$\widehat{f}(j)=\left\\{\begin{array}[]{ll}M_{\left|\alpha_{k}\right|}M_{\left<\alpha_{k}\right>}^{1/p-2},&\text{\thinspace\thinspace}j\in\left\\{M_{\left|\alpha_{k}\right|},...,M_{{}_{\left|\alpha_{k}\right|+1}}-1\right\\},\text{
}k=0,1,...\\\
0\,,&\text{\thinspace}j\notin\bigcup\limits_{i=0}^{\infty}\left\\{M_{{}_{\left|\alpha_{k}\right|}},...,M_{{}_{\left|\alpha_{k}\right|+1}}-1\right\\}.\text{
}\end{array}\right.$ (2.59)

Let $M_{\left|\alpha_{k}\right|}<j<\alpha_{k}.$ By using (2.59) we get that

$\displaystyle S_{j}f$ $\displaystyle=$ $\displaystyle
S_{M_{\left|\alpha_{k}\right|}}f+\sum_{v=M{{}_{\left|\alpha_{k}\right|}}}^{j-1}\widehat{f}(v)w_{v}$
$\displaystyle=$ $\displaystyle
S_{M_{\left|\alpha_{k}\right|}}f+M_{\left|\alpha_{k}\right|}M_{\left<\alpha_{k}\right>}^{1/p-2}\left(D_{j}-D_{M_{\left|\alpha_{k}\right|}}\right)$

Hence,

$\sigma_{{}_{\alpha_{k}}}f=\frac{1}{\alpha_{k}}\sum_{j=1}^{M_{\left|\alpha_{k}\right|}}S_{j}f+\frac{1}{\alpha_{k}}\sum_{j=M_{\left|\alpha_{k}\right|}+1}^{\alpha_{k}}S_{j}f$
(2.60) $\displaystyle=$
$\displaystyle\frac{M_{\left|\alpha_{k}\right|}}{\alpha_{k}}\sigma_{{}_{M_{\left|\alpha_{k}\right|}}}f+\frac{\left(\alpha_{k}-M_{\left|\alpha_{k}\right|}\right)S_{M_{\left|\alpha_{k}\right|}}f}{\alpha_{k}}$
$\displaystyle+$
$\displaystyle\frac{M_{\left|\alpha_{k}\right|}M_{\left<\alpha_{k}\right>}^{1/p-2)}}{\alpha_{k}}\sum_{j=M_{{}_{\left|\alpha_{k}\right|}}+1}^{\alpha_{k}}\left(D_{{}_{j}}-D_{M_{\left|\alpha_{k}\right|}}\right)$
$\displaystyle=$ $\displaystyle I+II+III.$

Since

$D_{j+M_{n}}=D_{M_{n}}+\psi_{M_{n}}D_{j},\text{ \thinspace when
\thinspace}j<M_{n+1}$

we obtain that

$\displaystyle\left|III\right|$ $\displaystyle=$
$\displaystyle\frac{M_{\left|\alpha_{k}\right|}M_{\left<\alpha_{k}\right>}^{1/p-2}}{\alpha_{k}}\left|\sum_{j=1}^{\alpha_{k}-M_{{\left|\alpha_{k}\right|}}}\left(D_{j+M_{{\left|\alpha_{k}\right|}}}-D_{M_{\left|\alpha_{k}\right|}}\right)\right|$
$\displaystyle=$
$\displaystyle\frac{M_{\left|\alpha_{k}\right|}M_{\left<n_{k}\right>}^{1/p-2}}{\alpha_{k}}\left|\sum_{j=1}^{\alpha_{k}-M_{\left|\alpha_{k}\right|}}D_{j}\right|$
$\displaystyle=$
$\displaystyle\frac{M_{\left|\alpha_{k}\right|}M_{\left<\alpha_{k}\right>}^{1/p-2}}{\alpha_{k}}\left(\alpha_{k}-M{{}_{\left|\alpha_{k}\right|}}\right)\left|K_{\alpha_{k}-M_{{}_{\left|\alpha_{k}\right|}}}\right|$
$\displaystyle\geq$ $\displaystyle
cM_{\left<\alpha_{k}\right>}^{1/p-2}\left(\alpha_{k}-M_{{}_{\left|\alpha_{k}\right|}}\right)\left|K_{\alpha_{k}-M_{{}_{\left|\alpha_{k}\right|}}}\right|.$

By combining (2.60) and (2.6) we can conclude that

$\displaystyle\left\|\sigma_{\alpha_{k}}f-f\right\|_{weak-
L_{p}}^{p}=\|I+II+III-f\|_{weak-L_{p}}^{p}$ $\displaystyle=$
$\displaystyle\|III+\frac{M_{\left|\alpha_{k}\right|}}{\alpha_{k}}\sigma_{{}_{M_{\left|\alpha_{k}\right|}}}f+\frac{\left(\alpha_{k}-M_{\left|\alpha_{k}\right|}\right)S_{M_{\left|\alpha_{k}\right|}}f}{\alpha_{k}}-f\|_{weak-
L_{p}}^{p}$ $\displaystyle=$
$\displaystyle\|III+\frac{M_{\left|\alpha_{k}\right|}}{\alpha_{k}}\left(\sigma_{{}_{M_{\left|\alpha_{k}\right|}}}f-f\right)+\frac{\alpha_{k}-M_{\left|\alpha_{k}\right|}}{\alpha_{k}}\left(S_{M_{\left|\alpha_{k}\right|}}f-f\right)\|_{weak-
L_{p}}^{p}$ $\displaystyle\geq$ $\displaystyle\|III\|_{weak-
L_{p}}^{p}-\left(\frac{M_{\left|\alpha_{k}\right|}}{\alpha_{k}}\right)^{p}\|\sigma_{M_{{}_{\left|\alpha_{k}\right|}}}f-f\|_{weak-
L_{p}}^{p}$ $\displaystyle-$
$\displaystyle\left(\frac{\alpha_{k}-M_{{}_{\left|\alpha_{k}\right|}}}{\alpha_{k}}\right)^{p}\|S_{M_{{}_{\left|\alpha_{k}\right|}}}f-f\|_{weak-
L_{p}}^{p}$ $\displaystyle\geq$ $\displaystyle\|III\|_{weak-
L_{p}}^{p}-\|\sigma_{M_{{}_{\left|\alpha_{k}\right|}}}f-f\|_{weak-
L_{p}}^{p}-\|S_{M_{{}_{\left|\alpha_{k}\right|}}}f-f\|_{weak-L_{p}}^{p}.$

If we combine (2.1) and (2.9) it is easy to show that

$\displaystyle\|\sigma_{M_{{}_{\left|\alpha_{k}\right|}}}f-f\|_{weak-
L_{p}}^{p}\rightarrow 0,\text{ \ \ as \ }k\rightarrow\infty,$
$\displaystyle\|S_{M_{\left|\alpha_{k}\right|}}f-f\|_{weak-
L_{p}}^{p}\rightarrow 0,\text{ \ \ as \ }k\rightarrow\infty,$

Hence, for sufficiently large $k,$ we can write that

$\displaystyle\left\|\sigma_{\alpha_{k}}f-f\right\|_{weak-L_{p}}^{p}$
$\displaystyle\geq$ $\displaystyle\frac{1}{2}\|III\|_{weak-
L_{p}}^{p}\geq\frac{M_{\left<\alpha_{k}\right>}^{1-2p}}{2}\|\left(\alpha_{k}-M_{\left|\alpha_{k}\right|}\right)K_{\alpha_{k}-M_{\left|\alpha_{k}\right|}}\|_{weak-
L_{p}}^{p}$

Let $x\in E_{\left<\alpha_{k}\right>}.$ By using Lemma 2.7 we have that

$\displaystyle\mu\left\\{x\in
G_{m}:\left(\alpha_{k}-M_{\left|\alpha_{k}\right|}\right)\left|K_{\alpha_{k}-M_{\left|\alpha_{k}\right|}}\right|\geq
cM_{\left<\alpha_{k}\right>}^{2}\right\\}$ $\displaystyle\geq$
$\displaystyle\mu\left(E_{\left<\alpha_{k}\right>}\right)\geq
c/M_{\left<\alpha_{k}\right>},$

and

$cM_{\left|\alpha_{k}\right|}^{2p}\mu\left\\{x\in
G:\left(\alpha_{k}-M_{{}_{\left|\alpha_{k}\right|}}\right)\left|K_{\alpha_{k}-M_{\left|\alpha_{k}\right|}}\right|\geq
cM_{\left|\alpha_{k}\right|}^{2p}\right\\}\geq
cM_{\left|\alpha_{k}\right|}^{2p-1}.$

Hence,

$\left\|\sigma_{\alpha_{k}}f-f\right\|_{weak-L_{p}}\nrightarrow
0,\,\,\,\text{as}\,\,\,k\rightarrow\infty$

and Theorem is proved.

On the other hand, the following unexpected new result is also obtained:

###### Corollary 2.22

a) Let $0<p<1/2,$ $f\in H_{p}$ and

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

Then

$\left\|\sigma_{M_{n_{k}}+1}f-f\right\|_{H_{p}}\rightarrow 0,\text{ \ \ as \
}k\rightarrow\infty.$

b) Let $\sup_{k\in\mathbb{N}}\rho\left(n_{k}\right)=\infty.$ Then there exists
a martingale $f\in H_{p}(G)$ $\left(0<p<1/2\right),$ for which

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

and

$\left\|\sigma_{{M_{n_{k}}+1}}f-f\right\|_{weak-L_{p}}\nrightarrow
0,\,\,\,\text{as\thinspace\thinspace\thinspace}k\rightarrow\infty.$

###### Corollary 2.23

a) Let $0<p<1/2,$ $f\in H_{p}$ and

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

Then

$\left\|\sigma_{M_{n_{k}}+M_{[n_{k}/2]}}f-f\right\|_{H_{p}}\rightarrow
0,\text{ \ \ as \ }k\rightarrow\infty.$

b) Let $\sup_{k\in\mathbb{N}}\rho\left(n_{k}\right)=\infty.$ Then there exists
a martingale $f\in H_{p}(G_{m})$ $\left(0<p<1/2\right),$ for which

$\omega_{H_{p}}\left(1/M_{\left|n_{k}\right|},f\right)=O\left(\frac{{M_{[n_{k}/2]}^{1/p-2}}}{M_{n_{k}}^{1/p-2}}\right),$

and

$\left\|\sigma_{M_{n_{k}}+M_{[n_{k}/2]}}f-f\right\|_{weak-L_{p}}\nrightarrow
0,\,\,\,\text{as\thinspace\thinspace\thinspace}k\rightarrow\infty.$

In the next corollary we state theorem for Walsh system only to clearly see
difference of divergence rates for the various subsequences:

###### Corollary 2.24

a) Let $0<p<1/2,$ $f\in H_{p}$ and

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

Then

$\left\|\sigma^{w}_{2^{n}+1}f-f\right\|_{H_{p}}\rightarrow 0,\text{ \ \ as \
}n\rightarrow\infty.$

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

$\omega_{H_{p}}\left(1/2^{n},f\right)=O\left(\frac{1}{2^{n(1/p-2)}}\right),$

and

$\left\|\sigma^{w}_{2^{n}+1}f-f\right\|_{weak-L_{p}}\nrightarrow
0,\,\,\,\text{as\thinspace\thinspace\thinspace}k\rightarrow\infty.$

###### Corollary 2.25

a) Let $0<p<1/2,$ $f\in H_{p}$ and

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

Then

$\left\|\sigma^{w}_{2^{n}+2^{[n/2]}}f-f\right\|_{H_{p}}\rightarrow 0,\text{ \
\ as \ }n\rightarrow\infty,$

where $[n/2]$ denotes integer part of $n/2$.

b) There exists a martingale $f\in H_{p}(G_{m})$ $\left(0<p<1/2\right),$ for
which

$\omega_{H_{p}}\left(1/M_{n},f\right)=O\left(\frac{1}{2^{n(1/p-2)/2}}\right)$

and

$\left\|\sigma^{w}_{2^{n}+2^{[n/2]}}f-f\right\|_{weak-L_{p}}\nrightarrow
0,\,\,\,\text{as\thinspace\thinspace\thinspace}n\rightarrow\infty.$

##  3 $T$ means of Vilenkin-Fourier series on martingale Hardy spaces

###  3.1 Some classical results on $T$ means of Vilenkin-Fourier series

It is well-known in the literature that the so-called $T$ means are
generalizations of the Fejér, Cesàro and logarithmic means. The $T$ summation
is a general summability method. Therefore it is of prior interest to study
the behavior of operators related to Nörlund means of Fourier series with
respect to orthonormal systems.

Since $T$ means are inverse of Nörlund means we state some interesting results
concerning Nörlund summability, which has high influence on the new results
for $T$ means of Vilenkin-Fourier series.

In [47] Goginava investigated the behavior of Cesàro means of Walsh-Fourier
series in detail. In the two-dimensional case approximation properties of
Nörlund and Cesàro means were considered by Nagy (see [83], [84] and [86]).
The maximal operator $\sigma^{\alpha,\ast}$ $\left(0<\alpha<1\right)$ of the
$\left(C,\alpha\right)$ means of Vilenkin systems was investigated by Weisz
[179]. In this paper Weisz proved that $\sigma^{\alpha,\ast}$ is bounded from
the martingale space $H_{p}$ to the Lebesgue space $L_{p}$ for
$p>1/\left(1+\alpha\right).$ Goginava [45] gave a counterexample which shows
that boundedness does not hold for $0<p\leq 1/\left(1+\alpha\right).$ Weisz
and Simon [123] showed that the maximal operator $\sigma^{\alpha,\ast}$ is
bounded from the Hardy space $H_{1/\left(1+\alpha\right)}$ to the space $weak-
L_{1/\left(1+\alpha\right)}$.

Strong convergence theorems and boundedness of weighted maximal operators of
of the $\left(C,\alpha\right)$ means of Vilenkin systems on the Hardy spaces,
when $0<p\leq 1/(1+\alpha)$ were considered by Blahota, Tephnadze [15] and
Blahota, Tephnadze, Toledo [16]. Summability of some general methods were
considered by Blahota, Nagy and Tephnadze [12], Weisz,

In Persson,Tephnadze and Wall [104] (see also [145]) considered maximal
operator of the Nörlund summation method (see (1.17)). In particular, the
maximal operator $t^{\ast}$ of the summability method (1.17) with non-
decreasing sequence $\\{q_{k}:k\in\mathbb{N}\\}$ is bounded from the Hardy
space $H_{1/2}$ to the space $weak-L_{1/2}.$

Moreover,for any $0<p<1/2$ and non-decreasing sequence
$\\{q_{k}:k\in\mathbb{N}\\}$ satisfying the condition

$\frac{q_{0}}{Q_{n}}\geq\frac{c}{n},\text{ \ \ }\left(c>0\right),$ (3.1)

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

$\underset{n\in\mathbb{N}}{\sup}\left\|t_{n}f\right\|_{weak-L_{p}}=\infty.$

In Persson, Tephnadze and Wall [106] was proved that if $0<p<1/2$ and the
sequence $\left\\{q_{k}:k\in\mathbb{N}\right\\}$ be non-decreasing, then the
maximal operator

$\widetilde{t}_{p,1}^{\ast}f:=\sup_{n\in\mathbb{N}}\frac{\left|t_{n}f\right|}{\left(n+1\right)^{1/p-2}}$

is bounded from the Hardy martingale space $H_{p}$ to the Lebesgue space
$L_{p}.$

On the other hand, according the fact that Fejér means are examples of Nörlund
means with non-decreasing sequence $\left\\{q_{k}:k\in\mathbb{N}\right\\}$ we
immediately obtain that the asymptotic behaviour of the sequence of weights

$\left\\{1/\left(k+1\right)^{1/p-2}:k\in\mathbb{N}\right\\}$

in Nörlund means can not be improved.

Let the sequence $\left\\{q_{k}:k\in\mathbb{N}\right\\}$ be non-decreasing.
Then the maximal operator

$\overset{\sim}{t}_{1}^{\ast}f:=\sup_{n\in\mathbb{N}}\frac{\left|t_{n}f\right|}{\log^{2}\left(n+1\right)}$

is bounded from the Hardy space $H_{1/2}$ to the Lebesgue space $L_{1/2}.$ On
the other hand, according the fact that Fejér means are examples of Nörlund
means with non-decreasing sequence $\left\\{q_{k}:k\in\mathbb{N}\right\\}$ we
immediately obtain that the asymptotic behaviour of the sequence of weights

$\left\\{1/\log^{2}\left(n+1\right):n\in\mathbb{N}\right\\}$

in Nörlund means can not be improved.

