# Approximation by Walsh-Kaczmarz-Fejér means on the Hardy space

George Tephnadze  G. Tephnadze, Department of Mathematics, Faculty of Exact
and Natural Sciences, Tbilisi State University, Chavchavadze str. 1, Tbilisi
0128, Georgia and Department of Engineering Sciences and Mathematics, Luleå
University of Technology, SE-971 87, Luleå, Sweden<EMAIL_ADDRESS>
###### Abstract.

The main aim of this paper is to find necessary and sufficient conditions for
the convergence of Walsh-Kaczmarz-Fejér means in the terms of the modulus of
continuity on the Hardy spaces $H_{p},$ when $0<p\leq 1/2.$

The research was supported by Shota Rustaveli National Science Foundation
grant no.13/06 (Geometry of function spaces, interpolation and embedding
theorems).

##  1\. Introduction

2000 Mathematics Subject Classification. 42C10.

Key words and phrases: Walsh-Kaczmarz system, Fejér means, martingale Hardy
space, modulus of continuity.

The a.e. convergence of Walsh-Fejér means $\sigma_{n}f$ was proved by Fine
[1]. Yano [35] has proved that $\left\|K_{n}\right\|\leq 2$
($n\in\mathbb{N}$).Consequently, $\left\|\sigma_{n}f-f\right\|_{p}\rightarrow
0$, as $n\rightarrow\infty$ ($f\in L_{p},$ $1\leq p\leq\infty$). However (see
[13, 19]) the rate of convergence can not be better then
$O\left(n^{-1}\right)$ $\left(n\rightarrow\infty\right)$ for nonconstant
functions. Fridli [2] use dyadic modulus of continuity to characterize the set
of functions in the space $L_{p}$, whose Walsh-Fejér means converge at the
given rate.

In 1975 Schipp [20] showed that the maximal operator $\sigma^{\ast}$ is of
weak type $(1,1)$ and of type $(p,p)$ for $1<p\leq\infty$. The boundedness
fails to hold for $p=1$. But, Fujii [4] proved that $\sigma^{\ast}$ is bounded
from the dyadic Hardy space $H_{1}$ to the space $L_{1}$. The theorem of Fujii
was extended by Weisz [33], he showed that the maximal operator
$\sigma^{\ast}$ is bounded from the martingale Hardy space $H_{p}$ to the
space $L_{p}$ for $p>1/2.$ Simon gave a counterexample [22], which shows that
the boundedness does not hold for $0<p<1/2.$ The counterexample for $p=1/2$
due to Goginava [9] (see also [26]). In the endpoint case $p=1/2$ Weisz [34]
proved that $\sigma^{\ast}$ is bounded from the Hardy space $H_{1/2}$ to the
space weak-$L_{1/2}$.

In 1948 $\breve{\text{S}}$neider [25] introduced the Walsh-Kaczmarz system and
showed that the inequality

$\limsup_{n\rightarrow\infty}\frac{D_{n}^{\kappa}(x)}{\log n}\geq C>0$

holds a.e. In 1974 Schipp [21] and Young [36] proved that the Walsh-Kaczmarz
system is a convergence system. In 1981, Skvortsov [24] showed that the Fejér
means with respect to the Walsh-Kaczmarz system converge uniformly to $f$ for
any continuous functions $f$. Gát [5] proved that, for any integrable
functions, the Fejér means with respect to the Walsh-Kaczmarz system converge
almost everywhere to the function. He showed that the maximal operator of
Walsh-Kaczmarz-Fejér means $\sigma^{\kappa,\ast}$ is weak type $(1,1)$ and of
type $(p,p)$ for all $1<p\leq\infty$. Gát’s result was generalized by Simon
[23], who showed that the maximal operator $\sigma^{\kappa,\ast}$ is of type
$(H_{p},L_{p})$ for $p>1/2$. In the endpoint case $p=1/2$ Goginava [8] (see
also [7, 6]) proved that maximal operator $\sigma^{\kappa,\ast}$ is not of
type $(H_{1/2},L_{1/2})$ and Weisz [34] showed that the maximal operator is of
weak type $(H_{1/2},L_{1/2})$. In [7] and [29] it is proved that the maximal
operator $\widetilde{\sigma}$ ${}^{\kappa,\ast\,}$defined by

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

where $0<p\leq 1/2$ and $\left[1/2+p\right]$ denotes integer part of $1/2+p,$
is bounded from the Hardy space $H_{p}$ to the space $L_{p}$. As a corollary
we get

(1) $\left\|\sigma_{n}^{\kappa}f\right\|_{p}\leq
c_{p}\left(n+1\right)^{1/p-2}\log^{2\left[1/2+p\right]}\left(n+1\right)\left\|f\right\|_{H_{p}}.$

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

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

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

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

For Walsh system analogical theorems are proved in [10], [27] and [28].

