# Modeling the impact of civilian firearm ownership in the evolution of
violent crimes

Nuno Crokidakis<EMAIL_ADDRESS>Instituto de Física, Universidade Federal
Fluminense, Niterói, Rio de Janeiro, Brazil

###### Abstract

We propose a simple mathematical model to describe the evolution of violent
crimes. For such purpose, we built a model based on ordinary differential
equations that take into account the number of violent crimes and the number
of legal and illegal guns. The dynamics is governed by probabilities, modeling
for example the police action, the risk perception regarding crimes that leads
to increase of ownership of legal guns, and so on. Our analytical and
numerical results show that, in addition to the rise of criminality due to the
presence of illegal guns, the increase of legal guns leads to a fast increase
of violent crimes, suggesting that the access of firearms by civilians is not
a good option regarding the control of crimes.

###### keywords:

Dynamics of social systems , Collective phenomena

††journal: Elsevier

##  1 Introduction

All countries in the world face problems with criminality. The control of the
number of committed violent crimes is one of the greatest problems of
governments. The problem of criminality was studied by a series of researchers
like mathematicians, physicists and computer scientists. For a recent review,
see [1].

In some countries, like United States, firearms are one of the central
flashpoints in people’s life [2]. The contemporary American desire to own a
gun, but it is a desire with consequences. Indeed, USA has more civilian-owned
guns than any other developed nation and as a direct result has a level of
gun-related deaths unmatched by any other developed nation [2]. Surveys
revealed that about four-in-ten US adults live in a gun-owing househould.
Thus, roughly $40\%$ of American households own a gun (about $30\%$ of
American adults), and half of those Americans who do not currently own a gun
can see themselves doing so in the future [3]. Thus, taking the population
size of US as about 334 million [4], $40\%$ means about 134 million guns. It
was discussed that a record number of firearm background checks were completed
in USA at the onset of the COVID-19 pandemic and during the protests following
the murder of George Floyd [5]. Gun ownership is an expression of fear, but
some analysis did suggest that people who own guns tend to exhibit lower
levels of fear than nongun owners [6].

The usual questions are how and why criminals acquire, carry, and use guns.
Key findings were that criminals rarely acquired firearms through conventional
retail markets, and that criminals were for various reasons adept at finding
ways to circumvent stricter gun controls [7]. The impact of not identifying
the source of a weapon is vital to combating gun-enabled crime; failure to do
so means that supply routes or “armourers” are not identified, allowing
further supply of firearms into the criminal world [8].

Data from 13 European Union countries suggest that the number of illegally
held firearms in the European Union is estimated to be up to 67 million. It is
estimated that illicit trafficking has been directly responsible for at least
10,000-15,000 firearms-related deaths in European Union member states over the
past decade [9]. An analysis of the relationship between gun ownership rates
and violent death rates in Europe indicates a strong positive correlation
between gun ownership rates in a country and the rate of firearms-related
deaths. [9]. Regarding the civilian gun ownership in Europe, it was estimated
in 93 million. The countries with more firearms in civilians’ hands are
Germany (25 million) and France (19 million) [9].

Unfortunately, research on the effect of gun levels on homicide and other
crime rates has generally been of poor quality [10]. Mathematical models can
be useful to analyze some aspects of the evolution of crimes due to legal
and/or illegal firearms. Indeed, we can cite some recent works [11, 12, 13,
14, 15]. In order to study specifically the impact of legal firearms on the
evolution of violent crimes, i.e., crimes commited by civilians using legal
guns, we propose an extension of the model presented in [11]. Our analytical
and numerical results show that the increase of legal guns leads to a fast
increase of violent crimes, suggesting that the access of firearms by
civilians is not a good option regarding the control of crimes.

##  2 The Model

We consider the evolution of the number of violent crimes committed by an
individual or a group of individuals. These crimes can be, for example,
assaults, robberies and homicides, and the individual can be a gang member or
a thief. These crimical activities are committed by using illegal guns, and it
can be inhibited by the police action. In a recent work, the impact of illegal
guns in the evolution of violent crimes was studied, and the author analyzed
how legally-obtained guns can inhibit such crimes [11]. However, it was not
considered the impact of such legally-obtained guns by citizens in the
evolution of violent crimes, specifically in homicides. Recent studies
analyzed the impact of homicides comitted by nonstragers, like husband, wife,
mother, father, son, boyfriend, and others. For example, in USA, a parent was
the perpetrator in the majority of homicides of children under age 5: $63\%$
were killed by a parent (1980 - 2008), $28\%$ by a friend/acquaintance and
just $3\%$ by a stranger [16]. In another work, it was observed that the
average proportion of firearm homicides that were committed by strangers
throughout the United States was only $21.9\%$ [17]. Motivated by those data,
we also consider in the model crimes (firearm homicides) committed by citizens
that own legal firearms.

Following the notation of Ref. [11], let $x(t)$ be the prevalence of violent
crimes per capita, $y(t)$ the number of legal firearms per capita and $z(t)$
the number of illegal firearms per capita. The following system of
differential equations governs the dynamics of the model:

$\displaystyle\frac{dx}{dt}$ $\displaystyle=$
$\displaystyle\alpha\,z-\nu\,x+\lambda\,y$ (1) $\displaystyle\frac{dy}{dt}$
$\displaystyle=$ $\displaystyle\sigma+\beta\,x-\theta\,y-\gamma\,y\,z$ (2)
$\displaystyle\frac{dz}{dt}$ $\displaystyle=$
$\displaystyle\gamma\,y\,z+\delta\,z-\epsilon\,z^{2}$ (3)

Let us elaborate on the model’s parameters. Regarding Eq. (1), the parameter
$\alpha$ models the increase of violent crimes due to the presence of illegal
guns. Typically we are talking about assaults, robberies and homicides. The
negative term with the parameter $\nu$ represents the decrease of crimes due
to the police’s work. Finally, the parameter $\lambda$ is the novelty of the
model, and it intends to take into account the rise of violent crimes due to
legal firearms, as above discussed. We are considering that civilians only buy
legal firearms ($y$). Thus, in this sense $y$ quantifies the civilian gun
ownership, since more legal guns mean more guns in civilians’ hands. In such a
case, the increase of civilian gun ownership ($y$) leads to the increase of
crimes ($x$), in addition to the crimes committed by illegal firearms ($z$).
This justifies the positive term $+\lambda\,y$ in Eq. (1). As a first approach
to the problem of commitment of violent crimes by the use of legal guns by
civilians, we considered the term as linear, but nonlinearities can also be
considered in future versions of the model. In other words, the legal
ownership of firearms by civilians leads to the commitment of crimes, for
example a traffic fight can lead to a homicide if at least one of the
civilians is a firearm owner, or the end of a marriage can lead the husband to
kill his wife (or the wife to kill her husband) for not accepting the end of
the relationship, if the husband/wife is a owner of a legal gun.

Looking now for Eq. (2), the parameter $\sigma$ represents the input of legal
firearms, that cannot depend on the present number of legal guns. The
parameter $\beta$ models the legal guns’ acquisition by the civil population,
due mainly to the risk perception. In other words, if the percepection of
violence is high, more guns are bought by civilians. Some governments have
programs to control the guns’ sales, like USA [18]. Thus, this gun control by
the State is modeled by the parameter $\theta$, that of course only act on the
legal firearms. Indeed, fear of crime is correlated with attitude toward gun
control [19]. All those terms are considered linear for simplicity. On the
other hand, the last term in Eq. (2) with the parameter $\gamma$ models the
situation in which a civilian with a legal gun’s possession is robbed, and
then that legal gun passes into the hands of an robber. In this way, this
weapon is no longer a legal firearm because it changed its owner [20]. This
term should be nonlinear since it models a kind of interaction among legal and
illegal firearms.

Finally, we have Eq. (3). The first term is related to the above-mentioned
conversion of legal firearms into illegal ones. The term $\gamma\,y\,z$ enters
in Eq. (2) as a negative term (decrease of legal firearms) and as a positive
term in Eq. (3) (increase of illegal firearms). Finally, the two last terms
with parameters $\delta$ and $\epsilon$ represent the increase of illegal
guns. The increase of weapons cannot by unlimited, which justifies the two
terms, one positive with parameter $\delta$ and other negative with
$\epsilon$, that leads to the maximum value $\delta/\epsilon$ [11]. The terms
can also be understood as follows. The positive term $+\delta\,z$ models in a
simple way (linear function) how the current illegal firearms can be used to
commitment of crimes such as assaults, robberies and homicides. These crimes
lead the criminals to acquire money, that can be used to buy more illegal
firearms. On the other hand, the negative term $-\epsilon\,z^{2}$ represents
the confiscation of illegal firearms in police’s actions, that is considered
nonlinear in order to garantee a sustainable amount of illegal arms.

From the above Equations (1) - (3), we took into account only $z$ presents a
saturation, according to the logistic form considered in Eq. (3). Of course
also the time evolution of variables $x$ and $y$ can be modeled considering
similar saturation terms. However, we are looking for a minimal model, i.e.,
the simplest form for the equations. In such a case, we are not considering
bounds for $x$ and $y$, only for $z$. However, we will see in the following
that the set of equations (1) - (3) leads to time evolutions that achieve
equilibrium values for $x,y$ and $z$. Notice that the proposed dynamical
system of Eqs. (1) - (3) can be the starting point for the development of more
complex models relating civilian gun ownership and the commitment of violent
crimes.

For simplicity, we did not considered the fear of a criminal encountering an
armed civilian. This kind of social interaction can be modeled by an extra
term $-\mu\,y\,z$ in Eq. (1), as investigated by Monteiro [11]. However, as
discussed in [21], in the case of USA, only about $1\%$ of the guns in private
hands are used for defensive purposes. In this case, the mentioned parameter
$\mu$ should be negligible.

