# Theoretical systematics in testing the Cosmological Principle with the
kinematic quasar dipole

Caroline Guandalin School of Physical & Chemical Sciences, Queen Mary
University of London, London E1 4NS, UK  Jade Piat  Aix-Marseille University,
Marseille, France  School of Physical & Chemical Sciences, Queen Mary
University of London, London E1 4NS, UK  Chris Clarkson School of Physical &
Chemical Sciences, Queen Mary University of London, London E1 4NS, UK
Department of Physics & Astronomy, University of Western Cape, Cape Town 7535,
South Africa  Department of Mathematics and Applied Mathematics, University of
Cape Town 7701, South Africa  Roy Maartens Department of Physics & Astronomy,
University of Western Cape, Cape Town 7535, South Africa  Institute of
Cosmology & Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK
National Institute for Theoretical & Computational Sciences (NITheCS), Cape
Town 7535, South Africa

###### Abstract

The Cosmological Principle is part of the foundation that underpins the
standard model of the Universe. In the era of precision cosmology, when stress
tests of the standard model are uncovering various tensions and possible
anomalies, it is critical to check the viability of this principle. A key test
is the consistency between the kinematic dipoles of the cosmic microwave
background and of the large-scale matter distribution. Results using radio
continuum and quasar samples indicate a rough agreement in the directions of
the two dipoles, but a larger than expected amplitude of the matter dipole.
The resulting tension with the radiation dipole has been estimated at $\sim
5\sigma$ for some cases, suggesting a potential new cosmological tension and a
possible violation of the CP. However, the standard formalism for predicting
the dipole in the two-dimensional projection of sources overlooks possible
evolution effects in the luminosity function. In fact, radial information from
the luminosity function is necessary for a correct projection of the three-
dimensional source distribution. Using a variety of current models of the
quasar luminosity function, we show that neglecting redshift evolution can
significantly overestimate the relative velocity amplitude. While the models
we investigate are consistent with each other and with current data, the
dipole derived from these, which depends on derivatives of the luminosity
function, can disagree by more than $3\sigma$. This theoretical systematic
bias needs to be resolved before robust conclusions can be made about a new
cosmic tension.

††facilities: This work uses data from the extended Baryon Oscillation
Spectroscopic Survey of the Sloan Digital Sky Survey (SDSS-IV/eBOSS), as
extracted from Table A.1 of PD2016.††software: We made extensive use of emcee
(Foreman-Mackey et al., 2013), CLASS (Blas et al., 2011), numpy (Harris et
al., 2020), scipy (Virtanen et al., 2020), and matplotlib (Hunter, 2007).

##  1 Introduction

The Cosmological Principle (CP), i.e. that the spatial distribution of matter
and radiation in the Universe is statistically homogeneous and isotropic on
large enough scales, is perhaps the most fundamental assumption in modern
cosmology. It is the basis for modelling the Universe using the Friedmann-
Lemaître-Robertson-Walker (FLRW) metric (see, e.g., Ehlers et al., 1968; Ellis
et al., 1983; Stoeger et al., 1995; Clarkson & Maartens, 2010; Clarkson,
2012). The CP implies that there is a unique cosmic frame defined by observers
who measure statistical isotropy and homogeneity of the cosmic microwave
background (CMB) and of the matter distribution on sufficiently large scales.

Earth observers are not at rest in the cosmic frame, but are moving with a
velocity $\bm{v}_{o}^{\rm CMB}$ relative to the CMB rest frame, i.e., the
frame in which the CMB dipole vanishes and the CMB is statistically isotropic
(Stewart & Sciama, 1967; Peebles & Wilkinson, 1968). This leads to a dipole of
$3362.08\pm 0.99\,\,\mu\,$K, much larger than the $\ell\geq 2$ multipoles,
which is used to extract a velocity of $v_{o}^{\rm CMB}=369.82\pm 0.11\,{\rm
km/s}$ towards $(l,b)=(264.021\pm 0.011,48.253\pm 0.005)^{\circ}$ (Aghanim et
al., 2020).

At leading order, the boosted temperature contrast is

$\displaystyle\tilde{\delta}_{\rm T}({\bm{n}})=\delta_{\rm T}(\bm{n})+{\cal
D}_{\rm CMB}\,\bm{n}\cdot{\bm{v}}_{o}^{\rm CMB},$ (1)

where $\bm{n}$ is a unit vector in the direction of observation. Using units
with $c=1$, the dimensionless CMB dipole factor is ${\cal D}_{\rm CMB}=1$.

The CP requires the large-scale distribution of sources on the sky to have the
same dipole – i.e., the relative velocity $\bm{v}_{o}$ extracted from these
sources should agree with that from the CMB as

$\displaystyle\bm{\hat{v}}_{o}\approx\bm{\hat{v}}_{o}^{\rm
CMB},~{}~{}{v}_{o}\approx{v}_{o}^{\rm CMB}.$ (2)

The boosted two-dimensional (2D) number density contrast is

$\displaystyle\tilde{\delta}_{2\rm D}({\bm{n}})=\delta_{2\rm D}(\bm{n})+{\cal
D}\,\bm{n}\cdot{\bm{v}}_{o}.$ (3)

The dimensionless number count dipole factor ${\cal D}$ is independent of
$v_{o}$, but it does depend on the sample.

Ellis & Baldwin (1984) estimated the dipole factor for the 2D projection of
radio sources as

${\cal D}_{\rm EB}=2+x\,(1+\alpha),$ (4)

in which the power-law indices $x$ and $\alpha$ are constants that determine
the number counts per solid angle (${\cal N}_{\Omega}\propto S_{c}^{-x}$, with
$S_{c}$ being the flux threshold of the survey) and the flux density of
sources in their rest frame
($S(\nu_{\mathrm{obs}})\propto\nu_{\mathrm{obs}}^{-\alpha}$) at fixed observed
frequency $\nu_{\mathrm{obs}}$. This is potentially the most powerful
consistency test of the CP, but it faces major observational obstacles.

First, the galaxy sample must cover a wide sky area and a deep redshift range,
with a high number density. High redshifts are required to extract the cosmic
dipole and the lowest-redshift sources need to be removed to avoid nonlinear
contamination (Tiwari & Nusser, 2016; Bengaly et al., 2019). Second,
measurements of large-scale features face well-known systematic errors
(effects of the mask, instrumental/survey systematics, stellar contamination,
etc.). In this paper, we focus on a key theoretical systematic error that
arises when estimating the dimensionless dipole factor ${\cal D}$ without
using radial information.

Different groups have applied Equation (4) to extract the relative velocity
from the measured dipole in wide-area radio continuum surveys (see, e.g.,
Blake & Wall, 2002; Gibelyou & Huterer, 2012; Rubart & Schwarz, 2013; Tiwari
et al., 2015; Ghosh et al., 2016; Colin et al., 2017; Bengaly et al., 2018). A
general conclusion of these analyses is that the dipole direction is typically
consistent with that of the CMB, but the relative velocity amplitude is
significantly larger:111Note that Darling (2022) finds no tension of this
form.

$\displaystyle\bm{\hat{v}}_{o}^{\rm EB}\approx\bm{\hat{v}}_{o}^{\rm
CMB},~{}~{}~{}v_{o}^{\rm EB}>v_{o}^{\rm CMB}.$ (5)

Recently, one of the most thorough analyses has used the very large
CatWISE2020 sample of quasars to apply the Ellis–Baldwin test (Secrest et al.,
2021, 2022). Their conclusion reinforces Equation (5), finding a tension with
the CMB velocity amplitude at $\sim 5\sigma$.

Some works have attempted to resolve this tension by invoking superhorizon
effects (for example, see Das et al., 2021; Domènech et al., 2022; Tiwari et
al., 2022). In this work, we do not investigate alternative cosmological
models. Instead, we consider a possible theoretical systematic in the velocity
estimates. We extend the pioneering analysis of Dalang & Bonvin (2022), who
showed that the presence of parameter evolution can lead to significant
corrections on the Ellis–Baldwin approximation. We investigate the impact of
different quasar luminosity function (QLF) models on the predicted amplitude
of the kinematic dipole. In Section 2, we review the theoretical predictions
for the expected kinematic dipole amplitude in three dimensions, ${\cal
D}_{3\rm D}$, and how it can be projected into the 2D dipole factor ${\cal D}$
of Equation (3). In Section 3, we introduce the QLF data and the theoretical
modelling used for the fits. The Markov Chain Monte Carlo (MCMC) analysis and
the goodness-of-fit criterion are described in Section 4. The results and
conclusion are presented in Section 5 and Section 6, respectively.

##  2 Correction to the standard dipole

In all of the results characterised by Equation (5), the dipole is measured
and then Equation (4) is used to extract the relative speed. However, ${\cal
D}_{\rm EB}$ assumes that radial information can be neglected. In reality,
${\cal D}$ is a radial projection of the three-dimensional redshift-dependent
dipole factor (Maartens et al., 2018)

${\cal D}_{3\rm D}=2+\frac{\dot{\cal H}}{{\cal H}^{2}}+\frac{2}{{r}{\cal
H}}-\frac{5s}{r{\cal H}}-b_{e},$ (6)

which can be derived from linear perturbations.222There is also an intrinsic
dipole sourced by primordial perturbations (Tiwari & Nusser, 2016; Nadolny et
al., 2021), which is not relevant for our discussion.

The first three terms on the right are purely cosmological ($r$ is the
comoving line-of-sight distance) and the last two contain the magnification
($s$) and evolution ($b_{e}$) biases, which depend on the sample’s luminosity
function $\Phi$ through the background comoving number density (in the source
frame)

$\displaystyle n(z,M_{\rm c})=\int_{-\infty}^{M_{\rm c}(z)}{\rm
d}M\,\Phi(z,M).$ (7)

The absolute magnitude of a source (corresponding to its intrinsic luminosity)
is given by

$\displaystyle M=m(z)-\mu(z)-K(z),$ (8)

where $m$ is the apparent magnitude measured at redshift $z$ (corresponding to
observed flux), $\mu$ is the distance modulus (defined by the background
luminosity distance) and $K$ is the $K$-correction. The apparent magnitude cut
of the survey $m_{\rm c}$ leads to an absolute threshold $M_{\rm c}(z)$ that
depends on redshift via Equation (8). The magnification and evolution biases
become (Challinor & Lewis, 2011; Alonso et al., 2015; Maartens et al., 2021)

$\displaystyle s(z,M_{\rm c})$ $\displaystyle=\frac{\partial\log{n}(z,M_{\rm
c})}{\partial M_{\rm c}}={\frac{1}{\ln 10}\,\frac{\Phi(z,M_{\rm
c})}{{n}(z,M_{\rm c})},}$ (9) $\displaystyle b_{e}(z,M_{\rm c})$
$\displaystyle=-\frac{\partial\ln{n}(z,M_{\rm c})}{\partial\ln(1+z)}$
$\displaystyle={-\frac{(1+z)}{{n}(z,M_{\rm c})}\int_{-\infty}^{M_{\rm
c}(z)}{\rm d}M\,\frac{\partial\Phi(z,M)}{\partial z}\bigg{|}_{M},}$ (10)

in which $s$ determines whether sources will be included or excluded from the
sample due to lensing convergence and $b_{e}$ describes the deviation of the
comoving number density of sources from the conserved case ($b_{e}=0$). Note
the partial redshift derivative in $b_{e}$ must be taken at fixed magnitude
$M_{\rm c}$.

