# Effects of dark matter on the in-spiral properties of the binary neutron
stars

H. C. Das1,2 , Ankit Kumar1,2, S. K. Patra1,2,
1 Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India
2 Homi Bhabha National Institute, Training School Complex, Anushakti Nagar,
Mumbai 400094, India
E-mail<EMAIL_ADDRESS>
###### Abstract

Using the relativistic mean-field model, we calculate the properties of binary
neutron star (BNS) in the in-spiral phase. Assuming the dark matter (DM)
particles are accreted inside the neutron star (NS) due to its enormous
gravitational field, the mass $M$, radius $R$, tidal deformability $\lambda$
and dimensionless tidal deformability $\Lambda$ are calculated at different DM
fractions. The value of $M$, $R$, $\lambda$ and $\Lambda$ decreases with the
increase of DM percentage inside the NS. The in-spiral phase properties of the
BNS are explored within the post-Newtonian (PN) formalism, as it is suitable
up to the last orbits in the in-spiral phase. We calculate the strain
amplitude of the polarization waveform $h_{+}$ and $h_{\times}$, (2,2) mode
waveform $h_{22}$, orbital phase $\Phi$, frequency of the gravitational wave
$f$ and PN parameter $x$ with DM as an extra candidate inside the NS. The
magnitude of $f$, $\Phi$ and $x$ are almost the same for all assumed forces;
however, the in-spiral time duration in the last orbit is different. We find
that the BNS with soft equation of state and a high fraction of DM sustains
more time in their in-spiral phase. We suggest that one should take DM inside
the NS when they modelling the in-spiral waveforms for the BNS systems.

###### keywords:

neutron stars, dark matter theory, gravitational waves

##  1 Introduction

The detection of the binary neutron star (BNS) coalescence event GW170817 by
LIGO/Virgo collaborations gives a new insight to study the compact objects
(Abbott et al., 2017a, 2018). The gravitational wave (GW) instrument observed
this event about 40 Mpc distance from the source with a total mass of 2.74
$M_{\odot}$ of the binary (Abbott et al., 2017a). The gamma rays are detected
after 1.7 seconds of the binary merger (Abbott et al., 2017c; Goldstein et
al., 2017). In this set-up, the optical telescope identified an
electromagnetic counterpart after 12 hours of the GW event near the NGC 4993
Galaxy called AT2017gfo (Cowperthwaite et al., 2017; Nicholl et al., 2017;
Chornock et al., 2017). The X-rays and radio waves are also detected several
days after the merger (Troja et al., 2017; Margutti et al., 2017; Hallinan et
al., 2017). Hence, the BNS collision event witnesses the beginning of a new
era in multi-messenger astronomy (Abbott et al., 2017b). The GWs coming from
the in-spiral and merger of the compact binary stars are the most promising
sources for ground-based detectors such as LIGO (CALTECH, 2002), Virgo (INFN,
1993) and KAGRA (KAGRA, 2010). The GWs encode information about the tidal
deformability of the BNS system. The limits on the dimensionless tidal
deformability parameter $\Lambda$ for the canonical star is given by GW170817
as $\Lambda_{1.4}=190_{-120}^{+390}$, which translates into a stringent bound
$\Lambda\leq 580$ at the 90 % confidence level (Abbott et al., 2018). A
considerable amount of research have been dedicated to constrain the mass $M$,
radius $R$ and $\Lambda$ of the neutron star (NS) using the GW170817 data
(Margalit & Metzger, 2017; Radice et al., 2018; Zhang et al., 2018; Most et
al., 2018; Nandi et al., 2019; Annala et al., 2018). Hence, the tidal
deformability is the most promising parameter to constraint the EoSs, $M$ and
$R$ of the NS. The detailed study of the GWs emitting from the in-spiral phase
needs an accurate theoretical framework. This is because the tidal effects
influence the dynamics of the in-spiral orbits and modify the corresponding
gravitational waveforms. Substantial number of analysis have already been done
to study the properties of BNS with tidal interactions as dominant
contribution to the late in-spiral phase (Flanagan & Hinderer, 2008; Hinderer
et al., 2010; Vines et al., 2011; Bini et al., 2012; Read et al., 2013;
Favata, 2014; Wade et al., 2014; Agathos et al., 2015).

The post-Newtonian (PN) formalism is suitable up to the last orbit to study
the BNS systems in the in-spiral phase (Blanchet, 2006). The high accuracy PN
template for the in-spiral stage plays a key role in the analysis of GW data.
This framework re-expresses the general relativistic description of particles
into the standard equations of motion, which express in terms of acceleration
of the particles (Blanchet, 2006). The BNS in its in-spiral phase tidally
interacts with each other, which affects their stellar structure. Each star’s
tidal property depends on the macroscopic quantity, such as mass $M$, radius
$R$ and second Love number $k_{2}$, which are all model-dependent in the
evaluation (Hinderer, 2008). Here, we used the relativistic mean-field (RMF)
model to calculate the NS’s properties.

On the other hand, dark matter (DM) accounts for approximately 85% of the
matter in the Universe and about 27% of its total mass-energy density.
Theoretically, several DM particles have been hypothesised due to its unknown
character such as weakly interacting massive particle (WIMPs) (Kouvaris &
Tinyakov, 2011; Quddus et al., 2020), feebly interacting massive particles
(FIMPs) (Hall et al., 2010; Bernal et al., 2017), Axions (Duffy & van Bibber,
2009) etc. The WIMP is the most abundant DM particle, which is also considered
as the thermal relics of the Universe. It might have decayed in the dense
region of the Universe, which produce the standard model (SM) particles,
leptons, gamma-rays and neutrinos at freeze-out temperature (Ruppin et al.,
2014). Direct and indirect detection experiments have already been established
to know the mystery of DM (Bernabei et al., 2008; CDMS, 2010; Aprile et al.,
2016). Also, different models have been incorporated to study the effects of
DM inside compact objects such as NS and white dwarf (Bertone & Fairbairn,
2008; Kouvaris, 2008; Kouvaris & Tinyakov, 2010). The properties and origin of
DM particles are not yet fully understood. In some formulation, it is treated
as bosonic matter (Bertoni et al., 2013). However, in maximum cases, the DM
are fermions. In the present calculations, we consider Neutralino as a DM
candidate, super-symmetric particle and the best candidate of WIMP (MARTIN, ;
Panotopoulos & Lopes, 2017; Das et al., 2019, 2020). These particles interact
with nucleons by exchanging SM Higgs boson. A detailed formalism and notations
used in this paper can be found in Refs. (Das et al., 2020, 2021).

The accretion of self-annihilating DM particles inside the NS affects the
cooling properties of supernovae remnant (Kouvaris, 2008; Bhat & Paul, 2020).
On the other hand, the non-annihilating DM particles develop interaction with
baryons which affects the structure of the NS (De Lavallaz & Fairbairn, 2010;
Ciarcelluti & Sandin, 2011). Here, we assume that the WIMP is a DM particle
that is non-annihilating in character. The addition of DM inside the NS
generally reduces the macroscopic properties such as $M$, $R$ and $\lambda$
(Das et al., 2020; Quddus et al., 2020; Das et al., 2021). This is because the
EoS becomes softer with the increase of DM percentage. In the present case, we
study the BNS properties in the in-spiral phase for DM admixed NS. The
emission of GWs and its properties such as the (2,2) mode waveform $h_{22}$,
polarization waveforms $h_{+}$ & $h_{\times}$, GW frequency $f$, orbital phase
$\Phi$ and PN parameter $x$ are calculated for equal mass binary NS. We take
three different equation of states (EoSs) to see the parametric dependence,
spanning from soft to stiff. All the in-spiral properties defined above are
calculated in the retarded time interval.

The manuscript is organized as follow: In Sec. 2.1, we outline the basic RMF
formalism required for the calculation of NS properties with the admixture of
DM inside the NS. The in-spiral phase of the BNS in quasi-circular orbits are
detailed in sub-sec. 2.2. In Sec. 3, we present our numerical results.
Finally, we concluded our findings in Sec. 4.

##  2 THEORETICAL FORMALISM

###  2.1 Equation of state

In the extended RMF (E-RMF) formalism, the $\sigma-$meson is mainly
responsible for the strong attraction at the intermediate range of the nuclear
force, but the self-interaction of this scalar meson produces a weak repulsion
at long-range (Biswal et al., 2015). Because of the self-interaction of the
$\sigma-$meson, the incompressibility of nuclear matter at nuclear saturation
$K_{\infty}$ comes down to the empirical range of $K_{\infty}=210\pm{30}$ MeV
(Blaizot, 1980). The $\omega-$meson responsible for the strong hard-core
repulsion of the nuclear potential and its self-interaction generates
attraction at very short range and makes the nuclear EoS softer (Bodmer, 1991;
Gmuca, 1992; Sugahara & Toki, 1994; Bunta & Gmuca, 2003). We take one RMF set
NL3 (Lalazissis et al., 1997) and two E-RMF sets such as G3 (Kumar et al.,
2017) and IOPB-I (Kumar et al., 2018) to calculate the NS properties in the
present work.

Inside the NS, the neutron decays to proton, electron and anti-neutrino. For
the stability of the system, the inverse $\beta$-decay takes place to maintain
both $\beta$ equilibrium and charge neutrality. Hence, the EoS of NS is
calculated from the contributions of both nucleons and leptons. The energy
density and pressure of a NS is written as (Das et al., 2020, 2021):

$\displaystyle{\cal{E}}_{NS}={\cal{E}}_{NM}+{\cal{E}}_{l},\quad\mathrm{and}\quad
P_{NS}=P_{NM}+P_{l}.$ (1)

Where ${\cal{E}}_{NM}$ and $P_{NM}$ are the energy density and pressure of the
nucleon; ${\cal{E}}_{l}$ and $P_{l}$ are for leptons, respectively.

After adding the DM to the NS, the total Lagrangian density is written as (Das
et al., 2019, 2020):

$\displaystyle{\cal{L}}$ $\displaystyle=$
$\displaystyle{\cal{L}}_{NS}+\bar{\chi}\left[i\gamma^{\mu}\partial_{\mu}-M_{\chi}+yh\right]\chi+\frac{1}{2}\partial_{\mu}h\partial^{\mu}h$
(2)
$\displaystyle-\frac{1}{2}M_{h}^{2}h^{2}+f\frac{M_{nucl.}}{v}\bar{\psi}h\psi,$

where ${\cal{L}}_{NS}$ is the NS Lagrangian. The $\psi$ is the nucleon field
and $\chi$ is the DM wave function. The parameters $y$, $f$ and $v$ are the
DM-Higgs coupling, nucleon-Higgs form factor and the vacuum expectation value
of Higgs field, respectively. The values of $y$, $f$ and $v$ are 0.07, 0.35
and 246 GeV, respectively (Das et al., 2020, 2021).