##  3 Results

Let us start analyzing the time evolution of the previous defined quantities
$x,y$ and $z$. We numerically integrated the Eqs. (1) - (3), and the results
are exhibited in Figure 1. The parameters are given in the caption of the
figure. All parameters are rates, i.e., probabilities per unit time, and are
given in unities year-1. Specifically, as $\lambda$ is the new parameter of
the model, we fixed all the other parameters and we exhibit the time evolution
of $x,y$ and $z$ for two distinct values of $\lambda$, namely $\lambda=0.0$
and $\lambda=1.0$, in order to compare the model with crimes committed by
legal guns ($\lambda=1.0$) with the model of Ref. [11], where such crimes were
not considered ($\lambda=0.0$) 111For a detailed comparison of the models, we
need to take $\lambda=0.0$ in our model and $\mu=0.0$ in the model of Ref.
[11].. First of all, we can see that the quantities evolve in time and after a
transient the system reaches stationary states with stable values of $x,y$ and
$z$. These equilibrium values will be described analytically after an initial
analysis of the time evolution. From Figure 1 we can see that a nonzero value
of $\lambda$ has a great impact in the evolution of violent crimes $x(t)$.
First of all, we observe a fast increase of $x(t)$ for initial times for
$\lambda=1.0$. On the other hand, for $\lambda=0.0$, $x(t)$ decreases for
initial times and after it increases slowly and reaches the stationary state.
Numerically, we found for $\lambda=0.0$ the stationary value $x\approx
0.1643$, whereas for $\lambda=1.0$ we have the equilibrium value $x\approx
0.7431$. In other words, the consideration of a relatively small value as
$\lambda=1.0$ leads the equilibrium value of violent crimes per capita to
increase about $4.5$ times in comparison with the absence of violent crimes
committed by legal firearms [11]. This considerable difference is the first
impact of $\lambda$ in the model, in comparison with Ref. [11]. The other
quantities, $y$ and $z$, present a similar rate of increase for initial times,
but for $\lambda=1.0$ the stationary values are greater than the case
$\lambda=0.0$ [11]. Numerically, we found $(y,z)=(0.9945,0.6572)$ for
$\lambda=0.0$ and $(y,z)=(1.1249,0.7224)$ for $\lambda=1.0$.

Figure 1: (Color online) Illustration of the time evolution of the quantities
$x(t)$ (dotted lines), $y(t)$ (dashed lines) and $z(t)$ (full lines), for the
cases where crimes are committed by civilians that owner legal guns
($\lambda=1.0$, red curves) and where crimes are not committed by civilians,
only by illegal guns ($\lambda=0.0$, black curves). The parameters are
$\alpha=0.5,\beta=0.8,\gamma=2.5,\delta=0.8,\epsilon=5.0,\theta=0.5$ and
$\nu=\sigma=2.0$ (all parameters are given in unities year-1), and the initial
conditions are given by $x(0)=0.1,y(0)=0.5$ and $z(0)=0.2$.

As we observed in Figure 1, the quantities $x(t),y(t)$ and $z(t)$ evolve in
time, and after some time they stabilize. In such stationary states, the time
derivatives of Eqs. (1) - (3) are zero. In the $t\to\infty$ limit, Eq. (3)
gives us two solutions, namely $z=0$ (extinction of illegal firearms) or

$y=\frac{\epsilon\,z-\delta}{\gamma}~{}.$ (4)

If the solution $z=0$ is valid, Eq. (1) leads to

$y=(\nu/\lambda)\,x~{}.$ (5)

Again considering the solution $z=0$ and take into account the result (5), Eq.
(2) gives us

$x=\frac{\lambda\,\sigma}{\theta\,\nu-\lambda\,\beta}~{}.$ (6)

Substituting Eq. (6) in (5), we found

$y=\frac{\nu\,\sigma}{\theta\,\nu-\lambda\,\beta}~{}.$ (7)

This set of solutions given by Eqs. (6), (7) and $z=0$ represents the first
equilibrium point $(x_{1}^{*},y_{1}^{*},z_{1}^{*})$.

The second equilibrium point $(x_{2}^{*},y_{2}^{*},z_{2}^{*})$ is obtained
considering the solution $z\neq 0$. For such a case, we obtained above the
result (4) for $y$ such that $y=y(z)$. Taking the limit $t\to\infty$ in Eq.
(1) and considering the result (4), we obtain a second solution for $x$,
namely

$x=\frac{1}{\nu}\left(\alpha+\frac{\lambda\,\epsilon}{\gamma}\right)z-\frac{\lambda\,\delta}{\gamma\,\nu}~{}.$
(8)

In such a case, we have $x=x(z)$. Thus, Eqs. (8) and (4) give us the solutions
for the equilibrium values of $x$ and $y$ as functions of $z$, for the case
$z\neq 0$. This solution $z\neq 0$ can be obtained from Eq. (2) taking
$t\to\infty$. We obtain $\sigma+\beta\,x=(\gamma\,z+\theta)\,y$. Substituing
Eqs. (4) and (8) in this last result, we obtain a second order polynomial for
$z$, namely $c_{1}\,z^{2}+c_{2}\,z+c_{3}=0$, where

$\displaystyle c_{1}$ $\displaystyle=$ $\displaystyle\gamma\,\nu\,\epsilon$
(9) $\displaystyle c_{2}$ $\displaystyle=$
$\displaystyle(\epsilon\,\theta-\gamma\,\delta)\,\nu-(\alpha\,\gamma+\lambda\,\epsilon)\,\beta$
(10) $\displaystyle c_{3}$ $\displaystyle=$
$\displaystyle\beta\,\lambda\,\delta-(\gamma\,\sigma+\delta\,\theta)\,\nu$
(11)

In other words, we have
$z=\frac{-c_{2}+\sqrt{c_{2}^{2}-4c_{1}c_{3}}}{2c_{1}}$, that can be written as

$z=\frac{1}{2\gamma\nu\epsilon}\\{[(\alpha\gamma+\lambda\epsilon)\beta-(\epsilon\theta-\gamma\delta)\nu]+\sqrt{\Delta}\\}~{},$
(12)

where

$\Delta=[(\alpha\gamma+\lambda\epsilon)\beta-(\epsilon\theta-\gamma\delta)]^{2}-4\gamma\nu\epsilon[\beta\lambda\delta-(\gamma\sigma+\delta\theta)\nu]$
(13)

Eq. (12) completes the second equilibrium solution
$(x_{2}^{*},y_{2}^{*},z_{2}^{*})$. Summarizing, the 2 equilibrium solutions
are given by

$\displaystyle(x_{1}^{*},y_{1}^{*},z_{1}^{*})$ $\displaystyle=$
$\displaystyle\left(\frac{\lambda\,\sigma}{\theta\,\nu-\lambda\,\beta},\frac{\nu\,\sigma}{\theta\,\nu-\lambda\,\beta},0\right)$
(14) $\displaystyle(x_{2}^{*},y_{2}^{*},z_{2}^{*})$ $\displaystyle=$
$\displaystyle\left(\frac{1}{\nu}\left(\alpha+\frac{\lambda\,\epsilon}{\gamma}\right)z_{2}^{*}-\frac{\lambda\,\delta}{\gamma\,\nu},\frac{\epsilon\,z_{2}^{*}-\delta}{\gamma},z_{2}^{*}\right)$
(15)

In other words, the presence of the parameter $\lambda$, that models the
increase of violent crimes due to the presence of legal firearms in the
civilians’ hands, eliminates the crime-free equilibrium found in [11]. For our
solution 1, we observe that the equilibrium number of illegal firearms per
capita is zero ($z_{1}^{*}=0$) but the other quantities, namely the prevalence
of violent crimes per capita $x_{1}^{*}$ and the number of legal firearms per
capita $y_{1}^{*}$, are different from zero. Notice that if we take
$\lambda=0$ in solution $(x_{1}^{*},y_{1}^{*},z_{1}^{*})$ we recover the
result $(0,\frac{\sigma}{\theta},0)$, i.e., the crime-free equilibrium (since
$x_{1}^{*}=0$) analyzed in [11].

A simple stability analysis shows that the solution
$(x_{1}^{*},y_{1}^{*},z_{1}^{*})$ is unstable, whereas the another one
$(x_{2}^{*},y_{2}^{*},z_{2}^{*})$ is stable. Thus, let us analyze such
solution 2. We observe that $x_{2}^{*}$ grows proportional to $z_{2}^{*}$, as
in [11], but the parameter $\lambda$ amplifies such effect. Thus, more illegal
guns lead to more crimes, as observed in [11]. We can also see that as
$x_{2}^{*}$ grows with $\lambda$, i.e., the number of violent crimes increases
due to the presence of legal firearms in the civilians’ hands, since more
crimes can be committed by such legal guns (motivated by conflicting opinions
or prejudice, among other, as discussed above). This is a realistic result,
and it was not observed in the model of Ref. [11] since the author in [11] did
not considered crimes committed by legal guns. The police action also
decreases the number of crimes, since $x_{2}^{*}\propto 1/\nu$. In other
words, if the state wants to fight homicides, evidences show that removing
criminals from circulation with incarceration is the best option [22].

Figure 2: (Color online) Equilibrium values $x_{2}^{*}$ (dotted line),
$y_{2}^{*}$ (dashed line) and $z_{2}^{*})$ (dotted-dashed line) as functions
of $\lambda$. The lines are given by Eqs. (15) and (12). The parameters are
$\alpha=0.5,\beta=0.8,\gamma=2.5,\delta=0.8,\epsilon=5.0,\theta=0.5$ and
$\nu=\sigma=2.0$ (all parameters are given in unities year-1). In the inset we
show a zoom in the region of large $\lambda$, and the full (black) lines
exhibit the behaviors $x_{2}^{*}\propto\lambda^{2},y_{2}^{*}\propto\lambda$
and $z_{2}^{*}\propto\lambda$, as discussed in the text.

Regarding the mentioned dependence of the equilibrium solution
$(x_{2}^{*},y_{2}^{*},z_{2}^{*})$ on the new parameter $\lambda$, a simple
inspection of Eqs. (15) and (12) shows that for large $\lambda$ we have the
following dependencies: $z_{2}^{*}\propto\lambda,y_{2}^{*}\propto\lambda$ and
$x_{2}^{*}\propto\lambda^{2}$. Since $\lambda$ is the rate of increase of
violent crimes ($x$) due to the presence of legal guns in the civilian’ hands
($y$), the last results show that the civilian gun ownership leads to a fast
increase of violent crimes, despite the slower growth of legal and illegal
firearms. An illustration is given in Figure 2. We can see the fast growth of
violent crimes for increasing $\lambda$, and slower growth of both legal and
illegal firearms. Those results are in agreement with the previous
discussions. The inset in Figure 2 shows the same graphic for the larger
values of $\lambda$, exhibiting the mentioned behaviors
$z_{2}^{*}\propto\lambda,y_{2}^{*}\propto\lambda$ and
$x_{2}^{*}\propto\lambda^{2}$. We also plot in the inset for comparison curves
of the type $f(x)=ax^{2}+bx+c$ (for $x_{2}^{*}$) and $g(x)=ax+b$ (for
$y_{2}^{*}$ and $z_{2}^{*}$), to better visualization of the mentioned
quadratic and linear behaviors of the quantities of interest (see the full
lines).

We can also discuss about the control of firearms by government. For this
purpose, we can follow the strategy of Ref. [11] and consider two situations:

  * 1.