The observed source number density ${\cal N}$ (number per redshift per solid
angle), is not the same as the number density $n$ (number per comoving volume)
measured at the source (Bonvin & Durrer, 2011; Challinor & Lewis, 2011; Alonso
et al., 2015; Maartens et al., 2018, 2021). At the background level, these
quantities are related by

$\displaystyle{\cal N}=\frac{{\rm d}{N}}{{\rm d}z\,{\rm
d}\Omega}=\frac{r^{2}}{(1+z)\mathcal{H}}\,n,$ (11)

where ${N}$ is the number of sources, which is the same in the observer and
source frames, and ${\rm d}{N}={\cal N}{\rm d}z\,{\rm d}\Omega=n\,{\rm d}V$,
with $V$ the comoving volume. The observed number density projected on the sky
becomes

$\displaystyle{\cal N}_{\Omega}=\int_{0}^{\infty}{\rm d}z\,{\cal N}(z).$ (12)

The 2D dipole factor is found by projecting Equation (6) along the radial
direction, weighted by the number counts (Nadolny et al., 2021; Dalang &
Bonvin, 2022)333In these papers, their ${\cal D}_{\rm kin}$ equals our ${\cal
D}\,v_{o}$.:

${{\cal D}=\int_{0}^{\infty}{\rm d}z\,f(z)\,{\cal D}_{3\rm
D}(z)~{}~{}\mbox{where}~{}~{}f(z)=\frac{{\cal N}(z)}{{\cal N}_{\Omega}}.}$
(13)

The result is

${{\cal D}={\cal D}_{\rm cosmo}+{\cal D}_{\rm mag}+{\cal D}_{\rm evol},}$ (14)

where

$\displaystyle{\cal D}_{\rm cosmo}$ $\displaystyle=\int_{0}^{\infty}{\rm
d}z\,f\Big{[}2+\frac{2}{{r}{\cal H}}+\frac{\dot{\cal H}}{{\cal
H}^{2}}\Big{]},$ (15) $\displaystyle{\cal D}_{\rm mag}$
$\displaystyle=-2\int_{0}^{\infty}{\rm d}z\,f\,\frac{x}{{r}{\cal
H}}~{}~{}\mbox{with}~{}~{}x=\frac{5}{2}\,s,$ (16) $\displaystyle{\cal D}_{\rm
evol}$ $\displaystyle=-\int_{0}^{\infty}{\rm d}z\,f\,b_{e}.$ (17)

For a constant magnification bias, $x$ corresponds to the Ellis–Baldwin
parameter in Equation (4), but in general it evolves with redshift. The dipole
factor ${\cal D}$, in Equations (14) – (17), depends on the evolution bias
$b_{e}$, which is unique for the sample, but not on the variable source
spectral index $\alpha$ (Carballo et al., 1999; Secrest et al., 2021). This
important feature arises from the fact that the absolute magnitude cut,
$M_{\rm c}(z)$, does not depend on $\alpha$ at a constant redshift, but at a
fixed distance $r$ (Dalang & Bonvin, 2022).

The Ellis–Baldwin formula is an approximation to Equation (13), ${\cal D}_{\rm
EB}={\cal D}+\Delta{\cal D}$, that includes a correction $\Delta{\cal D}$
which does not affect the dipole direction. The dipole amplitude is extracted
directly from data and is equal to the dipole factor times the relative speed

$\displaystyle{\cal D}_{\rm EB}\,v_{o}^{\rm EB}={\cal D}\,v_{o}.$ (18)

When using the exact formula (Equation 13) for the dipole factor ${\cal D}$,
the correct relative speed $v_{o}$ is extracted from the measurements.
However, using the Ellis–Baldwin approximation ${\cal D}_{\rm EB}$ we extract
an estimate of the relative speed that may differ from $v_{o}$: $v_{o}^{\rm
EB}=v_{o}+\Delta v_{o}$. For example, if the relative speed is overestimated,
i.e., $v_{o}^{\rm EB}>v_{o}$, then ${\cal D}_{\rm EB}$ underestimates ${\cal
D}$, since ${\Delta v_{o}}/{v_{o}}=-{\Delta{\cal D}}/{\cal D}$ by Equation
(18).

##  3 Data

The QLF measurements used in this work are from Palanque-Delabrouille et al.
(2016) (PD2016 hereafter) and include 13876 quasars: 7900 were identified
through flux variability (Schmidt et al., 2010; Palanque-Delabrouille et al.,
2011) in the Sloan Digital Sky Survey (SDSS) photometric $\\{u,g,r,i,z\\}$
bands and 5976 were spectroscopically identified in the Stripe 82 region. The
data is part of the extended Baryon Oscillation Spectroscopic Survey (eBOSS)
in SDSS-IV and is available in Table A.1 of PD2016. The magnitudes have been
corrected for Galactic extinction with a magnitude limit of $g_{\rm
dered}=22.5$ in the dereddened $g$ band.

###  3.1 Theoretical modelling

The number counts of quasars have two main sources of uncertainty: the bright
end has a small number of objects, while the faint end is limited by the
survey strategy. The large uncertainties on both ends of the luminosity
function leads us to the adoption of a double power-law fit for the comoving
space density of quasars (Boyle et al., 2000; Croom et al., 2004, 2009; Ross
et al., 2013)

$\Phi(z,M_{g})=\frac{\Phi_{*}}{10^{0.4(a+1)(M_{g}-M_{*})}+10^{0.4(b+1)(M_{g}-M_{*})}}.$
(19)

Above, $\Phi_{*}$ is a normalisation factor related to the characteristic
number density of quasars, $M_{*}$ is the break magnitude, associated with
their characteristic luminosity, $a$ and $b$ describe, respectively, the
behaviour of the bright and faint ends, and

$M_{g}(z)=m_{g}-\mu(z)-[K(z)-K(z=2)]$ (20)

is the absolute magnitude corresponding to the observed apparent magnitude
$m_{g}$ in the $g$ band after the $K$-correction

$K(z)=-2.5(1+\alpha_{v})\log_{10}(1+z),\,\,\,\alpha_{v}\simeq-0.5$ (21)

has been applied ( ; , ).

The number density of quasars has a strong redshift dependence and the QLF may
evolve in different ways: in the pure-luminosity evolution (PLE) case, the
characteristic number density remains constant, but the break magnitude
evolves over time; in the pure-density evolution (PDE), the luminosity of
individual sources remain constant, while the number density varies with time;
or it could be combination of both luminosity and density evolution (LEDE).
Redshift dependence may also impact the bright- and faint-end slopes.

Since the QLF is very uncertain, this work aims at investigating different
evolution models for $\Phi_{*},M_{*},a$ and $b$, and to assess their impact on
the kinematic dipole ${\cal D}$. The models are described in detail in
Appendix A. They are (; ; ):

  * •

PLE – the QLF redshift evolution comes from the break magnitude $M_{*}$, with
the pivot scale $z_{p}=2.2$ allowing the bright and faint slopes to vary for
low ($z<z_{p}$) and high ($z>z_{p}$) redshifts.

  * •

LEDE – both luminosity and number density vary with time. We consider a linear
(LEDE7) and quadratic (LEDE8) redshift dependence on the break magnitude
$M_{*}(z)$ and their extension, in which both slopes evolve linearly with
redshift (LEDE7+2 and LEDE8+2).

  * •

PLE+LEDE – it combines the PLE functional form for $z<z_{p}$, and the LEDE7
model for $z>z_{p}$. We allow the bright slope $a$ to evolve with redshift.

##  4 Method

###  4.1 Maximum likelihood

We consider a maximum-likelihood approach to sample the parameter space of the
models of Section 3.1. Because the QLF is estimated from the number counts of
quasars inside each magnitude bin, its error can be approximately modelled by
Poissonian error bars. Therefore, we model the log-likelihood function as
(Pozzetti et al., 2016)

$\ln{\cal L}=\sum_{i,j}\frac{\Delta^{2}_{i,j}}{\sigma_{i,j}^{2}},$ (22)

In this equation, $\Delta^{2}_{i,j}\equiv
1-\Phi_{\theta}(z_{i},M_{j})/\Phi_{\rm
obs}(z_{i},M_{j})+\ln[\Phi_{\theta}(z_{i},M_{j})/\Phi_{\rm
obs}(z_{i},M_{j})]$, $\sigma_{i,j}^{2}=1/N_{\rm obs}(z_{i},M_{j})$,
$\Phi_{\theta}$ is given by the model of Equation (19). Finally, $\Phi_{\rm
obs}$ and $N_{\rm obs}$ are, respectively, the corresponding QLF measurements
and the angle averaged number counts in each redshift $z_{i}$ and magnitude
$M_{j}$ bin. To maximise Equation (22), we used the MCMC sampler emcee
(Foreman-Mackey et al., 2013) with flat priors for the fitting parameters.

###  4.2 Goodness of fit

To assess the goodness of fit of the models considered in Section 3.1 to the
eBOSS data, in Table LABEL:tab:MCMCresults we present the Bayesian information
criterion (BIC)

${\rm BIC}=p\ln(n)-2\ln{\cal L}^{*},$ (23)

in which $p$ is the number of parameters in the model, $n$ is the number of
data points used, and $\ln{\cal L}^{*}$ is the log-likelihood computed at the
best-fitting values (e.g. Liddle, 2004).

##  5 Results

Figure 1: Comoving number density (top), evolution (middle) and magnification (bottom) biases derived from the QLF with different absolute magnitude thresholds, assuming Planck 2015 cosmology (Ade et al., 2016). The pivot redshift, $z_{p}=2.2$, entering the PLE and PLE+LEDE models is indicated by the vertical dashed lines. Table 1: 1$\sigma$ constraints for ${\cal D}_{\rm kin}={\cal D}\,v_{o}^{\rm CMB}/c$, in $10^{-3}$ units, obtained from a random subset of 5000 MCMC samples, with Planck 2015 and $w$CDM cosmologies. We present the results with different absolute magnitude thresholds of $-25$, $-24.6$ and a varying threshold $M_{\rm c}(z)$. The effect of neglecting the redshift evolution of the magnification and evolution biases (Equation 24) is shown in the last column for the Planck cosmology and absolute magnitude threshold $M_{{\rm c},g}=-25$. | ${\cal D}_{\rm kin}\,\,[10^{-3}]$
---|---
|  $w$CDM | Planck 2015 | $+\,\\{b_{e}^{\rm eff},s^{\rm eff}\\}$
| $M_{{\rm c},g}=-25$ | $M_{{\rm c},g}=-25$ | $M_{{\rm c},g}=-24.6$ | $M_{{\rm c},g}=M_{{\rm c},g}(z)$ | $M_{{\rm c},g}=-25$
| Best fit | Chains
PLE | $0.72^{+0.10}_{-0.10}$ | $0.75^{+0.10}_{-0.10}$ | $1.42^{+0.10}_{-0.10}$ | $0.39_{-0.05}^{+0.05}$ | $0.81$ | $0.84_{-0.12}^{+0.12}$
LEDE7 | $0.32^{+0.03}_{-0.03}$ | $0.34^{+0.03}_{-0.03}$ | $0.90^{+0.04}_{-0.04}$ | $0.53_{-0.06}^{+0.06}$ | $0.49$ | $0.28_{-0.04}^{+0.04}$
LEDE7+2 | $0.59^{+0.06}_{-0.06}$ | $0.62^{+0.06}_{-0.06}$ | $1.14^{+0.06}_{-0.06}$ | $0.67_{-0.05}^{+0.05}$ | $0.72$ | $0.75_{-0.07}^{+0.07}$
LEDE8 | $0.39^{+0.03}_{-0.03}$ | $0.42^{+0.03}_{-0.03}$ | $1.01^{+0.04}_{-0.04}$ | $0.44_{-0.06}^{+0.06}$ | $0.54$ | $0.38_{-0.04}^{+0.04}$
LEDE8+2 | $0.40^{+0.03}_{-0.05}$ | $0.42^{+0.03}_{-0.05}$ | $0.99^{+0.04}_{-0.04}$ | $0.50_{-0.07}^{+0.07}$ | $0.53$ | $0.42_{-0.07}^{+0.05}$
PLE+LEDE | $0.60^{+0.08}_{-0.08}$ | $0.62^{+0.08}_{-0.08}$ | $1.31^{+0.08}_{-0.09}$ | $0.47_{-0.05}^{+0.05}$ | $0.72$ | $0.77_{-0.11}^{+0.11}$
Figure 2: 1$\sigma$ constraints for ${\cal D}_{\rm kin}={\cal D},v_{o}^{\rm
CMB}/c$ obtained from the same random subset of 5000 MCMC samples used to
derive the constraints of Table 1 with the $\Lambda$CDM cosmology. The grey
dashed line is a reference for the intrinsic dipole (Dalang & Bonvin, 2022).
The top panel shows the analysis for a varying magnitude threshold $M_{c}(z)$,
i.e., the corresponding absolute magnitude for $g=22.5$ at each redshift used
to compute $n(z)$. The second and third panels have a fixed threshold given by
the last redshift bin (see Figure 3 for illustration), and the last panel
neglects the last bin (i.e., $M_{c}(z)$ corresponds to $g=22.5$ at $z=3.25$).
The third panel shows the constraints after neglecting the redshift evolution
(Equation 24) with $s(z)$ and $b_{e}(z)$ projected in redshift for each MCMC
sample (points with error bars), and for projections using the best-fitting
functions (dashed lines). Neglecting redshift evolution (third panel) causes
significant tensions in some cases, and neglecting data from the last redshift
bin shifts the amplitude mainly due to the different redshift range used to
obtain ${\cal D}$ (last panel). Different QLF models can lead to a $\sim
3\sigma$ tension in ${\cal D}$.

