\left(2-1/p\right)}\left|\sigma_{n}\left(a\right)\right|\right)^{p}d\mu\leq
c_{p}<\infty,$

for every $p$-atom $a$, where $I$ denotes support of the atom.

Analogously to Theorem 3.28 we may assume that $a$ is $p$-atom with support $\
I=I_{M}$, $\mu\left(I_{M}\right)=2^{-M}$ and $n>2^{M}.$ Since
$\left\|a\right\|_{\infty}\leq 2^{M/p}$ we can conclude that

$\displaystyle 2^{d\left(n\right)\left(2-1/p\right)}\left|\sigma_{n}a\right|$
$\displaystyle\leq$ $\displaystyle
2^{d\left(n\right)\left(2-1/p\right)}\left\|a\right\|_{\infty}\int_{I_{M}}\left|K_{n}\left(x+t\right)\right|d\mu\left(t\right)$
$\displaystyle\leq$ $\displaystyle
2^{d\left(n\right)\left(2-1/p\right)}2^{M/p}\int_{I_{M}}\left|K_{n}\left(x+t\right)\right|d\mu\left(t\right).$

Let $x\in I_{l+1}\left(e_{k}+e_{l}\right),\,0\leq k,l\leq\left[n\right]\leq
M.$ Then, by applying Lemma 3.23 we get that $K_{n}\left(x+t\right)=0,$ where
$t\in I_{M}$ and hence,

$2^{d\left(n\right)\left(2-1/p\right)}\left|\sigma_{n}a\right|=0.$ (3.23)

Let $x\in I_{l+1}\left(e_{k}+e_{l}\right),\,\left[n\right]\leq k,l\leq M$ or
$k\leq\left[n\right]\leq l\leq M.$ Then Lemma 3.25 follows that

$\displaystyle 2^{d\left(n\right)\left(2-1/p\right)}\left|\sigma_{n}a\right|$
$\displaystyle\leq$ $\displaystyle
2^{d\left(n\right)\left(2-1/p\right)}2^{M\left(1/p-2\right)+k+l}$
$\displaystyle\leq$ $\displaystyle
c_{p}2^{\left[n\right]\left(1/p-2\right)+k+l}.$

By combining (2.1), (3.23) and (3.3) we can conclude that

$\displaystyle\int_{\overline{I_{M}}}\left|2^{d\left(n\right)\left(2-1/p\right)}\sigma_{n}a\left(x\right)\right|^{p}d\mu\left(x\right)$
$\displaystyle\leq$
$\displaystyle\left(\overset{\left[n\right]-2}{\underset{k=0}{\sum}}\overset{\left[n\right]-1}{\underset{l=k+1}{\sum}}+\overset{\left[n\right]-1}{\underset{k=0}{\sum}}\overset{M-1}{\underset{l=\left[n\right]}{\sum}}+\overset{M-2}{\underset{k=\left[n\right]}{\sum}}\overset{M-1}{\underset{l=k+1}{\sum}}\right)\int_{I_{l+1}\left(e_{k}+e_{l}\right)}\left|2^{d\left(n\right)\left(2-1/p\right)}\sigma_{n}a\left(x\right)\right|^{p}d\mu\left(x\right)$
$\displaystyle+$
$\displaystyle\overset{M-1}{\underset{k=0}{\sum}}\int_{I_{M}\left(e_{k}\right)}\left|2^{d\left(n\right)\left(2-1/p\right)}\sigma_{n}a\left(x\right)\right|^{p}d\mu\left(x\right)$
$\displaystyle\leq$ $\displaystyle
c_{p}\overset{M-2}{\underset{k=\left[n\right]}{\sum}}\overset{M-1}{\underset{l=k+1}{\sum}}\frac{1}{2^{l}}2^{\left[n\right]\left(2p-1\right)}2^{p\left(k+l\right)}$
$\displaystyle+$ $\displaystyle
c_{p}\overset{\left[n\right]}{\underset{k=0}{\sum}}\overset{M-1}{\underset{l=\left[n\right]+1}{\sum}}\frac{1}{2^{l}}2^{\left[n\right]\left(2p-1\right)}2^{p\left(k+l\right)}$
$\displaystyle+$
$\displaystyle\frac{c_{p}2^{\left[n\right]\left(2p-1\right)}}{2^{M}}\overset{\left[n\right]}{\underset{k=0}{\sum}}2^{p\left(k+M\right)}<c_{p}<\infty.$

Now, we prove part b) of Theorem 3.29. According to (3.22) there exists an
increasing sequence of natural numbers $\left\\{\alpha_{k}:\text{
}k\in\mathbb{N}_{+}\right\\}\subset\left\\{n_{k}:\text{
}k\in\mathbb{N}_{+}\right\\},$ such that $\alpha_{0}\geq 3$ and

$\sum_{\eta=0}^{\infty}u^{-p}\left(\alpha_{\eta}\right)<c_{p}<\infty,\text{ \
\ }\text{\
}u\left(\alpha_{k}\right)=2^{d\left(\alpha_{k}\right)\left(1/p-2\right)/2}/\Phi^{1/2}\left(\alpha_{k}\right).$
(3.25)

Let $f$ be martingale from Example 2.6, where

$\lambda_{k}=u^{-1}\left(\alpha_{k}\right),$

If we apply (3.25) we get that (2.3) is fulfilled and it follows that $f\in
H_{p}(G).$ According to (2.4) we have that

$\widehat{f}(j)=\left\\{\begin{array}[]{ll}2^{\left|\alpha_{k}\right|\left(1/p-1\right)}/u\left(\alpha_{k}\right),&j\in\left\\{2^{\left|\alpha_{k}\right|},...,2^{{}_{\left|\alpha_{k}\right|+1}}-1\right\\},\text{
}k\in\mathbb{N}_{+},\\\
0\,,&j\notin\bigcup\limits_{k=0}^{\infty}\left\\{2^{\left|\alpha_{k}\right|},...,2^{{}_{\left|\alpha_{k}\right|+1}}-1\right\\}.\end{array}\right.$
(3.26)

Let $2^{\left|\alpha_{k}\right|}<j<\alpha_{k}.$ Then, analogously to (3.19)
and (3.3), if we apply (3.26) we get that

$\displaystyle\frac{\sigma_{{}_{\alpha_{k}}}f}{\Phi\left(\alpha_{k}\right)}$
$\displaystyle=$