(a) a first situation where the government tightly controls arms’ sales. For
such a case, we should take the following limits for the model’s parameters:
$\theta\to\infty$, $\sigma\to 0$, $\beta\to 0$

  * 2.

(b) a second situation where there is no strong control of guns. For such a
case, we should take the following limits for the model’s parameters:
$\theta\to 0$, $\beta\to\infty$

Considering situation (a), the equilibrium point 2, Eq. (15), can be
approximated to

$(x_{2}^{*},y_{2}^{*},z_{2}^{*})\approx\left(\frac{\alpha\delta}{\nu\epsilon},0,\frac{\delta}{\epsilon}\right)$
(16)

that is the same result of Ref. [11]. It means that if the government really
control the sale of firearms, the legal guns can disappear of the population
after a long time, and they cannot contribute to violent crimes. If the
civilians do not own legal guns, the number of homicides can effectively
decreases. Considering such strategy (a), the violent crimes can be reduced if
the police action is increased in order to seize weapons. This is a difficult
point, of course. For example, firearms registrations increased $120\%$ in
2020 in Brazil, but seizures are down [23].

On the other hand, for the situation (b) we have:

$(x_{2}^{*},y_{2}^{*},z_{2}^{*})\approx\left(\frac{1}{\gamma\nu}\left[\frac{F^{2}\beta}{\gamma\nu\epsilon}-\lambda\delta\right],\frac{1}{\gamma^{2}\nu}[F\beta-\delta\gamma\nu],\frac{F\beta}{\gamma\nu\epsilon}\right)$
(17)

where

$F=\alpha\gamma+\lambda\epsilon$ (18)

For such second situation (b), one can see that the increase of police action
$\nu$ leads to the decrease of crimes $x_{2}^{*}$, as well as the decrease of
illegal firearms $z_{2}^{*}$. For rising $\lambda$, crimes, legal and illegal
guns increase, since the control of guns is weak. We also observe the same
behavior as before, i.e., $z_{2}^{*}\propto\lambda,y_{2}^{*}\propto\lambda$
and $x_{2}^{*}\propto\lambda^{2}$. In other words, the number of legal
firearms increases with $\beta$ and $\lambda$, which leads to $x_{2}^{*}\neq
0$. Comparing with Ref. [11], the author found $x_{2}^{*}=0$. For our model,
we see that the violent criminality cannot be eradicated in both situations
(a) and (b). Since even in this limit of weak gun control the quantities $x,y$
and $z$ depend on $\lambda$, it is clear that a substantial number of per
capita crimes will be committed by civilians if they own legal guns. This can
be an important result regarding firearms’ control [9, 10].

##  4 Final Remarks

In this work we study a simple model of violent crimes’ evolution. For this
purpose, we built a system of three ordinary differential equations for the
time evolution of three quantities of interest: violent crimes $x$, legal
firearms $y$ and illegal firearms $z$. Probabilities govern the evolution of
such equations.

The model deals with the key point of crimes committed by citizens that own
legal guns. As discussed in [24], firearms in the hands of the population
would increase the risk of any conflict or dispute ending in murder. Thus, the
mathematical modeling of crimes committed by civilian population by the use of
legal firearms is an important subject to be studied.

We found the following dependencies for the equilibrium point:
$z_{2}^{*}\propto\lambda,y_{2}^{*}\propto\lambda$ and
$x_{2}^{*}\propto\lambda^{2}$. As we considered that civilians only buy legal
firearms, $y_{2}^{*}$ represents the equilibrium point for the number of legal
firearms per capita, i.e., it quantifies the civilian gun ownership. Thus,
since $\lambda$ represents the rate of increase of violent crimes due to the
presence of legal guns, our results show that the increase of civilian gun
ownership leads to a fast increase of violent crimes, despite the slower grow
of legal and illegal firearms. Thus, more guns do not imply in less crimes,
but the opposite occurs. Indeed, regarding firearm incidents in American
schools, as observed by the author in [25], State gun ownership rates declined
between 1980 and 2019 while school firearm incidents generally ranged between
20 and 40 incidents before skyrocketing to 102 incidents in 2018 and 110
incidents in 2019. In addition, another recent study examinated the
association between gun ownership rates and homicide rates in the United
States from 1973 to 2016. The results show a very strong positive correlation
between gun ownership and homicide rates, which supports the hypothesis that
higher rates of gun ownership increase the likelihood of homicide [26]. The
authors in [26] found that, on average, a $1\%$ increase in gun ownership is
found to be associated with $2.6$ additional homicides per million people. In
agreement with [26], the authors in [27] found that gun ownership was a
significant predictor of firearm homicide rates: the results indicated that
for each percentage point increase in gun ownership, the firearm homicide rate
increased by $0.9\%$.

Our results show that a substantial number of per capita crimes will be
committed by civilians if they own legal guns. The model shows that the
violent criminality cannot be eradicated even considering strong and weak
controls of guns. This can turn the violence more complicated in countries
like Brazil, where the control of firearms adopted has not been shown to be
effective in fighting crime or in reducing violence, specially in the number
of homicides [28]. Thus, police action as well as stronger gun control is
mandatory to control the occurrence of violent crimes.

## Acknowledgments

The author acknowledges financial support from the Brazilian scientific
funding agencies Conselho Nacional de Desenvolvimento Científico e Tecnológico
(CNPq, Grant 310893/2020-8) and Fundação de Amparo à Pesquisa do Estado do Rio
de Janeiro (FAPERJ, Grant 203.217/2017).

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# A Note on General Statistics of Publicly Accessible
Knowledge Bases

Feixiang Wang, Yixiang Fang, Yan Song, Shuang Li, Xinyun Chen  The Chinese
University of Hong Kong, Shenzhen<EMAIL_ADDRESS>fangyixiang,
songyan, lishuang<EMAIL_ADDRESS>
###### Abstract.

Knowledge bases are prevalent in various domains and have been widely used in
a large number of real applications such as applications in online
encyclopedia, social media, biomedical fields, bibliographical networks. Due
to their great importance, knowledge bases have received much attention from
both the academia and industry community in recent years. In this paper, we
provide a summary of the general statistics of several open-source and
publicly accessible knowledge bases, ranging from the number of objects,
relations to the object types and the relation types. With such statistics,
this concise note can not only help researchers form a better and quick
understanding of existing publicly accessible knowledge bases, but can also
guide the general audience to use the resource effectively when they conduct
research with knowledge bases.

##  1\. Introduction

Knowledge bases are prevalent in various domains such as online encyclopedias
(e.g., DBpedia), social media (e.g., Twitter), biomedical fields (e.g.,
Geonames), and bibliographical networks (e.g., DBLP). Essentially, a knowledge
base is defined as a collection of interlinked descriptions of entities (e.g.,
real-world objects and persons) that enables storage, analysis, and reuse of
this knowledge in a machine-interpretable manner (Ehrlinger and Wöß, 2016).
With the fact that the knowledge bases describe various properties and
relationships between different facts, the complex relationships among
different entities are thus well preserved and represented.

Nowadays, knowledge bases have been widely used in a large number of real
applications. For example, Google uses the knowledge graph to enhance its
search engine. Specifically, by using the knowledge graph, Google search
engine can identify different meanings for the same word and summarize
relevant content of the word to provide better search results (e.g., the word
“Taj Mahal” may denote the monument or a musician, but Google search engine
can show their difference when searching the relevant information like “Taj
Mahal’s history”111https://blog.google/products/search/introducing-knowledge-
graph-things-not/). For another example, the knowledge graph in Siemens has
supported the knowledge-driven application across companies and generates new
knowledge (Hubauer et al., 2018).

Owing to their great importance, knowledge bases have received much attention
from both the academia and industry community. Very often, researchers need to
resort to publicly accessible knowledge bases to evaluate their methods.
Ideally, the knowledge bases need to offer insights to the devised models and
solutions, and meanwhile should be user friendly and enable researchers to
easily evaluate their methods. In light of this, in this paper, we provide a
note on general statistics of several publicly accessible knowledge bases,
which include their numbers of objects, relations, object types, relation
types, etc. Table 1 lists the publicly accessible knowledge bases that are
reviewed in this paper222The statistics data were extracted from the official
websites of the knowledge bases on June 28, 2021.. As shown in Table 1, we
group these knowledge bases into two categories, i.e., schema-rich and simple-
schema knowledge bases. Herein, the schema of a knowledge base refers to a
network describing all allowable relation types between entities types and
also the properties of entities (Thompson et al., 2001). For example, if an
entity is “Taylor Swif”, then the schema defines the type of this entity
(i.e., Person) and the properties of this entity in this type (i.e., her age,
job, country, etc.). Therefore, the schema-rich knowledge bases often refer to
the knowledge bases with a large number of object/relation types, while the
simple-schema knowledge bases only have a few object/relation types. We would
like to remark that apart from these, there are many other knowledge graphs
whose entire data are not publicly available. For example, for the enterprise
knowledge bases (e.g., Twitter, Flickr, and Google), only a small proportion
of their data are accessible. For these knowledge bases, we will not cover in
this paper.

Note that, Table 1 only includes several publicly accessible knowledge bases
according to the popularity in academic research. We will continue to update
this table by including more open-source knowledge bases in the future.

11footnotetext: https://blog.google/products/search/introducing-knowledge-graph-things-not/ Table 1. Publicly accessible knowledge bases reviewed in this paper. Name | Schema type | URL | Major object types | Objects/entities | Initially released
---|---|---|---|---|---
Yago | Schema-rich | https://yago-knowledge.org/ | 10K | 67M | 2008
DBpedia | Schema-rich | https://www.dbpedia.org/ | 484k | 6 M | 2007
Wikidata | Schema-rich | https://www.wikidata.org/ | 24M | 93 M | 2012
KBpedia | Schema-rich | https://kbpedia.org/ | 173 (ontology) | 40M | 2016
DBLP | Simple Schema | https://dblp.org/ | 4 | 8M | 1993
IMDb | Simple Schema | https://www.imdb.com/ | 7 | 186M | 1990
Geonames | Simple Schema | https://www.geonames.org/ | 9 | 12M | 2005

##  2\. General Statistics of Publicly Accessible Knowledge Bases

We divide the general statistics of the publicly accessible knowledge bases
into two parts. The first part contains the common statistics of all knowledge
bases, while the second part includes specific statistics. For the first part,
we mainly show the schema type, URL, initially released time, and the numbers
of objects/entities, relations, and object types of each knowledge base. Table
1 summarizes the common statistics of these knowledge bases, where all the
statistic numbers were collected from the official websites in June 2021 and
they may change as time goes on. Note that for some knowledge bases, the
entity types are hierarchically organized, and here we mainly count the number
of object types in high levels which is denoted by “Major object types” in
Table 1. The second part covers the specific statistics for each knowledge
base. Since various knowledge bases have different structures, this leads to
specified statistics. Besides, the same concepts may have different meanings
in each knowledge base. For example, in Table 1, the concept of entity types
in Yago means the types for all the entities and ontology classes, while it
refers to the type of entities for DBLP. Therefore, apart from the common
information, one needs to further elaborate on their specific statistics.