We fitted the QLF models described in Appendix A to the eBOSS data. In Table
1, we show the mean of random subsets of 5000 samples, obtained from the
complete sets of $\gtrsim$ 300000 MCMC chains used to derive the constraints
on the QLF models, and the 1$\sigma$ constraints obtained for the amplitude of
the kinematic dipole from those subsets.444We checked the constraints against
different subsamples, and also by varying the number of MCMC chains in the
subsets, finding no significant changes. The posteriors for the QLF are
presented in Table LABEL:tab:MCMCresults, together with their goodness of fit.

In Figure 1, the number density, evolution and magnification biases derived
from the best-fitting QLF are shown, after assuming a flat $\Lambda$CDM model
with parameters $h=0.679,n_{s}=0.9681,\sigma_{8}=0.8154,\Omega_{m}=0.3065$ and
$\Omega_{b}h^{2}=0.02227$ (Ade et al., 2016) (same cosmology employed in
PD2016 for the distance modulus $\mu$). To assess the dependence of our
results on the fiducial cosmology used to compute Equation (14), in Table 1
the constraints obtained with a flat $w$CDM cosmology with parameters
$h=0.72,n_{s}=0.963,\sigma_{8}=0.852,\Omega_{m}=0.275,\Omega_{b}h^{2}=0.02258,\Omega_{\rm
DE}=0.725$, and $w=-1.2$ (Rasera et al., 2022) are also shown. The constraints
are robust against this change as seen in Table 1.

We considered different absolute magnitude thresholds: $M_{{\rm
c},g}(z_{8})=-25$, corresponding to the apparent magnitude cut $g_{\rm
dered}=22.5$ at the centre $z_{8}=3.75$ of the last redshift bin (e.g. Wang et
al., 2020), $M_{{\rm c},g}(z_{7})=-24.6$, corresponding to $g_{\rm dered}$ at
$z_{7}=3.25$, and a varying $M_{{\rm c},g}(z)$ obtained from Equation (20) at
each redshift used to compute $n(z,M_{\rm c}),b_{e}(z,M_{\rm c})$, and
$s(z,M_{\rm c})$.

In Figure 2, we show the impact of the magnitude cut $M_{\rm c}$ on the
results (first, second and fourth panels). By fixing a threshold such as
$M_{\rm c}=-25$, we discard data at lower redshifts (see Figure 3). Hence, the
number density increases by allowing larger values for $M_{\rm c}$. By
considering $M_{\rm c}=-24.6$, we neglect the last redshift bin in the
analysis to avoid adding sources too faint to be seen at the highest $z$ bin.
Thus, we are probing the kinematic dipole with lower-$z$ sources and this
shifts ${\cal D}$ to larger values as the integration range changes (fourth
panel of Figure 2). Most of the impact comes from the ${\cal D}_{\rm cosmo}$
term.

The evolution bias derived from the LEDE7, LEDE8 and LEDE8+2 models are very
similar at high redshifts. While the evolution bias is rather consistent among
the different models, the magnification bias has a larger dependence on the
QLF. However, the contributions coming from Equation (16) exhibit a smaller
dispersion due to the $1/r{\cal H}$ suppression in ${\cal D}_{\rm mag}$. The
contributions from the evolution term (Equation 17) are more sensitive to the
differences in the QLF. This is manifested in the final 2D amplitude ${\cal
D}$: LEDE7, LEDE8 and LEDE8+2 agree within $\sim 1\sigma$, while the LEDE7+2,
PLE and PLE+LEDE have $\sim 1\sigma$ consistency among each other.

We also tested the robustness of our results against the minimum redshift
considered; the values presented in Table 1 were obtained by numerically
integrating Equations (15) – (17) from $z_{\rm min}\approx 0$ to $z_{\rm
max}=4$ (the maximum redshift of the sample). Because the number of sources
below $z=0.68$ (minimum redshift) is very small, we find no significant
changes in the constraints for the dipole amplitude. Still, it is worth
stressing the fact that the intrinsic dipole is considerable for low-$z$
sources; therefore, in principle it is better to remove $z\lesssim 0.5$ to
avoid nonlinear contamination in real data (Tiwari & Nusser, 2016; Bengaly et
al., 2019).

Finally, considering the Planck 2015 cosmology and $M_{{\rm c},g}=-25$, we
neglect the redshift evolution of the magnification and evolution biases,
i.e.,

$s^{\rm eff}=\int{\rm d}z\,f(z)s(z)\,\,\,\text{ and }\,\,\,b_{e}^{\rm
eff}=\int{\rm d}z\,f(z)b_{e}(z).$ (24)

This has been done for (a) each one of the 5000 random chains, i.e.,
calculating $f(z)$, $s(z)$ and $b_{e}(z)$ for each chain and then projecting
along $z$, and (b) by fixing them to the best-fitting functions. In the first
case, the results agree within the 1$\sigma$ errors; for the second case,
however, the amplitudes are overestimated by more than 3$\sigma$ for the LEDE7
and LEDE8 models.

Our results summarise the expected dipole in the number counts of quasars from
a pure kinematic origin. They are all above the intrinsic dipole expected from
the clustering anisotropy (grey dashed line in Figure 2), which comes from
fluctuations in the number density around the FLRW metric (Nadolny et al.,
2021) and can be distinguished given the 1$\sigma$ errors.

##  6 Conclusions

In this work, we extended the analysis of Dalang & Bonvin (2022) to include
uncertainties on the magnification $s$ and evolution $b_{e}$ biases (Equations
9 and 2). We used a maximum-likelihood approach on six QLF models (Equation
19) to derive the number density $n(z)$ (Equation 7), $s(z)$, and $b_{e}(z)$
(Wang et al., 2020), finding that models of QLF have a non-negligible impact
on the biases (Figure 1). From MCMC chains, we calculated the kinematic dipole
amplitude ${\cal D}$ (Equation 14), finding that the QLF introduces a large
model dependence on ${\cal D}$. In some cases, a $\sim 3\sigma$ tension on the
dipole amplitude computed with different models (Figure 2) is found.

The constraints on ${\cal D}$ are robust to a change of cosmology (see Table
1). Neglecting data from the last redshift bin does not degrade the
constraints, as the $1\sigma$ error bars on ${\cal D}$ are of the same order,
although the amplitude increases with higher magnitude thresholds $M_{{\rm
c},g}$, mainly due to a different redshift range.

We analysed the impact of neglecting the redshift evolution of the
magnification and evolution biases by computing their effective values,
$s^{\rm eff}$ and $b_{e}^{\rm eff}$ (Equation 24). When these parameters were
computed for each MCMC sample, we found a shift in the expected amplitude
within the $1\sigma$ region of the cases in which the redshift evolution was
accounted for; by obtaining them from the best-fitting functions, we found an
overestimation of the 2D amplitude ${\cal D}$ when redshift evolution was
neglected in quantities that evolve with $z$.

The QLF introduces a model dependence in the dipole amplitude, contrasting
significantly with the model-independent null hypothesis of Ellis & Baldwin
(1984) for testing the validity of the CP. As we have shown, with no single
description of quasar evolution and multiple models fitting the data equally
well, testing the CP with the quasar dipole is complex and not as
straightforward as assumed. The difficulty is exacerbated for radio continuum
catalogues, which have no redshift information.

Finally, conclusions from our work cannot be directly extrapolated to the
results of Secrest et al. (2022), given the different wavelengths, magnitude
thresholds and redshift information for the CatWISE2020 and eBOSS catalogues.
In addition, following Dalang & Bonvin (2022), we used an alternative
theoretical approach to Secrest et al. (2022) (and to previous work on radio
continuum samples) – which is based on the evolution bias of the sample,
rather than on the spectral indices of individual sources. We find, consistent
with Dalang & Bonvin (2022), that the magnification and evolution biases vary
with redshift for all the models considered for the eBOSS quasars, with
significant implications for the dipole amplitude. Therefore it is paramount
to investigate whether evolution effects are also present in the data
considered for the analyses leading to Equation (5), before strong claims
about violation of the CP are validated.

## Acknowledgements

We thank Charles Dalang, Carolina Queiroz, José Luis Bernal, Mike Shengbo
Wang, and Phil Bull for discussions and comments. C.G. and C.C. are supported
by the UK Science & Technology Facilities Council consolidated grant
ST/T000341/1. J.P. received support from the French government under the
France 2030 investment plan, as part of the Initiative d’Excellence d’Aix-
Marseille Université -A*MIDEX (AMX-19-IET-008). R.M. is supported by the South
African Radio Astronomy Observatory and the National Research Foundation
(grant No. 75415).

##  Appendix A Quasar luminosity function models

In this appendix we describe the QLF models considered for the main analysis.

###  A.1 Pure-Luminosity Evolution (PLE)

In this model, the luminosity function evolves solely through a redshift
evolution in the break magnitude $M_{*}$ (; )

$M_{*}(z)=M_{*}(z_{p})-2.5\left[k_{1}(z-z_{p})+k_{2}(z-z_{p})^{2}\right].$
(A1)

Early studies showed a preference for this model, but deeper data revealed
deviations from the PLE model at redshifts larger than $z\gtrsim 2$ (Ross et
al., 2013) for $z_{p}=0$. Therefore, the bright- and faint-end slopes are
allowed to vary for low ($z<z_{p}$) and high ($z>z_{p}$) redshifts,
$a_{\\{l,h\\}},b_{\\{l,h\\}}$, and so are $k_{1}$ and $k_{2}$. The pivot
redshift is $z_{p}=2.2$. In this model, there are 10 free parameters:
$\\{a_{l},b_{l},a_{h},b_{h},k_{1,l},k_{2,l},k_{1,h},k_{2,h},M_{*}(z_{p}),\log_{10}\Phi_{*}\\}$
(PD2016).

###  A.2 Luminosity and Density Evolution (LEDE)

We also consider the case in which both the luminosity and number density vary
with time: the so-called Luminosity Evolution + Density Evolution (LEDE) model
(; ). We analyse this particular case motivated by the fact that the
discontinuity present at the pivot redshift $z_{p}=2.2$ for the derived
quantities is removed, thus leading to a smooth behaviour with redshift for
the derived quantities.

We will take the density and magnitude evolution, respectively, as

$\log_{10}\Phi_{*}(z)=\log_{10}\Phi_{*}(z_{p})+c_{1}(z-z_{p})+c_{2}(z-z_{p})^{2},$
(A2) $M_{*}(z)=M_{*}(z_{p})+c_{3}(z-z_{p}).$ (A3)

Because it has seven free parameters $\\{a,b$, $c_{1},c_{2},c_{3}$,
$\log_{10}\Phi_{*}(z_{p})$, $M_{*}(z_{p})\\}$, we will label it LEDE7
hereafter. Notice that, in this case, the bright and faint ends are fixed for
the whole redshift range: $a(z)=a$, $b(z)=b$.

We shall also consider a quadratic redshift dependence for the break magnitude
(Boyle et al., 2000; Croom et al., 2009)

$M_{*}(z)=M_{*}(z_{p})+c_{3}(z-z_{p})+c_{4}(z-z_{p})^{2}.$ (A4)

We dub this case LEDE8 as it contains eight free parameters: $\\{a,b$,
$c_{1},c_{2},c_{3},c_{4}$, $\log_{10}\Phi_{*}(z_{p})$, $M_{*}(z_{p})\\}$.
Again, $a$ and $b$ are fixed.

Finally, we consider extensions to LEDE7 and LEDE8, in which both slopes are
allowed to evolve linearly with redshift

$a(z)=a(z_{p})+c_{a}(z-z_{p}),\,\,\,\,b(z)=b(z_{p})+c_{b}(z-z_{p}).$ (A5)

These will be called, respectively, LEDE7+2 and LEDE8+2, where the $+2$
indicates the extra $c_{a}$ and $c_{b}$ parameters introduced for the redshift
evolution in the bright and faint ends.

###  A.3 Composite Evolution Model (PLE+LEDE)

We also consider the combination of the PLE functional form for $z<z_{p}$, and
the LEDE7 model for $z>z_{p}$ (PD2016). Other combinations with LEDE8 and
redshift dependencies of the slopes are possible. However, we focus on this
particular combination of PLE + LEDE7 and allow only the bright slope to
evolve with redshift: $\alpha(z)=\alpha(z_{p})+c_{a}(z-z_{p})$. In total, we
have 10 free parameters:
$\\{\alpha(z_{p}),\beta,k_{1},k_{2},c_{1},c_{2},c_{3},c_{a},\log_{10}\Phi_{*}(z_{p}),M_{*}(z_{p})\\}$.

##  Appendix B MCMC results

In Table LABEL:tab:MCMCresults, we present the best-fitting values for the QLF
obtained for the six models described in Section 3.1, from the MCMC analysis
discussed in Section 4.1 by considering all eight redshift bins, removing the
last bin and neglecting data from the last two bins.