Persson, Tephnade and Wall [104] proved that for all Nörlund means with non-
increasing sequence $\left\\{q_{k}:k\in\mathbb{N}\right\\}$ there exists a
martingale $f\in H_{p}$ such that

$\underset{n\in\mathbb{N}}{\sup}\left\|t_{n}f\right\|_{weak-L_{p}}=\infty.$

It follows that for any $0<p<1/2\ \ $ and Nörlund means $t_{n}$ with non-
increasing sequence $\\{q_{k}:k\in\mathbb{N}\\},$ the maximal operator
$t^{\ast}$ is not bounded from the martingale Hardy space $H_{p}$ to the space
$weak-L_{p},$ that is there exists a martingale $f\in H_{p},$ such that

$\underset{n\in\mathbb{N}}{\sup}\left\|t^{\ast}f\right\|_{weak-L_{p}}=\infty.$

Persson, Tephnade and Wall [104] find necessary condition for the Nörlund
means with non-increasing sequence $\left\\{q_{k}:k\in\mathbb{N}\right\\}$,
when $1/2\leq p<1.$ In particular, for $0<p<1/\left(1+\alpha\right),$
$0<\alpha\leq 1,$ and non-increasing sequence $\\{q_{k}:k\in\mathbb{N}\\}$
satisfying the condition

$\overline{\lim_{n\rightarrow\infty}}\frac{n^{\alpha}}{Q_{n}}=c>0,\text{
}0<\alpha\leq 1,$ (3.2)

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

$\underset{n\in\mathbb{N}}{\sup}\left\|t_{n}f\right\|_{weak-L_{p}}=\infty.$
(3.3)

Moreover, for any non-increasing sequence $\\{q_{k}:k\in\mathbb{N}\\}$
satisfying the condition

$\overline{\lim_{n\rightarrow\infty}}\frac{n^{\alpha}}{Q_{n}}=\infty,\text{ \
\ \ }\left(0<\alpha\leq 1\right),$ (3.4)

there exists an martingale $f\in H_{1/\left(1+\alpha\right)},$ such that

$\underset{n\in\mathbb{N}}{\sup}\left\|t_{n}f\right\|_{weak-
L_{1/\left(1+\alpha\right)}}=\infty.$ (3.5)

It follows that for any $0<p<1/\left(1+\alpha\right),$ $0<\alpha\leq 1$ and
non-increasing sequence $\\{q_{k}:k\in\mathbb{N}\\}$ satisfying the condition
(3.2). Then there exists a martingale $f\in H_{p}$ such that

$\left\|t^{\ast}f\right\|_{weak-L_{p}}=\infty.$

Moreover, if $\\{q_{k}:k\in\mathbb{N}\\}$ be a non-increasing sequence
satisfying the condition (3.4), then there exists an martingale $f\in
H_{1/\left(1+\alpha\right)}$ such that

$\left\|t^{\ast}f\right\|_{weak-L_{1/\left(1+\alpha\right)}}=\infty.$

In [80] it was proved that the maximal operator $t^{\ast}$ of the Nörlund
summability method with non-increasing sequence $\\{q_{k}:k\in\mathbb{N}\\},$
satisfying the condition

$\frac{1}{Q_{n}}=O\left(\frac{1}{n^{\alpha}}\right),\text{ \ \ when \ \
}n\rightarrow\infty$ (3.6)

and

$q_{n}-q_{n+1}=O\left(\frac{1}{n^{2-\alpha}}\right),\text{ \ \ when \ \
}n\rightarrow\infty,$ (3.7)

is bounded from the Hardy space $H_{1/\left(1+\alpha\right)}$ to the space
$weak-L_{1/\left(1+\alpha\right)},$ for $0<\alpha\leq 1.$

Moreover, for $0<\alpha\leq 1$ and non-increasing sequence
$\\{q_{k}:k\in\mathbb{N}\\}$ satisfying the conditions

$\overline{\lim_{n\rightarrow\infty}}\frac{n^{\alpha}}{Q_{n}}\geq
c_{\alpha}>0$ (3.8)

and

$\left|q_{n}-q_{n+1}\right|\geq c_{\alpha}n^{\alpha-2},\text{ \
}n\in\mathbb{N}\text{.}$ (3.9)

there exists a martingale $f\in H_{1/\left(1+\alpha\right)}$ such that

$\underset{n\in\mathbb{N}}{\sup}\left\|t_{n}f\right\|_{{1/\left(1+\alpha\right)}}=\infty.$

In [104] (see also [14]) was proved that if $f\in H_{p},$ where
$0<p\,<1/\left(1+\alpha\right)$ for some $0<\alpha\leq 1,$ and
$\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-increasing numbers
satisfying conditions (3.6) and (3.7) the maximal operator

$\overset{\sim}{t}_{p,\alpha}^{\ast}:=\frac{\left|t_{n}f\right|}{\left(n+1\right)^{1/p-1-\alpha}}$

is bounded from the martingale Hardy space $H_{p}$ to the Lebesgue space
$L_{p}.$

Moreover, let $\left\\{\Phi_{n}:n\in\mathbb{N}_{+}\right\\}$ be any non-
decreasing sequence, satisfying the condition

$\overline{\lim_{n\rightarrow\infty}}\frac{\left(n+1\right)^{1/p-1-\alpha}}{\Phi_{n}}=+\infty,$
(3.10)

then there exists Nörlund means with non-increasing sequence
$\\{q_{k}:k\in\mathbb{N}\\}$ satisfying the conditions (3.8) and (3.9) such
that

$\sup_{k\in\mathbb{N}}\frac{\left\|\frac{t_{M_{{}_{2n_{k}}}+1}f_{k}}{\Phi_{M_{{}_{2n_{k}}}+1}}\right\|_{weal-
L_{p}}}{\left\|f_{k}\right\|_{H_{p}}}=\infty.$

It follows that if $0<p<1/\left(1+\alpha\right)$ and $f\in H_{p},$ then there
exists an absolute constant $c_{p,\alpha},$ depending only on $p$ and
$\alpha$, such that

$\left\|t_{n}f\right\|_{p}\leq
c_{p,\alpha}\left(n+1\right)^{1/p-1-\alpha}\left\|f\right\|_{H_{p}},\text{ \
}n\in\mathbb{N}_{+}.$

On the other hand, let $\left\\{\Phi_{n}:n\in\mathbb{N}\right\\}$ be any non-
decreasing sequence satisfying the condition (3.10), then there exists a
martingale $f\in H_{p}$ such that

$\sup_{n\in\mathbb{N}}\left\|\frac{t_{n}f}{\Phi_{n}}\right\|_{weak-
L_{p}}=\infty.$

Moreover, let $\left\\{\Phi_{n}:n\in\mathbb{N}\right\\}$ be any non-decreasing
sequence satisfying the condition (3.10), then the maximal operator

$\sup_{n\in\mathbb{N}}\frac{\left|t_{n}f\right|}{\Phi_{n}}$

is not bounded from the Hardy space $H_{p}$ to the space $weak-L_{p}.$

In [17] (see also [14]) was proved if $f\in H_{1/(1+\alpha)},$ where
$0<\alpha\leq 1$ and $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-
increasing numbers satisfying the conditions (3.6) and (3.7), then there
exists an absolute constant $c_{\alpha}$ depending only on $\alpha$ such that
the maximal operator

$\overset{\sim}{t}_{\alpha}^{\ast}:=\frac{\left|t_{n}f\right|}{\log^{1+\alpha}\left(n+1\right)}$

is bounded from the martingale Hardy space $H_{1/(1+\alpha)}$ to the Lebesgue
space $L_{1/(1+\alpha)}.$

Moreover, if $\left\\{\Phi_{n}:n\in\mathbb{N}_{+}\right\\}$ be any non-
decreasing sequence satisfying the condition

$\overline{\lim_{n\rightarrow\infty}}\frac{\log^{1+\alpha}\left(n+1\right)}{\Phi_{n}}=+\infty,$
(3.11)

then there exists Nörlund means with non-increasing sequence
$\\{q_{k}:k\in\mathbb{N}\\}$ satisfying the conditions (3.8) and (3.9) such
that

$\sup_{k\in\mathbb{N}}\frac{\left\|\sup_{n}\left|\frac{t_{n}f_{k}}{\Phi_{n}}\right|\right\|_{1/(1+\alpha)}}{\left\|f\right\|_{H_{1/(1+\alpha)}}}=\infty.$

In Persson, Tephnadze and Wall [106] was proved that if $0<p<1/2$, $f\in
H_{p}$ and the sequence $\left\\{q_{k}:k\in\mathbb{N}\right\\}$ be non-
decreasing, then there exists an absolute constant $c_{p}$ depending only on
$p$ such that

$\overset{\infty}{\underset{k=1}{\sum}}\frac{\left\|t_{k}f\right\|_{p}^{p}}{k^{2-2p}}\leq
c_{p}\left\|f\right\|_{H_{p}}^{p}.$

On the other hand, according the fact that Fejér means are examples of Nörlund
means with non-decreasing sequence $\left\\{q_{k}:k\in\mathbb{N}\right\\}$ we
immediately obtain that the asymptotic behaviour of the sequence of weights

$\left\\{1/k^{2-2p}:k\in\mathbb{N}\right\\}$

in Nörlund means can not be improved.

In Persson, Tephnadze and Wall [106] was proved if $f\in H_{1/2}$ and the
sequence $\left\\{q_{k}:k\in\mathbb{N}\right\\}$ be non-decreasing satisfying
condition (3.13), then there exists an absolute constant $c,$ such that

$\frac{1}{\log
n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|t_{k}f\right\|_{1/2}^{1/2}}{k}\leq
c\left\|f\right\|_{H_{1/2}}^{1/2}.$

In Blahota and Tephnadze [14] was investigated Nörlund means with non-
increasing sequence $\\{q_{k}:k\in\mathbb{N}\\}$ in the case
$0<p<1/\left(1+\alpha\right)$ where $0<\alpha<1$. In particular, if $f\in
H_{p},$ where $0<p\,<1/\left(1+\alpha\right),$ $0<\alpha\leq 1$ and
$\\{q_{k}:k\in\mathbb{N}\\},$ be a sequence of non-increasing numbers
satisfying the conditions (3.6) and (3.7), then there exists an absolute
constant $c_{\alpha,p},$ depending only on $\alpha$ and $p$ such that

$\overset{\infty}{\underset{k=1}{\sum}}\frac{\left\|t_{k}f\right\|_{H_{p}}^{p}}{k^{2-\left(1+\alpha\right)p}}\leq
c_{\alpha,p}\left\|f\right\|_{H_{p}}^{p}.$

In Blahota, Persson and Tephnadze [17] was proved that if $f\in
H_{1/(1+\alpha)}$ where $0<\alpha\leq 1$ and $\\{q_{k}:k\in\mathbb{N}\\}$ be a
sequence of non-increasing numbers satisfying the conditions (3.6) and (3.7),
then there exists an absolute constant $c_{\alpha}$ depending only on $\alpha$
such that

$\frac{1}{\log
n}\overset{n}{\underset{m=1}{\sum}}\frac{\left\|t_{m}f\right\|_{H_{1/\left(1+\alpha\right)}}^{1/\left(1+\alpha\right)}}{m}\leq
c_{\alpha}\left\|f\right\|_{H_{1/\left(1+\alpha\right)}}^{1/\left(1+\alpha\right)}.$

In [165] we investigate the maximal operator $T^{\ast}$ of the summability
method (1.19) with non-increasing sequence $\\{q_{k}:k\geq 0\\},$ is bounded
from the Hardy space $H_{1/2}$ to the space $weak-L_{1/2}.$

Moreover, for any $0<p<1/2$ and non-increasing sequence $\\{q_{k}:k\geq 0\\}$
satisfying the condition

$\frac{q_{n+1}}{Q_{n+2}}\geq\frac{c}{n},\text{ \ \ }\left(c\geq 1\right).$

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

$\underset{n\in\mathbb{N}}{\sup}\left\|T_{n}f\right\|_{weak-L_{p}}=\infty.$

We also proved that the maximal operator $T^{\ast}$ of the summability method
(1.19) with non-decreasing sequence $\\{q_{k}:k\geq 0\\}$ satisfying the
condition

$\frac{q_{n-1}}{Q_{n}}=O\left(\frac{1}{n}\right)$

is bounded from the Hardy space $H_{1/2}$ to the space $weak-L_{1/2}.$

Moreover, for any $0<p<1/2$ and non-decreasing sequence $\\{q_{k}:k\geq 0\\},$
there exists a martingale $f\in H_{p},$ such that

$\underset{n\in\mathbb{N}}{\sup}\left\|T_{n}f\right\|_{weak-L_{p}}=\infty.$

In [166] we proved that for $0<p\leq 1/2,$ $f\in H_{p}$ and non-decreasing
sequence $\\{q_{k}:k\geq 0\\}$ the maximal operator

$\widetilde{T}_{p}^{\ast}f:=\sup_{n\in\mathbb{N}_{+}}\frac{\left|T_{n}f\right|}{\left(n+1\right)^{1/p-2}\log^{2\left[1/2+p\right]}\left(n+1\right)}$
(3.12)

is bounded from the Hardy space $H_{p}$ to the space $L_{p}.$

On the other hand if $0<p\leq 1/2,$ $f\in H_{p}$ and $\\{q_{k}:k\geq 0\\}$ be
a sequence of non-increasing numbers,satisfying the condition

$\frac{q_{n-1}}{Q_{n}}=O\left(\frac{1}{n}\right),\text{ \ \ as \ \ }\
n\rightarrow\infty.$ (3.13)

, then the maximal operator

$\widetilde{T}_{p}^{\ast}f:=\sup_{n\in\mathbb{N}_{+}}\frac{\left|T_{n}f\right|}{\left(n+1\right)^{1/p-2}\log^{2\left[1/2+p\right]}\left(n+1\right)}$
(3.14)

is bounded from the Hardy space $H_{p}$ to the space $L_{p}.$

Since the maximal operator

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

is bounded from the martingale Hardy space $H_{p}$ to the space $L_{p}$ and
the rate of denominator $\left(n+1\right)^{1/p-2}\log^{2\left[1/2+p\right]}$
is in the sense sharp and Fejer means is example of $T$ means non-decreasing
sequence we obtain that this weights are also sharp in (3.12) and (3.14).

In [166] we also investigate strong convergence of $T$ means with respect to
Vilenkin systems. In particular, if $0<p<1/2,$ $f\in H_{p}$ and
$\\{q_{k}:k\geq 0\\}$ be a sequence of non-decreasing numbers, then there
exists an absolute constant $c_{p},$ depending only on $p,$ such that the
inequality

$\overset{\infty}{\underset{k=1}{\sum}}\frac{\left\|T_{k}f\right\|_{p}^{p}}{k^{2-2p}}\leq
c_{p}\left\|f\right\|_{H_{p}}^{p}$

holds.

Moreover, if $f\in H_{1/2}$ and $\\{q_{k}:k\geq 0\\}$ be a sequence of non-
increasing numbers, satisfying the condition (3.13), then there exists an
absolute constant $c,$ such that the inequality

$\frac{1}{\log
n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|T_{k}f\right\|_{1/2}^{1/2}}{k}\leq
c\left\|f\right\|_{H_{1/2}}^{1/2}$

holds.

###  3.2 Auxiliary lemmas

###### Lemma 3.1

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

$\displaystyle Q_{n}$ $\displaystyle:=$
$\displaystyle\overset{n-1}{\underset{j=0}{\sum}}q_{j}=\overset{n-2}{\underset{j=0}{\sum}}\left(q_{j}-q_{j+1}\right)j+q_{n-1}{(n-1)}$
(3.15)

$F_{n}=\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=0}{\sum}}\left(q_{j}-q_{j+1}\right)jK_{j}+q_{n-1}(n-1)K_{n-1}\right).$
(3.16)

$t_{n}=\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=0}{\sum}}\left(q_{j}-q_{j+1}\right)j\sigma_{j}+q_{n-1}(n-1)\sigma_{n-1}\right).$
(3.17)

Proof: If we invoke Abel transformation we immediately get identities (3.15),
(3.16) and (3.17). The proof is complete.

###### Lemma 3.2

Let $n\in\mathbb{N}$ and $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-
increasing numbers, or non-decreasing function satisfying condition (3.13).
Then

$\|F_{n}\|_{1}<c<\infty.$ (3.18)

Proof: Let $n\in\mathbb{N}$ and $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of
non-increasing numbers. By combining (1.12) with (3.15) and (3.17) we can
conclude that

$\displaystyle\|T_{n}\|_{1}$ $\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=0}{\sum}}\left|q_{j}-q_{j+1}\right|j\|\sigma_{j}\|_{1}+q_{n-1}(n-1)\|\sigma_{n-1}\|_{1}\right)$
$\displaystyle\leq$
$\displaystyle\frac{c}{Q_{n}}\left(\overset{n-2}{\underset{j=0}{\sum}}\left(q_{j}-q_{j+1}\right)j+q_{n-1}(n-1)\right)\leq
c<\infty.$

Let $n\in\mathbb{N}$ and $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence non-
decreasing function satisfying condition (3.13). Then By using again (1.12)
with (3.15) and (3.17) we find that

$\displaystyle\|T_{n}\|_{1}$ $\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=0}{\sum}}\left|q_{j}-q_{j+1}\right|j\|\sigma_{j}\|_{1}+q_{n-1}(n-1)\|\sigma_{n-1}\|_{1}\right)$
$\displaystyle\leq$
$\displaystyle\frac{c}{Q_{n}}\left(\overset{n-2}{\underset{j=0}{\sum}}\left(q_{j+1}-q_{j}\right)j+q_{n-1}(n-1)\right)$
$\displaystyle=$
$\displaystyle\frac{c}{Q_{n}}\left(2q_{n-1}(n-1)-\left(\overset{n-2}{\underset{j=0}{\sum}}\left(q_{j}-q_{j+1}\right)j+q_{n-1}(n-1)\right)\right)$
$\displaystyle=$ $\displaystyle\frac{c}{Q_{n}}(2q_{n-1}(n-1)-Q_{n})\leq
c<\infty.$

The proof is complete.

###### Lemma 3.3

Let $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-increasing numbers and
$n>M_{N}$. Then

$\left|\frac{1}{Q_{n}}\overset{n-1}{\underset{j=M_{N}}{\sum}}q_{j}D_{j}\left(x\right)\right|\leq\frac{c}{M_{N}}\left\\{\sum_{j=0}^{\left|n\right|}M_{j}\left|K_{M_{j}}\right|\right\\},$

where $c$ is an absolute constant.

Proof: Since sequence is non-increasing number we get that

$\displaystyle\frac{1}{Q_{n}}\left(q_{M_{N}}+\overset{n-2}{\underset{j=M_{N}}{\sum}}\left|q_{j}-q_{j+1}\right|+q_{n-1}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\left(q_{M_{N}}+\overset{n-2}{\underset{j=M_{N}}{\sum}}\left(q_{j}-q_{j+1}\right)+q_{n-1}\right)$
$\displaystyle\leq$
$\displaystyle\frac{2q_{M_{N}}}{Q_{n}}\leq\frac{2q_{M_{N}}}{Q_{M_{N}+1}}\leq\frac{c}{M_{N}}.$

If we apply (3.15) and (3.16) in Lemma 3.1 and Abel transformation we
immediately get that

$\displaystyle\left|\frac{1}{Q_{n}}\overset{n-1}{\underset{j=M_{N}}{\sum}}q_{j}D_{j}\left(x\right)\right|$
$\displaystyle=$
$\displaystyle\frac{1}{Q_{n}}\left(q_{M_{n}}\sigma_{{M_{n}-1}}+\overset{n-2}{\underset{j=M_{N}}{\sum}}\left(q_{j}-q_{j+1}\right)\sigma_{j}+q_{n-1}\sigma_{n-1}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\left(q_{M_{n}}+\overset{n-2}{\underset{j=M_{N}}{\sum}}\left|q_{j}-q_{j+1}\right|+q_{n-1}\right)\sum_{i=0}^{\left|n\right|}M_{i}\left|K_{M_{i}}\right|$
$\displaystyle\leq$
$\displaystyle\frac{c}{M_{N}}\sum_{i=0}^{\left|n\right|}M_{i}\left|K_{M_{i}}\right|.$

The proof is complete.

###### Lemma 3.4

Let $x\in I_{N}^{k,l},\ \ k=0,\dots,N-1,\ \ l=k+1,\dots,N$ and
$\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-increasing numbers. Then

$\int_{I_{N}}\left|\frac{1}{Q_{n}}\overset{n-1}{\underset{j=M_{N}}{\sum}}q_{j}D_{j}\left(x-t\right)\right|d\mu\left(t\right)\leq\frac{cM_{l}M_{k}}{M^{2}_{N}}.$

Here $c$ is an absolute constant.