In the following, we mainly focus on introducing the specific statistics of
each publicly accessible knowledge base. Note that this paper will not
extensively discuss the specific statistics of Wikidata, since many other
knowledge bases (e.g., DBpedia, Yago, and KBpedia) are built based on this
data.

###  2.1. Schema-rich Knowledge Bases

In general, schema-rich knowledge bases refer to the knowledge bases
consisting of entities with a large number of types. For example, Yago covers
the information of various areas such as people, cities, countries, movies,
and organizations (Tanon et al., 2020), making it a knowledge base with
various types of objects.

####  2.1.1. DBpedia

DBpedia was created by scientists at the Free University of Berlin and Leipzig
University in collaboration with OpenLink Software(Auer et al., 2007). Now
DBpedia is maintained by scientists at the University of Mannheim and Leipzig
University.

Figure 1. “Berlin” recorded in DBpedia.

DBpedia mines data from “Wikipedia”, “DBpedia common”, and “Wikidata”, and
updates the data around the 15th of every month, allowing users to find
answers to questions where the information is spread across multiple Wikipedia
articles. For example, the word “Berlin” has many different meanings like
Berlin, West-Berlin, and Ost-Berlin. By querying DBpedia with the keyword
“Berlin”, it will return descriptions about Berlin, West-Berlin, and Ost-
Berlin, even if you only search Berlin. It will also return many properties
and relationships to other entities, e.g., the information of universities and
famous people in Berlin in history. In Figure
1333https://dbpedia.org/page/Berlin, we choose some properties of Berlin and
relationships to other entities. For example, “Berlin” has property “areaCode”
and its value is “030”. “Berlin” has a relationship to “German” and the
relationship is “country”.

The specific statistics of DBpedia are presented in Table
2444https://www.dbpedia.org/blog/yeah-we-did-it-again-new-2016-04-dbpedia-
release/ and Table 3555https://www.dbpedia.org/blog/yeah-we-did-it-again-
new-2016-04-dbpedia-release/. Object property defines the relationship between
one entity and another entity. Datatype property defines the relationship
between one object and one of its values. For example, “mother” is an object
property because mother is an individual as an entity, while “age” is a
datatype property because a human’s age is a value belonging to this human,
not an entity. Languages symbolize the languages number knowledge base
supports. Mappings define the mappings from Wikidata templates to the DBpedia
classes. The mappings are classified by language. For example, in these 5800
mappings, there are 646 mappings are used in Dutch. Table 3 shows the
distribution of ontology entities of DBpedia. Ontology classes mean the number
of classes in ontology. Some classes are not in ontology but in knowledge
bases. There are 5.2M entities in ontology classes. The classes having the
largest size are Persons, Places, Works, Organizations, Species, and Diseases.
In Table 3, we show the numbers of entities of each class having a large size.

Table 2. Specific statistics of DBpedia. | Object
---
properties
| Datatype
---
properties
Languages | Mappings
1105 | 1754 | 136 | 5800
Table 3. The distribution of ontology entity of DBpedia. Item | Number
---|---
Ontology classes | 754
Entities in ontology classes | 5.2M
Persons | 1.5M
Places | 810k
Works | 490k
Organizations | 275k
Species | 301k
Diseases | 5k
Relationships/facts | 9.5B

####  2.1.2. Yago

Yago is an open-source knowledge base developed at the Max Planck Institute
for Computer Science in Saarbrücken. It is a knowledge graph containing
knowledge about people, cities, countries, movies, and organizations. Each
entity in Yago belongs to at least one class. The high classes in Yago are
taken from schema.org and bioschemas.org, and the lower classes are a
selection of classes from Wikidata. Yago has been used in the Watson
artificial intelligence system (Ferrucci et al., 2010).

Yago extracted facts/relationships from Wikidata and Wikipedia, cleaned these
data, and finally organized the cleaned data in the structures defined by the
anthologies of “Schema.org” and “Biosche-ma.org”, which define the number and
the kind of classes, properties, and relationships. The facts/relationships
that are not fitted in these classes were ignored. Yago used these classes as
high-level classes and further built the low-level classes by extracting
classes from Wikidata to make the whole ontology more clear. If Wikipedia
articles lack some entities’ classes and properties, Yago will check the
labels in Wikipedia. If Wikipedia cannot satisfy the need, Yago will further
extract data from Wikidata (Tanon et al., 2020).

Figure 2. Using Yago to search “Taylor Swift”.

Figure 2666https://yago-knowledge.org/graph/yago:Taylor_Swift shows an example
of the information of the famous singer “Taylor$\\_$Swift” in Yago. This graph
only shows a few of relationships and properties belonging to
“Taylor$\\_$Swift”. In terms of top classes, Taylor Swift is a singer, so it
belongs to the class “Human”, which is a subclass of class “Person” . Note
that “Person” is a subclass of the highest classes “Thing” from Schema.org.
For properties of “Taylor$\\_$Swift”, the symbol “(+x)” after every properties
or entities is the number of relationships in this relationship or property.
In Figure 2, we see “rdfs:label” and “rdfs:comment”, which are datatype
properties of “Taylor$\\_$Swift”. “Rdfs:label” means her different names in
different languages, e.g., Chinese name. In Figure 2, “schema:X” symbolizes a
relationship between entity“Taylor$\\_$Swift” and other entities where this X
refers to different contents, e.g., “schema:nationality(schema:national…)”
shows that the country of “Taylor$\\_$Swift” is another entity “United
States”. “Owl:sameAs” is a link to other knowledge bases. In this example,
“wd:Q26875” symbolizes “Taylor$\\_$Swift” in Wikidata, and “(+2)” means that
there are 1+2=3 connections to other knowledge bases. Figure 2 only shows a
part of relationships and properties about entity “Taylor Swif”. Figure 3 is
the page showing all related information about entity “Taylor Swift”.

Figure 3. Using Yago to search “Taylor Swift”.

Table 4777https://yago-knowledge.org/resource/yago:Taylor_Swift presents
specific statistics of Yago. Here, the domain constraints mean some
relationships only apply to some classes. For instance, “birthPlace” can only
apply to a person and a place. Object range constraints and Datatype range
constraints are similar to Domain constraints in function. Four of six major
top-level classes (i.e., “schema:BioChemicalEntity”, “schema:Event”,
“schema:Person”, and “schema:CreativeWork”) are pairwise disjoint, which
implies that these classes cannot have any instances in common. As a result,
the disjoint constraints can make these instances different(Tanon et al.,
2020). Note that these statistics are collected from the latest version of the
Yago, called Yago 4, which is an RDFS knowledge base released in 2020.

Table 4. Yago ontology statistics (Tanon et al., 2020). Item | Number
---|---
Schema.org classes | 235
Bioschema.org classes | 6
Object Properties | 100
Datatype properties | 41
Node shapes | 49
Property shapes | 247
Domain constraints | 247
Object range constraints | 132
Datatype range constraints | 57
Regex constraints | 21
Disjoint constraints | 18
Relationships/Facts | 343M

####  2.1.3. KBpedia

The co-editors of KBpedia are Mike Bergman and Frédérick Giasson. Partial
sponsorship of KBpedia has been provided by Cognonto Corporation. KBpedia was
constructed by combining seven core knowledge bases, i.e., Wikipedia,
Wikidata, schema.org, DBpedia, GeoNames, OpenCyc, and standard UNSPSC products
and services. Until now, KBpedia has covered 98$\%$ Wikidata and nearly
completely covers Wikipedia.

The upper structure of the KBpedia Knowledge Ontology (KKO) is informed by the
triadic logic and universal categories of Charles Sanders Peirce. There are
three categories in Peirce’s trichotomy. The first category is about
possibilities or potentials, the basic forces or qualities that combine or
interact in various ways to enable the real things we perceive in the world,
such as matter, life and ideas. The second category is about particular
realized things or concepts we can describe such as an event, individual,
entity. The third category is about laws, habits, regularities and
continuities that may be generalized from particulars or possibilities. All
generals — what are also known as classes, kinds or types — belong to this
category888https://kbpedia.org/docs/kko-upper-structure/.

Now, we use an example to introduce KBpedia structure. We use In KBpedia, each
record has eight sections. The eight sections are “Header”, “Core Structure”,
“Extended Linkages”, “Typologies”, “Entities”, “Aspect-related Entities”,
“Broader Concepts”, and“Narrower Concepts”. “Header” contains the alternative
names, definition of “Currency” and image about this class (concept) if it
has. The core structure contains sub-classes, super-classes and equivalent
classes. Each item in this class is a mapping to the class in core knowledge
bases. “Extended Linkages” is similar to “Core structure” but is connecting to
more knowledge bases. Core knowledge bases are OpenCyc, UMBEL, GeoNames,
DBpedia, Wikipedia and Wikidata. “Entities” is the same to the concept
“Entities” in other knowledge bases. This item contains many entities and each
entity has its properties and relationships with other entities. “Aspect-
related Entities” are the entities related to class “Currency” but it provides
a classifications for entities by situation for example related entities in
financial. “Typologies” contain the core and extended typologies (classes)
that the class “Currency” belongs to. “Typologies” concept is similar to
“classes”. KBpedia has 30 core typologies in total such as “visual
information”. Extended typologies indicate typologies that are not included in
core typologies. “Broader Concepts” and “Narrow Concepts” correspond to super-
classes and sub-classes. 999https://kbpedia.org/use-cases/browse-the-
knowledge-graph/

Table 5101010https://kbpedia.org/resources/statistics/ shows the ontology
statistics of KBpedia. Ontology classes indicate the number of classes in
ontology. Mapped vocabularies mean the number of mappings to those 7 knowledge
bases and 16 extended vocabularies. Properties mean the number of
relationships between different entities or entities and its properties.

Table 5. Some statistics of KBpedia ontology. | Number
---|---
ontology classes | 173
Number of mapped vocabularies | 23
No. of properties(Sum) | 5004
mappings to Wikidata | 3970
mappings to schema.org | 877

###  2.2. Simple-schema Knowledge Bases

Simple-schema knowledge bases are often designed for describing the
information of objects in one particular area, so the numbers of object types
and relation types are often very limited.