There is a small impact in removing high-redshift information from the MCMC
analysis for the LEDE7 model: the parameters $c_{1}$, $c_{2}$ and $c_{3}$ show
a slight degradation in the constraints, while the best-fitting values are in
agreement within the $2\sigma$ region. The same happens for the PLE analysis,
except for the high-redshift parameters, which are severely degraded with the
removal of the last two redshift bins (i.e., considering data between
$0.68<z<3.5$), as it is expected since there is no constraining power at the
high-$z$ end, even though the fits are improved due to the fact that there are
more data at low $z$ (which can also be seen by the decrease in the BIC
value). We can also observe larger errors for the PLE+LEDE $c_{1},c_{2},c_{3}$
and $c_{a}$ parameters, with a $2\sigma$ disagreement for $c_{1}$ and $c_{2}$
after removing the last two redshift bins (considering only the data between
$0.68<z<3$). On the other hand, LEDE8, LEDE8+2 and LEDE7+2 are reasonably
insensitive to high-redshift information, showcasing similar constraints for
the three redshift ranges considered.

In Figure 3, we show the best-fitting QLF from the corresponding MCMC analysis
with all eight redshift bins. Dashed lines mark the limiting magnitude cut
$M_{\mathrm{c}}=-25$ for the last redshift bin. When we consider all redshift
bins and a constant magnitude cut for the dipole analysis, we neglect data
contained in the shaded region. Discarding higher bins in the dipole analysis
is equivalent to moving the dashed line and region of exclusion to fainter
magnitudes.

Table 2: Best-fitting values for the parameters of the models described in Section 3.1, obtained from the MCMC analysis described in Section 4.1. The last column gives the Bayesian Information Criterion (BIC, Equation 23) for each analysis. Model |  $\bm{z}$-range | Parameters | BIC
---|---|---|---
PLE | 8 bins | $M_{*}(z_{p})$ | $\log_{10}\Phi(z_{p})$ |  |  |  |  |  | 176.97
$0.68-4.0$ | $-25.86^{+0.12}_{-0.12}$ | $-5.63^{+0.04}_{-0.05}$ |  |  |  |  |
| $a_{l}$ | $b_{l}$ | $k_{1,l}$ | $k_{2,l}$ |  |  |
$0.68-2.2$ | $-3.09^{+0.09}_{-0.10}$ | $-1.31^{+0.04}_{-0.04}$ | $-0.16^{+0.05}_{-0.05}$ | $-0.42^{+0.04}_{-0.04}$ |  |  |
| $a_{h}$ | $b_{h}$ | $k_{1,h}$ | $k_{2,h}$ |  |  |
$2.2-4.0$ | $-2.46^{+0.05}_{-0.05}$ | $-1.10^{+0.07}_{-0.06}$ | $-0.42^{+0.07}_{-0.07}$ | $0.01^{+0.05}_{-0.05}$ |  |  |
7 bins | $M_{*}(z_{p})$ | $\log_{10}\Phi(z_{p})$ |  |  |  |  |  | 171.88
$0.68-3.5$ | $-25.82^{+0.13}_{-0.12}$ | $-5.62^{+0.05}_{-0.05}$ |  |  |  |  |
| $a_{l}$ | $b_{l}$ | $k_{1,l}$ | $k_{2,l}$ |  |  |
$0.68-2.2$ | $-3.06^{+0.09}_{-0.10}$ | $-1.29^{+0.04}_{-0.04}$ | $-0.16^{+0.05}_{-0.05}$ | $-0.42^{+0.04}_{-0.04}$ |  |  |
| $a_{h}$ | $b_{h}$ | $k_{1,h}$ | $k_{2,h}$ |  |  |
$2.2-3.5$ | $-2.43^{+0.05}_{-0.06}$ | $-1.09^{+0.07}_{-0.07}$ | $-0.47^{+0.10}_{-0.10}$ | $0.05^{+0.08}_{-0.08}$ |  |  |
6 bins | $M_{*}(z_{p})$ | $\log_{10}\Phi(z_{p})$ |  |  |  |  |  | 151.77
$0.68-3.0$ | $-25.63^{+0.14}_{-0.14}$ | $-5.53^{+0.05}_{-0.06}$ |  |  |  |  |
| $a_{l}$ | $b_{l}$ | $k_{1,l}$ | $k_{2,l}$ |  |  |
$0.68-2.2$ | $-2.93^{+0.09}_{-0.10}$ | $-1.21^{+0.05}_{-0.05}$ | $-0.09^{+0.06}_{-0.06}$ | $-0.37^{+0.04}_{-0.04}$ |  |  |
| $a_{h}$ | $b_{h}$ | $k_{1,h}$ | $k_{2,h}$ |  |  |
$2.2-3.0$ | $-2.37^{+0.07}_{-0.07}$ | $-0.96^{+0.12}_{-0.10}$ | $-1.03^{+0.24}_{-0.26}$ | $0.90^{+0.34}_{-0.33}$ |  |  |
PLE+LEDE | 8 bins | $M_{*}(z_{p})$ | $\log_{10}\Phi(z_{p})$ | $\alpha(z_{p})$ | $\beta$ |  |  |  | 175.50
$0.68-4.0$ | $-25.52^{+0.16}_{-0.16}$ | $-5.48^{+0.06}_{-0.06}$ | $-2.80^{+0.08}_{-0.09}$ | $-1.16^{+0.06}_{-0.06}$ |  |  |
| $k_{1}$ | $k_{2}$ |  |  |  |  |
$0.68-2.2$ | $-0.10^{+0.05}_{-0.05}$ | $-0.38^{+0.04}_{-0.04}$ |  |  |  |  |
| $c_{1}$ | $c_{2}$ | $c_{3}$ | $c_{a}$ |  |  |
$2.2-4.0$ | $-0.62^{+0.06}_{-0.06}$ | $-0.06^{+0.05}_{-0.05}$ | $-0.67^{+0.13}_{-0.13}$ | $-0.01^{+0.11}_{-0.11}$ |  |  |
7 bins | $M_{*}(z_{p})$ | $\log_{10}\Phi(z_{p})$ | $\alpha(z_{p})$ | $\beta$ |  |  |  | 163.30
$0.68-3.5$ | $-25.53^{+0.16}_{-0.16}$ | $-5.47^{+0.06}_{-0.06}$ | $-2.80^{+0.09}_{-0.09}$ | $-1.15^{+0.07}_{-0.06}$ |  |  |
| $k_{1}$ | $k_{2}$ |  |  |  |  |
$0.68-2.2$ | $-0.08^{+0.05}_{-0.05}$ | $-0.37^{+0.04}_{-0.04}$ |  |  |  |  |
| $c_{1}$ | $c_{2}$ | $c_{3}$ | $c_{a}$ |  |  |
$2.2-3.5$ | $-0.73^{+0.07}_{-0.07}$ | $0.07^{+0.07}_{-0.07}$ | $-0.69^{+0.16}_{-0.15}$ | $0.09^{+0.13}_{-0.13}$ |  |  |
6 bins | $M_{*}(z_{p})$ | $\log_{10}\Phi(z_{p})$ | $\alpha(z_{p})$ | $\beta$ |  |  |  | 140.38
$0.68-3.0$ | $-25.58^{+0.16}_{-0.16}$ | $-5.47^{+0.06}_{-0.06}$ | $-2.84^{+0.09}_{-0.10}$ | $-1.16^{+0.07}_{-0.06}$ |  |  |
| $k_{1}$ | $k_{2}$ |  |  |  |  |
$0.68-2.2$ | $-0.03^{+0.06}_{-0.06}$ | $-0.34^{+0.04}_{-0.04}$ |  |  |  |  |
| $c_{1}$ | $c_{2}$ | $c_{3}$ | $c_{a}$ |  |  |
$2.2-3.0$ | $-1.08^{+0.12}_{-0.12}$ | $0.68^{+0.18}_{-0.18}$ | $-0.77^{+0.28}_{-0.26}$ | $0.36^{+0.22}_{-0.22}$ |  |  |
LEDE7 | 8 bins | $M_{*}(z_{p})$ | $\log_{10}\Phi(z_{p})$ |  |  |  |  |  | 169.05
$0.68-4.0$ | $-25.37^{+0.16}_{-0.16}$ | $-5.46^{+0.05}_{-0.05}$ |  |  |  |  |
$a$ | $b$ | $c_{1}$ | $c_{2}$ | $c_{3}$ |  |
$-2.66^{+0.07}_{-0.08}$ | $-1.03^{+0.08}_{-0.07}$ | $-0.43^{+0.01}_{-0.01}$ | $-0.30^{+0.01}_{-0.01}$ | $-0.96^{+0.04}_{-0.04}$ |  |
7 bins | $M_{*}(z_{p})$ | $\log_{10}\Phi(z_{p})$ |  |  |  |  |  | 159.27
$0.68-3.5$ | $-25.40^{+0.16}_{-0.16}$ | $-5.47^{+0.05}_{-0.06}$ |  |  |  |  |
$a$ | $b$ | $c_{1}$ | $c_{2}$ | $c_{3}$ |  |
$-2.67^{+0.07}_{-0.08}$ | $-1.04^{+0.08}_{-0.07}$ | $-0.44^{+0.01}_{-0.01}$ | $-0.30^{+0.01}_{-0.01}$ | $-0.97^{+0.04}_{-0.04}$ |  |
6 bins | $M_{*}(z_{p})$ | $\log_{10}\Phi(z_{p})$ |  |  |  |  |  | 130.82
$0.68-3.0$ | $-25.61^{+0.17}_{-0.17}$ | $-5.54^{+0.06}_{-0.06}$ |  |  |  |  |
$a$ | $b$ | $c_{1}$ | $c_{2}$ | $c_{3}$ |  |
$-2.74^{+0.08}_{-0.09}$ | $-1.10^{+0.07}_{-0.07}$ | $-0.49^{+0.02}_{-0.02}$ | $-0.32^{+0.01}_{-0.01}$ | $-1.08^{+0.04}_{-0.04}$ |  |
LEDE7+2 | 8 bins | $M_{*}(z_{p})$ | $\log_{10}\Phi(z_{p})$ |  |  |  |  |  | 144.39
$0.68-4.0$ | $-26.17^{+0.16}_{-0.16}$ | $-5.77^{+0.06}_{-0.07}$ |  |  |  |  |
$c_{1}$ | $c_{2}$ | $c_{3}$ |  |  |  |
$-0.94^{+0.06}_{-0.06}$ | $-0.4^{+0.02}_{-0.02}$ | $-1.99^{+0.1}_{-0.1}$ |  |  |  |
$a(z_{p})$ | $c_{a}$ | $b(z_{p})$ | $c_{b}$ |  |  |
$-2.84^{+0.08}_{-0.09}$ | $-0.16^{+0.07}_{-0.07}$ | $-1.33^{+0.05}_{-0.05}$ | $-0.5^{+0.04}_{-0.04}$ |  |  |
7 bins | $M_{*}(z_{p})$ | $\log_{10}\Phi(z_{p})$ |  |  |  |  |  | 137.44
$0.68-3.5$ | $-26.16^{+0.16}_{-0.16}$ | $-5.76^{+0.07}_{-0.07}$ |  |  |  |  |
$c_{1}$ | $c_{2}$ | $c_{3}$ |  |  |  |
$-0.92^{+0.06}_{-0.07}$ | $-0.39^{+0.02}_{-0.02}$ | $-1.95^{+0.11}_{-0.11}$ |  |  |  |
$a(z_{p})$ | $c_{a}$ | $b(z_{p})$ | $c_{b}$ |  |  |
$-2.84^{+0.08}_{-0.09}$ | $-0.14^{+0.07}_{-0.07}$ | $-1.33^{+0.05}_{-0.05}$ | $-0.47^{+0.05}_{-0.05}$ |  |  |
6 bins | $M_{*}(z_{p})$ | $\log_{10}\Phi(z_{p})$ |  |  |  |  |  | 119.68
$0.68-3.0$ | $-26.18^{+0.18}_{-0.17}$ | $-5.77^{+0.07}_{-0.07}$ |  |  |  |  |
$c_{1}$ | $c_{2}$ | $c_{3}$ |  |  |  |
$-0.87^{+0.08}_{-0.08}$ | $-0.38^{+0.02}_{-0.02}$ | $-1.84^{+0.14}_{-0.13}$ |  |  |  |
$a(z_{p})$ | $c_{a}$ | $b(z_{p})$ | $c_{b}$ |  |  |
$-2.85^{+0.09}_{-0.10}$ | $-0.09^{+0.06}_{-0.06}$ | $-1.33^{+0.06}_{-0.05}$ | $-0.4^{+0.06}_{-0.06}$ |  |  |
LEDE8 | 8 bins | $M_{*}(z_{p})$ | $\log_{10}\Phi(z_{p})$ |  |  |  |  |  | 141.64
$0.68-4.0$ | $-25.72^{+0.16}_{-0.16}$ | $-5.56^{+0.06}_{-0.06}$ |  |  |  |  |
$a$ | $b$ | $c_{1}$ | $c_{2}$ | $c_{3}$ | $c_{4}$ |
$-2.79^{+0.08}_{-0.09}$ | $-1.10^{+0.07}_{-0.06}$ | $-0.37^{+0.01}_{-0.01}$ | $-0.20^{+0.01}_{-0.01}$ | $-0.78^{+0.04}_{-0.04}$ | $0.34^{+0.04}_{-0.04}$ |
7 bins | $M_{*}(z_{p})$ | $\log_{10}\Phi(z_{p})$ |  |  |  |  |  | 133.70
$0.68-3.5$ | $-25.72^{+0.16}_{-0.16}$ | $-5.56^{+0.06}_{-0.06}$ |  |  |  |  |
$a$ | $b$ | $c_{1}$ | $c_{2}$ | $c_{3}$ | $c_{4}$ |
$-2.79^{+0.08}_{-0.09}$ | $-1.10^{+0.07}_{-0.06}$ | $-0.37^{+0.02}_{-0.02}$ | $-0.20^{+0.02}_{-0.02}$ | $-0.75^{+0.04}_{-0.05}$ | $0.38^{+0.05}_{-0.05}$ |
6 bins | $M_{*}(z_{p})$ | $\log_{10}\Phi(z_{p})$ |  |  |  |  |  | 118.76
$0.68-3.0$ | $-25.75^{+0.17}_{-0.16}$ | $-5.58^{+0.06}_{-0.06}$ |  |  |  |  |
$a$ | $b$ | $c_{1}$ | $c_{2}$ | $c_{3}$ | $c_{4}$ |
$-2.80^{+0.08}_{-0.09}$ | $-1.12^{+0.07}_{-0.06}$ | $-0.40^{+0.02}_{-0.02}$ | $-0.22^{+0.02}_{-0.02}$ | $-0.75^{+0.07}_{-0.07}$ | $0.38^{+0.07}_{-0.07}$ |
LEDE8+2 | 8 bins | $M_{*}(z_{p})$ | $\log_{10}\Phi(z_{p})$ |  |  |  |  |  | 147.19
$0.68-4.0$ | $-25.92^{+0.20}_{-0.20}$ | $-5.65^{+0.08}_{-0.09}$ |  |  |  |  |
$c_{1}$ | $c_{2}$ | $c_{3}$ | $c_{4}$ |  |  |
$-0.50^{+0.09}_{-0.19}$ | $-0.21^{+0.02}_{-0.06}$ | $-1.04^{+0.22}_{-0.42}$ | $0.32^{+0.06}_{-0.11}$ |  |  |
$a(z_{p})$ | $c_{a}$ | $b(z_{p})$ | $c_{b}$ |  |  |
$-2.83^{+0.09}_{-0.10}$ | $-0.01^{+0.09}_{-0.11}$ | $-1.22^{+0.08}_{-0.08}$ | $-0.17^{+0.09}_{-0.17}$ |  |  |
7 bins | $M_{*}(z_{p})$ | $\log_{10}\Phi(z_{p})$ |  |  |  |  |  | 138.06
$0.68-3.5$ | $-25.90^{+0.20}_{-0.20}$ | $-5.64^{+0.08}_{-0.08}$ |  |  |  |  |
$c_{1}$ | $c_{2}$ | $c_{3}$ | $c_{4}$ |  |  |
$-0.48^{+0.09}_{-0.14}$ | $-0.21^{+0.02}_{-0.04}$ | $-0.99^{+0.23}_{-0.30}$ | $0.35^{+0.06}_{-0.08}$ |  |  |
$a(z_{p})$ | $c_{a}$ | $b(z_{p})$ | $c_{b}$ |  |  |
$-2.81^{+0.09}_{-0.10}$ | $0.02^{+0.10}_{-0.10}$ | $-1.21^{+0.08}_{-0.08}$ | $-0.17^{+0.09}_{-0.12}$ |  |  |
6 bins | $M_{*}(z_{p})$ | $\log_{10}\Phi(z_{p})$ |  |  |  |  |  | 118.97
$0.68-3.0$ | $-25.95^{+0.23}_{-0.21}$ | $-5.67^{+0.09}_{-0.09}$ |  |  |  |  |
$c_{1}$ | $c_{2}$ | $c_{3}$ | $c_{4}$ |  |  |
$-0.55^{+0.11}_{-0.12}$ | $-0.24^{+0.03}_{-0.04}$ | $-1.06^{+0.29}_{-0.29}$ | $0.33^{+0.09}_{-0.09}$ |  |  |
$a(z_{p})$ | $c_{a}$ | $b(z_{p})$ | $c_{b}$ |  |  |
$-2.80^{+0.11}_{-0.12}$ | $0.05^{+0.11}_{-0.12}$ | $-1.24^{+0.08}_{-0.07}$ | $-0.20^{+0.09}_{-0.09}$ |  |  |
Figure 3: Best-fitting QLFs for the models described in Section 3.1. The
corresponding best-fitting parameters are shown in Table
LABEL:tab:MCMCresults. The data points and uncertainties are taken from
PD2016. The dashed line separates the region below the limiting magnitude cut
$M_{\mathrm{c}}=-25$ for the last redshift bin (Wang et al., 2020). The shaded
region is neglected when we consider all eight redshift bins for the dipole
analysis.