The two parameters $y$ and $f$ play significant role on the effect of DM in
NS. Therefore, we constraint these two parameters using available experimental
data. The expression of scattering cross-section for the DM is written as
(Bhat & Paul, 2020):

$\sigma_{SI}=\frac{y^{2}f^{2}M_{nucl.}^{2}}{4\pi}\frac{\mu_{r}}{v^{2}M_{h}^{2}},$
(3)

where $\mu_{r}$ is the reduced mass $\frac{M_{nucl.}\times
M_{\chi}}{M_{nucl.}+M_{\chi}}$ and $M_{\chi}$ is the mass of the DM particle.
The DM detection experiments, like XENON (Aprile et al., 2016), PANDA-II (Tan
et al., 2016) and LUX (Akerib et al., 2017) have put up their upper limit for
the spin-independent scattering cross-section $\sigma_{SI}$ as $1.6\times
10^{-47}$ cm-2 ($M_{\chi}=50$ GeV), $1.6\times 10^{-47}$ cm-2 ($M_{\chi}=40$
GeV) and $1.6\times 10^{-47}$ cm-2 ($M_{\chi}=50$ GeV), respectively. The
PANDA-II experiment has also put a limit on the mass of the WIMP as
$M_{\chi}=5-1000$ GeV. The LHC has put a limit on the WIMP-nucleon scattering
for Higgs portal DM in the range $10^{-40}-10^{-50}$ cm-2 (Tan et al., 2016;
Aad et al., 2015). We calculate the $\sigma_{SI}$ for three different masses
of DM $M_{\chi}=50$, 100, and 200 GeV and their corresponding cross-sections
are found to be 9.43, 9.60 and 9.70 in the order of ($10^{-46}$ cm2)
respectively. These data confirmed that the predicted values are consistent
with the data within 90% confidence level. This model also satisfies the LHC
limit. The value of $f$ is reported in Ref. (Djouadi et al., 2012) using both
lattice QCD (Czarnecki et al., 2010) and MILC results (Toussaint & Freeman,
2009) whose value is $0.33_{-0.07}^{+0.30}$ (Aad et al., 2015) and our used
$f$ (= 0.35) is well within the region.

The ${\cal{E}}$ and $P$ for NS with DM can be calculated by solving the Eq.
(2) (Das et al., 2020, 2021)

$\displaystyle{\cal{E}}$ $\displaystyle=$
$\displaystyle{\cal{E}}_{NS}+\frac{2}{(2\pi)^{3}}\int_{0}^{k_{f}^{DM}}d^{3}k\sqrt{k^{2}+(M_{\chi}^{\star})^{2}}+\frac{1}{2}M_{h}^{2}h_{0}^{2},$
(4)

and

$\displaystyle P$ $\displaystyle=$ $\displaystyle
P_{NS}+\frac{2}{3(2\pi)^{3}}\int_{0}^{k_{f}^{DM}}\frac{d^{3}k\hskip
2.84526ptk^{2}}{\sqrt{k^{2}+(M_{\chi}^{\star})^{2}}}-\frac{1}{2}M_{h}^{2}h_{0}^{2},$
(5)

where $k_{f}^{DM}$ and $M_{\chi}^{*}$ are the Fermi momentum and the effective
mass of the DM respectively. $M_{h}$ is the mass of the Higgs equal to 125
GeV, and $h_{0}$ is the Higgs field calculated by applying the mean-field
approximation (Das et al., 2019). In the present calculations, the mass of the
DM is taken as 200 GeV.

A large number of forces are available in the literature (Dutra et al., 2012;
Kumar et al., 2017, 2018). Among them, NL3, G3 and IOPB-I are very much used
both for finite nuclei as well as for infinite nuclear matter (Kumar et al.,
2017, 2018). The NL3 is one of the stiffest parameter set, which predicts a NS
of mass 2.775 $M_{\odot}$ and the G3 is the softest one with a predictive mass
of 1.979 $M_{\odot}$. The values of the models are available in Ref. (Kumar et
al., 2018). Once the EoS has solved, the M-R profile of a single NS is
calculated by solving the Tolmann-Oppenheimer-Volkoff (TOV) equation. When we
consider a BNS system, the individual NS gets distorted due to the external
field created by its companion. The tidal deformability parameter
($\lambda=\frac{2}{3}k_{2}R^{5}$) created by the deformation depends on the
structural/macroscopic properties. The dimensionless Love number $k_{2}$ can
be calculated by solving two coupled differential equations as it is done in
Refs. (Hinderer, 2008; Hinderer et al., 2010; Kumar et al., 2018). The values
of $R$ and $k_{2}$ are fixed for a given mass by the EoS for the NS. The
dimensionless tidal deformability $\Lambda$ is defined as
($\Lambda=\lambda/M^{5}$).

###  2.2 Binary neutron stars in quasi-circular orbits

####  2.2.1 Post-Newtonian expanded Taylor-T4

The PN formalism is a slow-motion, weakly stressed, and weak field
approximation to the general theory of relativity and valid for any mass ratio
(Blanchet, 2006; Buonanno & Sathyaprakash, 2014). In the PN method, the
equations of motion are obtained by systematically expanding the metric. The
Einstein’s field equations are expressed in powers of the dimensionless
parameter $\epsilon=\sqrt{GM_{t}/c^{2}d}\sim\frac{v}{c}$, where $M_{t}$ is the
total mass of the system, $d$ is the distance between two NS and $v$ is the
characteristic velocity of particles. The metric is then solved iteratively in
powers of $\epsilon$, and the equations of motion are evaluated from the
metric using the geodesic equation. An expansion containing terms up to
$\epsilon^{n}$ or equivalently $(1/c)^{n}$ is denoted an $\frac{n}{2}$PN
expansion (Blanchet, 2006; Boyle et al., 2007). This calculation takes the
energy and luminosity up to 3PN and 3.5PN order, respectively, for the in-
spiralling binary in quasi-circular orbits.

The BNSs emit GWs in their in-spiral phases. Because of the emission of the
GWs for a classical system, the BNS have to rotate in quasi-circular orbits.
The flux (luminosity) emitted by the system balances the rate of change of
energy with time in that orbit, and the energy balance equation is given as
(Boyle et al., 2007; Maggiore, 2008; Lackey, 2012):

${\cal{L}}=-\frac{dE}{dt}=-\frac{dE/dx}{dt/dx},$ (6)

where $E$ is the energy of the system, $t$ is the time taken and $x$ is the PN
parameter which is defined as (Boyle et al., 2007; Maggiore, 2008; Lackey,
2012)

$x=\Big{(}M_{t}\frac{d\Phi}{dt}\Big{)}^{2/3}=\Big{(}M_{t}\Omega\Big{)}^{2/3},$
(7)

where $M_{t}$ is the total mass of BNS systems, $\Phi$ is the orbital phase,
$\Omega=2\pi\omega$ is the orbital angular velocity ($\omega=f/2$ is the
orbital angular frequency), and $f$ is the frequency of the emitted GW. The
energy of the system calculated up to 3PN order in terms of $x$ is given as
(Blanchet, 2006; Maggiore, 2008; Lackey, 2012):

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{1}{2}M_{t}\nu
x\Bigg{\\{}1+\Bigg{(}-\frac{3}{4}-\frac{\nu}{12}\Bigg{)}x+\Bigg{(}-\frac{27}{8}+\frac{19\nu}{8}-\frac{\nu^{2}}{24}\Bigg{)}x^{2}$
(8)
$\displaystyle+\Bigg{[}-\frac{675}{64}+\Bigg{(}\frac{34445}{576}-\frac{205\pi^{2}}{96}\Bigg{)}\nu-\frac{155\nu^{2}}{96}-\frac{35\nu^{3}}{5184}\Bigg{]}x^{3}\Bigg{\\}},$

where $\nu=m_{1}m_{2}/M_{t}^{2}$ is the symmetric mass ratio, $m_{1}$ and
$m_{2}$ are the individual masses of the binary. The luminosity ${\cal{L}}$ is
calculated by the time derivative of Eq. (8) using PN expansions up to 3.5
order is given as (Blanchet, 2006; Maggiore, 2008; Lackey, 2012)

$\displaystyle{\cal{L}}$ $\displaystyle=$
$\displaystyle\frac{32}{5}\nu^{2}x^{5}\Bigg{\\{}1+\Bigg{(}-\frac{1247}{336}-\frac{35\nu}{12}\Bigg{)}x+4\pi
x^{3/2}+\Bigg{(}-\frac{44711}{9072}$ (9)
$\displaystyle+\frac{9271\nu}{504}+\frac{65\nu^{2}}{18}\Bigg{)}x^{2}+\Bigg{(}-\frac{8191}{672}-\frac{583\nu}{24}\Bigg{)}\pi
x^{5/2}$
$\displaystyle+\Bigg{[}\frac{6643739519}{69854400}+\frac{16\pi^{2}}{3}-\frac{1712\gamma_{E}}{015}-\frac{856}{105}\ln(16x)$
$\displaystyle+\Bigg{(}-\frac{134543}{7776}+\frac{41\pi^{2}}{48}\Bigg{)}\nu-\frac{94403\nu^{2}}{3024}-\frac{775\nu^{3}}{324}\Bigg{]}x^{3}$
$\displaystyle+\Bigg{(}-\frac{16285}{504}+\frac{214745\nu}{1728}+\frac{193385\nu^{2}}{3024}\Bigg{)}\pi
x^{7/2}\Bigg{\\}},$

where $\gamma_{E}\approx 0.5772$ is the Euler’s constant. The phase evolution
of the binary system can be calculated by solving the following equations:

$\displaystyle\frac{dx}{dt}=\frac{dE/dt}{dE/dx}=-\frac{{\cal{L}}}{dE/dx},$
(10) $\displaystyle\frac{d\Phi}{dt}=\frac{x^{3/2}}{M_{t}}.$ (11)

There are various methods to integrate this system of equations labelled as
TaylorT1-TaylorT4 (Boyle et al., 2007; Creighton & Anderson, 2011; Lackey,
2012). In the TaylorT1 method, the Eqs. (8) and (9) are inserted in Eq. (10)
and then the integration is performed by using the initial conditions
$x_{0}=(M_{t}\Omega_{0})^{2/3}$ and $\Phi_{0}$. In the TaylorT2 method, the
equations are written by starting with a parametric solution of energy balance
equations. Then, each expressions integrand is re-expanded as a single PN
parameter x and truncated at the appropriate order (Boyle et al., 2007;
Lackey, 2012):

$\displaystyle t(x)=t_{0}+\int_{x}^{x_{0}}dx\frac{(dE/dx)}{{\cal{L}}},$ (12)
$\displaystyle\Phi(x)=\Phi_{0}+\int_{x}^{x_{0}}dx\frac{x^{3/2}}{M_{t}}\frac{(dE/dx)}{{\cal{L}}}.$
(13)

In the present calculations, we take TaylorT4 approximation. In this method,
the right-hand side of the Eq. (12) is re-expanded as a single series and
truncated at 3.5PN order before doing the integration. After calculating $x$,
$t$ and $\Phi$, one can know the properties related to the in-spiralling
binaries for the point-particle system.