Proof: Let $x\in I_{N}^{k,l},$ for $0\leq k<l\leq N-1$ and $t\in I_{N}.$
First, we observe that $x-t\in$ $I_{N}^{k,l}.$ Next, we apply equality 1.9 and
Lemmas 3.4 to obtain that

$\displaystyle\int_{I_{N}}\left|\frac{1}{Q_{n}}\overset{n-1}{\underset{j=M_{N}}{\sum}}q_{j}D_{j}\left(x-t\right)\right|d\mu\left(t\right)$
$\displaystyle\leq$
$\displaystyle\frac{c}{M_{N}}\underset{i=0}{\overset{\left|n\right|}{\sum}}M_{i}\int_{I_{N}}\left|K_{M_{i}}\left(x-t\right)\right|d\mu\left(t\right)$
$\displaystyle\leq$
$\displaystyle\frac{c}{M_{N}}\int_{I_{N}}\overset{l}{\underset{i=0}{\sum}}M_{i}M_{k}d\mu\left(t\right)$
$\displaystyle\leq$ $\displaystyle\frac{cM_{k}M_{l}}{M^{2}_{N}}$

and the first estimate is proved.

Now, let $x\in I_{N}^{k,N}$. Since $x-t\in I_{N}^{k,N}$ for $t\in I_{N},$ by
combining (1.6) and (1.7) we have that

$\left|D_{i}\left(x-t\right)\right|\leq M_{k}$

and

$\displaystyle\int_{I_{N}}\left|\frac{1}{Q_{n}}\overset{n-1}{\underset{j=M_{N}}{\sum}}q_{j}D_{j}\left(x-t\right)\right|d\mu\left(t\right)$
$\displaystyle\leq$
$\displaystyle\frac{c}{Q_{n}}\underset{i=0}{\overset{\left|n\right|}{\sum}}q_{i}\int_{I_{N}}\left|D_{i}\left(x-t\right)\right|d\mu\left(t\right)$
$\displaystyle\leq$
$\displaystyle\frac{c}{Q_{n}}\overset{\left|n\right|-1}{\underset{i=0}{\sum}}q_{i}\int_{I_{N}}M_{k}d\mu\left(t\right)$
$\displaystyle\leq$ $\displaystyle\frac{cM_{k}}{M_{N}}.$

According to (3.2) and (3.2) the proof is complete.

###### Lemma 3.5

Let $n>M_{N}$ and $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-increasing
numbers, satisfying condition (3.43). Then

$\left|\frac{1}{Q_{n}}\overset{n-1}{\underset{j=M_{N}}{\sum}}q_{j}D_{j}\left(x\right)\right|\leq\frac{c}{n}\left\\{\sum_{j=0}^{\left|n\right|}M_{j}\left|K_{M_{j}}(x)\right|\right\\},$

where $c$ is an absolute constant.

Proof: Since sequence is non-increasing number we get that

$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=M_{N}}{\sum}}\left|q_{j}-q_{j+1}\right|+q_{n-1}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=M_{N}}{\sum}}\left(q_{j}-q_{j+1}\right)+q_{n-1}\right)$
$\displaystyle\leq$
$\displaystyle\frac{q_{M_{N}}}{Q_{n}}\leq\frac{c}{Q_{n}}\leq\frac{c}{n}.$

If we apply (3.15) and (3.16) in Lemma 3.1 we immediately get that

$\displaystyle\left|\frac{1}{Q_{n}}\overset{n-1}{\underset{j=M_{N}}{\sum}}q_{j}D_{j}\left(x\right)\right|$
$\displaystyle\leq$
$\displaystyle\left(\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=M_{N}+1}{\sum}}\left|q_{j}-q_{j+1}\right|+q_{n-1}\right)\right)\sum_{i=0}^{\left|n\right|}M_{i}\left|K_{M_{i}}(x)\right|$
$\displaystyle\leq$
$\displaystyle\frac{c}{n}\sum_{i=0}^{\left|n\right|}M_{i}\left|K_{M_{i}}(x)\right|.$

The proof is complete.

###### Lemma 3.6

Let $x\in I_{N}^{k,l},\ \ k=0,\dots,N-2,\ \ l=k+1,\dots,N-1$ and
$\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-increasing numbers,
satisfying condition (3.43). Then

$\int_{I_{N}}\left|\frac{1}{Q_{n}}\overset{n-1}{\underset{j=M_{N}}{\sum}}q_{j}D_{j}\left(x-t\right)\right|d\mu\left(t\right)\leq\frac{cM_{l}M_{k}}{nM_{N}}.$

Let $x\in I_{N}^{k,N},$ $k=0,\dots,N-1.$ Then

$\int_{I_{N}}\left|\frac{1}{Q_{n}}\overset{n-1}{\underset{j=M_{N}}{\sum}}q_{j}D_{j}\left(x-t\right)\right|d\mu\left(t\right)\leq\frac{cM_{k}}{M_{N}}.$

Here $c$ is an absolute constant.

Proof: Let $x\in I_{N}^{k,l},$ for $0\leq k<l\leq N-1$ and $t\in I_{N}.$
First, we observe that $x-t\in$ $I_{N}^{k,l}.$ Next, we apply equality 1.9 and
Lemma 3.5 to obtain that

$\displaystyle\int_{I_{N}}\left|\frac{1}{Q_{n}}\overset{n-1}{\underset{j=M_{N}}{\sum}}q_{j}D_{j}\left(x-t\right)\right|d\mu\left(t\right)$
$\displaystyle\leq$
$\displaystyle\frac{c}{n}\underset{i=0}{\overset{\left|n\right|}{\sum}}M_{i}\int_{I_{N}}\left|K_{M_{i}}\left(x-t\right)\right|d\mu\left(t\right)$
$\displaystyle\leq$
$\displaystyle\frac{c}{n}\int_{I_{N}}\overset{l}{\underset{i=0}{\sum}}M_{i}M_{k}d\mu\left(t\right)$
$\displaystyle\leq$ $\displaystyle\frac{cM_{k}M_{l}}{nM_{N}}$

and the first estimate is proved.

Now, let $x\in I_{N}^{k,N}$. Since $x-t\in I_{N}^{k,N}$ for $t\in I_{N},$ by
combining again equality (1.9) and Lemma 3.5 we have that

$\displaystyle\int_{I_{N}}\left|\frac{1}{Q_{n}}\overset{n-1}{\underset{j=M_{N}}{\sum}}q_{j}D_{j}\left(x-t\right)\right|d\mu\left(t\right)$
$\displaystyle\leq$
$\displaystyle\frac{c}{n}\underset{i=0}{\overset{\left|n\right|}{\sum}}M_{i}\int_{I_{N}}\left|K_{M_{i}}\left(x-t\right)\right|d\mu\left(t\right)$
$\displaystyle\leq$
$\displaystyle\frac{c}{n}\overset{\left|n\right|-1}{\underset{i=0}{\sum}}M_{i}\int_{I_{N}}M_{k}d\mu\left(t\right)$
$\displaystyle\leq$ $\displaystyle\frac{cM_{k}}{M_{N}}.$

By combining (3.2) and (3.2) we complete the proof.

###### Lemma 3.7

Let $n\geq M_{N},\ \ x\in I_{N}^{k,l},\ \ k=0,\dots,N-1,\ \ l=k+1,\dots,N$ and
$\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-increasing sequence,
satisfying condition (3.43). Then

$\int_{I_{N}}\left|\frac{1}{Q_{n}}\overset{n-1}{\underset{j=M_{N}}{\sum}}q_{j}D_{j}\left(x-t\right)\right|d\mu\left(t\right)\leq\frac{cM_{l}M_{k}}{M_{N}^{2}},$

where $c$ is an absolute constant.

Proof: Since $n\geq M_{N}$ if we apply Lemma 3.6 we immediately get the proof.

###### Lemma 3.8

Let $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-decreasing numbers
satisfying (3.13). Then

$\left|F_{n}\right|\leq\frac{c}{n}\left\\{\sum_{j=0}^{\left|n\right|}M_{j}\left|K_{M_{j}}\right|\right\\},$

where $c$ is an absolute constant.

Proof: Since sequence $\\{q_{k}:k\in\mathbb{N}\\}$ be non-decreasing. If we
apply condition (3.13) we can conclude that

$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=0}{\sum}}\left|q_{j}-q_{j+1}\right|+q_{n-1}\right)$
$\displaystyle\leq\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=0}{\sum}}\left(q_{j+1}-q_{j}\right)+q_{n-1}\right)$
$\displaystyle\leq$
$\displaystyle\frac{2q_{n-1}-q_{0}}{Q_{n}}\leq\frac{q_{n-1}}{Q_{n}}\leq\frac{c}{n}.$

If we apply (3.15) and (3.16) in Lemma 3.1 we immediately get that

$\displaystyle\left|F_{n}\right|$ $\displaystyle\leq$
$\displaystyle\left(\frac{1}{Q_{n}}\left(\overset{n-1}{\underset{j=1}{\sum}}\left|q_{j}-q_{j+1}\right|+q_{0}\right)\right)\sum_{i=0}^{\left|n\right|}M_{i}\left|K_{M_{i}}\right|$
$\displaystyle=$
$\displaystyle\left(\frac{1}{Q_{n}}\left(\overset{n-1}{\underset{j=1}{\sum}}\left(q_{j}-q_{j+1}\right)+q_{0}\right)\right)\sum_{i=0}^{\left|n\right|}M_{i}\left|K_{M_{i}}\right|$
$\displaystyle\leq$
$\displaystyle\frac{q_{n-1}}{Q_{n}}\sum_{i=0}^{\left|n\right|}M_{i}\left|K_{M_{i}}\right|\leq\frac{c}{n}\sum_{i=0}^{\left|n\right|}M_{i}\left|K_{M_{i}}\right|.$

The proof is complete.

###### Lemma 3.9

Let $x\in I_{N}^{k,l},\ \ k=0,\dots,N-2,\ \ l=k+1,\dots,N-1$ and
$\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-decreasing numbers,
satisfying condition (3.13). Then

$\int_{I_{N}}\left|F_{n}\left(x-t\right)\right|d\mu\left(t\right)\leq\frac{cM_{l}M_{k}}{nM_{N}}.$

Let $x\in I_{N}^{k,N},$ $k=0,\dots,N-1.$ Then

$\int_{I_{N}}\left|F_{n}\left(x-t\right)\right|d\mu\left(t\right)\leq\frac{cM_{k}}{M_{N}}.$

Here $c$ is an absolute constant.

Proof: The proof is quite analogously to Lemma 3.6. So we leave out the
details.

###### Lemma 3.10

Let $n\geq M_{N},\ \ x\in I_{N}^{k,l},\ \ k=0,\dots,N-1,\ \ l=k+1,\dots,N$ and
$\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-decreasing sequence,
satisfying condition (3.13). Then

$\int_{I_{N}}\left|F_{n}\left(x-t\right)\right|d\mu\left(t\right)\leq\frac{cM_{l}M_{k}}{M_{N}^{2}},$

where $c$ is an absolute constant.

Proof: Since $n\geq M_{N}$ if we apply Lemma 3.9 we immediately get the proof.

###  3.3 On the maximal operators of $T$ means with respect to Vilenkin
systems on the martingale Hardy spaces

First we state our main result concerning the maximal operator of the
summation method (1.19), which we also show is in a sense sharp.

###### Theorem 3.11

a) The maximal operator $T^{\ast}$ of the summability method (1.19) with non-
increasing sequence $\\{q_{k}:k\geq 0\\},$ is bounded from the Hardy space
$H_{1/2}$ to the space $weak-L_{1/2}.$

The statement in a) is sharp in the following sense:

b) Let $0<p<1/2$ and $\\{q_{k}:k\geq 0\\}$ be a non-increasing sequence,
satisfying the condition

$\frac{q_{n+1}}{Q_{n+2}}\geq\frac{c}{n},\text{ \ \ }\left(c\geq 1\right).$
(3.23)

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

$\underset{n\in\mathbb{N}}{\sup}\left\|T_{n}f\right\|_{weak-L_{p}}=\infty.$

Proof: a). Let the sequence $\\{q_{k}:k\geq 0\\}$ be non-increasing. By
combining (3.15) with (3.16) and using Abel transformation we get that

$\displaystyle\left|T_{n}f\right|$ $\displaystyle\leq$
$\displaystyle\left|\frac{1}{Q_{n}}\overset{n-1}{\underset{j=1}{\sum}}q_{j}S_{j}f\right|$
$\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=1}{\sum}}\left|q_{j}-q_{j+1}\right|j\left|\sigma_{j}f\right|+q_{n-1}(n-1)\left|\sigma_{n}f\right|\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=1}{\sum}}\left(q_{j}-q_{j+1}\right)j+q_{n-1}(n-1)\right)\sigma^{\ast}f\leq\sigma^{\ast}f$

so that

$T^{\ast}f\leq\sigma^{\ast}f.$ (3.24)

If we apply (3.24), according that $\sigma^{*}$ is bounded from the Hardy
space $H_{1/2}$ to the space $weak-L_{1/2},$ we can conclude that the maximal
operators $T^{\ast}$ of all $T$ means with non-increasing sequence
$\\{q_{k}:k\geq 0\\},$ are bounded from the Hardy space $H_{1/2}$ to the space
$weak-L_{1/2}.$ The proof of part a) of Theorem 1 is complete.

b) Let $0<p<1/2$ and $\left\\{\alpha_{k}:k\in\mathbb{N}\right\\}$ be an
increasing sequence of positive integers such that:

$\sum_{k=0}^{\infty}1/\alpha_{k}^{p}<\infty,$ (3.25)

$\lambda\sum_{\eta=0}^{k-1}\frac{M_{\alpha_{\eta}}^{1/p}}{\alpha_{\eta}}<\frac{M_{\alpha_{k}}^{1/p}}{\alpha_{k}},$
(3.26)

$\frac{32\lambda
M_{\alpha_{k-1}}^{1/p}}{\alpha_{k-1}}<\frac{M_{\alpha_{k}}^{1/p-2}}{\alpha_{k}},$
(3.27)

where $\lambda=\sup_{n}m_{n}.$

We note that such an increasing sequence
$\left\\{\alpha_{k}:k\in\mathbb{N}\right\\}$ which satisfies conditions
(3.25), (3.26) and (3.27) can be constructed.

Let

$f^{\left(A\right)}=\sum_{\left\\{k;\text{
}\lambda_{k}<A\right\\}}\lambda_{k}a_{k},$ (3.28)

where

$\lambda_{k}=\frac{\lambda}{\alpha_{k}}\ \ \ \ and\ \ \ \
a_{k}=\frac{M_{\alpha_{k}}^{1/p-1}}{\lambda}\left(D_{M_{\alpha_{k}+1}}-D_{M_{{}_{\alpha_{k}}}}\right).$

By using Lemma 1.3, it is easy to show that the martingale $f\in H_{1/2}.$
Moreover, it is easy to show that

$\widehat{f}(j)=\left\\{\begin{array}[]{ll}\frac{M_{\alpha_{k}}^{1/p-1}}{\alpha_{k}},&\text{if
\thinspace\thinspace}j\in\left\\{M_{\alpha_{k}},...,M_{\alpha_{k}+1}-1\right\\},\text{
}k=0,1,2...,\\\ 0,&\text{if \
\thinspace\thinspace\thinspace}j\notin\bigcup\limits_{k=1}^{\infty}\left\\{M_{\alpha_{k}},...,M_{\alpha_{k}+1}-1\right\\}.\end{array}\right.$
(3.29)

We can write

$T_{M_{\alpha_{k}}+2}f=\frac{1}{Q_{M_{\alpha_{k}}+2}}\sum_{j=0}^{M_{\alpha_{k}}}q_{j}S_{j}f+\frac{q_{M_{\alpha_{k}}+1}}{Q_{M_{\alpha_{k}}+2}}S_{M_{\alpha_{k}}+1}f:=I+II.$
(3.30)

Let $M_{\alpha_{s}}\leq$ $j\leq M_{\alpha_{s}+1},$ where $s=0,...,k-1.$
Moreover,

$\left|D_{j}-D_{M_{{}_{\alpha_{s}}}}\right|\leq
j-M_{{}_{\alpha_{s}}}\leq\lambda M_{{}_{\alpha_{s}}},\text{ \ \
}\left(s\in\mathbb{N}\right)$

so that, according to (1.3) and (3.29), we have that

$\displaystyle\left|S_{j}f\right|=\left|\sum_{v=0}^{M_{\alpha_{s-1}+1}-1}\widehat{f}(v)\psi_{v}+\sum_{v=M_{\alpha_{s}}}^{j-1}\widehat{f}(v)\psi_{v}\right|$
$\displaystyle\leq$
$\displaystyle\left|\sum_{\eta=0}^{s-1}\sum_{v=M_{\alpha_{\eta}}}^{M_{\alpha_{\eta}+1}-1}\frac{M_{\alpha_{\eta}}^{1/p-1}}{\alpha_{\eta}}\psi_{v}\right|+\frac{M_{\alpha_{s}}^{1/p-1}}{\alpha_{s}}\left|\left(D_{j}-D_{M_{{}_{\alpha_{s}}}}\right)\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{\eta=0}^{s-1}\frac{M_{\alpha_{\eta}}^{1/p-1}}{\alpha_{\eta}}\left(D_{M_{{}_{\alpha_{\eta}+1}}}-D_{M_{{}_{\alpha_{\eta}}}}\right)\right|+\frac{M_{\alpha_{s}}^{1/p-1}}{\alpha_{s}}\left|\left(D_{j}-D_{M_{{}_{\alpha_{s}}}}\right)\right|$
$\displaystyle\leq$
$\displaystyle\lambda\sum_{\eta=0}^{s-1}\frac{M_{\alpha_{\eta}}^{1/p}}{\alpha_{\eta}}+\frac{\lambda
M_{\alpha_{s}}^{1/p}}{\alpha_{s}}\leq\frac{2\lambda
M_{\alpha_{s-1}}^{1/p}}{\alpha_{s-1}}+\frac{\lambda
M_{\alpha_{s}}^{1/p}}{\alpha_{s}}\leq\frac{4\lambda
M_{\alpha_{k-1}}^{1/p}}{\alpha_{k-1}}.$

Let $M_{\alpha_{s-1}+1}+1\leq$ $j\leq M_{\alpha_{s}},$ where $s=1,...,k.$
Analogously to (2.3) we can prove that