It is well-known (see [31]) that

(3) $\|\sigma_{2^{n}}^{\kappa}(f)-f\|_{H_{p}}\rightarrow 0,\text{ \ when
}n\rightarrow\infty,\text{ \ }\left(f\in H_{p},\text{ }p>0\right).$

Móricz and Siddiqi [14] investigates the approximation properties of some
special Nörlund means of Walsh-Fourier series of $L_{p}$ function in norm.
Fridli, Manchanda and Siddiqi [3] improved and extended the results of Móricz
and Siddiqi [14] among them in $H_{p}$ norm, where $0<p<1.$ In [12] Goginava
investigated the behavior of Cesáro means of Walsh-Fourier series in detail.
Analogical results for Walsh-Kaczmarz-Fourier series were proved by Nagy in
[15, 16, 17, 18].

The main aim of this paper is to find necessary and sufficient conditions for
the convergence of Fejér means in the terms of the modulus of continuity on
the Hardy spaces $H_{p},$ when $0<p<1/2$ and $p=1/2,$ respectively. That is,
the most investigated end point case $p=1/2$ is also discussed.

The paper is organized as following: In Section 3 we present and discuss the
main results and in Section 4 the auxiliary propositions will be given. In
Section 5 the the proofs can be found. Moreover, in order not to disturb our
discussions in these Sections some definitions and notations are given in
Section 2.

##  2\. Definitions and Notations

Now, we give a brief introduction to the theory of dyadic analysis [19]. Let
$\mathbb{N}_{+}$ denote the set of the positive integers,
$\mathbb{N}:=\mathbb{N}_{+}\cup\\{0\\}.$ Denote ${\mathbb{Z}}_{2}$ the
discrete cyclic group of order 2, that is ${\mathbb{Z}}_{2}=\\{0,1\\},$ where
the group operation is the modulo 2 addition and every subset is open. The
Haar measure on ${\mathbb{Z}}_{2}$ is given such that the measure of a
singleton is 1/2. Let $G$ be the complete direct product of the countable
infinite copies of the compact groups ${\mathbb{Z}}_{2}.$ The elements of $G$
are of the form $x=\left(x_{0},x_{1},...,x_{k},...\right)$ with
$x_{k}\in\\{0,1\\}\left(k\in\mathbb{N}\right).$ The group operation on $G$ is
the coordinatewise addition, the measure (denoted by $\mu$) and the topology
are the product measure and topology. The compact Abelian group $G$ is called
the Walsh group. A base for the neighborhoods of $G$ can be given in the
following way:

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

where $x\in G$ and $n\in\mathbb{N}.$ These sets are called dyadic intervals.
Let $0=\left(0:i\in\mathbb{N}\right)\in G$ denote the null element of $G,$ and
$I_{n}:=I_{n}\left(0\right)\,\left(n\in\mathbb{N}\right).$ Set
$e_{n}:=\left(0,...,0,x_{n}=1,0,...\right)\in G$ and

$J_{N}^{m,l}:=I_{N}\left(x_{0},...,x_{m}=1,0,...,x_{l}=1,0,...,0\right),\text{
\ }l=0,...,N-1,\text{ \ }m=-1,...,l.$

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

$r_{k}\left(x\right):=\left(-1\right)^{x_{k}}$

the $k$-th Rademacher function.