## References

  * Ade et al. (2016) Ade, P. A. R., Aghanim, N., Arnaud, M., et al. 2016, A&A, 594, A7, doi: 10.1051/0004-6361/201525830
  * Aghanim et al. (2020) Aghanim, N., Akrami, Y., Arroja, F., et al. 2020, Astronomy & Astrophysics, 641, A1, doi: 10.1051/0004-6361/201833880
  * Alonso et al. (2015) Alonso, D., Bull, P., Ferreira, P. G., Maartens, R., & Santos, M. G. 2015, The Astrophysical Journal, 814, 145, doi: 10.1088/0004-637X/814/2/145
  * Bengaly et al. (2018) Bengaly, C. A., Maartens, R., & Santos, M. G. 2018, Journal of Cosmology and Astroparticle Physics, 2018, 031, doi: 10.1088/1475-7516/2018/04/031
  * Bengaly et al. (2019) Bengaly, C. A., Siewert, T. M., Schwarz, D. J., & Maartens, R. 2019, Monthly Notices of the Royal Astronomical Society, 486, 1350, doi: 10.1093/mnras/stz832
  * Blake & Wall (2002) Blake, C., & Wall, J. 2002, arXiv preprint astro-ph/0203385, doi: 10.1038/416150a
  * Blas et al. (2011) Blas, D., Lesgourgues, J., & Tram, T. 2011, J. Cosmology Astropart. Phys, 2011, 034–034, doi: 10.1088/1475-7516/2011/07/034
  * Bonvin & Durrer (2011) Bonvin, C., & Durrer, R. 2011, Phys. Rev. D, 84, 063505, doi: 10.1103/PhysRevD.84.063505
  * Boyle et al. (2000) Boyle, B. J., Shanks, T., Croom, S., et al. 2000, Monthly Notices of the Royal Astronomical Society, 317, 1014, doi: 10.1046/j.1365-8711.2000.03730.x
  * Carballo et al. (1999) Carballo, R., González-Serrano, J. I., Benn, C. R., Sánchez, S. F., & Vigotti, M. 1999, Monthly Notices of the Royal Astronomical Society, 306, 137, doi: 10.1023/A:1002189323147
  * Challinor & Lewis (2011) Challinor, A., & Lewis, A. 2011, Physical Review D, 84, 043516, doi: 10.1103/PhysRevD.84.043516
  * Clarkson (2012) Clarkson, C. 2012, Comptes Rendus Physique, 13, 682, doi: 10.1016/j.crhy.2012.04.005
  * Clarkson & Maartens (2010) Clarkson, C., & Maartens, R. 2010, Class. Quant. Grav., 27, 124008, doi: 10.1088/0264-9381/27/12/124008
  * Colin et al. (2017) Colin, J., Mohayaee, R., Rameez, M., & Sarkar, S. 2017, Monthly Notices of the Royal Astronomical Society, 471, 1045, doi: 10.1093/mnras/stx1631
  * Croom et al. (2004) Croom, S. M., Smith, R., Boyle, B., et al. 2004, Monthly Notices of the Royal Astronomical Society, 349, 1397, doi: 10.1111/j.1365-2966.2004.07619.x
  * Croom et al. (2009) Croom, S. M., Richards, G. T., Shanks, T., et al. 2009, Monthly Notices of the Royal Astronomical Society, 399, 1755, doi: 10.1111/j.1365-2966.2009.15398.x
  * Dalang & Bonvin (2022) Dalang, C., & Bonvin, C. 2022, Monthly Notices of the Royal Astronomical Society, 512, 3895, doi: 10.1093/mnras/stac726
  * Darling (2022) Darling, J. 2022, The Astrophysical Journal Letters, 931, L14, doi: 10.3847/2041-8213/ac6f08
  * Das et al. (2021) Das, K. K., Sankharva, K., & Jain, P. 2022, Journal of Cosmology and Astroparticle Physics, 2021, 035, doi: 10.1088/1475-7516/2021/07/035
  * Domènech et al. (2022) Domènech, G., Mohayaee, R., Patil, S. P., & Sarkar, S. 2022, Journal of Cosmology and Astroparticle Physics, 2022, 019, doi: 10.1088/1475-7516/2022/10/019
  * Ehlers et al. (1968) Ehlers, J., Geren, P., & Sachs, R. K. 1968, Journal of Mathematical Physics, 9, 1344, doi: 10.1063/1.1664720
  * Ellis & Baldwin (1984) Ellis, G., & Baldwin, J. 1984, Monthly Notices of the Royal Astronomical Society, 206, 377, doi: 10.1093/mnras/206.2.377
  * Ellis et al. (1983) Ellis, G., Treciokas, R., & Matravers, D. 1983, Annals of Physics, 150, 487, doi: 10.1016/0003-4916(83)90024-6
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306, doi: 10.1086/670067
  * Ghosh et al. (2016) Ghosh, S., Jain, P., Kashyap, G., et al. 2016, Journal of Astrophysics and Astronomy, 37, 1, doi: 10.1007/s12036-016-9395-8
  * Gibelyou & Huterer (2012) Gibelyou, C., & Huterer, D. 2012, Monthly Notices of the Royal Astronomical Society, 427, 1994, doi: 10.1111/j.1365-2966.2012.22032.x
  * Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357, doi: 10.1038/s41586-020-2649-2
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90, doi: 10.1109/MCSE.2007.55
  * Liddle (2004) Liddle, A. R. 2004, Monthly Notices of the Royal Astronomical Society, 351, L49, doi: 10.1111/j.1365-2966.2004.08033.x
  * Maartens et al. (2018) Maartens, R., Clarkson, C., & Chen, S. 2018, Journal of Cosmology and Astroparticle Physics, 2018, 013, doi: 10.1088/1475-7516/2018/01/013
  * Maartens et al. (2021) Maartens, R., Fonseca, J., Camera, S., et al. 2021, Journal of Cosmology and Astroparticle Physics, 2021, 009, doi: 10.1088/1475-7516/2021/12/009
  * Nadolny et al. (2021) Nadolny, T., Durrer, R., Kunz, M., & Padmanabhan, H. 2021, Journal of Cosmology and Astroparticle Physics, 2021, 009, doi: 10.1088/1475-7516/2021/11/009
  * Palanque-Delabrouille et al. (2011) Palanque-Delabrouille, N., Yèche, C., Myers, A., et al. 2011, Astronomy & Astrophysics, 530, A122, doi: 10.1051/0004-6361/201016254
  * Palanque-Delabrouille et al. (2016) Palanque-Delabrouille, N., Magneville, C., Yèche, C., et al. 2016, Astronomy & Astrophysics, 587, A41, doi: 10.1051/0004-6361/201527392
  * Peebles & Wilkinson (1968) Peebles, P., & Wilkinson, D. T. 1968, Physical Review, 174, 2168, doi: 10.1103/PhysRev.174.2168
  * Pozzetti et al. (2016) Pozzetti, L., Hirata, C., Geach, J., et al. 2016, Astronomy & Astrophysics, 590, A3, doi: 10.1051/0004-6361/201527081
  * Rasera et al. (2022) Rasera, Y., Breton, M.-A., Corasaniti, P.-S., et al. 2022, Astronomy & Astrophysics, 661, A90, doi: 10.1051/0004-6361/202141908
  * Ross et al. (2013) Ross, N. P., McGreer, I. D., White, M., et al. 2013, The Astrophysical Journal, 773, 14, doi: 10.1088/0004-637X/773/1/14
  * Rubart & Schwarz (2013) Rubart, M., & Schwarz, D. J. 2013, Astronomy & Astrophysics, 555, A117, doi: 10.1051/0004-6361/201321215
  * Schmidt et al. (2010) Schmidt, K. B., Marshall, P. J., Rix, H.-W., et al. 2010, The Astrophysical Journal, 714, 1194, doi: 10.1088/0004-637X/714/2/1194
  * Secrest et al. (2022) Secrest, N. J., von Hausegger, S., Rameez, M., Mohayaee, R., & Sarkar, S. 2022, The Astrophysical Journal Letters, 937, L31, doi: 10.3847/2041-8213/ac88c0
  * Secrest et al. (2021) Secrest, N. J., von Hausegger, S., Rameez, M., et al. 2021, The Astrophysical journal letters, 908, L51, doi: 10.3847/2041-8213/abdd40
  * Stewart & Sciama (1967) Stewart, J., & Sciama, D. 1967, Nature, 216, 748, doi: 10.1038/216748a0
  * Stoeger et al. (1995) Stoeger, W. R., Maartens, R., & Ellis, G. 1995, The Astrophysical Journal, 443, 1, doi: 10.1086/175496
  * Tiwari et al. (2015) Tiwari, P., Kothari, R., Naskar, A., Nadkarni-Ghosh, S., & Jain, P. 2015, Astroparticle Physics, 61, 1, doi: 10.1016/j.astropartphys.2014.06.004
  * Tiwari & Nusser (2016) Tiwari, P., & Nusser, A. 2016, Journal of Cosmology and Astroparticle Physics, 2016, 062, doi: 10.1088/1475-7516/2016/03/062
  * Tiwari et al. (2022) Tiwari, P., Kothari, R., & Jain, P. 2022, The Astrophysical Journal Letters, 2022, L36, doi: 10.3847/2041-8213/ac447a
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2
  * Wang et al. (2020) Wang, M. S., Beutler, F., & Bacon, D. 2020, Monthly Notices of the Royal Astronomical Society, 499, 2598, doi: 10.1093/mnras/staa2998