$\displaystyle\left|S_{j}f\right|=\left|\sum_{v=0}^{M_{\alpha_{s-1}+1}-1}\widehat{f}(v)\psi_{v}\right|=\left|\sum_{\eta=0}^{s-1}\sum_{v=M_{\alpha_{\eta}}}^{M_{\alpha_{\eta}+1}-1}\frac{M_{\alpha_{\eta}}^{1/p-1}}{\alpha_{\eta}}\psi_{v}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{\eta=0}^{s-1}\frac{M_{\alpha_{\eta}}^{1/p-1}}{\alpha_{\eta}}\left(D_{M_{{}_{\alpha_{\eta}+1}}}-D_{M_{{}_{\alpha_{\eta}}}}\right)\right|\leq\frac{2\lambda
M_{\alpha_{s-1}}^{1/p}}{\alpha_{s-1}}\leq\frac{4\lambda
M_{\alpha_{k-1}}^{1/p}}{\alpha_{k-1}}.$

Hence,

$\left|I\right|\leq\frac{1}{Q_{M_{\alpha_{k}}+2}}\sum_{j=0}^{M_{\alpha_{k}}}q_{j}\left|S_{j}f\right|\leq\frac{4\lambda
M_{\alpha_{k-1}}^{1/p}}{\alpha_{k-1}}\frac{1}{Q_{M_{\alpha_{k}}+2}}\sum_{j=0}^{M_{\alpha_{k}}}q_{j}\leq\frac{4\lambda
M_{\alpha_{k-1}}^{1/p}}{\alpha_{k-1}}.$ (3.32)

If we now apply (3.29) and (3.3) we get that

$\displaystyle\left|II\right|$ $\displaystyle=$
$\displaystyle\frac{q_{M_{\alpha_{k}}+1}}{Q_{M_{\alpha_{k}}+2}}\left|\frac{M_{\alpha_{k}}^{1/p-1}}{\alpha_{k}}\psi_{M_{\alpha_{k}}}+S_{M_{\alpha_{k}}}f\right|$
$\displaystyle=$
$\displaystyle\frac{q_{M_{\alpha_{k}}+1}}{Q_{M_{\alpha_{k}}+2}}\left|\frac{M_{\alpha_{k}}^{1/p-1}}{\alpha_{k}}\psi_{M_{\alpha_{k}}}+S_{M_{\alpha_{k-1}+1}}f\right|$
$\displaystyle\geq$
$\displaystyle\frac{q_{M_{\alpha_{k}}+1}}{Q_{M_{\alpha_{k}}+2}}\left(\left|\frac{M_{\alpha_{k}}^{1/p-1}}{\alpha_{k}}\psi_{M_{\alpha_{k}}}\right|-\left|S_{M_{\alpha_{k-1}+2}}f\right|\right)$
$\displaystyle\geq$
$\displaystyle\frac{q_{M_{\alpha_{k}}+1}}{Q_{M_{\alpha_{k}}+2}}\left(\frac{M_{\alpha_{k}}^{1/p-1}}{\alpha_{k}}-\frac{4\lambda
M_{\alpha_{k-1}}^{1/p}}{\alpha_{k-1}}\right)$ $\displaystyle\geq$
$\displaystyle\frac{q_{M_{\alpha_{k}}+1}}{Q_{M_{\alpha_{k}}+2}}\frac{M_{\alpha_{k}}^{1/p-1}}{4\alpha_{k}}.$

Without lost the generality we may assume that $c=1$ in (3.23). By combining
(3.32) and (3.3) we get

$\displaystyle\left|T_{M_{\alpha_{k}}+2}f\right|$ $\displaystyle\geq$
$\displaystyle\left|II\right|-\left|I\right|\geq\frac{q_{M_{\alpha_{k}}+1}}{Q_{M_{\alpha_{k}}+2}}\frac{M_{\alpha_{k}}^{1/p-1}}{4\alpha_{k}}-\frac{4\lambda
M_{\alpha_{k-1}}^{1/p}}{\alpha_{k-1}}$ $\displaystyle\geq$
$\displaystyle\frac{M_{\alpha_{k}}^{1/p-2}}{4\alpha_{k}}-\frac{4\lambda
M_{\alpha_{k-1}}^{1/p}}{\alpha_{k-1}}\geq\frac{M_{\alpha_{k}}^{1/p-2}}{16\alpha_{k}}.$

On the other hand,

$\mu\left\\{x\in
G_{m}:\left|T_{M_{\alpha_{k}}+2}f\left(x\right)\right|\geq\frac{M_{\alpha_{k}}^{1/p-2}}{16\alpha_{k}}\right\\}=\mu\left(G_{m}\right)=1.$
(3.35)

Let $0<p<1/2.$ Then

$\displaystyle\frac{M_{\alpha_{k}}^{1/p-2}}{16\alpha_{k}}\cdot\left(\mu\left\\{x\in
G_{m}:\left|T_{M_{\alpha_{k}}+2}f\left(x\right)\right|\geq\frac{M_{\alpha_{k}}^{1/p-2}}{16\alpha_{k}}\right\\}\right)^{1/p}$
$\displaystyle=$
$\displaystyle\frac{M_{\alpha_{k}}^{1/p-2}}{16\alpha_{k}}\rightarrow\infty,\text{
\ as }k\rightarrow\infty.$

The proof is complete.

A number of special cases of our results are of particular interest and give
both well-known and new information. We just give the following examples of
such $T$ means with non-increasing sequence $\\{q_{k}:k\geq 0\\}:$

###### Corollary 3.12

The maximal operators $U^{\alpha,*}$, $V^{\alpha,*}$ and $R^{*}$ are bounded
from the Hardy space $H_{1/2}$ to the space $weak-L_{1/2}$ but are not bounded
from $H_{p}$ to the space $weak-L_{p},$ when $0<p<1/2.$

Proof: Since $R_{n},U_{n}^{\alpha}\text{ and }V_{n}^{\alpha}$ are the $T$
means with non-increasing sequence $\\{q_{k}:k\geq 0\\},$ then the proof of
this corollary is direct consequence of Theorem 3.11.

###### Corollary 3.13

Let $f\in L_{1}$ and $T_{n}$ be the $T$ means with non-increasing sequence
$\\{q_{k}:k\geq 0\\}$. Then $\ T_{n}f\rightarrow f,\text{ \ \ a.e., \ \ as \
}n\rightarrow\infty.$

Proof: According to part a) of Theorem 3.11 and Lemma 1.5 we also have weak
$(1,1)$ type inequality and by well-known density argument due to
Marcinkiewicz and Zygmund [192] (see Lemma 1.1) we have $T_{n}f\rightarrow f,$
a.e., for all $f\in L_{1}.$ Which follows proof of Corollary 3.13.

###### Corollary 3.14

Let $f\in L_{1}$. Then

$\displaystyle R_{n}f$ $\displaystyle\rightarrow$ $\displaystyle f,\text{ \ \
\ a.e., \ \ \ \ as \ }n\rightarrow\infty,$ $\displaystyle U_{n}^{\alpha}f$
$\displaystyle\rightarrow$ $\displaystyle f,\text{ \ \ \ a.e., \ \ \ as \
}n\rightarrow\infty,\text{\ \ \ }$ $\displaystyle V_{n}^{\alpha}f$
$\displaystyle\rightarrow$ $\displaystyle f,\text{ \ \ \ a.e., \ \ \ as \
}n\rightarrow\infty,\text{\ \ \ }$

Proof: Since $R_{n},U_{n}^{\alpha}\text{ and }V_{n}^{\alpha}$ are the $T$
means with non-increasing sequence $\\{q_{k}:k\geq 0\\},$ then the proof of
this corollary is direct consequence of Corollary 3.13.

Our next main result reads:

###### Theorem 3.15

a) The maximal operator $T^{\ast}$ of the summability method (1.19) with non-
decreasing sequence $\\{q_{k}:k\geq 0\\}$ satisfying the condition

$\frac{q_{n-1}}{Q_{n}}=O\left(\frac{1}{n}\right)$ (3.37)

is bounded from the Hardy space $H_{1/2}$ to the space $weak-L_{1/2}.$

b) Let $0<p<1/2.$ For any non-decreasing sequence $\\{q_{k}:k\geq 0\\},$ there
exists a martingale $f\in H_{p},$ such that

$\underset{n\in\mathbb{N}}{\sup}\left\|T_{n}f\right\|_{weak-L_{p}}=\infty.$

Proof: Let the sequence $\\{q_{k}:k\geq 0\\}$ be non-decreasing. By combining
(3.15) with (3.16) in Lemma 3.1 and using Abel transformation we get that

$\displaystyle\left|T_{n}f\right|$ $\displaystyle\leq$
$\displaystyle\left|\frac{1}{Q_{n}}\overset{n-1}{\underset{j=1}{\sum}}q_{j}S_{j}f\right|$
$\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=1}{\sum}}\left|q_{j}-q_{j+1}\right|j\left|\sigma_{j}f\right|+q_{n-1}(n-1)\left|\sigma_{n}f\right|\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=1}{\sum}}-\left(q_{j}-q_{j+1}\right)j-q_{n-1}(n-1)+2q_{n-1}(n-1)\right)\sigma^{\ast}f$
$\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\left(2q_{n-1}(n-1)-Q_{n}\right)\sigma^{\ast}f\leq
c\sigma^{\ast}f$

so that

$T^{\ast}f\leq c\sigma^{\ast}f.$ (3.38)

If we apply (3.38), according that $\sigma^{*}$ is bounded from the Hardy
space $H_{1/2}$ to the space $weak-L_{1/2},$ we can conclude that the maximal
operators $T^{\ast}$ of all $T$ means with non-decreasing sequence
$\\{q_{k}:k\geq 0\\}$ satisfying the condition 3.37 are bounded from the Hardy
space $H_{1/2}$ to the space $weak-L_{1/2}.$ The proof of part a) is complete.

To prove part b) of Theorem 2 we use the martingale, defined by (3.28) where
$\alpha_{k}$ satisfy conditions (3.25), (3.26) and (3.27). It is easy to show
that for every non-increasing sequence $\\{q_{k}:k\geq 0\\}$ it automatically
holds that

$q_{M_{\alpha_{k}+1}}/Q_{M_{\alpha_{k}+2}}\geq c/M_{\alpha_{k}}.$

According to (3.3), (3.35) and (3.3) we can conclude that

$\left|T_{M_{\alpha_{k}}+2}f\right|\geq\left|II\right|-\left|I\right|\geq\frac{M_{\alpha_{k}}^{1/p-2}}{8\alpha_{k}}.$

Analogously to (3.35) we then get that

$\sup_{k\in\mathbb{N}}\left\|T_{M_{\alpha_{k}}+2}f\right\|_{weak-
L_{p}}=\infty.$

The proof is complete.

A number of special cases of our results are of particular interest and give
both well-known and new information. We just give the following examples of
such $T$ means with non-decreasing sequence $\\{q_{k}:k\geq 0\\}:$

###### Corollary 3.16

The maximal operator $B^{\alpha,\beta,*}$ is bounded from the Hardy space
$H_{1/2}$ to the space $weak-L_{1/2}$ but is not bounded from $H_{p}$ to the
space $weak-L_{p},$ when $0<p<1/2.$

Proof: Since $B^{\alpha,\beta,*}$ are the $T$ means with non-decreasing
sequence $\\{q_{k}:k\geq 0\\},$ then the proof of this corollary is direct
consequence of Theorem 3.15.

###### Corollary 3.17

Let $f\in L_{1}$ and $T_{n}$ be the $T$ means with non-decreasing sequence
$\\{q_{k}:k\geq 0\\}$ and satisfying condition (3.37). Then

$T_{n}f\rightarrow f,\text{ \ \ a.e., \ \ as \ }n\rightarrow\infty.$

Proof of Corollary 3.17. According to Theorem 3.15 and Lemma and 1.5 we can
conclude that $T^{\ast}$ has weak type-(1,1) and by well-known density
argument due to Marcinkiewicz and Zygmund [192] (see Lemma 1.1) we also have
$T_{n}f\rightarrow f,$ a.e.. Which follows proof of Corollary 3.17.

###### Corollary 3.18

Let $f\in L_{1}$. Then $\ B_{n}^{\alpha,\beta}f\rightarrow f,\text{ \ \ \
a.e., \ \ \ as \ }n\rightarrow\infty.$

Proof: Since $B^{\alpha,\beta,*}$ are the $T$ means with non-decreasing
sequence $\\{q_{k}:k\geq 0\\},$ then the proof of this corollary is direct
consequence of Corollary 3.17.

###### Theorem 3.19

Let $0<p\leq 1/2,$ $f\in H_{p}$ and $\\{q_{k}:k\geq 0\\}$ be a sequence of
non-increasing numbers. Then the maximal operator

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

is bounded from the Hardy space $H_{p}$ to the space $L_{p}.$

Proof: Let the sequence $\\{q_{k}:k\geq 0\\}$ be non-increasing. By combining
(3.15) and (3.17) we get that

$\displaystyle\widetilde{T}_{p}^{\ast}f$ $\displaystyle:=$
$\displaystyle\frac{\left|T_{n}f\right|}{{\left(n+1\right)^{1/p-2}\log^{2\left[1/2+p\right]}\left(n+1\right)}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{{\left(n+1\right)^{1/p-2}\log^{2\left[1/2+p\right]}\left(n+1\right)}}\left|\frac{1}{Q_{n}}\overset{n-1}{\underset{j=1}{\sum}}q_{j}S_{j}f\right|$
$\displaystyle\leq$
$\displaystyle\frac{1}{\left(n+1\right)^{1/p-2}\log^{2\left[1/2+p\right]}\left(n+1\right)}\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=1}{\sum}}\left|q_{j}-q_{j+1}\right|j\left|\sigma_{j}f\right|+q_{n-1}(n-1)\left|\sigma_{n}f\right|\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=1}{\sum}}\frac{\left|q_{j}-q_{j+1}\right|j\left|\sigma_{j}f\right|}{{\left(j+1\right)^{1/p-2}\log^{2\left[1/2+p\right]}\left(j+1\right)}}+\frac{q_{n-1}(n-1)\left|\sigma_{n}f\right|}{\left(n+1\right)^{1/p-2}\log^{2\left[1/2+p\right]}\left(n+1\right)}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=1}{\sum}}\left(q_{j}-q_{j+1}\right)j+q_{n-1}(n-1)\right)\sup_{n\in\mathbb{N}_{+}}\frac{\left|\sigma_{n}f\right|}{\left(n+1\right)^{1/p-2}\log^{2\left[1/2+p\right]}\left(n+1\right)}$
$\displaystyle\leq$
$\displaystyle\sup_{n\in\mathbb{N}_{+}}\frac{\left|\sigma_{n}f\right|}{\left(n+1\right)^{1/p-2}\log^{2\left[1/2+p\right]}\left(n+1\right)}:=\widetilde{\sigma}_{p}^{\ast}f,$

so that

$\widetilde{T}_{p}^{\ast}f\leq\widetilde{\sigma}_{p}^{\ast}f.$ (3.39)

If we apply (3.39), according (see Tephnadze [139, 140]) that
$\widetilde{\sigma}_{p}^{\ast}f$ is bounded from the Hardy space $H_{p}$ to
the space $L_{p}$ for $p\leq 1/2$ we can conclude that the maximal operators
$\widetilde{T}_{p}^{\ast}$ of $T$ means with non-increasing sequence
$\\{q_{k}:k\geq 0\\},$ are bounded from the Hardy space $H_{p}$ to the space
$L_{p}$.

The proof of theorem is complete.

###### Corollary 3.20

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

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

is bounded from the Hardy space $H_{p}$ to the space $L_{p}.$

###### Corollary 3.21

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

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

is bounded from the Hardy space $H_{p}$ to the space $L_{p}.$

###### Corollary 3.22

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

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

is bounded from the Hardy space $H_{p}$ to the space $L_{p}.$

###### Theorem 3.23

Let $0<p\leq 1/2,$ $f\in H_{p}$ and $\\{q_{k}:k\geq 0\\}$ be a sequence of
non-increasing numbers, satisfying the condition

$\frac{q_{n-1}}{Q_{n}}=O\left(\frac{1}{n}\right),\text{ \ \ as \ \ }\
n\rightarrow\infty.$ (3.40)

Then the maximal operator

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

is bounded from the Hardy space $H_{p}$ to the space $L_{p}.$

Proof: Let the sequence $\\{q_{k}:k\geq 0\\}$ be non-decreasing satisfying the
condition (3.40). By combining (3.15) and (3.17) we get that

$\displaystyle\frac{\left|T_{n}f\right|}{{\left(n+1\right)^{1/p-2}\log^{2\left[1/2+p\right]}\left(n+1\right)}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{{\left(n+1\right)^{1/p-2}\log^{2\left[1/2+p\right]}\left(n+1\right)}}\left|\frac{1}{Q_{n}}\overset{n-1}{\underset{j=1}{\sum}}q_{j}S_{j}f\right|$
$\displaystyle\leq$
$\displaystyle\frac{1}{\left(n+1\right)^{1/p-2}\log^{2\left[1/2+p\right]}\left(n+1\right)}\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=1}{\sum}}\left|q_{j}-q_{j+1}\right|j\left|\sigma_{j}f\right|+q_{n-1}(n-1)\left|\sigma_{n}f\right|\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=1}{\sum}}\frac{\left|q_{j}-q_{j+1}\right|j\left|\sigma_{j}f\right|}{{\left(j+1\right)^{1/p-2}\log^{2\left[1/2+p\right]}\left(j+1\right)}}+\frac{q_{n-1}(n-1)\left|\sigma_{n}f\right|}{\left(n+1\right)^{1/p-2}\log^{2\left[1/2+p\right]}\left(n+1\right)}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=1}{\sum}}\left(q_{j+1}-q_{j}\right)j+q_{n-1}(n-1)\right)\sup_{n\in\mathbb{N}_{+}}\frac{\left|\sigma_{n}f\right|}{\left(n+1\right)^{1/p-2}\log^{2\left[1/2+p\right]}\left(n+1\right)}$
$\displaystyle\leq$
$\displaystyle\frac{2q_{n-1}(n-1)-Q_{n}}{Q_{n}}\sup_{n\in\mathbb{N}_{+}}\frac{\left|\sigma_{n}f\right|}{\left(n+1\right)^{1/p-2}\log^{2\left[1/2+p\right]}\left(n+1\right)}$
$\displaystyle\leq$
$\displaystyle\sup_{n\in\mathbb{N}_{+}}\frac{\left|\sigma_{n}f\right|}{\left(n+1\right)^{1/p-2}\log^{2\left[1/2+p\right]}\left(n+1\right)}$

so that

$\sup_{n\in\mathbb{N}_{+}}\frac{\left|T_{n}f\right|}{{\left(n+1\right)^{1/p-2}\log^{2\left[1/2+p\right]}\left(n+1\right)}}\leq\sup_{n\in\mathbb{N}_{+}}\frac{\left|\sigma_{n}f\right|}{\left(n+1\right)^{1/p-2}\log^{2\left[1/2+p\right]}\left(n+1\right)}.$
(3.41)

If we apply (3.41), according (see Tephnadze [139, 140]) that
$\widetilde{\sigma}_{p}^{\ast}f$ is bounded from the Hardy space $H_{p}$ to
the space $L_{p}$ for $p\leq 1/2$ we can conclude that the maximal operators
$\widetilde{T}_{p}^{\ast}$ of $T$ means with non-decreasing sequence
$\\{q_{k}:k\geq 0\\},$ are bounded from the Hardy space $H_{p}$ to the space
$L_{p}$.