In the case of BNS, the tidal interaction comes in to picture, which has a
significant role in the in-spiral properties. Therefore, we have to include
the extra part of tidal interactions with this point-particle approximation.
The motion of the system is tidally interacting, which influences their
internal structures. Also, the tidal interactions affect the evolution of the
GW phase by a parameter $\lambda$. Hence, the BNS system’s tidal contributions
must be added on the right-hand side of Eq. (10), which is modified as
(Baiotti et al., 2011; Hotokezaka et al., 2013; Hotokezaka et al., 2016):

$\frac{dx}{dt}=\frac{64\nu}{5M_{t}}x^{5}\\{F_{3.5}^{\mathrm{Taylor}}(x)+F^{\mathrm{Tidal}}(x)\\},$
(14)

where $F_{3.5}^{\mathrm{Taylor}}$is PN-expanded expression for point-mass
contribution using Taylor-T4 approximation, given by (Blanchet et al., 2008;
Boyle et al., 2007; Baiotti et al., 2011; Hotokezaka et al., 2013; Hotokezaka
et al., 2016)

$\displaystyle
F_{3.5}^{\mathrm{Taylor}}(x)=1-\Bigg{(}\frac{743}{336}+\frac{11}{4}\nu\Bigg{)}x+4\pi
x^{3/2}+\Bigg{(}\frac{34103}{18144}+\frac{13661}{2016}\nu$
$\displaystyle+\frac{59}{18}\nu^{2}\Bigg{)}x^{2}-\Bigg{(}\frac{4159}{672}+\frac{189}{8}\nu\Bigg{)}\pi
x^{5/2}+\Bigg{[}\frac{16447322263}{139708800}-\frac{1712}{105}\gamma_{E}$
$\displaystyle-\frac{56198689}{217728}\nu+\frac{541}{896}\nu^{2}-\frac{5605}{2592}\nu^{3}+\frac{\pi^{2}}{48}(256+451\nu)$
$\displaystyle-\frac{856}{105}\ln(16x)\Bigg{]}x^{3}+\Bigg{(}-\frac{4415}{4032}+\frac{358675}{6048}\nu+\frac{91495}{1512}\nu^{2}\Bigg{)}\pi
x^{7/2},$ (15)

With the addition of the tidal part to the PN formalism, we adopt the method
developed by Vines et al. (Vines et al., 2011) up to 1PN accuracy, which is
expressed as

$\displaystyle F^{\mathrm{Tidal}}(x)$
$\displaystyle=\frac{32\chi_{1}\lambda_{2}}{5M_{t}^{6}}\Big{[}12(1+11\chi_{1})x^{10}+\Big{(}\frac{4421}{28}-\frac{12263}{28}\chi_{2}$
(16)
$\displaystyle+\frac{1893}{2}\chi_{2}^{2}-661\chi_{2}^{3}\Big{)}x^{11}+(1\leftrightarrow
2)\Big{]},$

where $\lambda_{2}$ is the tidal deformability of the second star and
$\chi_{1}$ is the mass fraction of the first star ($\chi_{1}=M_{1}/M_{t}$).
The numerical values of $k_{2}$ and $R$ are given in Table 1 for different
parameter sets with three DM fractions. The numerical values are different
from others because they are model dependent. Baiotti et al. (Baiotti et al.,
2011) re-expressed the Eq. (16) in terms of tidal Love number $k_{2}$ and
compactness parameter $C$ as

$F^{\mathrm{Tidal}}(x)=\sum_{I=1,2}F_{LO}(\chi_{I})x^{5}\big{(}1+F_{1}(\chi_{I})x\big{)},$
(17)

with

$F_{LO}(\chi_{I})=4\hat{k_{2}^{I}}\Bigg{(}\frac{12-11\chi_{I}}{\chi_{I}}\Bigg{)},$
(18)

and

$F_{1}(\chi_{I})=\frac{4421-12263\chi_{I}+26502\chi_{I}^{2}-18508\chi_{I}^{3}}{336(12-11\chi_{I})}.$
(19)

The expression for $\hat{k_{2}^{I}}$ is defined as:

$\hat{k_{2}^{I}}=k_{2}^{I}\Bigg{(}\frac{\chi_{I}}{C_{I}}\Bigg{)}^{5}\ \
I=1,2.$ (20)

For equal-mass binary, $\chi_{1}=\chi_{2}=\chi=1/2$ and $C=C_{1}=C_{2}$, the
Eq. (16) can be written as (Hotokezaka et al., 2013)

$F^{\mathrm{Tidal}}(x)=\frac{52}{5M_{t}}\frac{k_{2}}{C^{5}}x^{10}\Big{(}1+\frac{5203}{4368}x\Big{)}.$
(21)

Although the tidal interactions affect the NS-NS in-spiral only at 5PN order,
its coefficient is 104 order magnitude for a NS with a radius of 10–15 km,
$k_{2}\sim 0.1$, and $C\sim 0.14\textendash 0.20$. Hence, it plays an
important role in the late in-spiral stage.

####  2.2.2 Polarization waveforms

From the observational point of view, the GW detector detects the radiation in
the direction of source. Therefore, the polarization waveforms are measured in
the line-of-sight of source, which can be expressed in spherical co-ordinates
($R,\hat{\theta},\hat{\phi}$) as (Maggiore, 2008; Isoyama et al., 2020):

$\displaystyle
h_{+}(\mathrm{t})=\frac{4}{r}{\cal{M}}^{5/3}\omega^{2/3}\Bigg{(}\frac{1+\cos^{2}\theta}{2}\Bigg{)}\cos(2\omega
t_{ret}+2\phi),$ $\displaystyle\mathrm{and}$ $\displaystyle
h_{\times}(\mathrm{t})=\frac{4}{r}{\cal{M}}^{5/3}\omega^{2/3}\cos\theta\sin(2\omega
t_{ret}+2\phi),$

where the chirp mass ${\cal{M}}$ is defined as ${\cal{M}}=\nu^{3/5}M$. The
amplitudes of the GW polarization with a fixed $\omega$ depends on the binary
masses through ${\cal{M}}$, which can also be derived as
${\cal{M}}=\frac{(m_{1}m_{2})^{3/5}}{(m_{1}+m_{2})^{1/5}}$. It is measured by
LIGO with a good precision ${\cal{M}}=1.188_{-0.002}^{+0.004}M_{\odot}$
(Abbott et al., 2017a). If we see the orbit edge-on, $\theta=\pi/2$, then
$h_{\times}$ vanishes and the GW is linearly polarized. If we put, $\theta=0$,
then $h_{+}$ and $h_{\times}$ have the same amplitude, but only difference is
the phase and hence it is circularly polarized.

The polarization waveforms for an equal mass binary NS (EMBNS) systems along
the z-axis (optimally oriented observer) calculated up to 2.5PN are given by
(Arun et al., 2004, 2005; Kidder et al., 2007; Boyle et al., 2007)

$\displaystyle h_{+}^{(z)}$ $\displaystyle=$
$\displaystyle\frac{M_{t}}{2D}x\Bigg{(}\cos
2\Phi\bigg{\\{}-2+\frac{17}{4}x-4\pi x^{3/2}+\frac{15917}{2880}x^{2}+9\pi
x^{5/2}\bigg{\\}}$ (23) $\displaystyle+\sin
2\Phi\Big{\\{}\frac{59}{5}x^{5/2}\Big{\\}}\Bigg{)},$ $\displaystyle
h_{\times}^{(z)}$ $\displaystyle=$ $\displaystyle\frac{M_{t}}{2D}x\Bigg{(}\sin
2\Phi\bigg{\\{}-2+\frac{17}{4}x-4\pi x^{3/2}+\frac{15917}{2880}x^{2}+9\pi
x^{5/2}\bigg{\\}}$ (24) $\displaystyle+\cos
2\Phi\Big{\\{}-\frac{59}{5}x^{5/2}\Big{\\}}\Bigg{)}.$

The polarization waveforms are shown in the Fig. 5 for IOPB-I parameter sets.
The dominant (2,2) mode up to 3PN order is written as follow (Blanchet et al.,
2008; Kidder, 2008; Lackey, 2012)

$\displaystyle h_{22}$
$\displaystyle=-8\nu\sqrt{\frac{\pi}{5}}\frac{M_{t}}{D}e^{-2i\Phi}x\Bigg{\\{}1+\Bigg{(}-\frac{107}{42}+\frac{55\nu}{42}\Bigg{)}x+2\pi
x^{3/2}$ (25)
$\displaystyle+\Bigg{(}-\frac{2173}{1512}-\frac{1069\nu}{216}+\frac{2047\nu^{2}}{1512}\Bigg{)}x^{2}+\Bigg{[}\frac{-107\pi}{21}+\Bigg{(}\frac{34\pi}{21}$
$\displaystyle-24i\Bigg{)}\nu\Bigg{]}x^{5/2}+\Bigg{[}\frac{27027409}{646800}-\frac{856\gamma_{E}}{105}+\frac{2\pi^{2}}{3}+\frac{428i\pi}{105}$
$\displaystyle-\frac{428}{105}\ln(16x)+\Bigg{(}\frac{41\pi^{2}}{96}-\frac{278185}{33264}\nu\Bigg{)}-\frac{20261\nu^{2}}{2772}$
$\displaystyle+\frac{11463\nu^{3}}{99792}\Bigg{]}x^{3}\Bigg{\\}},$

where $D$ is the distance between source and observer taken as 100 Mpc in our
calculations.

##  3 RESULTS AND DISCUSSIONS

In this section, we discussed our calculated results with three different
parameter sets. On top of the baryonic and leptonic EoS, the contribution of
DM on it and subsequently its effects on M-R profile, tidal deformability and
GW properties are analyzed. The results are given in Table 1 and Figs. 1-7.