# On the Nörlund logarithmic means with respect to Vilenkin system in the
martingale Hardy space $H_{1}$

L.-E. Persson, G. Tephnadze, P. Wall  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.  Lars-
<EMAIL_ADDRESS>G. Tephnadze, The University of Georgia, school of
Informatics, Engineering and Mathematics, IV, 77a Merab Kostava St, Tbilisi,
0128, Georgia, & Department of Engineering Sciences and Mathematics,
LuleåUniversity of Technology, SE-971 87 Luleå, Sweden.
<EMAIL_ADDRESS>P. Wall, Department of Engineering Sciences and
Mathematics, LuleåUniversity of Technology, SE-971 87 Luleå, Sweden.
<EMAIL_ADDRESS>
###### Abstract.

In this paper we prove and discuss a new divergence result of Nörlund
logarithmic means with respect to Vilenkin system in Hardy space $H_{1}.$

The research was supported by Shota Rustaveli National Science Foundation
grant YS15-2.1.1-47, by a Swedish Institute scholarship no. 24155/2016

2000 Mathematics Subject Classification. 42C10.

Key words and phrases: Vilenkin system, Nörlund logarithmic means, partial
sums, modulus of continuity, Hardy space.

##  1\. Introduction

It is well-known that (see e.g. [1] and [5]) Vilenkin systems do not form
bases in the space $L_{1}.$ Moreover, there exists a function in the dyadic
Hardy space $H_{1},$ such that the partial sums of $f$ are not bounded in
$L_{1}$-norm.

In [10] (see also [13]) it was proved that the following is true:

Theorem T1: The maximal operator $\widetilde{S}^{\ast}$ defined by

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

is bounded from the Hardy space $H_{1}$ to the space $L_{1}.$ Here $S_{n}$
denotes the $n$-th partial sum with respect to the Vilenkin system. Moreover,
it was proved that the rate of the factor $log(n+1)$ is in a sense sharp.

Móricz and Siddiqi [6] investigate the approximation properties of some
special Nörlund means of Walsh-Fourier series of $L_{p}$ functions in norm.
Fridli, Manchanda and Siddiqi [3] improved and extended the results of Móricz
and Siddiqi [6] to Martingale Hardy spaces. However, the case when
$\left\\{q_{k}=1/k:k\in\mathbb{N}_{+}\right\\}$ was excluded, since the
methods are not applicable to Nörlund logarithmic means. In [4] Gát and
Goginava proved some convergence and divergence properties of Walsh-Fourier
series of the Nörlund logarithmic means of functions in the Lebesgue space
$L_{1}.$ In particular, they proved that there exists an function in the space
$L_{1},$ such that

$\sup_{n\in\mathbb{N}}\left\|L_{n}f\right\|_{1}=\infty.$

Analogical result for some unbounded Vilenkin systems was proved in [2].

In [8] (see also [13]) it was proved that there exists a martingale $f\in
H_{p},\ \ (0<p<1)$ such that

$\sup_{n\in\mathbb{N}}\left\|L_{n}f\right\|_{p}=\infty,$

Here $L_{n}$ is $n$-th Nörlund logarithmic means with respect to Vilenkin
system.

In this paper we prove an analogical result for the bounded Vilenkin systems
in the case when $p=1.$ Moreover, we discuss boundedness of weighted maximal
operators on the Hardy space $H_{1}.$

##  2\. Preliminaries

Let $\mathbb{N}_{+}$ denote the set of the positive integers,
$\mathbb{N}:=\mathbb{N}_{+}\cup\\{0\\}.$

Let $m:=(m_{0},m_{1},\dots)$ denote a sequence of the positive numbers not
less than 2.

Denote by

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

the additive group of integers modulo $m_{k},$ $k\in\mathbb{N}.$

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 direct product $\mu$ of the measures

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

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

If $\sup\limits_{n}m_{n}<\infty$, then we call $G_{m}$ a bounded Vilenkin
group. If the generating sequence $m$ is not bounded, then $G_{m}$ is said to
be an unbounded Vilenkin group. In this paper we discuss bounded Vilenkin
groups only.

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

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

It is easy to give a base for the neighbourhood of $G_{m}:$

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

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

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

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

If we define the so-called generalized powers system based on $m$ in the
following way:

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

then every $n\in\mathbb{N}$ can be uniquely expressed as
$n=\overset{\infty}{\underset{k=0}{\sum}}n_{k}M_{k},$ where $n_{k}\in
Z_{m_{k}}\ (k\in\mathbb{N})$ and only a finite number of $n_{k}`$s differ from
zero. Let $\left|n\right|:=\max\\{k\in\mathbb{N}:\ n_{k}\neq 0\\}.$

Next, we introduce on $G_{m}$ an orthonormal system, which is called the
Vilenkin system. First define the complex valued functions
$r_{k}\left(x\right):G_{m}\rightarrow\mathbb{C},$ the generalized Rademacher
functions, by

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

Now, define the Vilenkin systems $\psi:=(\psi_{n}:n\in\mathbb{N})$ on $G_{m}$
as:

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

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

The Vilenkin systems are orthonormal and complete in $L_{2}\left(G_{m}\right)$
(see e.g. [1, 14]).

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

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

Let $\left\\{q_{k}:k\geq 0\right\\}$ be a sequence of nonnegative numbers. The
$n$-th Nörlund mean for the Fourier series of $f$ is defined by

$t_{n}f=\frac{1}{{l_{n}}}\sum_{k=0}^{n-1}q_{n-k}S_{k}f.$

If $q_{k}={1}/{k}$, then we get the Nörlund logarithmic means:

$L_{n}f:=\frac{1}{l_{n}}\sum_{k=0}^{n}\frac{S_{k}f}{n-k},\ \
l_{n}:=\sum_{k=1}^{n}\frac{1}{k},$

The kernel of the Nörlund logarithmic means is defined by

$F_{n}:=\frac{1}{l_{n}}\sum_{k=0}^{n}\frac{D_{k}}{n-k}.$

In the special case when $\\{q_{k}=1:k\in N\\},$ we get the Fejér means

$\sigma_{n}f:=\frac{1}{n}\sum_{k=1}^{n}S_{k}f\,.$

We define $n$-th Fejér kernel

$K_{n}f:=\frac{1}{n}\sum_{k=1}^{n}D_{k}.$

Let $f\in L_{1}(G_{m}).$ Then the maximal function is given by

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

The Hardy martingale spaces $H_{1}$ consist of all martingales for which

$\left\|f\right\|_{H_{1}}:=\left\|f^{\ast}\right\|_{1}<\infty.$

It it well-known (see e.g. [9] and [16]) that if $f\in L_{1},$ then

(1)
$\left\|f\right\|_{H_{1}}\sim\left\|\sup_{n\in\mathbb{N}}\left|S_{M_{n}}f\right|\right\|_{1}.$

A bounded measurable function $a$ is a 1-atom if either $a=1$ or

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

##  3\. Auxiliary propositions

The Hardy martingale space $H_{1}\left(G_{m}\right)$ has an atomic
characterization (see [16], [17]):

######  Lemma 1.

A function $f\in H_{1}$ if and only if there exist a sequence
$\left(a_{k},k\in\mathbb{N}\right)$ of 1-atoms and a sequence
$\left(\mu_{k},k\in\mathbb{N}\right)$ of real numbers such that

(2) $\qquad\sum_{k=0}^{\infty}\mu_{k}a_{k}=f,\text{ \ a.e.,}$

where

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

Moreover,

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

where the infimum is taken over all decomposition of $f$ of the form (2).

In Blahota, Gát [2] the following lemma (see Lemma 5) was proved for unbounded
Vilenkin systems:

######  Lemma 2.

Let $q_{A}=M_{2A}+M_{2A-2}+...+M_{0}.$ If $\log m_{n}=O(n^{\beta}),$ for some
$0<\delta<1/2,$ then

$\left\|F_{q_{A}}\right\|_{1}\geq c(\log q_{A})^{\beta},$

for some $0<\beta<1-\delta.$

For the proof of main result we also need the following new Lemma of
independent interest:

######  Lemma 3.

Let $G_{m}$ be a bounded Vilenkin system and
$q_{A}=M_{2A}+M_{2A-2}+...+M_{0}.$ Then

$\left\|F_{q_{A}}\right\|_{1}\geq c\log q_{A}.$

###### Proof.

During the proof we use some method of Blahota, Gát [2] (see Lemma 5) and Gát,
Goginava [4] (see Lemma 2).