####  2.2.1. DBLP

As a computer science bibliography website, the DBLP was created in 1993 at
Universität Trier in Germany. DBLP grew and became an organization hosting a
programming bibliography database site from a collection of HTML files (Ley,
2009). DBLP became a branch of Schloss Dagstuhl – Leibniz-Zentrum für
Informatik (LZI) in Novemberm 2018.

The typical types of objects in DBLP are publications, authors, conferences,
journals, research topics, etc. Figure
4111111https://dblp.org/search?q=hyperdoc2vec shows an example of a research
paper record in DBLP, which includes five authors, paper title, conference
name, published year, and page number. Table
6121212https://dblp.org/statistics/index.html and Table
7131313https://dblp.org/statistics/distributionofpublicationtype.html report
some specific statistics of
DBLP.141414https://dblp.org/statistics/distributionofpublicationtype.html
Table 6 shows 4 top classes and the number of entities belong to each of them.
For publications, DBLP has more detailed classification as shown in Table 7.

Table 6. The numbers of objects in each type in DBLP. Publications | Authors | Conferences | Journals
---|---|---|---
5633863 | 2773696 | 5438 | 1764
Table 7. DBLP articles distribution. Type | Percentage
---|---
References work | 0.48$\%$
Books and thesis | 1.74$\%$
Parts in Book or Collections | 0.70$\%$
Journal articles | 39.19$\%$
Informal publications | 6.84$\%$
Editorship | 0.88$\%$
Conference and workshop papers | 50.17$\%$
Figure 4. An example research paper in DBLP.

In Figure 4, we present an example of the article titled “Dynamic Mosaics”,
which was written by Rahul Garg and Steven M, Seitz. This article was
published in conference 3DIMPVT 2012 and its page number is 65-72. Note that
if this article was published in another venue like a journal, then the
conference name would be replaced by the name of the venue.

####  2.2.2. IMDb

IMDb is an online database containing information related to films, television
programs, home videos, video games, and streaming content online. IMDb was
established in 1990 as a movie database on the Usenet group “rec.arts.movies”.
It moved to the web in 1993. Now, IMDb is owned and operated by Amazon.

Table 8151515https://www.imdb.com/pressroom/stats/ shows the specific
statistics for IMDb. IMDb mainly consist of seven types of entities, i.e.,
titles, genres, filmography, names, images, and videos. More precisely, the
titles contain plot outline, literature, and keyword; The genres contain
drama, talk show, and romance; The names contain article, birth name, etc.;
The filmography contains different people in different works such as actors,
actresses, producers, writers, etc.; Images contain images, publicity images,
posters, etc.; Videos contain trailers, clips, feature films, etc.

To illustrate the data, we show a part of the information for the record of
the singer “Taylor Swift” in Figure
5161616https://www.imdb.com/find?q=Taylor+Swift&ref_=nv_sr_sm. Here, “Taylor
Swift” is an entity, and the introduction of “Taylor Swift” is a property of
“Taylor Swift”. The relationship could be considered as “rdfs:comments” in
Yago. Entity “Taylor Swift” has other properties and relationships. In the
complete record of “Taylor Swift”, the entity has three relationships to other
entities, i.e., “Photos”, “Known For”, “Filmography”. For example, each photo
is an entity and every single photo has properties like titles or some other
properties.

Table 8. Some specific statistics of IMDb. Item | Number | subclass types
---|---|---
Titles | 7766299 | 19
Genres | 12951717 | 28
Filmography | 142530261 | 32
Names | 10827285 | 12
Images | 11856447 | 8
Videos | 471449 | 7
Figure 5. An example of search result in IMDb.

####  2.2.3. GeoNames

GeoNames is a user-editable geographical knowledge graph founded in 2005.
GeoNames contains over 25,000,000 geographical names corresponding to over
11,800,000 unique features. Geonames is editable for everyone. In other words,
if some false information entered the knowledge base and the information is
not accessed frequently, it is difficult to find and fix it(Ahlers, 2013),
which is one of the drawbacks of Geonames. However, according to some studies,
there is a way to adjust such information in the Geonames (Singh and Rafiei,
2018).

Geonames has nine feature classes and 645 feature codes. All coordinates in
Geonames use the World Geodetic System 1984 (WGS84). Each feature in Geonames
has its corresponding URI, which provides access to HTML wiki page, or a RDF
description of the feature through content negotiation, using elements of the
GeoNames ontology. This ontology describes the GeoNames features properties
using the Web Ontology Language (OWL), the feature classes and codes being
described in the SKOS language. GeoNames data are linked to DBpedia data and
other RDF Linked Data through Wikipedia url in RDF descriptions.

Figure 6. Example of “New York City” in GeoNames.

We use the entity of “New York City” as an example to show the data of
GeoNames. As shown in Figure 6171717https://www.geonames.org/5128581/new-york-
city.html, the population is a property of “New York City” and its value is
8175133. Its first order administrative division is “New York” and its country
is “United States”, which are entities in GeoNames.

Table 9181818https://www.geonames.org/statistics/total.html presents some
specific statistics data for GeoNames. Specifically, Geonames has 9 top
classes (Maltese and Farazi, 2013): Administrative Boundary Features describes
the ways as how to define a complete Administrative boundary and its levels;
Hydrographic Features define the classes of hydrographic items like “lake”,
“well”, and “stream”; Area Features contain “park”, “locality”, etc.;
Populated Place Features describe the features in area with many people such
as populated locality and religious populated place; Road/Railroad Features
contain “road junction”; Spot features contain specified spot like “farm” and
“school”; Hypsographic Features contain “mountain”, “hill”, “island”, etc.;
Undersea Features contains “shoals”, “canyon”, etc.; Vegetation Features
contains “forest”, “grove”, etc. A more detailed introduction of these
statistics could be found from (Maltese and Farazi, 2013).

Table 9. Some specific statistics of GeoNames. Item | Number | Subfeatures Types
---|---|---
Administrative Boundary Features | 452363 | 25
Hydrographic Features | 2236741 | 135
Area Features | 417387 | 49
Populated Place Features | 4793777 | 19
Road/Railroad Features | 52112 | 21
Spot Features | 2440626 | 254
Hypsographic Features | 1598944 | 99
Undersea Features | 14899 | 63
Vegetation Features | 48709 | 18
Undefined | 5058 |

##  3\. Conclusions and Future Work

In this paper, we present the general statistics of several knowledge bases,
which are publicly accessible and receive much attention from both academic
area and industry area. In particular, we divide the knowledge graphs into
schema-rich knowledge bases and simple-schema knowledge bases. We introduce
their common statistics and also the specific statistics of each knowledge
base. We hope that by providing such statistics, the paper can make
researchers have a better understanding of existing open-source knowledge
bases.

In the future, we will continue to update the statistics of popular publicly
accessible knowledge bases. On the one hand, the existing knowledge bases are
growing as new information is included. On the other hand, more knowledge
bases will be released and made publicly accessible. Thus, we will keep
updating this paper continuously to ensure the statistics data in this paper
could provide up-to-date information.

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# A sharp boundedness result for restricted maximal operators of Vilenkin-
Fourier series on martingale Hardy spaces

I. Blahota, K. Nagy, L.E. Persson and G. Tephnadze  I. Blahota, Institute of
Mathematics and Computer Sciences, College of Nyíregyháza, P.O. Box 166,
Nyíregyháza, H-4400, Hungary.  K. Nagy, Institute of Mathematics and Computer
Sciences, College of Nyíregyháza, P.O. Box 166, Nyíregyháza, H-4400 Hungary,
L.E. Persson, Department of Engineering Sciences and Mathematics, Luleå
University of Technology, SE-971 87 Luleå, Sweden and UiT The Arctic
University of Norway, P.O. Box 385, N-8505, Narvik, Norway.  G. Tephnadze,
Department of Mathematics, Faculty of Exact and Natural Sciences, Ivane
Javakhishvili Tbilisi State University, Chavchavadze str. 1, Tbilisi 0128,
Georgia, & University of Georgia, IV, 77a Merab Kostava St, Tbilisi, 0128,
Georgia, & Department of Engineering Sciences and Mathematics, LuleåUniversity
of Technology, SE-971 87 Luleå, Sweden.

###### Abstract.

The restricted maximal operators of partial sums with respect to bounded
Vilenkin systems are investigated. We derive the maximal subspace of positive
numbers, for which this operator is bounded from the Hardy space $H_{p}$ to
the Lebesgue space $L_{p}$ for all $0<p\leq 1.$ We also prove that the result
is sharp in a particular sense.

Research was supported by project TÁMOP-4.2.2.A-11/1/KONV-2012-0051, by Shota
Rustaveli National Science Foundation grants DI/9/5-100/13, DO/24/5-100/14,
YS15-2.1.1-47 and by a Swedish Institute scholarship no. 10374-2015.

2010 Mathematics Subject Classification. 42C10, 42B25.

Key words and phrases: Vilenkin system, partial sums, maximal operator,
Vilenkin-Fourier series, martingale Hardy space.

##  1\.  Introduction

Pointwise convergence problems are of fundamental importance in Fourier
analysis, and as it is well known they are closely related to studying
boundedness of associated maximal operators. In the present paper we will deal
with maximal operators. Let us first recall in brief a historical development
of this theory.

It is well-known (for details see e.g. [2] and [11]) that Vilenkin systems do
not form bases in the space $L_{1}\left(G_{m}\right).$ Moreover, (for details
see e.g. [18, 19]) there is a function in the martingale Hardy space
$H_{1}\left(G_{m}\right),$ such that the partial sums of $f$ are not bounded
in $L_{1}\left(G_{m}\right)$-norm, but Watari [17] (see also Gosselin [10] and
Young [20]) proved that there exist absolute constants $c$ and $c_{p}$ such
that, for $n=1,2,...,$

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

In [14] it was proved that there exists a martingale $f\in
H_{p}\left(G_{m}\right)\left(0<p<1\right),$ such that

$\underset{n\in\mathbb{N}}{\sup}\left\|S_{M_{n}+1}f\right\|_{L_{p,\infty}}=\infty.$

The reason of divergence of $S_{M_{n}+1}f$ is that the Fourier coefficients of
$f\in H_{p}\left(G_{m}\right)$ ($0<p<1$) are not bounded (see Tephnadze [13]).