###  3.1 EOS, mass-radius relation, tidal deformability

As it is stated earlier, the three RMF EoS, namely NL3 (stiff) (Lalazissis et
al., 1997), IOPB (less stiff) (Kumar et al., 2018) and the G3 (soft) (Kumar et
al., 2017) for the core part are calculated. The well known BCPM EoS (Sharma
et al., 2015) is used uniformly for all these sets to include the effect of
the lower dense regions, i.e. the crust part and the NS properties are
calculated with these unified EOS. We add the DM inside the NS, which
interacts with nucleons via SM Higgs (see Sub-Sec. 2.1). As a representative
case, the EoS of IOPB-I parameter set is chosen with DM momentum,
$k_{f}^{DM}$= 0, 0.03 and 0.05 GeV, which in short, written as
IOPB-I+DM0111same as IOPB-I because DM0 means no DM inside the NS, IOPB-I+DM3,
and IOPB-I+DM5 respectively. The NS properties such $R_{1.35}$,
$\lambda_{1.35}$, $\Lambda_{1.35}$, $k_{2,1.35}$ are given in Table 1. The
predicted mass-radius for DM admixed NS with different DM momenta are also
given for comparison. The maximum mass decreases with the increase of
$k_{f}^{DM}$. The total EoS of the NS is depicted in Fig. 1. From the right
panel of Fig. 1, it is clear that with the increase of DM momentum, the EoS
becomes softer, which is consistent with our previous results (Das et al.,
2020).

Figure 1: (colour online) Left: EoSs are shown for 3-parameter sets NL3,
IOPB-I and G3. Right: The EOSs are shown with the addition of DM with
$k_{f}^{DM}$= 0, 0.03 and 0.05 GeV for IOPB-I parameter set as representative
case. Figure 2: (colour online) Mass-Radius plot for IOPB-I parameter sets
with $k_{f}^{DM}$ is equal to 0, 0.03 and 0.05 GeV. Recently measured heavy
NS, PSR J0740+6620 (Cromartie et al., 2019) and PSR J0348+0432 (Antoniadis et
al., 2013) are shown with horizontal bars. The NICER results are also shown
with rectangles from two different analysis (Miller et al., 2019; Riley et
al., 2019).

The $M$-$R$ relation of a star is calculated by solving the TOV equations with
EoS as the main input with the boundary conditions $P=P_{c}$ at $r=0$ and
$P=0$ at $r=R$. The masses and radii with IOPB-I parameter sets are calculated
for DM admixed NS, which are shown in Fig. 2. Recently measured massive pulsar
PSR J0740+6620 (Cromartie et al., 2019), PSR J0348+0432 (Antoniadis et al.,
2013) and NICER (Miller et al., 2019; Riley et al., 2019) data are also
depicted. With the addition of DM, the maximum mass $M_{max}$ and radius
$R_{max}$ decrease. For $k_{f}^{DM}$=0.03 GeV, the $M_{max}$ satisfy the
constraint given by Cromartie et al. (Cromartie et al., 2019) as well as
Antoniadis et al. (Antoniadis et al., 2013) and the radius constraint given by
NICER (Miller et al., 2019; Riley et al., 2019). However, with $k_{f}^{DM}$=
0.05 GeV, the $M_{max}$ and $R_{max}$ become very less, which doesn’t satisfy
both massive pulsar and NICER data. Therefore, one can constraints the DM
percentage inside the NS with the help of the observational data.

The tidal deformability of the NS decreases with DM’s addition, as shown in
Fig. 3. The same trend also obtained for $\Lambda$. This is because, with the
addition of DM, the EoS becomes softer and the mass and radius change
accordingly, as shown in Fig. 2. Therefore, the softer EoS gives lower
$\lambda$ as compare to the stiffer one. The vertical double-headed line
represents the constraints from GW170817 with
$\Lambda_{1.4}=190_{-120}^{+390}$ (Abbott et al., 2018). The values of
$\Lambda_{1.4}$ are 473.22, 169.644 for DM momentum 0.03 and 0.05 GeV,
respectively. This implies that the value of $\Lambda_{1.4}$ with these DM
momenta satisfies the constraints given by GW170817.

Figure 3: (colour online) Left: The mass variation of tidal deformability of the NS is shown for IOPB-I set with two different DM momentum. Right: The dimensionless tidal deformability vs. masses of the NS with the addition of DM are shown. The vertical line with bar represents the GW170817 constraints on $\Lambda_{1.4}$ (Abbott et al., 2017a, 2018). Table 1: The maximum mass of the spherical NS $M_{max}$ obtained with different EOS, the radius $R_{1.35}$, the tidal deformability $\lambda_{1.35}$, the dimensionless tidal deformability $\Lambda_{1.35}$, the tidal love number $k_{2,1.35}$, the compactness $C_{1.35}$ for NS mass 1.35 $M_{\odot}$. The total mass of the BNS system’s is $M_{t}=2.7M_{\odot}$ and $M_{t}\Omega_{0}=M_{t}\pi f_{0}=0.0155$ are taken in this calculation with the minimum gravitational wave frequency $f_{0}$ $\approx$ 371 Hz. $\lambda_{1.35}^{*}$ is in $10^{36}$ g cm2 sec-2 unit. EOSs |  $M_{max}$ ($M_{\odot})$ |  $R_{1.35}$ (km) | $\lambda_{1.35}^{*}$ | $\Lambda_{1.35}$ | $k_{2,1.35}$ | $C_{1.35}$ | $M_{t}\Omega_{0}$
---|---|---|---|---|---|---|---
NL3 | 2.775 | 14.568 | 7.415 | 1575.239 | 0.1130 | 0.137 | 0.0155
G3 | 1.979 | 12.476 | 2.754 | 585.182 | 0.0911 | 0.159 | 0.0155
IOPB-I | 2.149 | 13.168 | 4.026 | 855.304 | 0.1017 | 0.151 | 0.0155
IOPB-I+DM3 | 2.051 | 12.047 | 2.717 | 577.237 | 0.1071 | 0.165 | 0.0155
IOPB-I+DM5 | 1.788 | 10.204 | 0.990 | 210.387 | 0.0896 | 0.195 | 0.0155

###  3.2 Gravitational waves properties

In this sub-sec., we describe the BNS’s in-spiral properties using the PN
formalism applied to quasi-circular orbit. For simplicity, we restrict our
calculations only to EMBNS. Again, we take IOPB-I as a representative EoS for
the DM admixed NS since it predicts the mass of the NS 2.149 $M_{\odot}$ which
is consistent with PSR J0740+6620 (Cromartie et al., 2019). Here, we calculate
the polarization waveforms and the GW properties such as amplitude, frequency,
orbital phase for the EMBNS. The observables are calculated at retarded time
defined by (Hotokezaka et al., 2013):

$\displaystyle t_{ret}=t-r_{*},$ (26)

where $r_{*}$ is the tortoise coordinate defined as

$\displaystyle r_{*}=r_{A}+2M_{t}\ln\Big{(}\frac{r_{A}}{2M_{t}}-1\Big{)},$
(27)

where $r_{A}=\sqrt{A/4\pi}$ and A is the proper surface area of the sphere. In
a simplify manner, $r_{A}$ is written as (Hotokezaka et al., 2013)

$\displaystyle r_{A}=r\Bigg{(}1+\frac{M_{t}}{2r}\Bigg{)}^{2},$ (28)

where $r\approx 200M_{t}$ is the finite spherical-coordinate radius.

Figure 4: (colour online) (2,2) mode waveforms are shown with and without
tidal contributions for the IOPB-I parameter sets as a representative case.
Figure 5: (colour online) Plus and cross polarization waveforms are shown for
the IOPB-I parameter sets. Figure 6: (colour online) (2,2) mode waveform,
frequency, phase and PN-parameters are shown for three parameter sets of the
EMBNS in the retarded time interval calculated for $D=100$ Mpc.

Fig. 6 shows the waveforms, frequencies, phases, and $x$ for NL3, G3 and
IOPB-I sets. The stiff EoS, like NL3, predicts higher mass and tidal
deformability as compared to the softest EoS like G3. Therefore, the massive
NS easily deformed by the presence of tidal fields created by its companion
star. Hence, a less massive BNS system sustains a longer time than a more
massive BNS system in the in-spiral phase. This is because the tidal
interactions accelerate the orbital evolution in the late in-spiral phases due
to increased interactions between two NSs. Some significant effect has seen in
Fig. 4 with and without tidal interactions.

To have a quantitative and comparative estimation for the in-spiral properties
of the BNS merger, we enumerate the numerical value of $t_{ret}$, $\Phi$, $f$
and $x$ in Table 2 using NL3, IOPB-I and G3. The last orbital time period is
0.0583 sec. for NL3, which is less than from other two sets and confirms the
model-dependent prediction. There is significant influence seen for $t_{ret}$,
$\Phi$ and $f$, except the value of $x$. The frequency and phase are also
increased for the BNS of soft EoS. The pictorial representations of $f$,
$\Phi$ and $x$ are compared in Fig. 6 with and without DM contains.

Table 2: The orbital phase ($\Phi$), gravitational wave frequency ($f$), post-Newtonian parameter ($x$) are given in the last in-spiral orbits of the BNS system. EOSs |  $t_{ret}$ (sec.) |  $\Phi$ (radians) |  $f$ (hZ) |  $x$
---|---|---|---|---
NL3 | 0.0583 | 102.505 | 2356.47 | 0.213
G3 | 0.0597 | 106.383 | 2377.13 | 0.214
IOPB-I | 0.0593 | 105.178 | 2343.66 | 0.212
IOPB-I+DM3 | 0.0597 | 106.429 | 2387.09 | 0.215
IOPB-I+DM5 | 0.0603 | 108.224 | 2365.90 | 0.214
Figure 7: (colour online) Same as Fig. 6, but for DM admixed NS with IOPB-I
parameter set as a representative case.

With the addition of DM to the NS EoS, the calculated macroscopic properties
($M$, $R$ and $\lambda$) decreases. As a result, the tidal interaction between
the two NSs becomes weaker as compared to the normal BNS combination. For
example, $M=2.149M_{\odot}$ without DM and it is 2.051 $M_{\odot}$ with
$k_{f}^{DM}=0.03$ GeV with IOPB-I set. Similarly, the radius of the star is
$R_{1.35}=13.168$ km without DM, and it is 12.047 km when DM is included (see
Table 1). The calculated values for $f$, $\Phi$ and $x$ are tabulated in Table
2 for IOPB-I parameter set as a representative case with DM momenta
$k_{f}^{DM}=0.03$ and $0.05$ GeV. The time period $t_{ret}$ is found to be
0.0603 sec. with IOPB-I+DM5, which is larger than the other two values
obtained with IOPB-I and IOPB-I+DM3. These results infer the more sustainable
time in the in-spiral orbits of the BNS system in the presence of DM particles
(see Fig. 7). The other quantities such as $f$ and $\Phi$ also affected by DM,
as shown in Fig. 7.