###### Corollary 3.24

Let $0<p\leq 1/2,$ $f\in H_{p}$ and $\\{q_{k}:k\geq 0\\}$ be a sequence of
non-decreasing numbers, such that

$\sup_{n\in\mathbb{N}}q_{n}<c<\infty.$ (3.42)

Then all such $T$ means are bounded from the Hardy space $H_{p}$ to the space
$L_{p}.$

Proof: By using (3.42) we get

$\frac{q_{n-1}}{Q_{n}}\leq\frac{c}{Q_{n}}\leq\frac{c}{q_{0}n}=\frac{c_{1}}{n}=O\left(\frac{1}{n}\right),\text{
as }n\rightarrow 0,$

It follows that condition (3.40) is satisfied and for such $T$ means is
bounded from the Hardy space $H_{p}$ to the space $L_{p}.$

###### Remark 3.25

Since (see Tephnadze [139, 140]) the maximal operator

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

is bounded from the martingale Hardy space $H_{p}$ to the space $L_{p}$ and
the rate of denominator $\left(n+1\right)^{1/p-2}\log^{2\left[1/2+p\right]}$
is in the sense sharp and Fejer means is example of $T$ means of as non-
increasing as non-decreasing sequences we obtain that this weights are also
sharp in Theorems 3.19 and 3.23.

###  3.4 Strong convergence of $T$ means with respect to Vilenkin systems on
the martingale Hardy spaces

###### Theorem 3.26

a) Let $0<p<1/2,$ $f\in H_{p}$ and $\\{q_{k}:k\geq 0\\}$ be a sequence of non-
increasing numbers. Then there exists an absolute constant $c_{p},$ depending
only on $p,$ such that the inequality

$\overset{\infty}{\underset{k=1}{\sum}}\frac{\left\|T_{k}f\right\|_{p}^{p}}{k^{2-2p}}\leq
c_{p}\left\|f\right\|_{H_{p}}^{p}$

holds.

b)Let $f\in H_{1/2}$ and $\\{q_{k}:k\geq 0\\}$ be a sequence of non-increasing
numbers, satisfying the condition

$\frac{1}{Q_{n}}=O\left(\frac{1}{n}\right),\text{ \ \ as \ \ }\
n\rightarrow\infty.$ (3.43)

Then there exists an absolute constant $c,$ such that the inequality

$\frac{1}{\log
n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|T_{k}f\right\|_{1/2}^{1/2}}{k}\leq
c\left\|f\right\|_{H_{1/2}}^{1/2}$ (3.44)

holds.

Proof: Let the sequence $\\{q_{k}:k\geq 0\\}$ be non-increasing. By Lemma 1.3
(see also (1.4)), the proof of part a) will be complete, if we show that

$\frac{1}{\log^{\left[1/2+p\right]}n}\overset{n}{\underset{m=1}{\sum}}\frac{\left\|T_{m}a\right\|_{H_{p}}^{p}}{m^{2-2p}}\leq
c_{p},$ (3.45)

for every $p$-atom $a,$ with support$\ I$, $\mu\left(I\right)=M_{N}^{-1}.$ We
may assume that $I=I_{N}.$ It is easy to see that
$S_{n}\left(a\right)=T_{n}\left(a\right)=0,$ when $n\leq M_{N}$. Therefore, we
can suppose that $n>M_{N}$.

Let $x\in I_{N}.$ Since $T_{n}$ is bounded from $L_{\infty}$ to $L_{\infty}$
(boundedness follows from Lemma 3.2) and $\left\|a\right\|_{\infty}\leq
M_{N}^{1/p}$ we obtain that

$\int_{I_{N}}\left|T_{m}a\right|^{p}d\mu\leq\frac{\left\|a\right\|_{\infty}^{p}}{M_{N}}\leq
c<\infty,\text{ \ }0<p<1/2.$

Hence,

$\displaystyle\frac{1}{\log^{\left[1/2+p\right]}n}\overset{n}{\underset{m=1}{\sum}}\frac{\int_{I_{N}}\left|T_{m}a\right|^{p}d\mu}{m^{2-2p}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\log^{\left[1/2+p\right]}n}\overset{n}{\underset{k=1}{\sum}}\frac{1}{m^{2-2p}}\leq
c<\infty.$

It is easy to see that

$\displaystyle\left|T_{m}a\left(x\right)\right|$ $\displaystyle=$
$\displaystyle\left|\int_{I_{N}}a\left(t\right)F_{n}\left(x-t\right)d\mu\left(t\right)\right|$
$\displaystyle=$
$\displaystyle\left|\int_{I_{N}}a\left(t\right)\frac{1}{Q_{n}}\overset{n}{\underset{j=M_{N}}{\sum}}q_{j}D_{j}\left(x-t\right)d\mu\left(t\right)\right|$
$\displaystyle\leq$
$\displaystyle\left\|a\right\|_{\infty}\int_{I_{N}}\left|\frac{1}{Q_{n}}\overset{n}{\underset{j=M_{N}}{\sum}}q_{j}D_{j}\left(x-t\right)\right|d\mu\left(t\right)$
$\displaystyle\leq$ $\displaystyle
M_{N}^{1/p}\int_{I_{N}}\left|\frac{1}{Q_{n}}\overset{n}{\underset{j=M_{N}}{\sum}}q_{j}D_{j}\left(x-t\right)\right|d\mu\left(t\right)$

Let $T_{n}$ be $T$ means, with non-decreasing coefficients $\\{q_{k}:k\geq
0\\}$ and $x\in I_{N}^{k,l},\,0\leq k<l\leq N.$ Then, in the view of Lemma 2.5
we get that

$\left|T_{m}a\left(x\right)\right|\leq cM_{l}M_{k}M_{N}^{1/p-2},\text{ for
}0<p\leq 1/2.$ (3.48)

Let $0<p<1/2.$ By using (1.1), (3.4), (3.48) we find that

$\displaystyle\int_{\overline{I_{N}}}\left|T_{m}a\right|^{p}d\mu$
$\displaystyle=$
$\displaystyle\overset{N-2}{\underset{k=0}{\sum}}\overset{N-1}{\underset{l=k+1}{\sum}}\sum\limits_{x_{j}=0,\text{
}j\in\\{l+1,\dots,N-1\\}}^{m_{j-1}}\int_{I_{N}^{k,l}}\left|T_{m}a\right|^{p}d\mu$
$\displaystyle+$
$\displaystyle\overset{N-1}{\underset{k=0}{\sum}}\int_{I_{N}^{k,N}}\left|T_{m}a\right|^{p}d\mu$
$\displaystyle\leq$ $\displaystyle
c\overset{N-2}{\underset{k=0}{\sum}}\overset{N-1}{\underset{l=k+1}{\sum}}\frac{m_{l+1}\dotsm
m_{N-1}}{M_{N}}\left(M_{l}M_{k}\right)^{p}M_{N}^{1-2p}$ $\displaystyle+$
$\displaystyle\overset{N-1}{\underset{k=0}{\sum}}\frac{1}{M_{N}}M_{k}^{p}M_{N}^{1-p}$
$\displaystyle\leq$ $\displaystyle
cM_{N}^{1-2p}\overset{N-2}{\underset{k=0}{\sum}}\overset{N-1}{\underset{l=k+1}{\sum}}\frac{\left(M_{l}M_{k}\right)^{p}}{M_{l}}$
$\displaystyle+$
$\displaystyle\overset{N-1}{\underset{k=0}{\sum}}\frac{M_{k}^{p}}{M_{N}^{p}}\leq
cM_{N}^{1-2p}.$

Moreover, according to (3.4), we get that

$\overset{\infty}{\underset{m=M_{N}+1}{\sum}}\frac{\int_{\overline{I_{N}}}\left|T_{m}a\right|^{p}d\mu}{m^{2-2p}}\leq\overset{\infty}{\underset{m=M_{N}+1}{\sum}}\frac{cM_{N}^{1-2p}}{m^{2-2p}}<c<\infty,\text{
\ }\left(0<p<1/2\right).$

The proof of part a) is complete.

Let $p=1/2$ and $T_{n}$ be $T$ means, with non-increasing coefficients
$\\{q_{k}:k\geq 0\\}$, satisfying condition (3.43). By Lemma 1.3, the proof of
part b) will be complete, if we show that

$\frac{1}{\log
n}\overset{n}{\underset{m=1}{\sum}}\frac{\left\|T_{m}a\right\|_{H_{1/2}}^{1/2}}{m}\leq
c_{p},$ (3.50)

for every $p$-atom $a,$ with support$\ I$, $\mu\left(I\right)=M_{N}^{-1}.$ We
may assume that $I=I_{N}.$ It is easy to see that
$S_{n}\left(a\right)=T_{n}\left(a\right)=0,$ when $n\leq M_{N}$. Therefore, we
can suppose that $n>M_{N}$.

Let $x\in I_{N}.$ Since $T_{n}$ is bounded from $L_{\infty}$ to $L_{\infty}$
(boundedness follows from Lemma 3.2) and $\left\|a\right\|_{\infty}\leq
M_{N}^{2}$ we obtain that

$\int_{I_{N}}\left|T_{m}a\right|^{1/2}d\mu\leq\frac{\left\|a\right\|_{\infty}^{1/2}}{M_{N}}\leq
c<\infty,\text{ \ }0<p\leq 1/2.$

Hence,

$\frac{1}{\log
n}\overset{n}{\underset{m=1}{\sum}}\frac{\int_{I_{N}}\left|T_{m}a\right|^{1/2}d\mu}{m}\leq\frac{1}{\log
n}\overset{n}{\underset{k=1}{\sum}}\frac{1}{m}\leq c<\infty.$ (3.51)

It is easy to see that

$\displaystyle\left|T_{m}a\left(x\right)\right|$ $\displaystyle=$
$\displaystyle\left|\int_{I_{N}}a\left(t\right)\frac{1}{Q_{n}}\overset{n}{\underset{j=M_{N}}{\sum}}q_{j}D_{j}\left(x-t\right)d\mu\left(t\right)\right|$
$\displaystyle\leq$
$\displaystyle\left\|a\right\|_{\infty}\int_{I_{N}}\left|F_{m}\left(x-t\right)\right|d\mu\left(t\right)$
$\displaystyle\leq$ $\displaystyle
M_{N}^{2}\int_{I_{N}}\left|F_{m}\left(x-t\right)\right|d\mu\left(t\right).$

Let $x\in I_{N}^{k,l},\,0\leq k<l<N.$ Then, in the view of Lemma 3.6 we get
that

$\left|T_{m}a\left(x\right)\right|\leq\frac{cM_{l}M_{k}M_{N}}{m}.$ (3.53)

Let $x\in I_{N}^{k,N}.$ Then, according to Lemma 3.6 we obtain that

$\left|T_{m}a\left(x\right)\right|\leq cM_{k}M_{N}.$ (3.54)

By combining (1.1), (3.4), (3.53) and (3.54) we obtain that

$\displaystyle\int_{\overline{I_{N}}}\left|T_{m}a\left(x\right)\right|^{1/2}d\mu\left(x\right)$
$\displaystyle\leq$ $\displaystyle
c\overset{N-2}{\underset{k=0}{\sum}}\overset{N-1}{\underset{l=k+1}{\sum}}\frac{m_{l+1}\dotsm
m_{N-1}}{M_{N}}\frac{\left(M_{l}M_{k}\right)^{1/2}M_{N}^{1/2}}{m^{1/2}}$
$\displaystyle+$
$\displaystyle\overset{N-1}{\underset{k=0}{\sum}}\frac{1}{M_{N}}M_{k}^{1/2}M_{N}^{1/2}$
$\displaystyle\leq$ $\displaystyle
M_{N}^{1/2}\overset{N-2}{\underset{k=0}{\sum}}\overset{N-1}{\underset{l=k+1}{\sum}}\frac{\left(M_{l}M_{k}\right)^{1/2}}{m^{1/2}M_{l}}+\overset{N-1}{\underset{k=0}{\sum}}\frac{M_{k}^{1/2}}{M_{N}^{1/2}}$
$\displaystyle\leq$ $\displaystyle\frac{cM_{N}^{1/2}N}{m^{1/2}}+c.$

It follows that

$\displaystyle\frac{1}{\log
n}\overset{n}{\underset{m=M_{N}+1}{\sum}}\frac{\int_{\overline{I_{N}}}\left|T_{m}a\left(x\right)\right|^{1/2}d\mu\left(x\right)}{m}$
$\displaystyle\leq$ $\displaystyle\frac{1}{\log
n}\overset{n}{\underset{m=M_{N}+1}{\sum}}\left(\frac{cM_{N}^{1/2}N}{m^{3/2}}+\frac{c}{m}\right)<c<\infty.$

The proof of part b) is completed by just combining (3.51) and (3.4).

###### Corollary 3.27

Let $0<p\leq 1/2$ and $f\in H_{p}.$ Then there exists absolute constant
$c_{p},$ depending only on $p,$ such that the following inequality holds:

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

###### Corollary 3.28

Let $0<p\leq 1/2$ and $f\in H_{p}.$ Then there exists an absolute constant
$c_{p},$ depending only on $p,$ such that the following inequality holds:

$\frac{1}{\log^{\left[1/2+p\right]}n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|U_{k}^{\alpha}f\right\|_{p}^{p}}{k^{2-2p}}\leq
c_{p}\left\|f\right\|_{H_{p}}^{p}.$

###### Corollary 3.29

Let $0<p\leq 1/2$ and $f\in H_{p}.$ Then there exists an absolute constant
$c_{p},$ depending only on $p,$ such that the following inequality holds:

$\frac{1}{\log^{\left[1/2+p\right]}n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|V_{k}^{\alpha}f\right\|_{p}^{p}}{k^{2-2p}}\leq
c_{p}\left\|f\right\|_{H_{p}}^{p}.$

###### Corollary 3.30

Let $0<p\leq 1/2$ and $f\in H_{p}.$ Then there exists an absolute constant
$c_{p},$ depending only on $p,$ such that the following inequality holds:

$\overset{\infty}{\underset{k=1}{\sum}}\frac{\left\|R_{k}^{\alpha}f\right\|_{p}^{p}}{k^{2-2p}}\leq
c_{p}\left\|f\right\|_{H_{p}}^{p}.$

###### Theorem 3.31

a) Let $0<p<1/2,$ $f\in H_{p}$ and $\\{q_{k}:k\geq 0\\}$ be a sequence of non-
decreasing numbers. Then there exists an absolute constant $c_{p},$ depending
only on $p,$ such that the inequality

$\overset{\infty}{\underset{k=1}{\sum}}\frac{\left\|T_{k}f\right\|_{p}^{p}}{k^{2-2p}}\leq
c_{p}\left\|f\right\|_{H_{p}}^{p}$

holds.

b)Let $f\in H_{1/2}$ and $\\{q_{k}:k\geq 0\\}$ be a sequence of non-increasing
numbers, satisfying the condition (3.13). Then there exists an absolute
constant $c,$ such that the inequality

$\frac{1}{\log
n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|T_{k}f\right\|_{1/2}^{1/2}}{k}\leq
c\left\|f\right\|_{H_{1/2}}^{1/2}$ (3.56)

holds.

Proof:[Proof of Theorem 3.31] If we use Lemmas 3.9 and 3.10 and follows
analogical steps of Theorem 3.26 we immediately get the proof of Theorem 3.31.
So, we leave out the details.

###### Corollary 3.32

Let $0<p\leq 1/2,$ $f\in H_{p}$ and $\\{q_{k}:k\geq 0\\}$ be a sequence of
non-decreasing numbers, such that

$\sup_{n\in\mathbb{N}}q_{n}<c<\infty.$

Then condition (3.13) is satisfied and for such $T$ means there exists an
absolute constant $c,$ such that the inequality (3.56) holds.

We have already considered the case when the sequence $\\{q_{k}:k\geq 0\\}$ is
bounded. Now, we consider some Nörlund means, which are generated by a
unbounded sequence $\\{q_{k}:k\geq 0\\}.$

###### Corollary 3.33

Let $0<p\leq 1/2$ and $f\in H_{p}.$ Then there exists an absolute constant
$c_{p},$ depending only on $p,$ such that the following inequality holds:

$\frac{1}{\log^{\left[1/2+p\right]}n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|B^{\alpha,\beta}_{k}f\right\|_{p}^{p}}{k^{2-2p}}\leq
c_{p}\left\|f\right\|_{H_{p}}^{p}.$

##  4 Reisz and Nörlund logarithmoic means means on $H_{p}$ spaces

###  4.1 Introduction

Riesz logarithmic means with respect to the trigonometric system was studied
by a lot of authors. We mention, for instance, the paper by Szasz [135] and
Yabuta [181]. These means with respect to the Walsh and Vilenkin systems were
investigated by Simon [119] and Gàt [38]. Blahota and Gát [19] considered norm
summability of Nörlund logarithmic means and showed that Riesz logarithmic
means $R_{n}$ have better approximation properties on some unbounded Vilenkin
groups than the Fejér means. Moreover, in [147] it was proved that the maximal
operator of Riesz means is bounded from the Hardy space $H_{p}$ to the
Lebesgue space $L_{p}$ for $p>1/2$ but not when $0<p\leq 1/2.$ Strong
convergence theorems and boundedness of weighted maximal operators of Riesz
logarithmic means was considered in Lukkassen, Persson, Tutberidze, Tephnadze
[76] and Tephnadze [147].