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

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

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

The Walsh-Kaczmarz functions are defined by $\kappa_{0}=1$ and for $n\geq 1$

$\kappa_{n}\left(x\right):=r_{\left|n\right|}\left(x\right)\prod\limits_{k=0}^{\left|n\right|-1}\left(r_{\left|n\right|-1-k}\left(x\right)\right)^{n_{k}}=r_{\left|n\right|}\left(x\right)\left(-1\right)^{\sum\limits_{k=0}^{\left|n\right|-1}n_{k}x_{{}_{\left|n\right|-1-k}}}.$

Skvortsov (see [24]) gave a relation between the Walsh-Kaczmarz functions and
Walsh-Paley functions by the of the transformation $\tau_{A}:G\rightarrow G$
defined by

$\tau_{A}\left(x\right):=\left(x_{A-1},x_{A-2},...,x_{1},x_{0,}x_{A},x_{A+1},...\right),$

for $A\in\mathbb{N}.$ By the definition we have

$\kappa_{n}\left(x\right)=r_{\left|n\right|}\left(x\right)w_{n-2^{\left|n\right|}}\left(\tau_{\left|n\right|}\left(x\right)\right),\qquad\left(n\in\mathbb{N}\mathbf{,}\text{
}x\in G\right).$

The Dirichlet kernels are defined

$D_{n}^{\alpha}:=\sum_{k=0}^{n-1}\alpha_{k\text{ }},$

where $\alpha_{n\text{ }}=w_{n}$ or $\kappa_{n}$
$\left(n\in\mathbb{N}\right),$ $D_{0}^{\alpha}:=0.$ the $2^{n}$-th Dirichlet
kernels have a closed form (see e.g. [19])

(4)
$D_{2^{n}}^{w}(x)=D_{2^{n}}^{\kappa}(x)=D_{2^{n}}(x)=\left\\{\begin{array}[]{ll}2^{n}&\text{\thinspace\thinspace}\,\text{if\thinspace\thinspace\thinspace\
}x\in I_{n},\\\ 0&\text{ \thinspace if}\,\ \,x\notin I_{n}\end{array}\right.$

The norm (or quasi-norm) of the space $L_{p}(G)$ is defined by

$\left\|f\right\|_{p}:=\left(\int_{G}\left|f\right|^{p}d\mu\right)^{1/p}\qquad\left(0<p<\infty\right).$

The space $L_{p,\infty}(G)$ consists of all measurable functions $f$ for which

$\left\|f\right\|_{L_{p,\infty}}:=\underset{\lambda>0}{\sup}\lambda\mu\left\\{\left|f\right|>\lambda\right\\}^{1/p}\leq
c<\infty.$

The $\sigma$-algebra generated by the dyadic intervals of measure $2^{-k}$
will be denoted by $F_{k}$ $\left(k\in\mathbb{N}\right).$ Denote by
$f=\left(f^{\left(n\right)},n\in\mathbb{N}\right)$ a martingale with respect
to $\left(F_{n},n\in\mathbb{N}\right)$ (for details see, e. g. [31]). The
maximal function of a martingale $f$ is defined by

$f^{\ast}=\sup\limits_{n\in\mathbb{N}}\left|f^{\left(n\right)}\right|.$

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

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

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

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

The best approximation of $f\in L_{p}(G)$ ($1\leq p\in\infty$) is defined as

$E_{n}\left(f,L_{p}\right)=\inf_{\psi\in\emph{p}_{n}}\left\|f-\psi\right\|_{p}$

where $\emph{p}_{n}$ is set of all Walsh-Kaczmarz polynomials of order less
than $n\in\mathbb{N}$.

The integrated modulus of continuity of the function $f\in L_{p}(G)$, is
defined by

$\omega_{p}\left(1/2^{n},f\right)=\sup\limits_{h\in
I_{n}}\left\|f\left(\cdot+h\right)-f\left(\cdot\right)\right\|_{p}.$

The concept of modulus of continuity in $H_{p}(G)$ $\left(0<p\leq 1\right)$ is
defined in the following way

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

Since $\left\|f\right\|_{H_{p}}\sim\left\|f\right\|_{p}$, when $p>1$, we
obtain that

$\omega_{H_{p}}\left(1/2^{n},f\right)\sim\left\|f-S_{2^{n}}f\right\|_{p},\text{
\ \ }p>1.$

On the ather hand Watari [30] showed that there are strong connection among
$\omega_{p}\left(1/2^{n},f\right),$ $E_{2^{n}}\left(f,L_{p}\right)$ and
$\left\|f-S_{2^{n}}f\right\|_{p},$ $p\geq 1,$ $n\in\mathbb{N}$. In particular