In general, with the addition of DM, the EOS becomes softer, which affects the
macroscopic properties of the BNS, i.e. the mass of the NS decreases with DM.
However, a lot of ambiguity prevails in the post-merger remnants of BNS. This
has been settled partially after the analysis of GW170817 and GW190425 events.
Some of the possible pathways can be found in the Ref. (Sarin & Lasky, 2021).
The likely theoretical probabilities for a post-merger NS remnant basically
depend on (i) the remnant mass ($M_{R}$), and (ii) the EOS of the NS which
dictates TOV mass ($M_{TOV}$);

  1. 1.

if $M_{R}$ $\gtrsim 1.5\ M_{TOV}$, the system promptly collapses to a black
hole.

  2. 2.

for $1.2\ M_{TOV}\lesssim M\lesssim 1.5\ M_{TOV}$, a hypermassive neutron star
survives the collision, but collapses to form a black hole on dynamical
timescales.

  3. 3.

in the case, $M_{TOV}<M_{R}\lesssim 1.2\ M_{TOV}$, a supramassive NS is
survived the collision and will collapses to form a black hole on secular
timescales.

  4. 4.

if $M_{R}\leq M_{TOV}$, a stable NS is formed after the merger.

We refer the readers to see Fig. 1 of this Ref. (Sarin & Lasky, 2021) for a
clear picture.

The post-merger remnant of DM admixed BNS merger depends on the percentage of
DM inside the NS. In Ref. (Ellis et al., 2018), it is reported that for a
contribution of DM $\sim$ 5-10% of the NS mass, an extra peak in the power
spectral density of the GW signal is found. Also, for a self-interacting DM of
about $\lesssim 10\%$, leads to the early formation of a black hole (Pollack
et al., 2015), which can be subsequent seed for supermassive black hole.
Therefore, to understand the nature of post-merger remnant of DM admixed BNS,
one has to know (i) the exact nature of DM particles and (ii) the percentage
of DM present inside the NS.

##  4 SUMMARY AND CONCLUSIONS

In summary, we study the effects of DM of the EMBNS system in the in-spiral
phase. The RMF model is used to calculate the EoSs which are input to the TOV
equations. In our calculations, the relatively old NL3 parameter set and the
two recently developed IOPB-I and G3 forces are used to see the model-
dependent nature of the BNS properties. Finally, the calculations are repeated
with and without dark matter contain to see the effects at various DM
percentage.

The waveforms of the BNS merger in their in-spiral phase are compared with and
without tidal interactions. We find that if the tidal interactions are
switched off, the BNS sustain more time in their in-spiral phases. The
polarization waveform $h_{+}$ and $h_{\times}$ and the strain amplitude of
$h_{22}$ mode are calculated for the three-parameter sets. The calculations
are done in the retarded time interval at a distance of 100 Mpc between the
source and the observer. Here, we find that the binary system favours a longer
time for softer EoS (G3) as compared to the stiffer NL3 case. The other
quantities (frequencies and phase) are almost similar to all the three-
parameter sets. However, the in-spiral times in the last orbit are force-
dependent.

The DM affects the NS’s properties by assuming that the nucleons interact with
DM through SM Higgs. The coupling parameters are constrained from both direct
detection experiments and LHC searches. The macroscopic properties such as
mass, radius, and tidal deformability decrease with the increase of DM
percentage inside the NS. The in-spiralling properties are also studied for
the DM admixed NS. We find that with the addition of DM, the binary system
becomes less deformed, and it sustains more time in their in-spiral phases.
The other properties are also affected by the DM as compared to the ordinary
NS. Hence, the DM effects on the in-spiral properties of the BNS is
significant. Thus, we suggest that the people have to take DM inside the
compact objects when they modelling the in-spiral waveforms for the BNS
systems.

## Acknowledgement

HCD would like to thanks Dr. Bharat Kumar and Dr. S. K. Biswal for the
discussion on the post-Newtonian formalism and waveform of the gravitational
waves.

## DATA AVAILABILITY

This manuscript has no associated data or the data will not be deposited. Data
sharing not applicable to this article as no data sets were generated or
analysed during the current study.

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# An example related to the Erdős–Falconer question over arbitrary finite
fields

Brendan Murphy and Giorgis Petridis

###### Abstract

There exists an infinite family of examples of subsets of $\mathbb{F}_{q}^{2}$
with $q^{4/3}$ elements whose distance sets are not the whole of
$\mathbb{F}_{q}$.

††footnotetext: The first author is supported by the Leverhulme grant RPG
2017-371. The second author is supported by the NSF Award 1723016 and
gratefully acknowledges the support from the RTG in Algebraic Geometry,
Algebra, and Number Theory at the University of Georgia, and from the NSF RTG
grant DMS-1344994.

##  1 Introduction

Iosevich and Rudnev were the first to investigate in [6] the Erdős–Falconer
distance set question ([2, 3]) over finite fields. In this context distance is
defined algebraically and the _distance set_ of a set $E$ is

$\Delta(E)=\\{(x-y)\cdot(x-y):x,y\in E\\}.$

The _distance number_ of $E$ is $|\Delta(E)|$. Iosevich and Rudnev studied
large subsets of vector spaces over finite fields and sought to find a
condition on the cardinality of the subset that guarantees that its distance
set is the entire finite field. They proved the following lower bound
($\mathbb{F}_{q}$ denotes a finite field of $q$ elements).

######  Theorem 1 (Iosevich–Rudnev).

For all $d\geq 2$ and all odd $q$, if $E\subseteq\mathbb{F}_{q}^{d}$ satisfies
$|E|>4q^{(d+1)/2}$, then $\Delta(E)=\mathbb{F}_{q}$.

The exponent $(d+1)/2$ cannot be improved for odd dimensions $d$. Hart,
Iosevich, Koh, and Rudnev proved in [5] that there exists an absolute $c>0$
and subsets $E\subset\mathbb{F}_{q}^{d}$ for any odd $d$, such that $|E|\geq
cq^{(d+1)/2}$ and $\Delta(E)\neq\mathbb{F}_{q}$. Theorem 1 remains the state-
of-the-art for even $d$.

In this short note, we construct an infinite family of examples for $d=2$,
which shows that the exponent in Theorem 1 must be at least as big as $4/3$.
Our examples disprove Conjecture 1.1 in [6] for $d=2$.

######  Theorem 2.

Let $p$ be a prime. There exist infinitely many powers $q$ of $p$ and a subset
$E\subset\mathbb{F}_{q}^{2}$ such that $|E|=q^{4/3}$ and
$\Delta(E)\neq\mathbb{F}_{q}$. In fact, as $q\to\infty$, $|\Delta(E)|/q\leq
1/2$.

The construction is very similar to a construction in [7]. Some remarks.

  1. 1.

The examples in Theorem 2 stem from properties of vector spaces over
subfields. This means that they cannot be extended to prime order fields. They
show that vector spaces over subfields (and not only subfields) can obstruct
growth in geometric questions.

  2. 2.

The examples in Theorem 2 complement results in [1, 4], which show that if
$E\subseteq\mathbb{F}_{q}^{2}$ and $|E|>q^{4/3}$, then both the distance
number and, the intuitively defined, pinned distance number of $E$ is a
constant multiple of $q$. Theorem 2 shows that for such $E$ $\Delta(E)$ does
not always contain, say, $99\%$ of the elements of $\mathbb{F}_{q}$.

  3. 3.

Theorem 2 does not rule out the existence of a positive $\delta>0$ with the
property that, for all sufficiently large $q$, any
$E\subseteq\mathbb{F}_{q}^{2}$ with $|E|>q^{4/3-\delta}$ elements satisfies
$|\Delta(E)|>q/2$. A result along these lines was proved in [7] for an
analogous question.

##  2 Proof of Theorem 2

We prove Theorem 2 for odd primes $p$ that are congruent to 3 modulo 4. The
cases where $p=2$ or $p$ is congruent to $1$ modulo 4 are similar.

We start with $\mathbb{F}_{p}$ and adjoin a square root of -1, denoted by $i$,
to get the quadratic extension $\mathbb{F}_{p}[i]$. Let $F$ be any finite
extension of $\mathbb{F}_{p}[i]$ and $\mathbb{F}_{q}$ be any cubic extension
of $F$. In other words we may take $q=p^{6r}$ for every positive integer $r$.

Consider $\mathbb{F}_{q}$ as a vector space over $F$ and let $V$ be any
2-dimensional vector subspace. It is shown in [7, Proposition 11] that
$VV\neq\mathbb{F}_{q}$ and that $|VV|/q\to 1/2$ as ($r$ and hence)
$q\to\infty$.

We are now in position to describe $E$. Set

$E=\\{(u,iv):u,v\in V\\}\subset\mathbb{F}_{q}^{2}.$

Finally note that, because of closure of $V$ under addition and subtraction,

$\displaystyle\Delta(E)$
$\displaystyle=\\{(u_{1}-u_{2})^{2}+(iv_{1}-iv_{2})^{2}:u_{1},u_{2},v_{1},v_{2}\in
V\\}$ $\displaystyle=\\{u^{2}-v^{2}:u,v\in V\\}$
$\displaystyle=\\{(u-v)(u+v):u,v\in V\\}$ $\displaystyle\subseteq VV.$

Therefore $\Delta(E)\neq\mathbb{F}_{q}$ while $|E|=|V|^{2}=|F|^{4}=q^{4/3}$.

Note that when $q$ is odd we have $\Delta(E)=VV$.

## References

  * [1] J. Chapman, M. B. Erdoğan, D. Hart, A. Iosevich, and D. Koh.  Pinned distance sets, $k$-simplices, Wolff’s exponent in finite fields and sum-product estimates.  Math. Z., 271(1-2):63–93, 2012.
  * [2] P. Erdős.  On sets of distances of $n$ points.  Amer. Math. Monthly, 53:248–250, 1946.
  * [3] K. J. Falconer.  On the Hausdorff dimensions of distance sets.  Mathematika, 32:206–212, 1985.
  * [4] B. Hanson, B. Lund, and O. Roche-Newton.  On distinct perpendicular bisectors and pinned distances in finite fields.  Finite Fields Appl., 37:240–264, 2016.
  * [5] D. Hart, A. Iosevich, D. Koh, and M. Rudnev.  Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdős-Falconer distance conjecture.  Trans. Amer. Math. Soc., 363:3255–3275, 2011.
  * [6] A. Iosevich and M. Rudnev.  Erdős distance problem in vector spaces over finite fields.  Trans. Amer. Math. Soc., 359:6127–6142, 2007.
  * [7] B. Murphy and G. Petridis.  Products of differences over arbitrary finite fields.  Discrete Analysis, 18:42 pp., 2018.