In [148] Tephnadze proved that the maximal operator of Riesz logarithmic means
$R^{\ast}$ is bounded from the Hardy space $H_{1/2}$ to the space $weak-
L_{1/2}.$ Moreover, there exists a martingale $f\in H_{p},$ where $0<p\leq
1/2$ such that

$\left\|R^{\ast}f\right\|_{p}=+\infty.$

In [147] Tephnadze proved that for any $0<p<1/2,$ the maximal operator

$\overset{\sim}{R}_{p}^{\ast}:=\sup_{n\in\mathbb{N}}\frac{\log
n\left|R_{n}f\right|}{\left(n+1\right)^{1/p-2}}$

is bounded from the Hardy space $H_{p}$ to the space $L_{p}.$

Moreover, for $0<p<1/2$ and non-decreasing function
$\varphi:\mathbb{N}_{+}\rightarrow[1,\infty)$ satisfying the condition

$\frac{\left(n+1\right)^{1/p-2}}{\log\left(n+1\right)\varphi\left(n\right)}=\infty,$
(4.1)

the maximal operator

$\sup_{n\in\mathbb{N}}\frac{\left|R_{n}f\right|}{\varphi\left(n\right)}$

is not bounded from the Hardy space $H_{p}$ to the space $weak-L_{p}.$

In the case $p=1/2$ he also proved that the maximal operator

$\widetilde{R}^{\ast}f:=\sup_{n\in\mathbb{N}}\frac{\left|R_{n}f\right|}{\log\left(n+1\right)}$

is bounded from the Hardy space $H_{1/2}$ to the space $L_{1/2}.$

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

$\overline{\lim_{n\rightarrow\infty}}\frac{\log\left(n+1\right)}{\varphi\left(n\right)}=+\infty,$

the maximal operator

$\sup_{n\in\mathbb{N}}\frac{\left|R_{n}f\right|}{\varphi\left(n\right)}$

is not bounded from the Hardy space $H_{1/2}$ to the space $L_{1/2}.$

In this thesis (see also [76]) we also proved that if $0<p<1/2$ and $f\in
H_{p}(G_{m}),$ there exists an absolute constant $c_{p},$ depending only on
$p,$ such that the inequality

$\overset{\infty}{\underset{n=1}{\sum}}\frac{\log^{p}n\left\|R_{n}f\right\|_{H_{p}}^{p}}{n^{2-2p}}\leq
c_{p}\left\|f\right\|_{H_{p}}^{p}$

holds, where $R_{n}f$ denotes the $n$-th Reisz logarithmic mean with respect
to the Vilenkin-Fourier series of $f.$

Móricz and Siddiqi [82] investigate the approximation properties of some
special Nörlund means of Walsh-Fourier series of $L_{p}$ functions in norm.
The case when $\left\\{q_{k}=1/k:k\in\mathbb{N}\right\\}$ was excluded, since
the methods of Móricz and Siddiqi are not applicable to Nörlund logarithmic
means. Fridli, Manchanda and Siddiqi [34] improved and extended results of
Móricz and Siddiqi [82] to dyadic homogeneous Banach spaces and Martingale
Hardy spaces. In [39] Gát and Goginava proved some convergence and divergence
properties of the Nörlund logarithmic means of functions in the class of
continuous functions and in the Lebesgue space $L_{1}.$ In particular, they
gave a negative answer to the question of Móricz and Siddiqi [82]. Gát and
Goginava [40] proved that for each measurable function satisfying
$\phi\left(u\right)=o\left(u\log^{1/2}u\right),\ as\ u\rightarrow\infty,$
there exists an integrable function $f$ such that

$\int_{G_{m}}\phi\left(\left|f\left(x\right)\right|\right)d\mu\,\left(x\right)<\infty$

and that there exists a set with positive measure such that the Walsh-
logarithmic means of the function diverges on this set. It follows that that
weak-(1,1) type inequality does not hold for the maximal operator of Nörlund
logarithmic means:

$\displaystyle L^{\ast}f$ $\displaystyle:=$
$\displaystyle\sup_{n\in\mathbb{N}}\left|L_{n}f\right|$

On the other hand, there exists an absolute constant $c_{p}$ such that

$\left\|L^{\ast}f\right\|_{p}\leq c_{p}\left\|f\right\|_{p},\text{ \ when \
}f\in L_{p},\text{ \ }p>1.$

If we consider the following restricted maximal operator

$\displaystyle\widetilde{L}_{\\#}^{\ast}f$ $\displaystyle:=$
$\displaystyle\sup_{n\in\mathbb{N}}\left|L_{M_{n}}f\right|,\ \ \
(M_{k}:=m_{0}...m_{k-1},\ \ \ k=0,1...)$

then

$\displaystyle\lambda\mu\left\\{\widetilde{L}_{\\#}^{\ast}f>\lambda\right\\}\leq
c\left\|f\right\|_{1},\ \ \ f\in L_{1}(G_{m}),\ \ \lambda>0.$

Hence, if $f\in L_{1}(G_{m})$ then

$L_{M_{n}}f\to f,\ \ \text{a.e. on }\ \ G_{m}.$

In this thesis we prove that if $f\in L_{1}(G_{m})$ then $L_{M_{n}}f(x)\to
f(x)$ for all Lebesgue points.

In [141] it was proved that there exists a martingale $f\in H_{p},$ $(0<p\leq
1),$ such that the maximal operator of Nörlund logarithmic means $L^{\ast}$ is
not bounded in the Lebesgue space $L_{p}.$ In particular, it was proved that
there exists a martingale $f\in H_{p}$ such that

$\left\|L^{\ast}f\right\|_{p}=+\infty.$

Boundedness of weighted maximal operators of Nörlund logarithmic means was
considered Persson, Tephnadze, Wall [105]. In particular, maximal operator

$\overset{\sim}{L}^{\ast}f:=\sup_{n\in\mathbb{N}}\frac{\left|L_{n}f\right|}{\log\left(n+1\right)}$

is bounded from the Hardy space $H_{1}\left(G_{m}\right)$ to the space
$L_{1}\left(G_{m}\right).$

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

$\overline{\lim_{n\rightarrow\infty}}\frac{\log\left(n+1\right)}{\varphi\left(n\right)}=+\infty,$
(4.2)

then there exists a martingale $f\in H_{1}\left(G_{m}\right),$ such that the
maximal operator

$\sup_{n\in\mathbb{N}}\frac{\left|L_{n}f\right|}{\varphi\left(n\right)}$

is not bounded from the Hardy space $H_{1}\left(G_{m}\right)$ to the Lebesgue
space $L_{1}\left(G_{m}\right).$

In Tephnadze and Tutberidze [167] was proved the maximal operator

$\overset{\sim}{L}_{p}^{*}f:=\sup_{n\in\mathbb{N}}\frac{\left|L_{n}f\right|}{\left(n+1\right)^{1/p-1}}$

is bounded from the Hardy space $H_{p}\left(G_{m}\right)$ to the space
$L_{p}\left(G_{m}\right).$

We also proved that for $0<p<1$ and a non-decreasing function
$\varphi:\mathbb{N}_{+}\rightarrow[1,\infty)$ satisfying the condition

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

then there exists a martingale $f\in H_{p}\left(G_{m}\right),$ such that the
maximal operator

$\sup_{n\in\mathbb{N}}\frac{\left|L_{n}f\right|}{\varphi\left(n+1\right)}$

is not bounded from the Hardy space $H_{p}\left(G_{m}\right)$ to the space
$L_{p}\left(G_{m}\right).$

In this paper also state the following open problem:

Open Problem. For any $0<p<1$ let find non-decreasing function
$\Theta:\mathbb{N}_{+}\rightarrow[1,\infty)$ suth that the following maximal
operator

$\overset{\sim}{L}_{p}^{*}f:=\sup_{n\in\mathbb{N}}\frac{\left|L_{n}f\right|}{\Theta\left(n+1\right)}$

is bounded from the Hardy space $H_{p}\left(G_{m}\right)$ to the Lebesgue
space $L_{p}\left(G_{m}\right)$ and the rate of
$\Theta:\mathbb{N}_{+}\rightarrow[1,\infty)$ is sharp, that is, for any non-
decreasing function $\varphi:\mathbb{N}_{+}\rightarrow[1,\infty)$ satisfying
the condition

$\overline{\lim_{n\rightarrow\infty}}\frac{\Theta\left(n\right)}{\varphi\left(n\right)}=+\infty,$

then there exists a martingale $f\in H_{p}\left(G_{m}\right),$ such that the
maximal operator

$\sup_{n\in\mathbb{N}}\frac{\left|L_{n}f\right|}{\varphi\left(n+1\right)}$

is not bounded from the Hardy space $H_{p}\left(G_{m}\right)$ to the space
$L_{p}\left(G_{m}\right).$

According to Theorems above we can conclude that there exist absolute
constants $C_{1}$ and $C_{2}$ such that

$\frac{C_{1}n^{1/p-1}}{\log(n+1)}\leq\Theta\left(n\right)\leq C_{2}n^{1/p-1}.$

Later on, Memic generalized result of Tephnadze and Tutberidze [167] and
proved that the maximal operator

$\sup_{n\in\mathbb{N}}\frac{\log
n\left|L_{n}f\right|}{\left(n+1\right)^{1/p-1}}$

is bounded from the Hardy space $H_{p}\left(G_{m}\right)$ to the space
$L_{p}\left(G_{m}\right).$

Sharpness of this result immediately follows negative result of Tephnadze and
Tutberidze [167], which is already stated above.

###  4.2 Auxiliary lemmas

We need the following lemma of independent interest:

###### Lemma 4.1

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

$Y_{n}=\frac{1}{l_{n}}\overset{n-1}{\underset{j=1}{\sum}}\frac{K_{j}}{j+1}+\frac{K_{n}}{l_{n}}.$
(4.3)

Moreover,

$\|Y_{n}\|_{1}<c<\infty.$ (4.4)

Proof: To rewrite the kernels of the Riesz logarithmic means as in equality
(4.3) we have to just use Abel transformation. On the other hand, equality
(4.4) immediately follows equality (4.3) and (1.12).

To prove our main results we need the following lemma, which is proved in
Tephnadze [144]. Here we also give some short proof.

###### Lemma 4.2

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

$\int_{I_{N}}\underset{j=M_{N}+1}{\overset{n}{\sum}}\frac{\left|K_{j}\left(x-t\right)\right|}{j+1}d\mu\left(t\right)\leq\frac{cM_{k}M_{l}}{M_{N}^{2}}.$

Let $x\in I_{N}\left(x_{k}e_{k}\right),$ $1\leq x_{k}\leq m_{k}-1,$ $\
k=0,...,N-1.$ Then

$\int_{I_{N}}\underset{j=M_{N}+1}{\overset{n}{\sum}}\frac{\left|K_{j}\left(x-t\right)\right|}{j+1}d\mu\left(t\right)\leq\frac{cM_{k}}{M_{N}}l_{n}.$

proof: Let $x\in I_{N}\left(x_{k}e_{k}+x_{l}e_{l}\right),$ $\ 1\leq x_{k}\leq
m_{k}-1,$ $1\leq x_{l}\leq m_{l}-1,$ $\ k=0,...,N-2,$ $l=k+1,...,N-1.$ By
using Lemma 2.4 we have that

$\displaystyle\int_{I_{N}}\underset{j=M_{N}+1}{\overset{n}{\sum}}\frac{\left|K_{j}\left(x-t\right)\right|}{j+1}d\mu\left(t\right)$
$\displaystyle\leq$
$\displaystyle\underset{j=M_{N}+1}{\overset{n}{\sum}}\frac{cM_{k}M_{l}}{\left(j+1\right)jM_{N}}$
$\displaystyle\leq$
$\displaystyle\frac{cM_{k}M_{l}}{M_{N}}\underset{j=M_{N}+1}{\overset{\infty}{\sum}}\left(\frac{1}{j}-\frac{1}{j+1}\right)$
$\displaystyle\leq$ $\displaystyle\frac{cM_{k}M_{l}}{M_{N}^{2}}.$

Let $x\in I_{N}\left(x_{k}e_{k}\right),$ $1\leq x_{k}\leq m_{k}-1,$ $\
k=0,...,N-1.$ Then

$\displaystyle\int_{I_{N}}\underset{j=M_{N}+1}{\overset{n}{\sum}}\frac{\left|K_{j}\left(x-t\right)\right|}{j+1}d\mu\left(t\right)$
$\displaystyle\leq$
$\displaystyle\underset{j=M_{N}+1}{\overset{n}{\sum}}\frac{cM_{k}}{\left(j+1\right)M_{N}}$
$\displaystyle\leq$ $\displaystyle\frac{cM_{k}}{M_{N}}l_{n}.$

Combining (4.2) and (4.2) we complete the proof of Lemma.

Next we study some special consequences of kernels of Nörlund logarithmic
means:

###### Lemma 4.3

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

$\displaystyle
P_{M_{n}}(x)=D_{M_{n}}(x)-\psi_{M_{n}-1}(x)\overline{Y}_{M_{n}}(x)$ (4.7)

Moreover,

$\displaystyle\|P_{M_{n}}(x)\|_{1}<c<\infty.$ (4.8)

Proof: By using 1.2 we get that

$\displaystyle P_{M_{n}}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{Q_{M_{n}}}\overset{M_{n}}{\underset{k=1}{\sum}}\frac{D_{k}(x)}{M_{n}-k}=\frac{1}{Q_{M_{n}}}\overset{M_{n}-1}{\underset{k=0}{\sum}}\frac{D_{M_{n}-k}(x)}{k}$
$\displaystyle=$
$\displaystyle\frac{1}{Q_{M_{n}}}\overset{M_{n}-1}{\underset{k=0}{\sum}}\frac{1}{k}\left(D_{M_{n}}(x)-\psi_{M_{n}-1}(x)\overline{D}_{k}(x)\right)$
$\displaystyle=$ $\displaystyle
D_{M_{n}}(x)-\psi_{M_{n}-1}(x)\overline{Y}_{M_{n}}(x).$

which compete proof of identity (4.7).

On the other hand, if we combine (1.8) and (1.24) we also get proof of (4.8).
The proof is complete.

The proof of next Lemma can be found in Tephnadze [143]:

###### Lemma 4.4

Let $x\in I_{s}\backslash I_{s+1},\ \ s=0,...,N-1.$ Then

$\int_{I_{N}}\left|P_{n}\left(x-t\right)\right|d\mu\left(t\right)\leq\frac{cM_{s}}{M_{N}},$

where $c$ is an absolute constant.

Proof: The proof is direct consequence of Lemma 2.1. So, we leave out the
details.

###  4.3 strong convergence of Reisz means means with respect to Vilenkin
systems on the martingale Hardy spaces

Our first main result reads:

###### Theorem 4.5

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

$\overset{\infty}{\underset{n=1}{\sum}}\frac{\log^{p}n\left\|R_{n}f\right\|_{H_{p}(G_{m})}^{p}}{n^{2-2p}}\leq
c_{p}\left\|f\right\|_{H_{p}(G_{m})}^{p}$ (4.9)

holds, where $Rf$ denotes the $n$-th Reisz logarithmic mean with respect to
the Vilenkin-Fourier series of $f.$

Proof: According to (4.4) in Lemma 4.1 we get that

$\sup_{n\in\mathbb{N}}\int_{G_{m}}\left|Y_{n}a\right|d\mu\leq c<\infty.$

and it follows that $R_{n}$ is bounded from $L_{\infty}$ to $L_{\infty}.$ By
Proposition 1.4, the proof of theorem will be complete, if we show that

$\overset{\infty}{\underset{n=1}{\sum}}\frac{\log^{p}n\int\limits_{\overset{-}{I}}\left|R_{n}a\right|^{p}d\mu}{n^{2-2p}}\leq
c_{p}<\infty,\ \ \text{for}\ \ 0<p<1/2,$ (4.10)

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

Let $a$ be an arbitrary p-atom with support $I$ and
$\mu\left(I\right)=M_{N}^{-1}.$ We may assume that $I=I_{N}.$ It is easy to
see that $R_{n}a=\sigma_{n}\left(a\right)=0,$ when $n\leq M_{N}$. Therefore we
suppose that $n>M_{N}.$

Since $\left\|a\right\|_{\infty}\leq cM_{N}^{2}$ if we apply (4.3) in Lemma
4.1, we can conclude that

$\displaystyle\left|R_{n}a\left(x\right)\right|$ $\displaystyle=$
$\displaystyle\int_{I_{N}}\left|a\left(t\right)\right|\left|Y_{n}\left(x-t\right)\right|d\mu(t)$
$\displaystyle\leq$
$\displaystyle\left\|a\right\|_{\infty}\int_{I_{N}}\left|Y_{n}\left(x-t\right)\right|d\mu\left(t\right)$
$\displaystyle\leq$
$\displaystyle\frac{cM_{N}^{1/p}}{l_{n}}\underset{I_{N}}{\int}\underset{j=M_{N}+1}{\overset{n-1}{\sum}}\frac{\left|K_{j}\left(x-t\right)\right|}{j+1}d\mu\left(t\right)$
$\displaystyle+$
$\displaystyle\frac{cM_{N}^{1/p}}{l_{n}}\int_{I_{N}}\left|K_{n}\left(x-t\right)\right|d\mu\left(t\right).$

Let $x\in I_{N}\left(x_{k}e_{k}+x_{l}e_{l}\right),$ $\ 1\leq x_{k}\leq
m_{k}-1,$ $1\leq x_{l}\leq m_{l}-1,$ $k=0,...,N-2,$ $l=k+1,...,N-1.$ From
Lemma 4.2 it follows that

$\left|R_{n}a\left(x\right)\right|\leq\frac{cM_{l}M_{k}M_{N}^{1/p-2}}{\log(n+1)}.$
(4.12)

Let $x\in I_{N}\left(x_{k}e_{k}\right),$ $1\leq x_{k}\leq m_{k}-1,$
$k=0,...,N-1.$ Applying Lemma 4.2 we can conclude that

$\left|R_{n}a\left(x\right)\right|\leq{M_{N}^{1/p-1}M_{k}}.$ (4.13)

By combining (1.1) and (4.3-4.13) we obtain that

$\displaystyle\int_{\overline{I_{N}}}\left|R_{n}a\left(x\right)\right|^{p}d\mu\left(x\right)$
$\displaystyle=$
$\displaystyle\overset{N-2}{\underset{k=0}{\sum}}\overset{N-1}{\underset{l=k+1}{\sum}}\sum\limits_{x_{j}=0,j\in\\{l+1,...,N-1}^{m_{j-1}}\int_{I_{N}^{k,l}}\left|R_{n}a\right|^{p}d\mu$
$\displaystyle+$
$\displaystyle\overset{N-1}{\underset{k=0}{\sum}}\int_{I_{N}^{k,N}}\left|R_{n}a\right|^{p}d\mu$
$\displaystyle\leq$ $\displaystyle
c\overset{N-2}{\underset{k=0}{\sum}}\overset{N-1}{\underset{l=k+1}{\sum}}\frac{m_{l+1}\dots
m_{N-1}}{M_{N}}\frac{\left(M_{l}M_{k}\right)^{p}M_{N}^{1-2p}}{{\log}^{p}(n+1)}$
$\displaystyle+$
$\displaystyle\overset{N-1}{\underset{k=0}{\sum}}\frac{1}{M_{N}}M_{k}^{p}M_{N}^{1-p}$
$\displaystyle\leq$
$\displaystyle\frac{cM_{N}^{1-2p}}{{\log^{p}(n+1)}}\overset{N-2}{\underset{k=0}{\sum}}\overset{N-1}{\underset{l=k+1}{\sum}}\frac{\left(M_{l}M_{k}\right)^{p}}{M_{l}}+\overset{N-1}{\underset{k=0}{\sum}}\frac{M_{k}^{p}}{M_{N}^{p}}$
$\displaystyle=$
$\displaystyle\frac{cM_{N}^{1-2p}}{\log^{p}(n+1)}\overset{N-2}{\underset{k=0}{\sum}}\frac{1}{M_{k}^{1-2p}}\overset{N-1}{\underset{l=k+1}{\sum}}\frac{M_{k}^{1-p}}{M_{l}^{1-p}}+\overset{N-1}{\underset{k=0}{\sum}}\frac{M_{k}^{p}}{M_{N}^{p}}$
$\displaystyle\leq$ $\displaystyle\frac{cM_{N}^{1-2p}}{\log^{p}(n+1)}+c_{p}.$

It is easy to see that

$\overset{\infty}{\underset{n=M_{N}+1}{\sum}}\frac{1}{n^{2-2p}}\leq\frac{c}{M_{N}^{1-2p}},\
\ \text{ for }\ \ 0<p<1/2.$ (4.15)

By combining (4.3) and (4.15) we get that

$\displaystyle\overset{\infty}{\underset{n=M_{N}+1}{\sum}}\frac{\log^{p}n\int_{\overline{I_{N}}}\left|R_{n}a\right|^{p}d\mu}{n^{2-2p}}$
$\displaystyle\leq$
$\displaystyle\overset{\infty}{\underset{n=M_{N}+1}{\sum}}\left(\frac{c_{p}M_{N}^{1-2p}}{n^{2-p}}+\frac{c_{p}}{n^{2-p}}\right)+c_{p}$
$\displaystyle\leq$ $\displaystyle
c_{p}M_{N}^{1-2p}\overset{\infty}{\underset{n=M_{N}+1}{\sum}}\frac{1}{n^{2-2p}}$
$\displaystyle+$
$\displaystyle\overset{\infty}{\underset{n=M_{N}+1}{\sum}}\frac{1}{n^{2-p}}+c_{p}\leq
C_{p}<\infty.$

It means that (4.10) holds true and the proof is complete.