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

and

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

If $f\in L_{1}\left(G\right)$, then it is easy to show that the sequence
$\left(S_{2^{n}}f:n\in\mathbb{N}\right)$ is a martingale. If $f$ is a
martingale, that is $f=(f^{\left(0\right)},f^{\left(1\right)},...)$ then the
Walsh-(Kaczmarz)-Fourier coefficients must be defined in a little bit
different way:

$\widehat{f}\left(i\right)=\lim\limits_{k\rightarrow\infty}\int\limits_{G}f^{\left(k\right)}\left(x\right)\alpha_{i}\left(x\right)d\mu\left(x\right),\
\ (\alpha_{i}=w_{i}\text{ or }\kappa_{i}).$

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

The partial sums and Fejér means of the Walsh-(Kaczmarz)-Fourier v of
martingale $f\in H_{p}$ are defined by

$S_{n}^{\alpha}(f):=\sum\limits_{i=0}^{n-1}\widehat{f}\left(i\right)\alpha_{i},\quad\sigma_{n}^{\alpha}(f)=\frac{1}{n}\sum\limits_{j=1}^{n}S_{j}^{\alpha}(f),\text{
\ \ \ \ }(\alpha=w\text{ or }\kappa)$

respectively.

The Fejér kernel of order $n$ of the Walsh-(Kaczmarz)-Fourier series defined
by

$K_{n}^{\alpha}:=\frac{1}{n}\sum\limits_{k=1}^{n}D_{k}^{\alpha},\text{
\qquad}(\alpha=w\text{ or }\kappa).$

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

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

For the martingale $f=\sum_{n=0}^{\infty}\left(f_{n}-f_{n-1}\right)$ the
conjugate transforms are defined as

$\widetilde{f^{\left(t\right)}}=\overset{\infty}{\underset{n=0}{\sum}}r_{n}\left(t\right)\left(f_{n}-f_{n-1}\right),$

where $t\in G$ is fixed. Note that $\widetilde{f^{\left(0\right)}}=f.$ As is
well known (see [31])

(5)
$\left\|\widetilde{f^{\left(t\right)}}\right\|_{H_{p}}=\left\|f\right\|_{H_{p}},\text{
\ \ \
}\left\|f\right\|_{H_{p}}^{p}\sim\int_{G}\left\|\widetilde{f^{\left(t\right)}}\right\|_{p}^{p}dt,\text{
\ \ \ \ \ \
}\widetilde{\sigma_{n}^{\kappa}f^{\left(t\right)}}=\sigma_{n}^{\kappa}\widetilde{f^{\left(t\right)}}.$

##  3\. Formulation of Main Results

######  Theorem 1.

a) Let $0<p<1/2$ and

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

Then

(6) $\left\|\sigma_{n}^{\kappa}\left(f\right)-f\right\|_{H_{p}}\rightarrow
0,\text{ when }n\rightarrow\infty.$

b) For any $p\in\left(0,1/2\right)$ there exists a martingale $f\in H_{p}(G)$
for which

(7)
$\omega_{H_{p}}\left(1/2^{k},f\right)=O\left(1/2^{k\left(1/p-2\right)}\right),\text{
\ as }k\rightarrow\infty.$

and

$\left\|\sigma_{n}^{\kappa}\left(f\right)-f\right\|_{L_{p,\infty}}\nrightarrow
0,\,\,\,\text{as\thinspace\thinspace\thinspace}n\rightarrow\infty.$

######  Theorem 2.

a) Let

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

Then

$\left\|\sigma_{n}^{\kappa}\left(f\right)-f\right\|_{H_{1/2}}\rightarrow
0,\text{ when }n\rightarrow\infty.$

b) There exists a martingale $f\in H_{1/2}(G),$ for which

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

and

$\left\|\sigma_{n}\left(f\right)-f\right\|_{1/2}\nrightarrow
0\,\,\,\text{as\thinspace\thinspace\thinspace}n\rightarrow\infty.$

##  4\. AUXILIARY PROPOSITIONS

######  Lemma 1.

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

(9) $\qquad\sum_{k=0}^{\infty}\mu_{k}S_{M_{n}}a_{k}=f_{n},\text{ \ \ a.e.}$

and

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

Moreover,
$\left\|f\right\|_{H_{p}}\backsim\inf\left(\sum_{k=0}^{\infty}\left|\mu_{k}\right|^{p}\right)^{1/p},$
where the infimum is taken over all decomposition of $f$ of the form (9).