Department of Mathematics, University of Bristol, and Heilbronn Institute of
Mathematical Research, Bristol, UK.

Department of Mathematics, University of Georgia, Athens, GA 30602, USA.

Email addresses<EMAIL_ADDRESS>and<EMAIL_ADDRESS>


# Weighted Inequalities for Fractional maximal functions on the infinite
rooted $k$-ary tree

Abhishek Ghosh and Ezequiel Rela  Tata Institute of Fundamental Research,
Centre for Applicable Mathematics, Bangalore–560065, Karnataka, India.
<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Mathematics,
University of Buenos Aires, Ciudad Universitaria Pabellón I, Buenos Aires
(1428), Argentina<EMAIL_ADDRESS>
###### Abstract.

In this article we introduce the fractional Hardy-Littlewood maximal function
on the infinite rooted $k$-ary tree and study its weighted boundedness. We
also provide examples of weights for which the fractional Hardy-Littlewood
maximal function satisfies strong type $(p,q)$ estimates on the infinite
rooted $k$-ary tree.

###### Key words and phrases:

Infinite rooted $k$-ary tree, fractional Hardy-Littlewood maximal function,
weights

###### 2020 Mathematics Subject Classification:

42B25, 05C63

E.R. is partially supported by grants UBACyT 20020170200057BA and PIP
(CONICET) 11220110101018

##  1\. Introduction and Preliminaries

Weighted estimates for classical operators in Harmonic analysis has always
been an active and growing area of research. The seminal work of Muckenhoupt
[Muc72] introduced the $A_{p}$ class of weights and characterized the weighted
inequalities for the Hardy-Littlewood maximal function. Subsequently,
Muckenhoupt and Wheeden in [MW74] studied weighted inequalities for the
fractional Hardy-Littlewood maximal function and introduced the $A_{p,q}$
class of weights. In this article we initiate the study of weighted estimates
for the fractional Hardy-Littlewood maximal function on infinite graphs. In
recent times there are several substantial works devoted to the study of the
discrete analogues of the Hardy-Littlewood maximal function and the fractional
Hardy-Littlewood maximal function in various frameworks, for instance we refer
[CH12, GRM21, GRM22]. Our main motivation for studying this object originates
from the recent works [ORRS21] and [ORR21]. In [ORRS21], Ombrosi, Rivera-Ríos
and Safe have proved a sharp analog of Fefferman-Stein inequality for the
Hardy-Littlewood maximal operator on the infinite rooted $k$-ary tree and
subsequently in [ORR21] weighted inequalities for the Hardy-Littlewood maximal
function were investigated by Ombrosi and Rivera-Ríos. We also refer the
articles [ST16, ST19] where the authors studied connections between
geometrical properties of infinite graphs and the boundedness of the Hardy-
Littlewood maximal operator. To elaborate the results let us start with
preliminaries.

For $k\geq 2$, we denote $\mathcal{T}^{k}$ to be the infinite rooted $k$-ary
tree i.e., the infinite tree where each vertex has $k$ successors.
$(\mathcal{T}^{k},d,|.|)$ is a metric measure space where $d$ is the standard
tree distance, i.e., $d(x,y)$ is the number of edges in the minimal unique
path joining $x$ and $y$ and $|A|$ denotes the counting measure of any set
$A\subset\mathcal{T}^{k}$. Given any function $g$,
$\int_{A}g(y)\,dy:=\sum_{y\in A}g(y)$ and
$B(x,r):=\\{y\in\mathcal{T}^{k}:d(x,y)\leq r\\}$ will denote the ball of
radius $r$ centred at the point $x\in\mathcal{T}^{k}$. Set
$\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$. Now let us define the fractional
maximal operator in this setting.

######  Definition 1.1.

Fix $0\leq\alpha<1$. Let $f$ be any locally integrable function then the
fractional Hardy-Littlewood maximal operator is defined as follows

$\displaystyle
M_{\alpha}f(x)=\sup_{r\in\mathbb{N}_{0}}\frac{1}{|B(x,r)|^{1-\alpha}}\int_{B(x,r)}|f(y)|\,dy.$

For $\alpha=0$, $M_{0}$ is the classical Hardy-Littlewood maximal function and
will be simply denoted by $M$. Let $S(x,r):=\\{y:d(x,y)=r\\}$ denote the
sphere, centred at $x$, of radius $r\in\mathbb{N}_{0}$. We also define the
spherical fractional maximal operator as following

$\displaystyle\mathcal{S}_{\alpha}f(x)=\sup_{r\in\mathbb{N}_{0}}\frac{1}{|S(x,r)|^{1-\alpha}}\int_{S(x,r)}|f(y)|\,dy.$

We will denote by
$A_{r,\alpha}f(x):=\frac{1}{|S(x,r)|^{1-\alpha}}\int_{S(x,r)}|f|$ to be the
spherical fractional average of $f$ at $x$ over the sphere $S(x,r)$. Unlike
the Euclidean setting, it is easy to see that the operators $M_{\alpha}$ and
$\mathcal{S}_{\alpha}$ are pointwise equivalent i.e.,
$C_{\alpha}\mathcal{S}_{\alpha}f(x)\leq M_{\alpha}f(x)\leq
c_{\alpha}\mathcal{S}_{\alpha}f(x)$ for a.e. $x\in\mathcal{T}^{k}$. This is
due to the fact that $|S(x,r)|\simeq k^{r}$ and $\frac{|B(x,r)|}{|S(x,r)|}\leq
2$ (see [ORRS21]). Keeping this in mind we will only write our results for the
spherical fractional maximal function which can be easily transferred to the
fractional Hardy-Littlewood maximal function.

Let us review some unweighted boundedness for the spherical fractional maximal
operator $\mathcal{S}_{\alpha}$. Note that
$\mathcal{S}_{\alpha}f(x)\lesssim\|f\|_{L^{1}(\mathcal{T}^{k})}^{\alpha}(Mf(x))^{1-\alpha}$
which implies that $\mathcal{S}_{\alpha}$ maps $L^{1}(\mathcal{T}^{k})$ to
$L^{\frac{1}{1-\alpha},\infty}(\mathcal{T}^{k})$ boundedly, where we have used
the fact that $M$ is of weak type $(1,1)$. Hölder’s inequality implies that
$\mathcal{S}_{\alpha}:L^{1/\alpha}(\mathcal{T}^{k})\to
L^{\infty}(\mathcal{T}^{k})$ boundedly. Interpolating the above two, we obtain
that $\mathcal{S}_{\alpha}:L^{p}(\mathcal{T}^{k})\to L^{q}(\mathcal{T}^{k})$,
where $\frac{1}{q}=\frac{1}{p}-\alpha$ and $1<p<\frac{1}{\alpha}$. In this
article we will provide sufficient conditions on weights $w$ for which
$\mathcal{S}_{\alpha}$ maps $L^{p}(w)$ to $L^{q}(w)$ with $1<p\leq q<\infty$.

We would like to point out that in [ORR21] the authors showed that there exist
weights for which the Hardy-Littlewood maximal function is bounded from
$L^{p}(\mathcal{T}^{k})$ to itself but the obvious Muckenhoupt-type condition
does not hold. This suggests that the classical $A_{p}$-theory is not
applicable for the infinite rooted $k$-ary tree. To overcome this difficulty,
let us start with obtaining some useful necessary conditions for the
boundedness of the spherical fractional Hardy-Littlewood maximal function on
the infinite rooted $k$-ary tree.

For any set $A\subset\mathcal{T}^{k}$ and any weight $w$, denote
$w(A):=\sum_{x\in A}w(x)$. It is easy to see that if $\mathcal{S}_{\alpha}$
maps $L^{p}(w)$ to $L^{q}(w)$ then for any two finite subsets $E$ and $F$ of
$\mathcal{T}^{k}$ and any $r\in\mathbb{N}_{0}$ we have

$\displaystyle\sum_{x\in E}\frac{w(F\cap S(x,r))}{|S(x,r)|^{1-\alpha}}$
$\displaystyle=\sum_{x\in E}A_{r,\alpha}(w\chi_{F})(x)$
$\displaystyle=\int_{\mathcal{T}^{k}}\chi_{E}(x)A_{r,\alpha}(w\chi_{F})(x)$
$\displaystyle=\int_{\mathcal{T}^{k}}A_{r,\alpha}(\chi_{E})(x)w(x)\chi_{F}(x)$
$\displaystyle\leq\|A_{r,\alpha}\chi_{E}\|_{L^{q}(w)}\|\chi_{F}\|_{L^{q^{\prime}}(w)}$
$\displaystyle\lesssim\|\chi_{E}\|_{L^{p}(w)}\|\chi_{F}\|_{L^{q^{\prime}}(w)}\lesssim
w(E)^{\frac{1}{p}}w(F)^{1-\frac{1}{q}},$

where we have used the self-adjointness of the operator $A_{r,\alpha}$. Hence
the condition

(1.1) $\sum_{x\in E}w(F\cap S(x,r))\lesssim
k^{r(1-\alpha)}w(E)^{\frac{1}{p}}w(F)^{1-\frac{1}{q}}$

is a necessary condition for the boundedness of $\mathcal{S}_{\alpha}$ from
$L^{p}(w)$ to $L^{q}(w)$. However, the condition (1.1) will not be sufficient
for the boundedness of $\mathcal{S}_{\alpha}$, but assuming little better
decay on $k$ we are able to obtain the boundedness. Motivated by this we
define the following class of weights.

######  Definition 1.2.

Let $1<p,q<\infty$ and $0<\alpha<1$. We say
$w\in\mathcal{Z}_{p,q}^{\epsilon,\alpha}$ if there exist $\epsilon\in(0,1)$,
and a constant $C>0$ such that for all $r\in\mathbb{N}$ and all measurable
sets $E,F\subset\mathcal{T}^{k}$ we have

(1.2) $\displaystyle\sum_{x\in E}w(F\cap S(x,r))\leq C\,k^{\epsilon
r(1-\alpha)}w(E)^{\frac{1}{p}}w(F)^{1-\frac{1}{q}}.$

Now we are at a position to state the main result of this article.

######  Theorem 1.3.

Let $1<p\leq q<\infty$ and $0<\alpha<1$. Then for
$w\in\mathcal{Z}_{p,q}^{\epsilon,\alpha}$ we have
$\mathcal{S}_{\alpha}:L^{p}(w)\to L^{q}(w)$ boundedly.