Our next main result shows in particular that the inequality in Theorem 4.5 is
in a special sense sharp at least in the case of Walsh-Fourier series.

###### Theorem 4.6

Let $0<p<1/2$ and $\Phi:\mathbb{N}\rightarrow[1,\infty)$ be any non-decreasing
function, satisfying the condition

$\underset{n\rightarrow\infty}{\lim}\Phi\left(n\right)=+\infty.$ (4.16)

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

$\sum_{n=1}^{\infty}\frac{\log^{p}n\left\|R^{w}_{n}f\right\|_{p}^{p}\Phi\left(n\right)}{n^{2-2p}}=\infty,$
(4.17)

where $R^{w}_{n}f$ denotes the $n$-th Reisz logarithmic means with respect to
Walsh-Fourier series of $f.$

Proof: It is evident that if we assume that $\Phi\left(n\right)\geq cn,$ where
$c$ is some positive constant then

$\frac{\log^{p}n\Phi\left(n\right)}{n^{2-2p}}\geq
n^{1-2p}\log^{p}n\rightarrow\infty,\ \text{as}\ n\rightarrow\infty,$

and also (4.17) holds. So, without lost the generality we may assume that
there exists an increasing sequence of positive integers
$\left\\{\alpha^{\prime}_{k}:k\in\mathbb{N}\right\\}$ such that

$\Phi\left(\alpha^{\prime}_{k}\right)=o(\alpha^{\prime}_{k}),\ \text{as}\
k\rightarrow\infty.$ (4.18)

Let
$\left\\{\alpha_{k}:k\in\mathbb{N}\right\\}\subseteq\left\\{\alpha^{\prime}_{k}:k\in\mathbb{N}\right\\}$
be an increasing sequence of positive integers such that $\alpha_{0}\geq 2$
and

$\sum_{k=0}^{\infty}\frac{1}{\Phi^{1/2}(2^{2\alpha_{k}})}<\infty,$ (4.19)

$\sum_{\eta=0}^{k-1}\frac{2^{2\alpha_{\eta}/p}}{\Phi^{1/2p}(2^{2\alpha_{\eta}})}\leq\frac{2^{2\alpha_{k-1}/p+1}}{\Phi^{1/2p}(2^{2\alpha_{k-1}})},$
(4.20)

$\frac{2^{2\alpha_{k-1}/p+1}}{\Phi^{1/2p}(2^{2\alpha_{k-1}})}\leq\frac{1}{128\alpha_{k}}\frac{2^{2\alpha_{k}(1/p-2)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}.$
(4.21)

We note that under condition (4.18) we can conclude that

$\frac{2^{2\alpha_{\eta}/p}}{\Phi^{1/2p}(2^{2\alpha_{\eta}})}\geq{\left(\frac{2^{2\alpha_{\eta}}}{\Phi(2^{2\alpha_{\eta}})}\right)}^{1/2p}\rightarrow\infty,\
\text{as}\ \eta\rightarrow\infty$

and it immediately follows that such an increasing sequence
$\left\\{\alpha_{k}:k\in\mathbb{N}\right\\},$ which satisfies conditions
(4.19)-(4.21), can be constructed.

Let

$f^{\left(n\right)}\left(x\right):=\sum_{\left\\{k;\text{
}2\alpha_{k}<n\right\\}}\lambda_{k}a_{k},$

where

$\lambda_{k}=\frac{1}{\Phi^{1/2p}(2^{2\alpha_{k}})}$

and

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

From (4.19) and Lemma 1.3 we can conclude that
$f=\left(f^{\left(n\right)},n\in\mathbb{N}\right)\in H_{p}(G_{2}).$

It is easy to show that

$\widehat{f}^{w}(j)=\left\\{\begin{array}[]{l}\frac{2^{2\alpha_{k}(1/p-1)}}{\Phi^{1/2p}(2^{2\alpha_{k}})},\,\,\text{
if \thinspace\thinspace}j\in\left\\{2^{2\alpha_{k}},...,\text{
~{}}2^{2\alpha_{k}+1}-1\right\\},\text{ }k\in\mathbb{N},\\\ 0,\text{
\thinspace\thinspace\thinspace if
\thinspace\thinspace\thinspace}j\notin\bigcup\limits_{k=1}^{\infty}\left\\{2^{2\alpha_{k}},...,\text{
~{}}2^{2\alpha_{k}+1}-1\right\\}.\end{array}\right.$ (4.22)

For $n=\sum_{i=1}^{s}2^{n_{i}},$ $n_{1}<n_{2}<...<n_{s}$ we denote

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

Let $2^{2\alpha_{k}}\leq j\leq 2^{2\alpha_{k}+1}-1$ and
$j\in\mathbb{A}_{0,2}.$ Then

$\displaystyle R^{w}_{j}f$ $\displaystyle=$
$\displaystyle\frac{1}{l_{j}}\sum_{n=1}^{2^{2\alpha_{k}}-1}\frac{S_{n}f}{n}+\frac{1}{l_{j}}\sum_{n=2^{2\alpha_{k}}}^{j}\frac{S_{n}f}{n}:=I+II.$
(4.23)

Let $n<2^{2\alpha_{k}}.$ Then from (4.20), (4.21) and (4.22) we have that

$\displaystyle\left|S^{w}_{n}f\left(x\right)\right|$ $\displaystyle\leq$
$\displaystyle\sum_{\eta=0}^{k-1}\sum_{v=2^{2\alpha_{\eta}}}^{2^{2\alpha_{\eta}+1}-1}\left|\widehat{f}^{w}(v)\right|$
$\displaystyle\leq$
$\displaystyle\sum_{\eta=0}^{k-1}\sum_{v=2^{2\alpha_{\eta}}}^{2^{2\alpha_{\eta}+1}-1}\frac{2^{2\alpha_{\eta}(1/p-1)}}{\Phi^{1/2p}(2^{2\alpha_{\eta}})}$
$\displaystyle\leq$
$\displaystyle\sum_{\eta=0}^{k-1}\frac{2^{2\alpha_{\eta}/p}}{\Phi^{1/2p}(2^{2\alpha_{\eta}})}$
$\displaystyle\leq$
$\displaystyle\frac{2^{2\alpha_{k-1}/p+1}}{\Phi^{1/2p}(2^{2\alpha_{k-1}})}$
$\displaystyle\leq$
$\displaystyle\frac{1}{128\alpha_{k}}\frac{2^{2\alpha_{k}(1/p-2)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}.$

Consequently,

$\displaystyle\left|I\right|$ $\displaystyle\leq$
$\displaystyle\frac{1}{l_{j}}\underset{n=1}{\overset{2^{2\alpha_{k}}-1}{\sum}}\frac{\left|S^{w}_{n}f\left(x\right)\right|}{n}$
$\displaystyle\leq$
$\displaystyle\frac{1}{l_{2^{2\alpha_{k}}}}\frac{1}{128\alpha_{k}}\frac{2^{2\alpha_{k}(1/p-2)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}\sum_{n=1}^{2^{2\alpha_{k}}-1}\frac{1}{n}$
$\displaystyle\leq$
$\displaystyle\frac{1}{128\alpha_{k}}\frac{2^{2\alpha_{k}(1/p-2)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}.$

Let $2^{2\alpha_{k}}\leq n\leq 2^{2\alpha_{k}+1}-1.$ Then we have the
following

$\displaystyle S^{w}_{n}f$ $\displaystyle=$
$\displaystyle\sum_{\eta=0}^{k-1}\sum_{v=2^{2\alpha_{\eta}}}^{2^{2\alpha_{\eta}+1}-1}\widehat{f}^{w}(v)w_{v}+\sum_{v=2^{2\alpha_{k}}}^{n-1}\widehat{f}^{w}(v)w_{v}$
$\displaystyle=$
$\displaystyle\sum_{\eta=0}^{k-1}\frac{2^{{2\alpha_{\eta}}\left(1/p-1\right)}}{\Phi^{1/2p}(2^{2\alpha_{\eta}})}\left(D^{w}_{2^{2\alpha_{\eta}+1}}-D^{w}_{2^{2\alpha_{\eta}}}\right)$
$\displaystyle+$
$\displaystyle\frac{2^{{2\alpha_{k}}\left(1/p-1\right)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}\left(D^{w}_{n}-D^{w}_{2^{2\alpha_{k}}}\right).$

This gives that

$\displaystyle II$ $\displaystyle=$
$\displaystyle\frac{1}{l_{j}}\underset{n=2^{2\alpha_{k}}}{\overset{2^{2\alpha_{k}+1}}{\sum}}\
\frac{1}{n}\left(\sum_{\eta=0}^{k-1}\frac{2^{2\alpha_{\eta}\left(1/p-1\right)}}{\Phi^{1/2p}(2^{2\alpha_{\eta}})}\left(D^{w}_{2^{2\alpha_{\eta}+1}}-D^{w}_{2^{2\alpha_{\eta}}}\right)\right)$
$\displaystyle+$
$\displaystyle\frac{1}{l_{j}}\frac{2^{2\alpha_{k}\left(1/p-1\right)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}\sum_{n=2^{2\alpha_{k}}}^{j}\frac{\left(D^{w}_{n}-D^{w}_{2^{2\alpha_{k}}}\right)}{n}$
$\displaystyle:=$ $\displaystyle II_{1}+II_{2}.$

Let $x\in I_{2}(e_{0}+e_{1})\in I_{0}\backslash I_{1}.$ According to (1.38)
and (1.39) we conclude that

$D^{w}_{n}\left(x\right)=\left\\{\begin{array}[]{ll}w_{n},&\,\text{if\thinspace\thinspace\thinspace}n\
\ \text{is odd number,}\\\ 0,&\text{if}\,\,n\ \ \text{is even
number.}\end{array}\right.$

Since $\alpha_{0}\geq 2,\ \ k\in\mathbb{N}$ we obtain that $2\alpha_{k}\geq
4,$ for all $k\in\mathbb{N}$ and if we apply (1.38) we get that

$II_{1}=0$ (4.26)

and

$\displaystyle II_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{l_{j}}\frac{2^{2\alpha_{k}(1/p-1)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}\sum_{n=2^{2\alpha_{k}-1}}^{(j-1)/2}\frac{w_{2n+1}}{2n+1}$
$\displaystyle=$
$\displaystyle\frac{1}{l_{j}}\frac{2^{2\alpha_{k}(1/p-1)}r_{1}}{\Phi^{1/2p}(2^{2\alpha_{k}})}\sum_{n=2^{2\alpha_{k}-1}}^{(j-1)/2}\frac{w_{2n}}{2n+1}.$

Let $x\in I_{2}(e_{0}+e_{1}).$ Then, by the definition of Walsh functions, we
get that

$w_{4n+2}=r_{1}w_{4n}=-w_{4n}$

and

$\displaystyle\left|II_{2}\right|=\frac{1}{l_{j}}\frac{2^{2\alpha_{k}(1/p-1)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}\left|\sum_{n=2^{2\alpha_{k}-1}}^{(j-1)/2}\frac{w_{2n}}{2n+1}\right|$
$\displaystyle=$
$\displaystyle\frac{1}{l_{j}}\frac{2^{2\alpha_{k}(1/p-1)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}\left|\frac{w_{j-1}}{j}+\sum_{n=2^{2\alpha_{k}-2}+1}^{(j-1)/4}\left(\frac{w_{4n-4}}{4n-3}+\frac{w_{4n-2}}{4n-1}\right)\right|$
$\displaystyle=$
$\displaystyle\frac{1}{l_{j}}\frac{2^{2\alpha_{k}(1/p-1)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}\left|\frac{w_{j-1}}{j}+\sum_{n=2^{2\alpha_{k}-2}+1}^{(j-1)/4}\left(\frac{w_{4n-4}}{4n-3}-\frac{w_{4n-2}}{4n-1}\right)\right|$
$\displaystyle\geq$
$\displaystyle\frac{c}{\log({2^{2\alpha_{k}+1}})}\frac{2^{2\alpha_{k}(1/p-1)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}\left(\left|\frac{w_{j-1}}{j}\right|-\sum_{n=2^{2\alpha_{k}-2}+1}^{(j-1)/4}\left|{w_{4n-4}}\right|\left(\frac{1}{4n-3}-\frac{1}{4n-1}\right)\right)$
$\displaystyle\geq\frac{1}{4\alpha_{k}}\frac{2^{2\alpha_{k}(1/p-1)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}\left(\frac{1}{j}-\sum_{n=2^{2\alpha_{k}-2}+1}^{(j-1)/4}\left(\frac{1}{4n-3}-\frac{1}{4n-1}\right)\right).$

Hence, by simple calculation we can conclude that

$\displaystyle\sum_{n=2^{2\alpha_{k}-2}+1}^{(j-1)/4}\left(\frac{1}{4n-3}-\frac{1}{4n-1}\right)$
$\displaystyle=$
$\displaystyle\sum_{n=2^{2\alpha_{k}-2}+1}^{(j-1)/4}\frac{2}{(4n-3)(4n-1)}$
$\displaystyle\leq$
$\displaystyle\sum_{n=2^{2\alpha_{k}-2}+1}^{(j-1)/4}\frac{2}{(4n-4)(4n-2)}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{n=2^{2\alpha_{k}-2}+1}^{(j-1)/4}\frac{1}{(2n-2)(2n-1)}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\sum_{n=2^{2\alpha_{k}-2}+1}^{(j-1)/4}\frac{1}{(2n-2)(2n-2)}$
$\displaystyle=$
$\displaystyle\frac{1}{8}\sum_{n=2^{2\alpha_{k}-2}+1}^{(j-1)/4}\frac{1}{(n-1)(n-1)}$
$\displaystyle\leq$
$\displaystyle\frac{1}{8}\sum_{n=2^{2\alpha_{k}-2}+1}^{(j-1)/4}\frac{1}{(n-1)(n-2)}$
$\displaystyle=$
$\displaystyle\frac{1}{8}\sum_{l=2^{2\alpha_{k}-2}+1}^{(j-1)/4}\left(\frac{1}{n-2}-\frac{1}{n-1}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{8}\left(\frac{1}{2^{2\alpha_{k}-2}-1}-\frac{4}{j-5}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{8}\left(\frac{1}{2^{2\alpha_{k}-2}-1}-\frac{4}{j}\right).$

Since $2^{2\alpha_{k}}\leq j\leq 2^{2\alpha_{k}+1}-1,$ where $\alpha_{k}\geq
2,$ we obtain that

$\frac{2}{2^{2\alpha_{k}}-4}\leq\frac{2}{2^{4}-4}=\frac{1}{6}$

and

$\displaystyle\left|II_{2}\right|$ $\displaystyle\geq$
$\displaystyle\frac{1}{4\alpha_{k}}\frac{2^{2\alpha_{k}(1/p-1)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}\left(\frac{1}{j}-\frac{1}{8}\left(\frac{1}{2^{2\alpha_{k}-2}-1}-\frac{4}{j}\right)\right)$
$\displaystyle\geq$
$\displaystyle\frac{1}{4\alpha_{k}}\frac{2^{2\alpha_{k}(1/p-1)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}\left(\frac{3}{2j}-\frac{1}{2^{2\alpha_{k}+1}-8}\right)$
$\displaystyle\geq$
$\displaystyle\frac{1}{4\alpha_{k}}\frac{2^{2\alpha_{k}(1/p-1)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}\left(\frac{3}{4}\frac{1}{2^{2\alpha_{k}}}-\frac{1}{2}\frac{1}{2^{2\alpha_{k}}-4}\right)$
$\displaystyle\geq$
$\displaystyle\frac{1}{4\alpha_{k}}\frac{2^{2\alpha_{k}(1/p-1)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}\left(\frac{1}{4}\frac{1}{2^{2\alpha_{k}}}+\frac{1}{2}\frac{1}{2^{2\alpha_{k}}}-\frac{1}{2}\frac{1}{2^{2\alpha_{k}}-4}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{4\alpha_{k}}\frac{2^{2\alpha_{k}(1/p-1)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}\left(\frac{1}{4}\frac{1}{2^{2\alpha_{k}}}-\frac{2}{2^{2\alpha_{k}}(2^{2\alpha_{k}}-4)}\right)$
$\displaystyle\geq$
$\displaystyle\frac{1}{4\alpha_{k}}\frac{2^{2\alpha_{k}(1/p-1)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}\left(\frac{1}{4}\frac{1}{2^{2\alpha_{k}}}-\frac{1}{6}\frac{1}{2^{2\alpha_{k}}}\right)$
$\displaystyle\geq$
$\displaystyle\frac{1}{48\alpha_{k}}\frac{2^{2\alpha_{k}(1/p-2)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}$
$\displaystyle\geq$
$\displaystyle\frac{1}{64\alpha_{k}}\frac{2^{2\alpha_{k}(1/p-2)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}.$