Uniform and point-wise convergence and some approximation properties of the
partial sums in $L_{1}\left(G_{m}\right)$ norms were investigated by Goginava
[8, 9] and Avdispahić, Memić [1]. Some related results can also be found in
the recent PhD thesis by Tephnadze [15]. Moreover, Fine [4] obtained
sufficient condition for the uniform convergence it is in complete analogy
with the Dini-Lipschitz condition. Guličev [12] estimated the rate of uniform
convergence of a Walsh-Fourier series using Lebesgue constants and modulus of
continuity. Uniform convergence of subsequences of partial sums was also
investigated by Goginava and Tkebuchava [7]. This problem was considered for
the Vilenkin group $G_{m}$ by Fridli [5], Blahota [3] and Gát [6].

In [14] the following maximal operator was considered:

$\widetilde{S}_{p}^{\ast}f:=\sup_{n\in\mathbb{N}}\frac{\left|S_{n}f\right|}{\left(n+1\right)^{1/p-1}\log^{\left[p\right]}\left(n+1\right)},\quad
0<p\leq 1$

(where $\left[x\right]$ denotes integer part of $x$). It was proved that the
maximal operator $\widetilde{S}_{p}^{\ast}$ is bounded from the Hardy space
$H_{p}\left(G_{m}\right)$ to the space $L_{p}\left(G_{m}\right).$ Moreover, if
$0<p\leq 1$ and $\varphi:\mathbb{N}_{+}\rightarrow[1,\infty)$ is a non-
decreasing function satisfying the condition

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

then

$\sup_{n\in\mathbb{N}}\left\|\frac{S_{n}f}{\varphi\left(n\right)}\right\|_{L_{p,\infty}\left(G_{m}\right)}=\infty,\text{
for }0<p<1,$

and

$\sup_{n\in\mathbb{N}}\left\|\frac{S_{n}f}{\varphi\left(n\right)}\right\|_{1}=\infty.$

It is also known (for details see e.g. Weisz [19]) that

$\left\|S_{n_{k}}f\right\|_{1}\leq c\left\|f\right\|_{1}$

holds if and only if

$\sup_{k\in\mathbb{N}}\left\|D_{n_{k}}\right\|_{1}<c<\infty,$

where $D_{n_{k}}$ denotes the $n_{k}$th Dirichlet kernel with respect to
Vilenkin system. Moreover, the corresponding subsequence $S_{n_{k}}$ of the
partial sums $S_{{n}}$ are bounded from the Hardy space
$H_{p}\left(G_{m}\right)$ to the Hardy space $H_{p}\left(G_{m}\right),$ for
all $p>0.$

It is also well-known (for details see e.g. Weisz [19] and Tephnadze [15])
that the following restricted maximal operator

${S}^{\\#}f:=\sup_{n\in\mathbb{N}}\left|S_{M_{n}}f\right|$

is bounded from the martingale Hardy space $H_{p}\left(G_{m}\right)$ to the
Lebesgue space $L_{p}\left(G_{m}\right),$ for all $p>0$.

In this paper we find the maximal subspace of positive numbers, for which the
restricted maximal operator of partial sums with respect to Vilenkin systems
in this subspace is bounded from the Hardy space $H_{p}$ to the Lebesgue space
$L_{p}$ for all $0<p\leq 1.$ As applications, both some well-known and new
results are pointed out.

The paper is organized as follows: Some Preliminaries (definitions, notations
and basic facts) are presented in Section 2. The main result (Theorem 1) and
some of its consequences (Corollaries 1-5) are presented and discussed in
Section 3\. Theorem 1 is proved in Section 5. For this proof we need some
Lemmas, one of them is new and of independent interest (see Section 4).

##  2\. Preliminaries

Let $\mathbb{N}_{+}$ denote the set of the positive integers,
$\mathbb{N}:=\mathbb{N}_{+}\cup\\{0\\}$ and assume that
$m:=(m_{0},m_{1},\ldots)$ is a sequence of positive integers not less than 2.

Denote by

$Z_{m_{k}}:=\\{0,1,\ldots,m_{k}-1\\}$

the additive group of integers modulo $m_{k}.$

Define the group $G_{m}$ as the complete direct product of the group
$Z_{m_{k}}$ with the product of the discrete topologies of $Z_{m_{k}}$‘s.

The product measure $\mu$ of the measures

$\mu_{k}\left(\\{j\\}\right):=1/m_{k}\quad(j\in Z_{m_{k}})$

is a Haar measure on $G_{m}$ with $\mu\left(G_{m}\right)=1$.

If the sequence $m:=(m_{0},m_{1},\ldots)$ is bounded, then $G_{m}$ is called a
bounded Vilenkin group, otherwise it is called an unbounded one. In the
present paper we deal only with bounded Vilenkin groups.

The elements of $G_{m}$ are represented by sequences

$x:=(x_{0},x_{1},\ldots,x_{k},\ldots)\quad\left(x_{k}\in Z_{m_{k}}\right).$

A base for the neighbourhood of $G_{m}$ can be given as follows:

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

Denote $I_{n}:=I_{n}\left(0\right)$ for $n\in\mathbb{N}$ and
$\overline{I_{n}}:=G_{m}\backslash I_{n}$.

It is evident that

(1) $\overline{I_{N}}=\overset{N-1}{\underset{s=0}{\bigcup}}I_{s}\backslash
I_{s+1}.$

The generalized number system based on $m$ is defined in the following way

$M_{0}:=1,{\ \quad}M_{k+1}:=m_{k}M_{k}\quad(k\in\mathbb{N}),$

Every $n\in\mathbb{N}$ can be uniquely expressed as

$n=\sum_{j=0}^{\infty}n_{j}M_{j},\quad\text{ where }\quad n_{j}\in Z_{m_{j}}\
(j\in\mathbb{N})$

and only a finite number of $n_{j}$‘s differ from zero.

Let

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

that is $M_{\left|n\right|}\leq n\leq M_{\left|n\right|+1}.$ Set

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

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

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

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

$\left\|f\right\|_{L_{p},\infty}:=\sup_{\lambda>0}\lambda\mu\left(f>\lambda\right)^{1/p}<+\infty.$

Next, we introduce on $G_{m}$ an orthonormal system which is called Vilenkin
system.

First, we define the complex valued function $r_{k}\left(x\right)\colon
G_{m}\rightarrow\mathbb{C},$ the generalized Rademacher functions as

$r_{k}\left(x\right):=\exp\left(2\pi\imath
x_{k}/m_{k}\right)\quad\left(\imath^{2}=-1,\ x\in G_{m},\
k\in\mathbb{N}\right).$

Let us define the Vilenkin system $\psi:=(\psi_{n}:n\in\mathbb{N})$ on $G_{m}$
as:

$\psi_{n}(x):=\prod_{k=0}^{\infty}r_{k}^{n_{k}}\left(x\right)\quad\left(n\in\mathbb{N}\right).$

Specifically, we call this system the Walsh-Paley one if $m\equiv 2$.

The Vilenkin system is orthonormal and complete in $L_{2}\left(G_{m}\right)$
(see e.g. [2, 16]).

Now, we present the usual definitions in Fourier analysis.

If $f\in L_{1}\left(G_{m}\right)$ we can establish Fourier coefficients, the
partial sums of Fourier series, Dirichlet kernels with respect to the Vilenkin
system in the usual manner:

$\displaystyle\widehat{f}\left(k\right)$ $\displaystyle:=$
$\displaystyle\int_{G_{m}}f\overline{\psi}_{k}d\mu,\quad\left(k\in\mathbb{N}\right),$
$\displaystyle S_{n}f$ $\displaystyle:=$
$\displaystyle\sum_{k=0}^{n-1}\widehat{f}\left(k\right)\psi_{k},\quad\left(n\in\mathbb{N}_{+},\
S_{0}f:=0\right),$ $\displaystyle D_{n}$ $\displaystyle:=$
$\displaystyle\sum_{k=0}^{n-1}\psi_{k},\quad\left(n\in\mathbb{N}_{+}\right).$

Recall that (see [2])

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

and

(3)
$D_{n}\left(x\right)=\psi_{n}(x)\left(\underset{j=0}{\overset{\infty}{\sum}}D_{M_{j}}\left(x\right)\overset{m_{j}-1}{\underset{u=m_{j}-n_{j}}{\sum}}r_{j}^{u}\left(x\right)\right).$

The $\sigma$-algebra generated by the intervals
$\left\\{I_{n}\left(x\right):x\in G_{m}\right\\}$ will be denoted by
$\digamma_{n}$ $\left(n\in\mathbb{N}\right)$. Let us denote a martingale with
respect to $\digamma_{n}\left(n\in\mathbb{N}\right)$ by
$f=\left(f_{n}:n\in\mathbb{N}\right)$ (for details see e.g. [18]). The maximal
function of a martingale $f$ is defined by

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

In the case $f\in L_{1}\left(G_{m}\right),$ the maximal function is also given
by

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

For $0<p<\infty$ the Hardy martingale spaces $H_{p}\left(G_{m}\right)$ consist
of all martingales, for which

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

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

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

The Hardy martingale spaces $H_{p}\left(G_{m}\right)$ have an atomic
characterization for $0<p\leq 1$. In fact the following theorem is true (see
e.g. Weisz [18, 19]):

######  Theorem W.

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

(4) $\sum_{k=0}^{\infty}\mu_{k}S_{M_{n}}a_{k}=f_{n}$

and

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

Moreover,

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

where the infimum is taken over all decomposition of $f$ of the form (4).

If $f\in L_{1}\left(G_{m}\right),$ then it is easily shown that the sequence
$\left(S_{M_{n}}f:n\in\mathbb{N}\right)$ is a martingale.

If $f=\left(f_{n},n\in\mathbb{N}\right)$ is a martingale, then Vilenkin-
Fourier coefficients are defined in a slightly different manner:

$\widehat{f}\left(i\right):=\lim_{k\rightarrow\infty}\int_{G_{m}}f_{k}\left(x\right)\overline{\psi}_{i}\left(x\right)d\mu\left(x\right).$

Vilenkin-Fourier coefficients of $f\in L_{1}\left(G_{m}\right)$ are the same
as the martingale $\left(S_{M_{n}}f:n\in\mathbb{N}\right)$ obtained from $f$.