Set

$\theta_{n}:=l_{n}F_{n}=\overset{n-1}{\underset{k=1}{\sum}}\frac{D_{n-k}}{k}.$

Then we have that

$\displaystyle\theta_{q_{A}}(x)$ $\displaystyle=$
$\displaystyle\overset{M_{2A-2}+...+M_{0}-1}{\underset{k=1}{\sum}}\frac{1}{k}D_{M_{2A}+M_{2A-2}+...+M_{0}-k}(x)$
$\displaystyle+$
$\displaystyle\overset{M_{2A}+...+M_{0}-1}{\underset{k=M_{2A-2}+...+M_{0}}{\sum}}\frac{1}{k}D_{M_{2A}+M_{2A-2}+...+M_{0}-k}(x)$
$\displaystyle:=$ $\displaystyle I+II.$

We first discuss $I.$ Since $k<M_{2A-2}+...+M_{0},$ then

$\displaystyle
D_{M_{2A}+...+M_{0}-k}(x)=D_{M_{2A}}(x)+r_{2A}D_{M_{2A-2}+...+M_{0}-k}(x).$

This gives that

(3) $\displaystyle
I=l_{q_{A-1}}D_{M_{2A}}(x)+r_{2A}G_{M_{2A-2}+...+M_{0}}(x).$

Moreover, by using Abel transformation, we get that

$\displaystyle II$ $\displaystyle=$
$\displaystyle\overset{M_{2A}+...+M_{0}-1}{\underset{k=M_{2A-2}+...+M_{0}}{\sum}}\frac{1}{k}D_{M_{2A}+M_{2A-2}+...+M_{0}-k}$
$\displaystyle=$
$\displaystyle\frac{K_{1}}{M_{2A}+...+M_{0}-1}-\frac{\left(M_{2A}+...+M_{0}-2\right)K_{M_{2A}+...+M_{0}-2}}{M_{2A}+...+M_{0}-1}$
$\displaystyle+$
$\displaystyle\overset{M_{2A}+...+M_{0}-2}{\underset{k=M_{2A-2}+...+M_{0}}{\sum}}\frac{M_{2A}+M_{2A-2}+...+M_{0}-k}{k(k+1)}K_{M_{2A}+M_{2A-2}+...+M_{0}-k}.$

Since (for details see e.g. [1]) $\left\|K_{n}\right\|_{1}\leq 2,$ for all
$n\in\mathbb{N},$ we obtain that

$\displaystyle\left\|II\right\|_{1}\leq\frac{2}{M_{2A}+...+M_{0}-1}+\frac{2\left(M_{2A}+...+M_{0}-2\right)}{M_{2A}+...+M_{0}-1}$
$\displaystyle+$ $\displaystyle
2\overset{M_{2A}+...+M_{0}-2}{\underset{k=M_{2A-2}+...+M_{0}}{\sum}}\frac{M_{2A}+M_{2A-2}+...+M_{0}-k}{k(k+1)}<c<\infty.$

Hence, by (3) and (3), we get that

$\left\|\theta_{q_{A}}\right\|_{1}\geq\left\|l_{q_{A-1}}D_{M_{2A}}+r_{2A}\theta_{q_{A-1}}\right\|_{1}-c.$

We now discuss the right-hand side of this inequality, more exactly we give a
lower bound for it. First, we consider the integral on the set $G_{m}\setminus
I_{2A}$:

$\displaystyle\int_{G_{m}\setminus
I_{2A}}{\left|l_{q_{A-1}}D_{M_{2A}}(x)+r_{2A}\theta_{q_{A-1}}(x)\right|}d\mu(x)$
$\displaystyle=$ $\displaystyle\int_{G_{m}\setminus
I_{2A}}{\left|\theta_{q_{A-1}}(x)\right|}d\mu(x)=\left\|\theta_{q_{A-1}}\right\|_{1}-\int_{I_{2A}}{\left|\theta_{q_{A-1}}(x)\right|}d\mu(x)$
$\displaystyle=$
$\displaystyle\left\|\theta_{q_{A-1}}\right\|_{1}-\frac{\theta_{q_{A-1}}(0)}{M_{2A}}.$

Next, we note that on the set $I_{2A}$ we have that

$\displaystyle\int_{I_{2A}}{\left|l_{q_{A-1}}D_{M_{2A}}(x)+r_{2A}\theta_{q_{A-1}}(x)\right|}d\mu(x)=l_{q_{A-1}}-\frac{\theta_{q_{A-1}}(0)}{M_{2A}}.$

It follows that

(5)
$\left\|\theta_{q_{A}}\right\|_{1}\geq\left\|\theta_{q_{A-1}}\right\|_{1}+l_{q_{A-1}}-\frac{2\theta_{q_{A-1}}(0)}{M_{2A}}-c.$

Moreover, according to simple estimation,

$\theta_{n}(0)=\overset{n-1}{\underset{k=1}{\sum}}\frac{n-k}{k}=n\overset{n-1}{\underset{k=1}{\sum}}\frac{1}{k}-n+1=nl_{n}+O(n).$

Therefore, since

$q_{A-1}\leq
M_{2A-2}(1+\frac{1}{4}+\frac{1}{16}+...)=\frac{4}{3}M_{2A-2}\leq\frac{1}{3}M_{2A}$

we obtain that

(6)
$\frac{2}{l_{q_{A}}M_{2A}}\theta_{q_{A-1}}(0)=\frac{2q_{A-1}}{M_{2A}}\frac{l_{q_{A-1}}}{l_{q_{A}}}+o(1)\leq\frac{2}{3}\frac{l_{q_{A-1}}}{l_{q_{A}}}+o(1).$

Finally, by using (5) and (6), we conclude that

$\displaystyle\left\|F_{q_{A}}\right\|_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{l_{q_{A}}}\left\|\theta_{q_{A}}\right\|_{1}$
$\displaystyle\geq$
$\displaystyle\frac{1}{l_{q_{A}}}\left\|\theta_{q_{A-1}}\right\|_{1}+\frac{l_{q_{A-1}}}{l_{q_{A}}}-\frac{2}{3}\frac{l_{q_{A-1}}}{l_{q_{A}}}-o(1)$
$\displaystyle\geq$
$\displaystyle\frac{1}{l_{q_{A}}}\left\|\theta_{q_{A-1}}\right\|_{1}+\frac{1}{3}\frac{l_{q_{A-1}}}{l_{q_{A}}}-o(1)$
$\displaystyle\geq$
$\displaystyle\frac{1}{l_{q_{A}}}\left\|\theta_{q_{A-1}}\right\|_{1}+\frac{1}{6}-o(1)$
$\displaystyle\geq$
$\displaystyle\frac{l_{q_{A-1}}}{l_{q_{A}}}\left\|F_{q_{A-1}}\right\|_{1}+\frac{1}{6}-o(1)$
$\displaystyle\geq$
$\displaystyle\frac{l_{q_{A-1}}}{l_{q_{A}}}\left\|F_{q_{A-1}}\right\|_{1}+\frac{1}{8}$

By iterating this estimate we get

$\left\|F_{q_{A}}\right\|_{1}\geq\frac{1}{8}\overset{A-1}{\underset{k=1}{\sum}}\frac{l_{q_{k}}}{l_{q_{A}}}\geq\frac{c}{8A}\overset{A-1}{\underset{k=1}{\sum}}k\geq
cA\geq c\log q_{A}.$

The proof is complete. ∎

##  4\. Formulation of the Main Results

Our main results read:

######  Theorem 1.

Let $G_{m}$ be a bounded Vilenkin system. Then there exist a martingale $f\in
H_{1}$ such that

(7) $\sup\limits_{n\in\mathbb{N}}\left\|L_{n}f\right\|_{1}=+\infty.$

######  Remark 1.

In one point of view Theorem 1 of Blahota and Gát [2] is better then our resut
(see Theorem 1) because in their result boundedness of the group $G_{m}$ is
not necessary and in the other point of view theorem of Blahota and Gát is a
slightly weaker since they construct function in the Lebesgue space $L_{1}$.

The next result can be found in [13] and [8], respectively:

######  Corollary 1.

Let $G_{m}$ be a bounded Vilenkin system. Then there exists a martingale $f\in
H_{1}$ such that

$\left\|L^{\ast}f\right\|_{1}=+\infty.$

######  Corollary 2.

Let $0<p<1$ and $G_{m}$ be a bounded Vilenkin system. Then there exists a
martingale $f\in H_{p}$ such that

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

######  Theorem 2.

a) Let $G_{m}$ be a bounded Vilenkin system. Then the 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
$L_{1}\left(G_{m}\right).$

b) Let $\varphi:\mathbb{N}_{+}\rightarrow[1,\infty)$ be a nondecreasing
function satisfying the condition

(8)
$\overline{\lim_{n\rightarrow\infty}}\frac{\log\left(n+1\right)}{\varphi\left(n\right)}=+\infty.$

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 $L_{1}\left(G_{m}\right).$

######  Remark 2.

Note that b) shows that the statement in part a) of Theorem 2 is in the sense
sharp with respect to the logarithmic factor in (2).

##  5\. Proofs of the Main Results

###### Proof of Theorem 1..

Let $\left\\{\alpha_{k}:k\in\mathbb{N}\right\\}$ be an increasing sequence of
positive integers such that

(9)
$\sum_{\eta=0}^{k-1}\frac{M_{2\alpha_{\eta}}}{\alpha^{1/2}_{\eta}}<\frac{M_{2\alpha_{k}}}{\alpha^{1/2}_{k}},$

(10) $\frac{M_{2\alpha_{k-1}}}{\alpha^{1/2}_{k-1}}<\alpha_{k}.$

Since

$\alpha_{k}\rightarrow\infty,\ \
\sum_{\eta=0}^{k-1}\frac{M_{2\alpha_{\eta}}}{\alpha^{1/2}_{\eta}}\rightarrow\infty,\
\ \frac{M_{2\alpha_{k}}}{\alpha^{1/2}_{k}}\rightarrow\infty,\ \ \text{as}\ \
k\rightarrow\infty$

and all of three sequences are strightly increasing, we can construct lacunar
increasing sequence $\left\\{\alpha_{k}:k\in\mathbb{N}\right\\}$ of natural
number $\mathbb{N}$, for which

$\sum_{\eta=0}^{k-1}\frac{M_{2\alpha_{\eta}}}{\alpha^{1/2}_{\eta}}<\frac{M_{2\alpha_{k}}}{\alpha^{1/2}_{k}}<\alpha_{k+1}.$

We also note that under condition (10) we get that

$M_{2\alpha_{k-1}}<\alpha_{k}\alpha^{1/2}_{k-1}\leq\alpha_{k}^{3/2}$

and

(11)
$\sum_{k=0}^{\infty}\alpha_{k}^{-1/2}<\alpha_{0}+\sum_{k=0}^{\infty}\frac{1}{M_{2\alpha_{k-1}}^{1/3}}<c<\infty.$

Similar consructions of martingales can be found in [10]-[13] (see also [7]).

Let

$f\left(x\right)=\sum_{k=0}^{\infty}\lambda_{k}a_{k},$

where

$\lambda_{k}=\frac{m_{2\alpha_{k}}}{\alpha_{k}},\ \
a_{k}=\frac{1}{m_{2\alpha_{k}}}\left(D_{M_{2\alpha_{k}+1}}-D_{M_{{}_{2\alpha_{k}}}}\right).$

From (11) and Lemma 1 we can conclude that $f\in H_{1}\left(G_{m}\right).$ It
is easy to show that

(12)
$\widehat{f}(j)=\left\\{\begin{array}[]{l}\frac{1}{\alpha^{1/2}_{k}},\,\,\text{
if \thinspace\thinspace}j\in\left\\{M_{2\alpha_{k}},...,\text{
~{}}M_{2\alpha_{k}+1}-1\right\\},\text{ }k=0,1,2...,\\\ 0,\text{
\thinspace\thinspace\thinspace if
\thinspace\thinspace\thinspace}j\notin\bigcup\limits_{k=1}^{\infty}\left\\{M_{2\alpha_{k}},...,\text{
~{}}M_{2\alpha_{k}+1}-1\right\\}\text{ .}\end{array}\right.$

We can write that

$\displaystyle L_{q_{\alpha_{k}}}f$ $\displaystyle=$
$\displaystyle\frac{1}{l_{q_{\alpha_{k}}}}\underset{j=1}{\overset{q_{\alpha_{k}}}{\sum}}\frac{S_{j}f}{q_{\alpha_{k}}-j}$
$\displaystyle=$
$\displaystyle\frac{1}{l_{q_{\alpha_{k}}}}\underset{j=1}{\overset{M_{2\alpha_{k}}-1}{\sum}}\frac{S_{j}f}{q_{\alpha_{k}}-j}+\frac{1}{q_{\alpha_{k}}}\underset{j=M_{2\alpha_{k}}}{\overset{q_{\alpha_{k}}}{\sum}}\frac{S_{j}f}{q_{\alpha_{k}}-j}$
$\displaystyle=$
$\displaystyle\frac{1}{l_{q_{\alpha_{k}}}}\underset{j=1}{\overset{M_{2\alpha_{k-1}}+1}{\sum}}\frac{S_{j}f}{q_{\alpha_{k}}-j}+\frac{1}{q_{\alpha_{k}}}\underset{j=M_{2\alpha_{k}}}{\overset{q_{\alpha_{k}}}{\sum}}\frac{S_{j}f}{q_{\alpha_{k}}-j}$
$\displaystyle:=$ $\displaystyle I+II.$

Let $j<M_{2\alpha_{k-1}+1}.$ By combining (9) and (12) we get that

$\displaystyle\left|S_{j}f\right|$ $\displaystyle\leq$
$\displaystyle\sum_{\eta=0}^{k-1}\sum_{v=M_{2\alpha_{\eta}}}^{M_{2\alpha_{\eta}+1}-1}\left|\widehat{f}(v)\right|$
$\displaystyle\leq$
$\displaystyle\sum_{\eta=0}^{k-1}\sum_{v=M_{2\alpha_{\eta}}}^{M_{2\alpha_{\eta}+1}-1}\frac{1}{\sqrt{\alpha_{\eta}}}\leq
c\sum_{\eta=0}^{k-1}\frac{M_{2\alpha_{\eta}}}{\sqrt{\alpha_{\eta}}}$
$\displaystyle\leq$ $\displaystyle
c\sum_{\eta=0}^{k-2}\frac{M_{2\alpha_{\eta}}}{\sqrt{\alpha_{\eta}}}+\frac{cM_{2\alpha_{k-1}}}{\sqrt{\alpha_{k-1}}}$
$\displaystyle\leq$
$\displaystyle\frac{2cM_{2\alpha_{k-1}}}{\sqrt{\alpha_{k-1}}}.$