######  Lemma 2.

[11] Let $2<A\in\mathbb{N}$ and $q_{A}=2^{2A}+2^{2A-2}+...+2^{2}+2^{0}.$ Then

$q_{A-1}\left|K_{q_{A-1}}^{w}\left(x\right)\right|\geq 2^{2m+2s-3},$

for $x\in
I_{2A}\left(0,...,0,x_{2m}=1,0...,0,x_{2s}=1,x_{2s+1},...,x_{2A-1}\right),$
$m=0,1,...,A-3,$ $s=m+2,$ $m+3,...,A-1.$

##  5\. Proof of the Theorems

Proof of Theorem 1. Let $0<p<1/2.$ Combining (1) and (5) we conclude that

$\displaystyle\left\|\sigma_{n}^{\kappa}f\right\|_{H_{p}}^{p}=\int_{G}\left\|\widetilde{\sigma_{n}^{\kappa}f^{\left(t\right)}}\right\|_{p}^{p}dt$
$\displaystyle=$
$\displaystyle\int_{G}\left\|\sigma_{n}^{\kappa}\widetilde{f^{\left(t\right)}}\right\|_{p}^{p}dt\leq
c_{p}n^{1-2p}\int_{G}\left\|\widetilde{f^{\left(t\right)}}\right\|_{H_{p}}^{p}dt$
$\displaystyle\sim$ $\displaystyle
c_{p}n^{1-2p}\int_{G}\left\|f\right\|_{H_{p}}^{p}dt=c_{p}n^{1-2p}\left\|f\right\|_{H_{p}}^{p}.$

Let $2^{m}<n\leq 2^{m+1}.$ Using (5) we have that

$\displaystyle\left\|\sigma_{n}^{\kappa}f-f\right\|_{H_{p}}^{p}$
$\displaystyle\leq$
$\displaystyle\left\|\sigma_{n}^{\kappa}f-\sigma_{n}^{\kappa}S_{2^{m}}f\right\|_{H_{p}}^{p}+\left\|\sigma_{n}^{\kappa}S_{2^{m}}f-S_{2^{m}}f\right\|_{H_{p}}^{p}+\left\|S_{2^{m}}f-f\right\|_{H_{p}}^{p}$
$\displaystyle=$
$\displaystyle\left\|\sigma_{n}^{\kappa}\left(S_{2^{m}}f-f\right)\right\|_{H_{p}}^{p}+\left\|S_{2^{m}}f-f\right\|_{H_{p}}^{p}+\left\|\sigma_{n}^{\kappa}S_{2^{m}}f-S_{2^{m}}f\right\|_{H_{p}}^{p}$
$\displaystyle\leq$ $\displaystyle
c_{p}\left(n^{1-2p}+1\right)\omega_{H_{p}}^{p}\left(1/2^{m},f\right)+\left\|\sigma_{n}^{\kappa}S_{2^{m}}f-S_{2^{m}}f\right\|_{H_{p}}^{p}.$

By simple calculation we get that

$\displaystyle\sigma_{n}^{\kappa}S_{2^{m}}f-S_{2^{m}}f=\frac{2^{m}}{n}\left(S_{2^{m}}\sigma_{2^{m}}f-S_{2^{m}}f\right)$
$\displaystyle=$
$\displaystyle\frac{2^{m}}{n}S_{2^{m}}\left(\sigma_{2^{m}}f-f\right).$

Let $p>0.$ By applying (3) we obtain that

$\displaystyle\left\|\sigma_{n}^{\kappa}S_{2^{m}}f-S_{2^{m}}f\right\|_{H_{p}}^{p}$
$\displaystyle\leq$
$\displaystyle\frac{2^{pm}}{n^{p}}\left\|S_{2^{m}}\left(\sigma_{2^{m}}f-f\right)\right\|_{H_{p}}^{p}$
$\displaystyle\leq$
$\displaystyle\left\|\sigma_{2^{m}}f-f\right\|_{H_{p}}^{p}\rightarrow 0,\text{
when }k\rightarrow\infty\text{.}$

It follows that if

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

then

$\left\|\sigma_{n}^{\kappa}f-f\right\|_{H_{p}}\rightarrow 0,\text{ when
}n\rightarrow\infty.$