##  3\. The Main Result

Our main theorem reads as follows:

######  Theorem 1.

a) Let $0<p\leq 1$ and $\left\\{\alpha_{k}:k\in\mathbb{N}\right\\}$ be a
subsequence of positive natural numbers, such that

(5) $\sup_{k\in\mathbb{N}}\rho\left(\alpha_{k}\right)=:\varkappa<\infty.$

Then the maximal operator

$\widetilde{S}^{\ast,\vartriangle}f:=\sup_{k\in\mathbb{N}}\left|S_{\alpha_{k}}f\right|$

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

b) Let $0<p<1$ and $\left\\{\alpha_{k}:k\in\mathbb{N}\right\\}$ be a
subsequence of positive natural numbers satisfying the condition

(6) $\sup_{k\in\mathbb{N}}\rho\left(\alpha_{k}\right)=\infty.$

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

$\sup_{k\in\mathbb{N}}\left\|S_{\alpha_{k}}f\right\|_{L_{p,\infty}}=\infty.$

######  Remark 1.

Since $L_{p}\subset L_{p,\infty}$ part b) means in particular that the
statement in part a) is sharp in a special sense for the case $0<p<1.$

We also mention the following well-known consequences (for details see e.g.
the books [18, 19] and [14]):

######  Corollary 1 (Tephnadze [14]).

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

$\sup_{n\in\mathbb{N_{+}}}\left|S_{M_{n}+1}f\right|$

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

In fact, we only have to notice that

$\left|M_{n}+1\right|=n,\quad\left\langle
M_{n}+1\right\rangle=0,\quad\rho\left(M_{n}+1\right)=n.$

The second part of Theorem 1 implies our Corollary.

######  Corollary 2.

Let $p>0$ and $f\in H_{p}$. Then the maximal operator

$\sup_{n\in\mathbb{N_{+}}}\left|S_{M_{n}+M_{n-1}}f\right|$

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

We notice that

$\left|M_{n}+M_{n-1}\right|=n,\quad\left\langle
M_{n}+M_{n-1}\right\rangle=n-1,\quad\rho\left(M_{n}+M_{n-1}\right)=1.$

Thus, the second part of Theorem 1 gives again Corollary 2.

######  Corollary 3.

Let $p>0$ and $f\in H_{p}$. Then the maximal operator

${S}^{\\#}f:=\sup_{n\in\mathbb{N}}\left|S_{M_{n}}f\right|$

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

We find that $\left|M_{n}\right|=\left\langle M_{n}\right\rangle=n,$
$\rho\left(M_{n}\right)=0$. Using part b) of Theorem 1, we immediately get
Corollary 3.

Since $S_{n}P=P$ for every $P\in\mathcal{P}$, where $\mathcal{P}$ is the set
of all Vilenkin polynomials. The set $\mathcal{P}$ is dense in the space
$L_{1}(G_{m})$. Combining Lemma 1 and part a) of Theorem 1, we obtain that
under condition (5) the restricted maximal operator of partial sums is bounded
from the space $L_{1}(G_{m})$ to the space $weak-L_{1}(G_{m})$ It follows that

######  Corollary 4.

Let $f\in L_{1}$ and $\left\\{\alpha_{k}:k\in\mathbb{N}\right\\}$ be a
subsequence of positive natural numbers, satisfying condition (5). Then

$S_{\alpha_{k}}f\rightarrow f\text{\ a.e. when \ }k\rightarrow\infty.$

######  Corollary 5.

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

$S_{M_{n}}f\rightarrow f\text{ \ a.e. \ when \ }n\rightarrow\infty.$

##  4\. Lemmas

First, we note the following well-known result, which was proved in Weisz [18,
19]:

######  Lemma 1.

Suppose that an operator $T$ is sub-linear and, for some $0<p\leq 1$

$\int\limits_{\overline{I}}\left|Ta\right|^{p}d\mu\leq c_{p}<\infty$

for every $p$-atom $a$, where $I$ denotes the support of the atom $a$. If $T$
is bounded from $L_{\infty}$ to $L_{\infty},$ then

$\left\|Tf\right\|_{p}\leq c_{p}\left\|f\right\|_{H_{p}}.$

Moreover, if $p<1,$ then we have weak (1,1) type estimate, i.e. it holds that

$\lambda\mu\left\\{x\in G_{m}:\
\left|Tf\left(x\right)\right|>\lambda\right\\}\leq\left\|f\right\|_{1}$

for all $f\in L_{1}.$

The next Lemma can be found in Tephnadze [13]:

######  Lemma 2.

Let $n\in\mathbb{N}$ and $x\in I_{s}\backslash I_{s+1},$ $0\leq s\leq N-1$.
Then

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

We also need the following estimate of independent interest:

######  Lemma 3.

Let $n\in\mathbb{N}$, $\left|n\right|\neq\left\langle n\right\rangle$ and
$x\in I_{\left\langle n\right\rangle+1}\left(e_{\left\langle
n\right\rangle}\right)$ where $e_{k}:=(0,\dots,0,1,0,\dots)\in G_{m}$ (only
the $k$-th coordinate is one, the others are zero). Then

$\left|D_{n}\left(x\right)\right|=\left|D_{n-M_{\left|n\right|}}\left(x\right)\right|\geq
M_{\left\langle n\right\rangle}.$

###### Proof.

Let $x\in I_{\left\langle n\right\rangle+1}\left(e_{\left\langle
n\right\rangle}\right)$. Since

$n=n_{\left\langle n\right\rangle}M_{\left\langle
n\right\rangle}+\sum_{j=\left\langle
n\right\rangle+1}^{\left|n\right|-1}n_{j}M_{j}+n_{\left|n\right|}M_{\left|n\right|}$

and

$n-M_{\left|n\right|}=n_{\left\langle n\right\rangle}M_{\left\langle
n\right\rangle}+\sum_{j=\left\langle
n\right\rangle+1}^{\left|n\right|-1}n_{j}M_{j}+\left(n_{\left|n\right|}-1\right)M_{\left|n\right|},$

Applying (2) and (3) we can conclude that

$\displaystyle\left|D_{n-M_{\left|n\right|}}\right|$ $\displaystyle\geq$
$\displaystyle\left|\psi_{n-M_{n}}D_{M_{\left\langle
n\right\rangle}}\sum_{s=m_{\left\langle n\right\rangle}-n_{\left\langle
n\right\rangle}}^{m_{\left\langle n\right\rangle}-1}r_{\left\langle
n\right\rangle}^{s}\right|-\left|\psi_{n-M_{n}}\sum_{j=\left\langle
n\right\rangle+1}^{\left|n\right|}D_{M_{j}}\sum_{s=m_{j}-n_{j}}^{m_{j}-1}r_{j}^{s}\right|$
$\displaystyle=$ $\displaystyle\left|D_{M_{\left\langle
n\right\rangle}}\sum_{s=m_{\left\langle n\right\rangle}-n_{\left\langle
n\right\rangle}}^{m_{\left\langle n\right\rangle}-1}r_{\left\langle
n\right\rangle}^{s}\right|$ $\displaystyle=$
$\displaystyle\left|D_{M_{\left\langle n\right\rangle}}r_{\left\langle
n\right\rangle}^{m_{\left\langle n\right\rangle}-n_{\left\langle
n\right\rangle}}\sum_{s=0}^{n_{\left\langle n\right\rangle}-1}r_{\left\langle
n\right\rangle}^{s}\right|$ $\displaystyle=$ $\displaystyle D_{M_{\left\langle
n\right\rangle}}\left|\sum_{s=0}^{n_{\left\langle
n\right\rangle}-1}r_{\left\langle n\right\rangle}^{s}\right|.$

Let $x_{n}=1.$ Then we readily get for $s_{n}<m_{n}$ that

$\displaystyle\left|\sum_{u=0}^{s_{n}-1}r_{n}^{u}\left(x\right)\right|$
$\displaystyle=$
$\displaystyle\left|\frac{r_{n}^{s_{n}}\left(x\right)-1}{r_{n}\left(x\right)-1}\right|$
$\displaystyle=$ $\displaystyle\frac{\sin\left(\pi
s_{n}x_{n}/m_{n}\right)}{\sin\left(\pi x_{n}/m_{n}\right)}$ $\displaystyle=$
$\displaystyle\frac{\sin\left(\pi
s_{n}/m_{n}\right)}{\sin\left(\pi/m_{n}\right)}\geq 1.$

It follows that

$\left|D_{n-M_{\left|n\right|}}\left(x\right)\right|\geq D_{M_{\left\langle
n\right\rangle}}\left(x\right)\geq M_{\left\langle n\right\rangle}.$

Moreover, by using the same arguments as above it is easily seen that

$\left|D_{n}\left(x\right)\right|=\left|D_{n-M_{\left|n\right|}}\left(x\right)\right|,\
\text{ for }x\in I_{\left\langle n\right\rangle+1}\left(e_{\left\langle
n\right\rangle}\right),\ \left|n\right|\neq\left\langle n\right\rangle,\
n\in\mathbb{N}.$

The proof is complete. ∎

##  5\. Proof of the main theorem

###### Proof of Theorem 1.

First, we prove part a). Combining (2) and (3) we easily conclude that if
condition (5) holds, then

$\displaystyle\left\|D_{\alpha_{k}}\right\|_{1}$ $\displaystyle\leq$
$\displaystyle\overset{\left|\alpha_{k}\right|}{\underset{j=\left\langle\alpha_{k}\right\rangle}{\sum}}\left\|D_{M_{j}}\right\|_{1}m_{j}$
$\displaystyle\leq$ $\displaystyle
c\overset{\left|\alpha_{k}\right|}{\underset{j=\left\langle\alpha_{k}\right\rangle}{\sum}}1=c(\rho\left(\alpha_{k}\right)+1)\leq
c<\infty.$

It follows that $\widetilde{S}^{\ast,\vartriangle}$ is bounded from
$L_{\infty}$ to $L_{\infty}.$ By Lemma 1 we obtain that the proof of part a)
will be complete if we show that

$\int\limits_{\overline{I_{N}}}\left|\widetilde{S}^{\ast,\vartriangle}a\right|^{p}d\mu\leq
c<\infty$

for every $p$-atom $a$ with support $I=I_{N}$. Since
$S_{\alpha_{k}}\left(a\right)=0$ when $\alpha_{k}\leq M_{N},$ we can suppose
that $\alpha_{k}>M_{N}.$ (That is, $|\alpha_{k}|\geq N$.)

Let $t\in I_{N}$ and $x\in I_{s}\backslash I_{s+1},$ $1\leq s\leq N-1.$ If
$\left\langle\alpha_{k}\right\rangle\geq N,$ we get that
$s<N\leq\left\langle\alpha_{k}\right\rangle$ and since $x-t\in I_{s}\backslash
I_{s+1},$ by combining (2) and (3) we obtain that

(7) $D_{\alpha_{k}}\left(x-t\right)=0.$

Analogously, by combining again (2) and (3) we can conlude that (7) holds, for
$s<\left\langle\alpha_{k}\right\rangle\leq N-1$.