We note here that the exponents $p,q$ in the above theorem are not related to
the parameter $\alpha$ and this might look strange, compared to the unweighted
theory. The key is that in order to have a nontrivial example of a weight
belonging to the class $\mathcal{Z}_{p,q}^{\epsilon,\alpha}$ satisfying (1.2)
we will need to consider the classical condition
$\frac{1}{q}=\frac{1}{p}-\alpha$ (see Theorem 1.4 and Theorem 1.5).

Since the volume of the balls grows exponentially in the infinite rooted
$k$-ary tree, i.e., $\frac{|B(x,r)|}{|B(x,r/2)|}\simeq k^{r/2}\to\infty$ as
$r\to\infty$, the space $(\mathcal{T}^{k},d,|.|)$ is not even an upper-
doubling space. Therefore, covering arguments and Calderón-Zygmund
decomposition are not well suited here. However, overcoming these obstacles,
the weak type $(1,1)$ estimate for the Hardy-Littlewood maximal function on
infinite rooted $k$-ary tree was obtained in [NT10]. In [NT10], Naor and Tao
developed an ingenious strategy, based on “expander” properties of graphs and
combinatorial arguments, to obtain the weak type $(1,1)$ boundedness of the
Hardy-Littlewood maximal function with bounds independent of $k$ (see Theorem
1.5 in [NT10]). We would also like to note that, in [CMS10], the same result
was proved using a different approach. The idea of our proofs is inspired by
the works [ORRS21] and [ORR21] where they have brilliantly implemented the
techniques developed by Naor and Tao in the weighted setting. We also refer
the article [ST16] where the authors extend the result of Naor and Tao to more
general infinite graphs. We end this section by constructing examples of
appropriate weights.

Let us introduce some notations. Set $\mathcal{T}_{0}$ as the set containing
the root of the infinite tree and $\mathcal{T}_{i}$ denotes the set of
children of the vertices in $\mathcal{T}_{i-1}$ for all $i\geq 1$. Then
$\mathcal{T}^{k}=\bigcup\limits_{i=0}^{\infty}\mathcal{T}_{i}$. The examples
of weights belonging to the class $\mathcal{Z}_{p,q}^{\epsilon,\alpha}$ are
not easy to obtain since the condition (1.2) is generally difficult to verify.
One of the primary reason being the fact that the condition (1.2) involves
arbitrary sets and their possible interactions with multiple levels
$\mathcal{T}_{i}$. To overcome this difficulty, let us first provide a simpler
criterion which only depends on the behaviour of the weight on individual
scales $\mathcal{T}_{i}$. In that direction our result reads as follows.

######  Theorem 1.4.

Let $\frac{1}{q}=\frac{1}{p}-\alpha$ with $1<p<\frac{1}{\alpha}$ and
$0<\alpha<1$. Denote $\delta=\frac{1-\alpha p-\alpha p^{2}}{1-\alpha p}$ and
let $w$ be a weight satisfying the following condition: For
$r\in\mathbb{N}_{0}$ and $|i-j|\leq r$, we have that

(1.3) $w(\mathcal{T}_{i}\cap S(x,r))\lesssim
k^{\frac{r+i-j}{2}(p-\delta)}k^{r\delta}w(x)^{q/p}~{}~{}\text{for~{}all}~{}~{}x\in\mathcal{T}_{j}.$

Then $w\in\mathcal{Z}^{\epsilon,\alpha}_{p,q}$ for
$\epsilon=\frac{1-p\alpha}{1-\alpha}$.

The proof of the above theorem relies on an optimization technique developed
in [NT10, ORR21]. However, we modify it significantly since in our case we are
dealing with the much more general situation of off-diagonal estimates for the
fractional maximal function. Finally the following result provides some
concrete example of weights.

######  Theorem 1.5.

Let $0<\alpha<1$, $1<p<\frac{1}{\alpha}$ and $\frac{1}{q}=\frac{1}{p}-\alpha$.
Define

$w(x)=\sum_{j\geq 0}k^{\beta j}\chi_{\mathcal{T}_{j}},$

where $0\leq\beta\leq\frac{p(p-1)}{q}$. Then $w$ belongs to
$\mathcal{Z}^{\epsilon,\alpha}_{p,q}$ for
$\epsilon=\frac{1-p\alpha}{1-\alpha}$ and hence $\mathcal{S}_{\alpha}$ maps
$L^{p}(w)$ to $L^{q}(w)$ boundedly.

The article is organized as follows. In the next section we prove the main
boundedness result i.e., Theorem 1.3. Section 3 is devoted in proving Theorem
1.4 and Theorem 1.5. Throughout this article $A\lesssim B$ abbreviates that
$A\leq CB$ where the constant $C$ is independent of $A$ and $B$. We will also
be using the inequality
$\sum_{j=0}^{\infty}a_{j}^{\theta}\leq(\sum_{j=0}^{\infty}a_{j})^{\theta}$,
where $a_{j}\geq 0$ and $1\leq\theta<\infty$, frequently.

##  2\. Main results

Let us prove the following lemma.

######  Lemma 2.1.

Let $1<p\leq q<\infty$, $0<\alpha<1$ and
$w\in\mathcal{Z}_{p,q}^{\epsilon,\alpha}$ for some $\epsilon\in(0,1)$. Then
for $\lambda>0,r\in\mathbb{N}$, we have the following estimate

$w(A_{r,\alpha}f>\lambda)\lesssim\frac{1}{(2^{\beta}-1)^{q}}\sum\limits_{n\in\mathbb{N}_{0}:\,1\leq
2^{n}\leq
k^{r}}2^{nq}\left(\frac{k^{r}}{2^{n}}\right)^{q\beta}\frac{k^{rq\epsilon(1-\alpha)}}{k^{rq}}w\left(\Big{\\{}x\in\mathcal{T}^{k}:\frac{|f|k^{r\alpha}}{\lambda}\geq
2^{n-1}\Big{\\}}\right)^{\frac{q}{p}}$

for all $\beta\in(0,1)$.

###### Proof.

Without loss of generality let us assume $f$ to be a non-negative function. We
can write

$\displaystyle f\leq\frac{1}{2}+\sum\limits_{n\in\mathbb{N}_{0}:1\leq
2^{n}\leq k^{r}}2^{n}\chi_{E_{n}}+f\chi_{\\{f\geq\frac{1}{2}k^{r\alpha}\\}},$

where $E_{n}=\\{2^{n-1}\leq f<2^{n}\\}$. This implies

$\displaystyle
A_{r,\alpha}f(x)\leq\frac{k^{r\alpha}}{2}+\sum\limits_{n\in\mathbb{N}_{0}:1\leq
2^{n}\leq
k^{r}}2^{n}A_{r,\alpha}(\chi_{E_{n}})(x)+A_{r,\alpha}(f\chi_{\\{f\geq\frac{1}{2}k^{r\alpha}\\}})(x).$

Consider the set $E=\\{x:A_{r,\alpha}f(x)>k^{r\alpha}\\}$. Using the above
estimate, we obtain the following inequality for all $x\in E$

$\sum\limits_{n\in\mathbb{N}_{0}:1\leq 2^{n}\leq
k^{r}}2^{n}A_{r,\alpha}(\chi_{E_{n}})(x)+A_{r,\alpha}(f\chi_{\\{f\geq\frac{1}{2}k^{r\alpha}\\}})(x)\geq\frac{1}{2}k^{r\alpha},$

which implies that

$\sum\limits_{n\in\mathbb{N}_{0}:1\leq 2^{n}\leq
k^{r}}2^{n}A_{r,\alpha}\left(\frac{1}{k^{r\alpha}}\chi_{E_{n}}\right)(x)+A_{r,\alpha}\left(\frac{1}{k^{r\alpha}}f\chi_{\\{f\geq\frac{1}{2}k^{r\alpha}\\}}\right)(x)\geq\frac{1}{2}.$

Hence $w(E)\leq w(I)+w(II)$, where

$I:=\Bigg{\\{}x:\sum\limits_{n\in\mathbb{N}_{0}:1\leq 2^{n}\leq
k^{r}}2^{n}A_{r,\alpha}\left(\frac{1}{k^{r\alpha}}\chi_{E_{n}}\right)(x)\geq\frac{1}{4}\Bigg{\\}},$

and

$II:=\Big{\\{}x:A_{r,\alpha}\left(\frac{1}{k^{r\alpha}}f\chi_{\\{f\geq\frac{1}{2}k^{r\alpha}\\}}\right)(x)\geq
0\Big{\\}}.$

Let us fix some $\beta\in(0,1)$. For $x\in I$, there exist some
$n\in\mathbb{N}_{0}$ with $1\leq 2^{n}\leq k^{r}$ such that

$2^{n}A_{r,\alpha}\left(\frac{1}{k^{r\alpha}}\chi_{E_{n}}\right)(x)\geq\frac{1}{16}(2^{\beta}-1)\left(\frac{2^{n}}{k^{r}}\right)^{\beta}.$

Otherwise if we have
$2^{n}A_{r,\alpha}\left(\frac{1}{k^{r\alpha}}\chi_{E_{n}}\right)(x)<\frac{1}{16}(2^{\beta}-1)\left(\frac{2^{n}}{k^{r}}\right)^{\beta}$,
for all $n\in\mathbb{N}_{0}$ with $1\leq 2^{n}\leq k^{r}$, we will have a
contradiction.