Now, we are ready to prove second part of Theorem 1. We set

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

and

$f^{\left(n\right)}\left(x\right)=\underset{i=0}{\overset{n}{\sum}}\frac{1}{2^{\left(1/p-2\right)i}}a_{i}(x),\text{
}n\in\mathbb{N}.$

It is easy to show that
$f=\left(f^{\left(1\right)},f^{\left(2\right)},...,f^{\left(n\right)},...\right)\in
H_{p}.$ Indeed, since

(12)
$S_{2^{A}}a_{i}\left(x\right)=\left\\{\begin{array}[]{ll}a_{i}\left(x\right),&i<A,\\\
0,&i\geq A,\end{array}\right.$

$\text{supp}(a_{i})=I_{i},\text{\ \ \ }\int_{I_{i}}a_{i}d\mu=0\text{\ }$

and

$\text{\ }\left\|a_{i}\right\|_{\infty}\leq 2^{\left(1/p-1\right)i}\cdot
2^{i}=2^{i/p}=\left(\text{supp }a_{i}\right)^{-1/p},$

if we apply Lemma 1 we conclude that $f\in H_{p}.$

It is easy to show that

$\displaystyle f-S_{2^{n}}f$ $\displaystyle=$
$\displaystyle\left(f^{\left(1\right)}-S_{2^{n}}f^{\left(1\right)},...,f^{\left(n\right)}-S_{2^{n}}f^{\left(n\right)},...,f^{\left(n+k\right)}-S_{2^{n}}f^{\left(n+k\right)}\right)$
$\displaystyle=$
$\displaystyle\left(0,...,0,f^{\left(n+1\right)}-f^{\left(n\right)},...,f^{\left(n+k\right)}-f^{\left(n\right)},...\right)$
$\displaystyle=$
$\displaystyle\left(0,...,0,\underset{i=n}{\overset{n+k}{\sum}}\frac{a_{i}(x)}{2^{i\left(1/p-2\right)}},...\right),\text{
\ }k\in\mathbb{N}_{+}.$

is a martingale. By using (5) we get that

$\omega_{H_{p}}(1/2^{n},f)$
$\leq\sum\limits_{i=n+1}^{\infty}1/2^{\left(1/p-2\right)i}\leq
c/2^{\left(1/p-2\right)n}=O\left(1/2^{\left(1/p-2\right)n}\right).$

It is easy to show that

(14) $\widehat{f}(j)=2^{i},\text{ \ \ if
\thinspace\thinspace}j\in\left\\{2^{i},...,2^{i+1}-1\right\\},\text{
}i=0,1,...$

By applying (3) and (14) we have that

$\displaystyle\limsup\limits_{n\rightarrow\infty}\|\sigma_{2^{n}+1}^{\kappa}(f)-f\|_{L_{p,\infty}}$
$\displaystyle=$
$\displaystyle\limsup\limits_{n\rightarrow\infty}\|\frac{2^{n}\sigma_{2^{n}}^{\kappa}(f)}{2^{n}+1}+\frac{S_{2^{n}}(f)}{2^{n}+1}+\frac{2^{n}\kappa_{2^{n}}}{2^{n}+1}-\frac{2^{n}f}{2^{n}+1}-\frac{f}{2^{n}+1}\|_{L_{p,\infty}}$
$\displaystyle\geq$
$\displaystyle\limsup\limits_{n\rightarrow\infty}\frac{2^{n}}{2^{n}+1}\|\kappa_{2^{n}}\|_{L_{p_{,\infty}}}$
$\displaystyle-\limsup\limits_{n\rightarrow\infty}\frac{2^{n}}{2^{n}+1}\|\sigma_{2^{n}}^{\kappa}(f)-f\|_{L_{p,\infty}}$
$\displaystyle-\limsup\limits_{n\rightarrow\infty}\frac{1}{2^{n}+1}\|S_{2^{n}}(f)-f\|_{L_{p,\infty}}$
$\displaystyle>$
$\displaystyle\limsup\limits_{n\rightarrow\infty}\frac{2^{n}}{2^{n}+1}-o(1)>0.$

Theorem 1 is proved.