It follows that

(8) $\left|S_{\alpha_{k}}a\left(x\right)\right|=0,\ \ \text{either}\ \
\left\langle\alpha_{k}\right\rangle\geq N,\ \ \text{or}\ \
s<\left\langle\alpha_{k}\right\rangle\leq N-1.$

Let $0<p\leq 1$, $t\in I_{N}$ and $x\in I_{s}\backslash I_{s+1}$,
$\left\langle\alpha_{k}\right\rangle\leq s\leq N-1.$ Applying the fact that
$\left\|a\right\|_{\infty}\leq M_{N}^{1/p}$ and Lemma 2 we find that

$\displaystyle\left|S_{\alpha_{k}}\left(a\right)\right|$ $\displaystyle\leq$
$\displaystyle
M_{N}^{1/p}\int_{I_{N}}\left|D_{\alpha_{k}}\left(x-t\right)\right|d\mu\left(t\right)\leq
c_{p}M_{N}^{1/p-1}M_{s}.$

Let us set
$\varrho:=\min_{k\in\mathbb{N}}\left\langle\alpha_{k}\right\rangle.$ Then, in
view of (8) and (5) we can conclude that

(9) $\left|\widetilde{S}^{\ast,\vartriangle}a\left(x\right)\right|=0,\text{ \
for }\ x\in I_{s}\backslash I_{s+1},\ 0\leq s\leq\varrho$

and

(10) $\left|\widetilde{S}^{\ast,\vartriangle}a\left(x\right)\right|\leq
c_{p}M_{N}^{1/p-1}M_{s},\text{ for \ }x\in I_{s}\backslash I_{s+1},\
\varrho<s\leq N-1.$

By the definition of $\varrho$ there exists at least one index
$k_{{0}}\in\mathbb{N}_{+}$ such that
$\varrho=\left\langle\alpha_{k_{{0}}}\right\rangle.$ By using contition (5) we
can conclude that

$\displaystyle N-\varrho$ $\displaystyle=$ $\displaystyle
N-\left\langle\alpha_{k_{{0}}}\right\rangle\leq\left|\alpha_{k_{{0}}}\right|-\left\langle\alpha_{k_{{0}}}\right\rangle$
$\displaystyle\leq$
$\displaystyle\sup_{k\in\mathbb{N}}\rho\left(\alpha_{k}\right)=\varkappa<c<\infty.$

Let us set $m_{*}:=\sup_{k}m_{k}$.

Let $0<p<1.$ According to (1) and using (9), (10) and (5) we obtain that

$\displaystyle\int_{\overline{I_{N}}}\left|\widetilde{S}^{\ast,\vartriangle}a\left(x\right)\right|^{p}d\mu\left(x\right)$
$\displaystyle=$
$\displaystyle\overset{N-1}{\underset{s=\varrho+1}{\sum}}\int_{I_{s}\backslash
I_{s+1}}\left|\widetilde{S}^{\ast,\vartriangle}a\left(x\right)\right|^{p}d\mu\left(x\right)$
$\displaystyle\leq$ $\displaystyle
c_{p}M_{N}^{1-p}\overset{N-1}{\underset{s=\varrho+1}{\sum}}\frac{M_{s}^{p}}{M_{s}}$
$\displaystyle=$ $\displaystyle
c_{p}M_{N}^{1-p}\overset{N-1}{\underset{s=\varrho+1}{\sum}}\frac{1}{M_{s}^{1-p}}$
$\displaystyle\leq$
$\displaystyle\frac{c_{p}M_{N}^{1-p}}{M_{\varrho}^{1-p}}\leq
c_{p}m_{*}^{\varkappa(1-p)}\leq c_{p}<\infty.$

Let $p=1.$ We combine (9)-(5) and invoke identity (1) to obtain that

$\displaystyle\int_{\overline{I_{N}}}\left|\widetilde{S}^{\ast,\vartriangle}a\left(x\right)\right|d\mu\left(x\right)$
$\displaystyle=$
$\displaystyle\overset{N-1}{\underset{s=\varrho+1}{\sum}}\int_{I_{s}\backslash
I_{s+1}}\left|\widetilde{S}^{\ast,\vartriangle}a\left(x\right)\right|d\mu\left(x\right)$
$\displaystyle\leq$ $\displaystyle
c\overset{N-1}{\underset{s=\varrho+1}{\sum}}\frac{M_{s}}{M_{s}}$
$\displaystyle=$ $\displaystyle
c\overset{N-1}{\underset{s=\varrho+1}{\sum}}1\leq c\varkappa\leq c<\infty.$

The proof of part a) is complete.

Now, we prove the second part of our theorem. Since,

$\frac{M_{\left|\alpha_{k}\right|}}{M_{\left\langle\alpha_{k}\right\rangle}}\geq
2^{\rho\left(\alpha_{k}\right)},$

under condition (6), there exists an increasing subsequence $\left\\{n_{k}:\
k\in\mathbb{N}\right\\}\subset\left\\{\alpha_{k}:\ k\in\mathbb{N}\right\\}$
such that $n_{0}\geq 3$ and

(11)
$\lim_{k\rightarrow\infty}\frac{M_{\left|n_{k}\right|}^{\left(1-p\right)/2}}{M_{\left\langle
n_{k}\right\rangle}^{\left(1-p\right)/2}}=\infty$

and

(12) $\sum_{k=0}^{\infty}\frac{M_{\left\langle
n_{k}\right\rangle}^{\left(1-p\right)/2}}{M_{\left|n_{k}\right|}^{\left(1-p\right)/2}}<c<\infty.$

Let $f=\left(f_{n}:n\in\mathbb{N}\right)$ be a martingale defined by

$f_{n}:=\sum_{\left\\{k:\ \left|n_{k}\right|<n\right\\}}\lambda_{k}a_{k},$

where

$a_{k}:=\frac{M_{\left|n_{k}\right|}^{1/p-1}}{m_{*}}\left(D_{M_{\left|n_{k}\right|+1}}-D_{M_{{}_{\left|n_{k}\right|}}}\right)$

and

(13) $\lambda_{k}=\frac{m_{*}M_{\left\langle
n_{k}\right\rangle}^{\left(1/p-1\right)/2}}{M_{\left|n_{k}\right|}^{\left(1/p-1\right)/2}}.$

It is easily seen that $a$ is a $p$-atom. Under condition (12) we can conclude
that $f\in H_{p}$. (Theorem W immediately yields that $\|f\|_{H_{p}}\leq
c_{p}<\infty$.)

According to (13) we readily see that

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

Since, $M_{\left|n_{k}\right|}<n_{k}$, we get

$\displaystyle S_{n_{k}}f$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{M_{|n_{k}|}-1}\hat{f}(j)\psi_{j}+\sum_{j=M_{|n_{k}|}}^{n_{k}-1}\hat{f}(j)\psi_{j}$
$\displaystyle=$ $\displaystyle S_{M_{\left|n_{k}\right|}}f+M_{\left\langle
n_{k}\right\rangle}^{\left(1/p-1\right)/2}M_{\left|n_{k}\right|}^{\left(1/p-1\right)/2}\psi_{M_{\left|n_{k}\right|}}D_{n_{k}-M_{\left|n_{k}\right|}}$
$\displaystyle:=$ $\displaystyle I+II.$

According to part a) of Theorem 1 for $I$ we have that

$\left\|I\right\|_{L_{p},\infty}^{p}\leq\left\|S_{M_{\left|n_{k}\right|}}f\right\|_{L_{p},\infty}^{p}\leq
c_{p}\left\|f\right\|_{H_{p}}^{p}\leq c_{p}<\infty.$

Moreover, under condition (6) we can conclude that

$\left\langle n_{k}\right\rangle\neq\left|n_{k}\right|\text{ and }\left\langle
n_{k}-M_{\left|n_{k}\right|}\right\rangle=\left\langle n_{k}\right\rangle.$

Let $x\in I_{\left\langle n_{k}\right\rangle+1}\left(e_{\left\langle
n_{k}\right\rangle}\right).$ Applying Lemma 3 we obtain that

$\left|D_{n_{k}-M_{\left|n_{k}\right|}}\right|\geq M_{\left\langle
n_{k}\right\rangle}$

Thus, we immediately have

$\displaystyle\left|II\right|$ $\displaystyle=$ $\displaystyle M_{\left\langle
n_{k}\right\rangle}^{\left(1/p-1\right)/2}M_{\left|n_{k}\right|}^{\left(1/p-1\right)/2}\left|D_{n_{k}-M_{\left|n_{k}\right|}}\right|$
$\displaystyle\geq$ $\displaystyle M_{\left\langle
n_{k}\right\rangle}^{\left(1/p+1\right)/2}M_{\left|n_{k}\right|}^{\left(1/p-1\right)/2}.$

It follows that

$\displaystyle\left\|II\right\|_{L_{p,\infty}}^{p}$ $\displaystyle\geq$
$\displaystyle c_{p}\left(M_{\left\langle
n_{k}\right\rangle}^{\left(1/p+1\right)/2}M_{\left|n_{k}\right|}^{\left(1/p-1\right)/2}\right)^{p}\mu\left\\{x\in
G_{m}:\text{ }\left|II\right|\geq c_{p}M_{\left\langle
n_{k}\right\rangle}^{\left(1/p+1\right)/2}M_{\left|n_{k}\right|}^{\left(1/p-1\right)/2}\right\\}$
$\displaystyle\geq$ $\displaystyle
c_{p}M_{\left|n_{k}\right|}^{\left(1-p\right)/2}M_{\left\langle
n_{k}\right\rangle}^{\left(1+p\right)/2}\mu\left\\{I_{\left\langle
n_{k}\right\rangle+1}\left(e_{\left\langle
n_{k}\right\rangle}\right)\right\\}\geq\frac{c_{p}M_{\left|n_{k}\right|}^{\left(1-p\right)/2}}{M_{\left\langle
n_{k}\right\rangle}^{\left(1-p\right)/2}}.$

Hence, for large enough $k$,

$\displaystyle\left\|S_{n_{k}}f\right\|_{L_{p,\infty}}^{p}$
$\displaystyle\geq$
$\displaystyle\left\|II\right\|_{L_{p,\infty}}^{p}-\left\|I\right\|_{L_{p,\infty}}^{p}\geq\frac{1}{2}\left\|II\right\|_{L_{p,\infty}}^{p}$
$\displaystyle\geq$
$\displaystyle\frac{c_{p}M_{\left|n_{k}\right|}^{\left(1-p\right)/2}}{M_{\left\langle
n_{k}\right\rangle}^{\left(1-p\right)/2}}\rightarrow\infty,\ \text{ when }\
k\rightarrow\infty.$

The proof is complete. ∎

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