By combining (4.2), (4.23)-(4.3) for $\in I_{2}(e_{0}+e_{1})$ and $0<p<1/2$ we
find that

$\displaystyle\left|R^{w}_{j}f\left(x\right)\right|$ $\displaystyle\geq$
$\displaystyle\left|II_{2}\right|-\left|II_{1}\right|-\left|I\right|$
$\displaystyle\geq$
$\displaystyle\frac{1}{64\alpha_{k}}\frac{2^{2\alpha_{k}(1/p-2)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}-\frac{1}{128\alpha_{k}}\frac{2^{2\alpha_{k}(1/p-2)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}$
$\displaystyle=$
$\displaystyle\frac{1}{128\alpha_{k}}\frac{2^{2\alpha_{k}(1/p-2)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}.$

and

$\displaystyle\left\|R^{w}_{j}f\right\|_{weak-L_{p}(G_{2})}^{p}$
$\displaystyle\geq$
$\displaystyle\frac{1}{128\alpha_{k}^{p}}\frac{2^{2\alpha_{k}(1-2p)}}{\Phi^{1/2}(2^{2\alpha_{k}})}\mu\left\\{x\in
G_{2}:\left|R^{w}_{j}f\right|\geq\frac{1}{128\alpha_{k}}\frac{2^{2\alpha_{k}(1/p-2)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}\right\\}^{1/p}$
$\displaystyle\geq$
$\displaystyle\frac{1}{128\alpha_{k}^{p}}\frac{2^{2\alpha_{k}(1-2p)}}{\Phi^{1/2}(2^{2\alpha_{k}})}\mu\left\\{x\in
I_{2}(e_{0}+e_{1}):\left|R^{w}_{j}f\right|\geq\frac{1}{128\alpha_{k}}\frac{2^{2\alpha_{k}(1/p-2)}}{\Phi^{1/2p}(2^{2\alpha_{k}})}\right\\}$
$\displaystyle\geq$
$\displaystyle\frac{1}{128\alpha_{k}^{p}}\frac{2^{2\alpha_{k}(1-2p)}}{\Phi^{1/2}(2^{2\alpha_{k}})}(\mu\left(x\in
I_{2}(e_{0}+e_{1})\right))$ $\displaystyle>$
$\displaystyle\frac{1}{516\alpha_{k}^{p}}\frac{2^{2\alpha_{k}(1-2p)}}{\Phi^{1/2}(2^{2\alpha_{k}})}.$

Moreover,

$\displaystyle\underset{j=1}{\overset{\infty}{\sum}}\frac{\left\|R^{w}_{j}f\right\|_{weak-
L_{p}(G_{2})}^{p}\log^{p}{(j)}\Phi(j)}{j^{2-2p}}$ $\displaystyle\geq$
$\displaystyle\underset{\left\\{j\in\mathbb{A}_{0,2}:\text{
}2^{2\alpha_{k}}<j\leq
2^{2\alpha_{k}+1}-1\right\\}}{\sum}\frac{\left\|R^{w}_{j}f\right\|_{weak-
L_{p}}^{p}\log^{p}{(j)}\Phi(j)}{j^{2-2p}}$ $\displaystyle\geq$
$\displaystyle\frac{c}{\alpha_{k}^{p}}\frac{2^{2\alpha_{k}(1-2p)}}{\Phi^{p/2}(2^{2\alpha_{k})}}\underset{\left\\{j\in\mathbb{A}_{0,2}:\text{
}2^{2\alpha_{k}}<j\leq
2^{2\alpha_{k}+1}-1\right\\}}{\sum}\frac{\log^{p}{(j)}\Phi(j)}{j^{2-2p}}$
$\displaystyle\geq$
$\displaystyle\frac{c\Phi(2^{2\alpha_{k}})\log^{p}{(2^{2\alpha_{k}})}}{\alpha_{k}^{p}}\frac{2^{2\alpha_{k}(1-2p)}}{\Phi^{1/2}(2^{2\alpha_{k}})}\underset{\left\\{j\in\mathbb{A}_{0,2}:\text{
}2^{2\alpha_{k}}<j\leq 2^{2\alpha_{k}+1}-1\right\\}}{\sum}\frac{1}{j^{2-2p}}$
$\displaystyle\geq$
$\displaystyle\Phi^{1/2}(2^{2\alpha_{k}})\rightarrow\infty,\text{ \ as \ \
}k\rightarrow\infty.$

The proof is complete.

###### Theorem 4.7

Let $p\geq 1$ and $f\in L_{p}$. Then

$\|R_{n}f-f\|_{p}\to 0\ \ \text{as}\ \ n\to\infty$ (4.30)

for all $f\in L_{p}(G_{m})$. Moreover,

$\underset{n\rightarrow\infty}{\lim}R_{n}f(x)=f(x)$

for all Vilenkin-Lebesgue points of $f\in L_{p}(G_{m})$.

Proof: By using Abel transformation we get

$R_{n}=\frac{1}{l_{n}}\overset{n-1}{\underset{j=1}{\sum}}\frac{\sigma_{j}}{j+1}+\frac{\sigma_{n}}{l_{n}}.$
(4.31)

Since (for details see Yano [182])

$\|\sigma_{n}f-f\|_{p}\to 0\ \ \text{as}\ \ n\to\infty$

if we apply (4.31) we immediately get (4.30).

On the other hand, since (for details see [53])

$\underset{n\rightarrow\infty}{\lim}\sigma_{n}f(x)=f(x)$

for all Vilenkin-Lebesgue points of $f\in L_{p}(G_{m})$, if we invoke again
equality (4.31) we also get

$\underset{n\rightarrow\infty}{\lim}R_{n}f(x)=f(x)$

for all Vilenkin-Lebesgue points of $f\in L_{p}(G_{m})$.

###  4.4 Nörlund logarithmic means of Vilenkin-Fourier series on Martingale
Hardy spaces

###### Theorem 4.8

a) Let $0<p<1.$ Then the maximal operator

$\overset{\sim}{L}_{p}^{*}f:=\sup_{n\in\mathbb{N}}\frac{\left|L_{n}f\right|}{\left(n+1\right)^{1/p-1}}$

is bounded from the Hardy space $H_{p}\left(G_{m}\right)$ to the space
$L_{p}\left(G_{m}\right).$

b) Let $0<p<1$ and $\varphi:\mathbb{N}_{+}\rightarrow[1,\infty)$ be a non-
decreasing function satisfying the condition

$\overline{\lim_{n\rightarrow\infty}}\frac{n^{1/p-1}}{\log
n\varphi\left(n\right)}=+\infty.$ (4.32)

Then there exists a martingale $f\in H_{p}\left(G_{m}\right),$ such that the
maximal operator

$\text{
}\sup_{n\in\mathbb{N}}\frac{\left|L_{n}f\right|}{\varphi\left(n+1\right)}$

is not bounded from the Hardy space $H_{p}\left(G_{m}\right)$ to the space
$L_{p}\left(G_{m}\right).$

Proof: Since

$\displaystyle\frac{\left|L_{n}f\right|}{\left(n+1\right)^{1/p-1}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\left(n+1\right)^{1/p-1}}\underset{1\leq k\leq
n}{\sup}\left|S_{k}f\right|$ $\displaystyle\leq$ $\displaystyle\underset{1\leq
k\leq n}{\sup}\frac{\left|S_{k}f\right|}{\left(k+1\right)^{1/p-1}}$
$\displaystyle\leq$
$\displaystyle\underset{n\in\mathbb{N}}{\sup}\frac{\left|S_{n}f\right|}{\left(n+1\right)^{1/p-1}}$

and

$\underset{n\in\mathbb{N}}{\sup}\frac{\left|L_{n}f\right|}{\left(n+1\right)^{1/p-1}}\leq\underset{n\in\mathbb{N}}{\sup}\frac{\left|S_{n}f\right|}{\left(n+1\right)^{1/p-1}}$
(4.33)

On the other hand, Tephnadze [144] (see also [146] and [149]) proved that for
every $0<p<1,$ the maximal operator

$\widetilde{S}_{p}^{\ast}f:=\sup_{n\in\mathbb{N}}\frac{\left|S_{n}f\right|}{\left(n+1\right)^{1/p-1}}$

is bounded from the Hardy space $H_{p}$ to the Lebesgue space $L_{p}.$ Hence,
according to inequality (4.33) we can conclude that

$\left\|\underset{n\in\mathbb{N}}{\sup}\frac{\left|L_{n}f\right|}{\left(n+1\right)^{1/p-1}}\right\|_{p}\leq\left\|\underset{n\in\mathbb{N}}{\sup}\frac{\left|S_{n}f\right|}{\left(n+1\right)^{1/p-1}}\right\|_{p}\leq
c_{p}\left\|f\right\|_{H_{p}}.$

Now, prove part b) of the Theorem. Let

$f_{n_{k}}=D_{M_{2n_{k}+1}}-D_{M_{2n_{k}}}.$

It is evident

$\widehat{f}_{n_{k}}\left(i\right)=\left\\{\begin{array}[]{l}\text{
}1,\qquad\text{ if }\qquad i=M_{2n_{k}},...,M_{2n_{k}+1}-1,\\\ \text{
}0,\qquad\text{otherwise.}\end{array}\right.$ (4.34)

Then we can write

$S_{i}f_{n_{k}}=\left\\{\begin{array}[]{l}D_{i}-D_{M_{2n_{k}}},\quad\text{ if
}\quad i=M_{2n_{k}}+1,...,M_{2n_{k}+1}-1,\\\ f_{n_{k}},\qquad\qquad\quad\text{
if }\quad i\geq M_{2n_{k}+1},\\\ 0,\qquad\qquad\qquad\text{
otherwise.}\end{array}\right.$ (4.35)

From (1.5) we get

$\displaystyle\left\|f_{n_{k}}\right\|_{H_{p}}$ $\displaystyle=$
$\displaystyle\left\|\sup\limits_{n\in\mathbb{N}}S_{M_{n}}f_{n_{k}}\right\|_{p}$
$\displaystyle=$
$\displaystyle\left\|D_{M_{2n_{k}+1}}-D_{M_{{}_{2n_{k}}}}\right\|_{p}$
$\displaystyle\leq$
$\displaystyle\left\|D_{M_{2n_{k}+1}}\right\|_{p}+\left\|D_{M_{{}_{2n_{k}}}}\right\|_{p}$
$\displaystyle\leq$ $\displaystyle cM_{{}_{2n_{k}}}^{1-1/p}<c<\infty.$

Let $0<p<1$ and $\left\\{\lambda_{k}:k\in\mathbb{N}_{+}\right\\}$ be an
increasing sequence of the positive integers such that

$\lim_{k\rightarrow\infty}\frac{\lambda_{k}^{1/p-1}}{\varphi\left(\lambda_{k}\right)}=\infty.$

Let$\
\left\\{n_{k}:k\in\mathbb{N}_{+}\right\\}\subset\left\\{\lambda_{k}:k\in\mathbb{N}_{+}\right\\}$
such that

$\displaystyle\underset{k\rightarrow\infty}{\lim}\frac{\left(M_{{}_{2n_{k}}}+2\right)^{1/p-1}}{\log{\left(M_{2n_{k}}+2\right)}\varphi\left(M_{2n_{k}+2}\right)}\geq
c\lim_{k\rightarrow\infty}\frac{\lambda_{k}^{1/p-1}}{\varphi\left(\lambda_{k}\right)}=\infty.$

According to (4.35) we can conclude that

$\displaystyle\left|\frac{L_{M_{2n_{k}}+2}f_{n_{k}}}{\varphi\left(M_{2n_{k}+2}\right)}\right|$
$\displaystyle=$
$\displaystyle\frac{\left|D_{M_{{}_{2n_{k}}}+1}-D_{M_{{}_{2n_{k}}}}\right|}{l_{M_{2n_{k}}+1}\varphi\left(M_{2n_{k}+1}\right)}$
$\displaystyle=$
$\displaystyle\frac{\left|\psi_{M_{{}_{2n_{k}}}}\right|}{l_{M_{2n_{k}}+2}\varphi\left(M_{2n_{k}+1}\right)}$
$\displaystyle=$
$\displaystyle\frac{1}{l_{M_{2n_{k}}+1}\varphi\left(M_{2n_{k}+2}\right)}.$

Hence,

$\mu\left\\{x\in
G_{m}:\left|L_{M_{2n_{k}}+2}f_{n_{k}}\right|\geq\frac{1}{l_{M_{2n_{k}}+2}\varphi\left(M_{2n_{k}+2}\right)}\right\\}=\mu\left(G_{m}\right)=1.$
(4.37)

By combining (4.4) and (4.37) we get that

$\displaystyle\frac{\frac{1}{l_{M_{2n_{k}}+2}\varphi\left(M_{2n_{k}+2}\right)}\left(\mu\left\\{x\in
G_{m}:\quad\left|L_{M_{2n_{k}}+2}f_{n_{k}}\right|\geq\frac{1}{l_{M_{2n_{k}}+2}\varphi\left(M_{2n_{k}+2}\right)}\right\\}\right)^{1/p}}{\left\|f_{n_{k}}\right\|_{p}}$
$\displaystyle\geq$
$\displaystyle\frac{M_{{}_{2n_{k}}}^{1/p-1}}{l_{M_{2n_{k}}+2}\varphi\left(M_{2n_{k}+2}\right)}$
$\displaystyle\geq$
$\displaystyle\frac{c\left(M_{{}_{2n_{k}}}+2\right)^{1/p-1}}{\log{\left(M_{2n_{k}}+2\right)}\varphi\left(M_{2n_{k}+2}\right)}\rightarrow\infty,\text{\quad
as \quad}k\rightarrow\infty.$

Theorem is proved.

###### Corollary 4.9

Let $0<p<1\ $and $f\in H_{p}\left(G_{m}\right).$ Then there is an absolute
constant $c_{p},$ depends only $p,$ such that

$\overset{\infty}{\underset{k=1}{\sum}}\frac{\left\|L_{k}f\right\|_{p}^{p}}{k^{2-p}}\leq
c_{p}\left\|f\right\|_{H_{p}}^{p}.$

Proof: If we apply Lemma 1.4 we only have to prove that

$\overset{\infty}{\underset{k=1}{\sum}}\frac{\left\|L_{k}a\right\|_{p}^{p}}{k^{2-p}}\leq
c_{p}<\infty.$

for every $p$ atom $a$, with support $I_{N}$ and
$\mu\left(I_{N}\right)=M_{N}^{-1}$. Since $S_{k}a=0,$ for $k\leq M_{N},$ we
get that $L_{n}a=0,$ for $n\leq M_{N}.$ So, we may assume that $n>M_{N}.$

Let $x\in\overline{I_{N}}$ and $0<p<1.$ Applying (1.1) and Lemma 4.4 we have
that

$\displaystyle\overset{\infty}{\underset{k=M_{N}}{\sum}}\frac{\left\|L_{k}a\right\|_{p}^{p}}{k^{2-p}}$
$\displaystyle\leq$
$\displaystyle\overset{\infty}{\underset{k=M_{N}}{\sum}}\frac{1}{k}\int_{\overline{I_{N}}}\left|\frac{L_{k}a}{k^{1/p-1}}\right|^{p}d\mu$
$\displaystyle=$
$\displaystyle\overset{\infty}{\underset{k=M_{N}}{\sum}}\frac{1}{k}\overset{N-1}{\underset{s=0}{\sum}}\int_{I_{s}\backslash
I_{s+1}}\left|\frac{L_{k}a}{k^{1/p-1}}\right|^{p}d\mu$ $\displaystyle\leq$
$\displaystyle
c\overset{\infty}{\underset{k=M_{N}}{\sum}}\frac{1}{k}\overset{N-1}{\underset{s=0}{\sum}}\int_{I_{s}\backslash
I_{s+1}}\left|\frac{M_{N}^{1/p-1}M_{s}}{k^{1/p-1}}\right|^{p}d\mu$
$\displaystyle\leq$ $\displaystyle
c_{p}M_{N}^{1-p}\overset{\infty}{\underset{k=M_{N}}{\sum}}\frac{1}{k^{2-p}}\overset{N-1}{\underset{s=0}{\sum}}\int_{I_{s}\backslash
I_{s+1}}M_{s}^{p}$ $\displaystyle\leq$ $\displaystyle
c_{p}M_{N}^{1-p}\overset{\infty}{\underset{k=M_{N}}{\sum}}\frac{1}{k^{2-p}}\overset{N-1}{\underset{s=0}{\sum}}M_{s}^{p-1}d\mu$
$\displaystyle+$ $\displaystyle
c_{p}M_{N}^{1-p}\overset{\infty}{\underset{k=M_{N}}{\sum}}\frac{1}{k^{2-p}}\leq
c_{p}<\infty.$

Which complete the proof.

###### Theorem 4.10

Let $p\geq 1$ and $f\in L_{p}$. Then

$\|L_{M_{n}}f-f\|_{p}\to 0\ \ \text{as}\ \ n\to\infty$

for all $f\in L_{p}(G_{m})$. Moreover,

$\underset{n\rightarrow\infty}{\lim}L_{M_{n}}f(x)=f(x)$

for all Lebesgue points of $f\in L_{p}(G_{m})$.

Proof: To prove a.e convergence we use first identity in Lemma 4.3 to write
that

$\displaystyle L_{M_{n}}f\left(x\right)$ $\displaystyle=$
$\displaystyle\underset{G_{m}}{\int}f\left(t\right)P_{n}\left(x-t\right)d\mu\left(t\right)$
$\displaystyle=$
$\displaystyle\underset{G_{m}}{\int}f\left(t\right)D_{M_{n}}\left(x-t\right)d\mu\left(t\right)$
$\displaystyle-$
$\displaystyle\underset{G_{m}}{\int}f\left(t\right)\psi_{M_{n}-1}(x-t)\overline{Y}_{M_{n}}(x-t)$
$\displaystyle=$ $\displaystyle I-II.$

By applying (1.26) we can conclude that

$I=S_{M_{n}}f(x)\to f(x)$

for all Lebesgue points of $f\in L_{p}(G_{m})$. By using (1.2) we can conclude
that

$II=\psi_{M_{n}-1}(x)\underset{G_{m}}{\int}f\left(t\right)\overline{Y}_{M_{n}}(x-t)\overline{\psi}_{M_{n}-1}(t)d(t)$

By combining 4.8 in Lemma 4.3 and 1.26 we find that

$f\left(t\right)\overline{Y}_{M_{n}}(x-t)\in L_{p}\ \ \text{ where}\ \ p\geq
1\ \ \text{for any }\ \ x\in G_{m},$

and $II$ is Fourier coefficients of integrable function. According to Riemann-
Lebesgue Lemma it vanish as $n\to\infty,$

$II\to 0\ \ \text{for any }\ \ x\in G_{m},\ \ n\to\infty.$

The proof is complete.

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