Proof of Theorem 2. Analogously of first part of Theorem 1 we can show that if
modulus of continuity of the martingale $f\in H_{1/2}$ satisfies the condition
(8), then $\left\|\sigma_{n}^{\kappa}f-f\right\|_{H_{1/2}}\rightarrow\infty,$
when $n\rightarrow\infty.$

Now, we are ready to prove the second part of theorem 1. We set

$a_{i}(x)=2^{2^{i}}\left(D_{2^{2^{i}+1}}(x)-D_{2^{2^{i}}}(x)\right)$

and

$f^{\left(n\right)}\left(x\right)=\overset{n}{\sum_{i=1}}\frac{a_{i}(x)}{2^{2i}}.$

It is easy to show that $\
f=\left(f^{\left(1\right)},f^{\left(2\right)},...,f^{\left(n\right)},...\right)\in
H_{1/2}.$ Indeed, since

$S_{2^{A}}a_{i}\left(x\right)=\left\\{\begin{array}[]{ll}a_{i}\left(x\right),&2^{i}<A,\\\
0,&2^{i}\geq A\end{array}\right.$

$\text{supp}(a_{i})=I_{2^{i}},\text{ \ \ }\int_{I_{2^{i}}}a_{i}d\mu=0,\text{ \
\ \
}\left\|a_{i}\right\|_{\infty}\leq\left(2^{2^{i}}\right)^{2}=\left(\mu(\text{supp
}a_{i})\right)^{2},$

if we apply Lemma 1 we conclude that $f\in H_{1/2}.$

By simple calculation we get that

$\omega_{H_{1/2}}\left(1/2^{n},f\right)\leq\overset{\infty}{\sum_{i=\left[\log_{2}^{n}\right]}}\frac{1}{2^{2i}}=O\left(1/n^{2}\right),$

where $\left[\log_{2}^{n}\right]$ denotes the integer part of $\log_{2}^{n}.$
Hence

$\displaystyle\|\sigma_{q_{2^{k-1}}}^{\kappa}(f)-f\|_{1/2}$ $\displaystyle=$
$\displaystyle\|\frac{2^{2^{k}}\sigma_{2^{2^{k}}}^{\kappa}(f)}{q_{{}_{2^{k-1}}}}+\frac{1}{q_{{}_{2^{k-1}}}}\sum_{j=2^{2^{k}}+1}^{q_{2^{k-1}}}S_{j}^{\kappa}f-\frac{2^{2^{k}}f}{q_{{}_{2^{k-1}}}}-\frac{q_{{}_{2^{k-1}-1}}f}{q_{{}_{2^{k-1}}}}\|_{1/2}$

It is easy to show that

(16)
$\widehat{f}(j)=\left\\{\begin{array}[]{ll}\frac{2^{2^{i}}}{2^{2i}},&\text{if
\thinspace\thinspace}j\in\left\\{2^{2^{i}},...,2^{2^{i}+1}-1\right\\},\text{
}i=0,1,...\\\ 0,&\text{\thinspace if
\thinspace\thinspace\thinspace}j\notin\bigcup\limits_{n=0}^{\infty}\left\\{2^{2^{n}},...,2^{2^{n}+1}-1\right\\}.\text{
}\end{array}\right.$

Let $2^{2^{i}}<j\leq q_{{}_{2^{i-1}}}.$ Since

$D_{s+2^{2^{i}}}^{\kappa}=D_{2^{2^{i}}}+r_{2^{{}^{i}}}D_{s}^{w}\left(\tau_{2^{i}}\right),\text{
\qquad when \thinspace\thinspace}s<2^{2^{i}}$

by applying (16) we obtain that

$\displaystyle S_{j}^{\kappa}f$ $\displaystyle=$ $\displaystyle
S_{2^{2^{i}}}f+\sum_{v=2^{2^{i}}}^{j-1}\widehat{f}(v)\kappa_{v}=S_{2^{2^{i}}}f+\frac{2^{2^{i}}}{2^{2i}}\sum_{v=2^{i}}^{j-1}\kappa_{v}$
$\displaystyle=$ $\displaystyle
S_{2^{2^{i}}}f+\frac{2^{2^{i}}}{2^{2i}}\left(D_{{}_{j}}^{\kappa}-D_{2^{2^{i}}}\right)=S_{2^{2^{i}}}f+\frac{2^{2^{i}}r_{2^{{}^{i}}}D_{{}_{j-2^{2^{i}}}}^{w}\left(\tau_{2^{i}}\right)}{2^{2i}}.$