Since

$\displaystyle\sum_{j=0}^{M_{2\alpha_{k-1}}}\frac{1}{q_{\alpha_{k}}-j}$
$\displaystyle\leq$
$\displaystyle\sum_{j=M_{2\alpha_{k}-1}}^{M_{2\alpha_{k}}}\frac{1}{j}\leq\log
M_{2\alpha_{k}}-\log M_{2\alpha_{k}-1}$ $\displaystyle\leq$
$\displaystyle\log\frac{M_{2\alpha_{k}}}{M_{2\alpha_{k}-1}}<c<\infty$

according to (10) and (5) we can conclude that

$\displaystyle\left|I\right|\leq\frac{2c}{\alpha_{k}}\sum_{j=0}^{M_{2\alpha_{k-1}}}\frac{1}{q_{\alpha_{k}}-j}\frac{M_{2\alpha_{k-1}}}{\sqrt{\alpha_{k-1}}}$
$\displaystyle\leq$
$\displaystyle\frac{c}{\alpha_{k}}\frac{M_{2\alpha_{k-1}}}{\sqrt{\alpha_{k-1}}}<c<\infty.$

Hence,

$\left\|I\right\|_{1}<c<\infty.$

Let $M_{2\alpha_{k}}\leq j\leq q_{\alpha_{k}}.$ Then we have that

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

This gives that

$\displaystyle II$ $\displaystyle=$
$\displaystyle\frac{1}{l_{q_{\alpha_{k}}}}\sum_{j=M_{2\alpha_{k}}}^{q_{\alpha_{k}}}\frac{1}{q_{\alpha_{k}}-j}\left(\sum_{\eta=0}^{k-1}\frac{1}{\sqrt{\alpha_{\eta}}}\left(D_{M_{{}_{2\alpha_{\eta}+1}}}-D_{M_{{}_{2\alpha_{\eta}}}}\right)\right)$
$\displaystyle+\frac{1}{l_{q_{\alpha_{k}}}}\frac{1}{\sqrt{\alpha_{k}}}\sum_{j=M_{2\alpha_{k}}}^{q_{\alpha_{k}}}\frac{\left(D_{j}-D_{M_{{}_{2\alpha_{k}}}}\right)}{q_{\alpha_{k}}-j}:=II_{1}+II_{2}.$

Applying (9) in $II_{1}$ we have that

$\displaystyle\left\|II_{1}\right\|_{1}$ $\displaystyle\leq$
$\displaystyle\frac{1}{l_{q_{\alpha_{k}}}}\sum_{j=M_{2\alpha_{k}}}^{q_{\alpha_{k}}}\frac{1}{q_{\alpha_{k}}-j}\sum_{\eta=0}^{k-1}\frac{1}{\sqrt{\alpha_{\eta}}}\left\|D_{M_{{}_{2\alpha_{\eta}+1}}}-D_{M_{{}_{2\alpha_{\eta}}}}\right\|_{1}$
$\displaystyle\leq$
$\displaystyle\frac{1}{l_{q_{\alpha_{k}}}}\sum_{j=M_{2\alpha_{k}}}^{q_{\alpha_{k}}}\frac{1}{q_{\alpha_{k}}-j}\sum_{\eta=0}^{\infty}\frac{1}{\sqrt{\alpha_{\eta}}}<c<\infty.$

Since

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

for $II_{2}$ we obtain that

$\displaystyle II_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{l_{q_{\alpha_{k}}}}\frac{1}{\sqrt{\alpha_{k}}}\psi_{M_{2\alpha_{k}}}\sum_{j=0}^{q_{\alpha_{k}-1}-1}\frac{D_{j}}{q_{\alpha_{k}-1}-j}$
$\displaystyle=$
$\displaystyle\frac{l_{q_{\alpha_{k}}-1}}{l_{q_{\alpha_{k}}}}\frac{1}{\sqrt{\alpha_{k}}}\psi_{M_{2\alpha_{k}}}F_{q_{\alpha_{k}}-1}.$

Hence, if we apply Lemma 2 in the case when $\delta=1/8$ and $\beta=3/4$ for
the sufficiently large $k$ we can estimate as follows

$\displaystyle\left\|L_{q_{\alpha_{k}}}f\right\|_{1}$ $\displaystyle\geq$
$\displaystyle\left\|II_{2}\right\|_{1}-\left\|II_{1}\right\|_{1}-\left\|I\right\|_{1}$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\left\|II_{2}\right\|_{1}\geq\frac{c\left\|F_{q_{\alpha_{k}}-1}\right\|_{1}}{\sqrt{\alpha_{k}}}$
$\displaystyle\geq$
$\displaystyle\frac{c\alpha_{k}^{3/4}}{\sqrt{\alpha_{k}}}\geq
c\alpha_{k}^{1/4}\rightarrow\infty,\text{ as }k\rightarrow\infty.$

The proof is complete. ∎

###### Proof of Theorem 2..

It is obvious that

$\underset{n\in\mathbb{N}}{\sup}\frac{\left|L_{n}f\right|}{\log\left(n+1\right)}\leq\frac{1}{l_{n}}\sum_{k=0}^{n}\frac{1}{(n-k)}\underset{k\in\mathbb{N}}{\sup}\frac{\left|S_{k}f\right|}{\log\left(k+1\right)}\leq\sup_{n\in\mathbb{N}}\frac{\left|S_{n}f\right|}{\log\left(n+1\right)}.$

By using Theorem T1 we can conclude that

$\left\|\underset{n\in\mathbb{N}}{\sup}\frac{\left|L_{n}f\right|}{\log\left(n+1\right)}\right\|_{1}\leq\left\|\sup_{n\in\mathbb{N}}\frac{\left|S_{n}f\right|}{\log\left(n+1\right)}\right\|_{1}\leq
c\left\|f\right\|_{H_{1}}.$

The proof of part a) is complete.

Under condition (8), there exists a positive integers $\
\left\\{n_{k};k\in\mathbb{N}_{+}\right\\}\subset\left\\{\lambda_{k};k\in\mathbb{N}_{+}\right\\}$
such that

$\lim_{k\rightarrow\infty}\frac{n_{k}}{\varphi\left(q_{n_{k}}\right)}=\infty.$

Set

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

It is evident

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

Then we have that

(16)
$S_{i}f_{n_{k}}=\left\\{\begin{array}[]{l}D_{i}-D_{M_{{}_{2n_{k}}}},\text{
}\text{ if }i=M_{{}_{2n_{k}}}+1,...,M_{2n_{k}+1}-1,\\\ \text{
}f_{n_{k}},\text{ \ \ }\text{ \ \ }\text{ \ \ }\text{ \ \ }\text{ if }i\geq
M_{2n_{k}+1},\\\ 0,\text{ \ \ }\text{ \ \ }\text{ \ \ }\text{ \qquad
otherwise.}\end{array}\right.$

From (1) and (16) we get that

$\left\|f_{n_{k}}\right\|_{H_{1}}<c<\infty.$

It is easy to show that

$\displaystyle L_{q_{n_{k}}}f_{n_{k}}$ $\displaystyle=$
$\displaystyle\frac{1}{l_{q_{n_{k}}}}\underset{j=1}{\overset{q_{n_{k}}}{\sum}}\frac{S_{j}f_{n_{k}}}{q_{n_{k}}-j}$
$\displaystyle=$
$\displaystyle\frac{1}{l_{q_{n_{k}}}}\underset{j=M_{{}_{2n_{k}}}+1}{\overset{q_{n_{k}}}{\sum}}\frac{D_{j}-D_{M_{{}_{2n_{k}}}}}{q_{n_{k}}-j}$
$\displaystyle=$
$\displaystyle\frac{1}{l_{q_{n_{k}}}}\underset{j=1}{\overset{q_{n_{k}-1}-1}{\sum}}\frac{D_{j+M_{{}_{2n_{k}}}}-D_{M_{{}_{2n_{k}}}}}{q_{n_{k}-1}-j}.$

By applying (15) we find that

$\left|L_{q_{n_{k}}}f_{n_{k}}\right|=\frac{1}{l_{q_{n_{k}}}}\left|\underset{j=1}{\overset{q_{n_{k}-1}}{\sum}}\frac{D_{j}}{q_{n_{k}-1}-j}\right|=\frac{l_{q_{n_{k}-1}}}{l_{q_{n_{k}}}}\left|F_{q_{n_{k}-1}}\right|.$

By now using Lemma 3 we can conclude that

$\int_{G_{m}}\frac{\left|L_{q_{n_{k}}}f_{n_{k}}\right|}{\varphi\left({q_{n_{k}}}\right)}d\mu\geq\frac{c\left\|F_{q_{n_{k}-1}}\right\|}{\varphi\left(q_{n_{k}}\right)}\geq\frac{cn_{k}}{\varphi\left(q_{n_{k}}\right)}\rightarrow\infty,\text{
as }k\rightarrow\infty.$

The proof is complete.

Acknowledgment: The author would like to thank the referee for helpful
suggestions, which absolutely improve the final version of our paper. ∎

## References

  * [1] G. N. Agaev, N. Ya. Vilenkin, G. M. Dzhafarly and A. I. Rubinstein, Multiplicative systems of functions and harmonic analysis on zero-dimensional groups, Baku, Ehim, 1981 (in Russian).
  * [2] I. Blahota and G. Gàt, Norm summability of Nörlund logarithmic means on unbounded Vilenkin groups, Anal. Theory Appl., 24 (2008) no 1, 1-17.
  * [3] S. Fridli, P. Manchanda and A.H. Siddiqi, Approximation by Walsh-Nörlund means, Acta Sci. Math.(Szeged) 74 (2008), no. 3-4, 593-608.
  * [4] G. Gát and U. Goginava, Uniform and $L$ -convergence of logarithmic means of Walsh-Fourier series, Acta Math. Sin. 22 (2006), no. 2, 497–506.
  * [5] B. I. Golubov, A. V. Efimov and V. A. Skvortsov, Walsh series and transforms. (Russian) Nauka, Moscow, 1987, English transl. in Mathematics and its Applications (Soviet Series), 64. Kluwer Academic Publishers Group, Dordrecht, 1991.
  * [6] F. Móricz and A. Siddiqi, Approximation by Nörlund means of Walsh-Fourier series, J. Approx. Theory 70 (1992), no. 3, 375–389.
  * [7] K. Nagy, G. Tephnadze, Kaczmarz-Marcinkiewicz means and Hardy spaces, Acta math. Hung., 149, 2 (2016), 346-374.
  * [8] L. E. Persson, G. Tephnadze and P. Wall, On the maximal operators of Vilenkin-Nörlund means, J. Fourier Anal. Appl., 21, 1 (2015), 76-94.
  * [9] F. Schipp, W.R. Wade, P. Simon and J. Pál, Walsh series, An Introduction to Dyadic Harmonic Analysis, Akadémiai Kiadó, (Budapest-Adam-Hilger (Bristol-New-York)), 1990.
  * [10] G. Tephnadze, On the partial sums of Vilenkin-Fourier series, 49 (2014), no. 1, 60–72, J. Contemp. Math. Anal. 49 (2014), no. 1, 23-32.
  * [11] G. Tephnadze, Strong convergence theorems of Walsh-Fejér means, Acta Math. Hung., 142, 1 (2014), 244-259.
  * [12] G. Tephnadze, On the maximal operators of Walsh-Kaczmarz-Fejér means, Period. Math. Hung., 67, 1 (2013), 33-45.
  * [13] G. Tephnadze, Martingale Hardy Spaces and Summability of the One Dimensional Vilenkin-Fourier Series, PhD thesis, Department of Engineering Sciences and Mathematics, Luleå University of Technology, Oct. 2015 (ISSN 1402-1544).
  * [14] N. Ya. Vilenkin, A class of complete orthonormal systems, Izv. Akad. Nauk. U.S.S.R., Ser. Mat., 11 (1947), 363-400.
  * [15] C. Watari, Best approximation by Walsh polynomials, Tohoku Math. J., 15 (1963), 1-5.
  * [16] F. Weisz, Martingale Hardy spaces and their applications in Fourier Analysis, Springer, Berlin-Heidelberg-New York, 1994.
  * [17] F. Weisz, Hardy spaces and Cesàro means of two-dimensional Fourier series, Bolyai Soc. Math. Studies, (1996), 353-367.