Hence

$\displaystyle\sum_{j=2^{2^{i}}+1}^{q_{2^{i-1}}}S_{j}^{\kappa}f=\frac{q_{2^{i-1}-1}S_{2^{2^{i}}}f}{q_{2^{i-1}}}+\frac{2^{2^{i}}r_{2^{{}^{i}}}}{q_{2^{i}}2^{2i}}\sum_{j=2^{2^{i}}+1}^{q_{2^{i-1}}}D_{{}_{j-2^{2^{k}}}}^{w}\left(\tau_{2^{i}}\right)$
$\displaystyle=$
$\displaystyle\frac{q_{2^{i-1}-1}S_{2^{2^{i}}}f}{q_{2^{i-1}}}+\frac{2^{2^{i}}r_{2^{{}^{i}}}}{q_{2^{i-1}}2^{2i}}\sum_{j=1}^{q_{2^{i-1}-1}}D_{{}_{j}}^{w}\left(\tau_{2^{i}}\right)$
$\displaystyle=$
$\displaystyle\frac{q_{2^{i-1}-1}S_{2^{2^{i}}}f}{q_{2^{i-1}}}+\frac{2^{2^{i}}r_{2^{{}^{i}}}q_{2^{i-1}-1}K_{q_{2^{i-1}-1}}^{w}\left(\tau_{2^{i}}\right)}{q_{2^{i-1}}2^{2i}}.$

By using (5) we get that

$\displaystyle\|\sigma_{q_{2^{i-1}}}^{\kappa}(f)-f\|_{1/2}^{1/2}$
$\displaystyle\geq$
$\displaystyle\frac{c}{2^{i}}\|q_{2^{i-1}-1}K_{q_{2^{i-1}-1}}^{w}\left(\tau_{2^{i}}\right)\|_{1/2}^{1/2}-\left(\frac{2^{2^{i}}}{q_{2^{i-1}}}\right)^{1/2}\|\sigma_{{}_{2^{2^{i}}}}f-f\|_{1/2}^{1/2}$
$\displaystyle-\left(\frac{q_{2^{i-1}-1}}{q_{2^{i-1}}}\right)^{1/2}\|S_{2^{2^{i}}}f-f\|_{1/2}^{1/2}=IV-
V-VI.$

Let $x\in J_{2^{i}}^{2^{i}-2s-1,2^{i}-2\eta-1},$ where
$\eta=1,...,2^{i}-3,\,s=\eta+3,...,2^{i}-1.$ By applying Lemma 2 we have that

$q_{2^{i-1}-1}\left|K_{q_{2^{i-1}-1}}^{w}\left(\tau_{2^{i}}\left(x\right)\right)\right|\geq
2^{2\eta+2s-3}.$

We can write

$\displaystyle\int_{G}\left|q_{2^{i-1}-1}K_{q_{2^{i-1}-1}}^{w}\left(\tau_{2^{i}}\left(x\right)\right)\right|^{1/2}d\mu\left(x\right)$
$\displaystyle\geq$ $\displaystyle
c\sum_{\eta=1}^{2^{i}-3}\sum_{s=\eta+2}^{2^{i}-1}\sum_{x_{2s+1}=0}^{1}...\sum_{x_{{}_{2^{i}-1}}=0}^{1}\int_{J_{2^{i}}^{2^{i}-2s-1,2^{i}-2\eta-1}}\left|q_{2^{i-1}-1}K_{q_{2^{i-1}-1}}^{w}\left(\tau_{2^{i}}\left(x\right)\right)\right|^{1/2}d\mu\left(x\right)$
$\displaystyle\geq$ $\displaystyle
c\sum_{\eta=1}^{2^{i}-3}\sum_{s=\eta+2}^{2^{i}-1}\frac{1}{2^{2s}}\sqrt{2^{2\eta+2s}}\geq\sum_{\eta=1}^{2^{i}-3}\sum_{s=\eta+2}^{2^{i}-1}\sqrt{\frac{2^{2\eta}}{2^{2s}}}\geq
c\sum_{\eta=1}^{2^{i}-3}\geq c2^{i}.$

By using (5) we have that

$\limsup\limits_{i\rightarrow\infty}\|\sigma_{q_{2^{i-1}}}^{\kappa}(f)-f\|_{1/2}\geq
c>0.$

Theorem 2 is proved.

Acknowledgment: The author would like to thank the referees for their helpful
suggestions.

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