Let us define

$F_{n}:=\Bigg{\\{}x:A_{r,\alpha}(\chi_{E_{n}})(x)\geq\frac{1}{2^{n+4}}(2^{\beta}-1)k^{r\alpha}\left(\frac{2^{n}}{k^{r}}\right)^{\beta}\Bigg{\\}}.$

Thus to estimate $w(I)$, it is enough to estimate
$\sum\limits_{n\in\mathbb{N}_{0}:1\leq 2^{n}\leq k^{r}}w(F_{n})$. Let us
estimate each $w(F_{n})$. Using the self-adjointness of the operator
$A_{r,\alpha}$ and since $F_{n}$ is finite we obtain

$\displaystyle\sum_{x\in E_{n}}\frac{w(F_{n}\cap
S(x,r))}{|S(x,r)|^{1-\alpha}}$ $\displaystyle=$ $\displaystyle\sum_{x\in
E_{n}}A_{r,\alpha}(w\chi_{F_{n}})(x)$ $\displaystyle=$
$\displaystyle\int_{\mathcal{T}^{k}}\chi_{E_{n}}A_{r,\alpha}(w\chi_{F_{n}})$
$\displaystyle=$
$\displaystyle\int_{\mathcal{T}^{k}}A_{r,\alpha}(\chi_{E_{n}})w\chi_{F_{n}}$
$\displaystyle\geq$
$\displaystyle\frac{1}{2^{n+4}}(2^{\beta}-1)k^{r\alpha}\left(\frac{2^{n}}{k^{r}}\right)^{\beta}w(F_{n}).$

Recall that the condition on the weight is as follows:

(2.1) $\sum_{x\in E}w(F\cap S(x,r))\lesssim
k^{r\epsilon(1-\alpha)}w(F)^{1/q^{\prime}}w(E)^{1/p}.$

From the above two estimates we obtain

$\frac{1}{2^{n+4}}(2^{\beta}-1)k^{r\alpha}\left(\frac{2^{n}}{k^{r}}\right)^{\beta}w(F_{n})\lesssim\frac{k^{r\epsilon(1-\alpha)}}{k^{r(1-\alpha)}}w(F_{n})^{1/q^{\prime}}w(E_{n})^{1/p},$

which is equivalent to

$w(F_{n})^{1/q}\lesssim\frac{1}{2^{\beta}-1}2^{n}\left(\frac{k^{r}}{2^{n}}\right)^{\beta}\frac{k^{r\epsilon(1-\alpha)}}{k^{r}}w(E_{n})^{1/p}.$

We then obtain that

(2.2)
$w(F_{n})\lesssim\frac{1}{(2^{\beta}-1)^{q}}2^{nq}\left(\frac{k^{r}}{2^{n}}\right)^{q\beta}\frac{k^{rq\epsilon(1-\alpha)}}{k^{rq}}w\left(\\{f\geq
2^{n-1}\\}\right)^{q/p}.$

Thus we have the following estimate

(2.3) $w(I)\leq\sum\limits_{n\in\mathbb{N}_{0}:1\leq 2^{n}\leq
k^{r}}\frac{1}{(2^{\beta}-1)^{q}}2^{nq}\left(\frac{k^{r}}{2^{n}}\right)^{q\beta}\frac{k^{rq\epsilon(1-\alpha)}}{k^{rq}}w\left(\\{f\geq
2^{n-1}\\}\right)^{q/p}.$

Now let us estimate $w(II)$.

$\displaystyle w(II)$
$\displaystyle:=w\left(\Big{\\{}x:A_{r,\alpha}\left(\frac{1}{k^{r\alpha}}f\chi_{\\{f\geq\frac{1}{2}k^{r\alpha}\\}}\right)(x)\geq
0\Big{\\}}\right)$
$\displaystyle=w\left(\Big{\\{}x:A_{r,\alpha}\left(f\chi_{\\{f\geq\frac{1}{2}k^{r\alpha}\\}}\right)(x)\geq
0\Big{\\}}\right)$ $\displaystyle\leq
w\left(\bigcup_{y\in\\{f\geq\frac{1}{2}k^{r\alpha}\\}}S(y,r)\right)$ (2.4)
$\displaystyle\leq\sum_{y\in\\{f\geq\frac{1}{2}k^{r\alpha}\\}}w(S(y,r)).$

If we take $E=\\{y\\}$ and $F=S(y,r)$ in (2.1), we obtain

$w(S(y,r))\leq k^{r\epsilon(1-\alpha)}w(S(y,r))^{1/q^{\prime}}w(y)^{1/p},$

yielding

$w(S(y,r))\leq k^{r\epsilon(1-\alpha)q}w(y)^{q/p}.$

Plugging the above estimate in (2.4) we obtain

$\displaystyle
w(II)\leq\sum_{y\in\\{f\geq\frac{1}{2}k^{r\alpha}\\}}k^{r\epsilon(1-\alpha)q}w(y)^{q/p}\leq
k^{r\epsilon(1-\alpha)q}w\left(\Big{\\{}f\geq\frac{1}{2}k^{r\alpha}\Big{\\}}\right)^{q/p},$

where we have used the fact that $q\geq p$ in the last inequality. Observe
that the estimate for $w(II)$ is same as the final term in the summation in
(2.3). Combining the above estimates we obtain

$\displaystyle
w(\\{x:A_{r,\alpha}f>k^{r\alpha}\\})\leq\frac{1}{(2^{\beta}-1)^{q}}\sum\limits_{n\in\mathbb{N}_{0}:1\leq
2^{n}\leq
k^{r}}2^{nq}\left(\frac{k^{r}}{2^{n}}\right)^{q\beta}\frac{k^{rq\epsilon(1-\alpha)}}{k^{rq}}w(\\{f\geq
2^{n-1}\\})^{q/p}.$

Using the above estimate and homogeneity we obtain

$w(\\{A_{r,\alpha}f>\lambda\\})\leq\frac{1}{(2^{\beta}-1)^{q}}\sum\limits_{n\in\mathbb{N}_{0}:1\leq
2^{n}\leq
k^{r}}2^{nq}\left(\frac{k^{r}}{2^{n}}\right)^{q\beta}\frac{k^{rq\epsilon(1-\alpha)}}{k^{rq}}w\left(\Big{\\{}x\in\mathcal{T}^{k}:\frac{fk^{r\alpha}}{\lambda}\geq
2^{n-1}\Big{\\}}\right)^{q/p}.$

This completes the proof of the lemma. ∎

Now we prove the main theorem of this article.

###### Proof of Theorem 1.3.

We start with estimating the norm $\|A_{r,\alpha}\|_{L^{q}(w)}^{q}$ by using
the well know layer-cake formula

$\|A_{r,\alpha}\|_{L^{q}(w)}^{q}=q\int_{0}^{\infty}\lambda^{q-1}w(A_{r,\alpha}f(x)>\lambda)\,d\lambda.$

Then,

$\displaystyle\|A_{r,\alpha}\|_{L^{q}(w)}^{q}$ $\displaystyle\leq$
$\displaystyle
c_{\beta,q}\int_{0}^{\infty}\lambda^{q-1}\sum\limits_{n\in\mathbb{N}_{0};1\leq
2^{n}\leq
k^{r}}2^{nq}\left(\frac{k^{r}}{2^{n}}\right)^{q\beta}\frac{k^{rq\epsilon(1-\alpha)}}{k^{rq}}w\left(\frac{|f|k^{r\alpha}}{\lambda}\geq
2^{n-1}\right)^{q/p}\,d\lambda$ $\displaystyle\leq$ $\displaystyle
c_{\beta,q}\sum\limits_{n\in\mathbb{N}_{0};1\leq 2^{n}\leq
k^{r}}2^{nq}\left(\frac{k^{r}}{2^{n}}\right)^{q\beta}\frac{k^{rq\epsilon(1-\alpha)}}{k^{rq}}\int_{0}^{\infty}\lambda^{q-1}w\left(\frac{|f|k^{r\alpha}}{\lambda}\geq
2^{n-1}\right)^{q/p}\,d\lambda$ $\displaystyle=$ $\displaystyle
c_{\beta,q}\sum\limits_{n\in\mathbb{N}_{0};1\leq 2^{n}\leq
k^{r}}2^{nq}\left(\frac{k^{r}}{2^{n}}\right)^{q\beta}\frac{k^{rq\epsilon(1-\alpha)}}{k^{rq}}\displaystyle\int_{0}^{\infty}\lambda^{q-1}\left(\sum_{x\in\mathcal{T}^{k}}\chi_{\\{\frac{|f|k^{r\alpha}}{\lambda}\geq
2^{n-1}\\}}(x)w(x)\right)^{q/p}\,d\lambda$

Using Minkowski’s inequality the above is dominated by the following

$\displaystyle c_{\beta,q}\sum\limits_{n\in\mathbb{N}_{0};1\leq 2^{n}\leq
k^{r}}2^{nq}\left(\frac{k^{r}}{2^{n}}\right)^{q\beta}\frac{k^{rq\epsilon(1-\alpha)}}{k^{rq}}\left(\sum_{x\in\mathcal{T}^{k}}\left(\int_{0}^{\infty}\chi_{\\{x:\frac{|f(x)|k^{r\alpha}}{\lambda}\geq
2^{n-1}\\}}\lambda^{q-1}\ d\lambda\right)^{p/q}w(x)\ dx\right)^{q/p}$
$\displaystyle\leq c_{\beta,q}\sum\limits_{n\in\mathbb{N}_{0};1\leq 2^{n}\leq
k^{r}}2^{nq}\left(\frac{k^{r}}{2^{n}}\right)^{q\beta}\frac{k^{rq\epsilon(1-\alpha)}}{k^{rq}}\left(\sum_{x\in\mathcal{T}^{k}}\frac{|f(x)|^{p}k^{r\alpha
p}}{2^{np}}w(x)\right)^{q/p}$ $\displaystyle\leq
c_{\beta,q}\frac{1}{k^{rq(1-\beta-\epsilon(1-\alpha)-\alpha)}}\sum_{n\geq
0}\frac{1}{2^{nq\beta}}\left(\sum_{x\in\mathcal{T}^{k}}|f(x)|^{p}w(x)\right)^{q/p}$
$\displaystyle\leq
c_{\beta,q}\frac{1}{k^{rq(1-\beta-\epsilon(1-\alpha)-\alpha)}}\|f\|_{L^{p}(w)}^{q}.$

Now

$\displaystyle\|\mathcal{S}_{\alpha}\|_{L^{q}(w)}$
$\displaystyle\leq\sum_{r=0}^{\infty}\|A_{r,\alpha}f\|_{L^{q}(w)}$
$\displaystyle\leq
c_{\beta,q}\sum_{r=0}^{\infty}\frac{1}{k^{r(1-\beta-\epsilon(1-\alpha)-\alpha)}}\|f\|_{L^{p}(w)}$
$\displaystyle\lesssim\|f\|_{L^{p}(w)},$

provided we choose $\beta$ such that $(1-\beta-\epsilon(1-\alpha)-\alpha)>0$,
i.e., $\beta<(1-\epsilon)(1-\alpha)$. This completes the theorem. ∎

##  3\. Examples

This section is devoted in constructing examples of weights belonging to the
class $\mathcal{Z}^{\epsilon,\alpha}_{p,q}$.

###### Proof of Theorem 1.4:.

Let $E$ and $F$ be any two subsets of $\mathcal{T}^{k}$. Define
$E_{j}=E\cap\mathcal{T}_{j}$ and $F_{i}=F\cap\mathcal{T}_{i}$. Then

(3.1) $\displaystyle\sum_{x\in E}w(F\cap
S(x,r))=\sum_{m=0}^{\infty}\,\sum_{\begin{subarray}{c}i,j\in\mathbb{N}_{0}\\\
i=j+r-2m\end{subarray}}\,\sum_{x\in E_{j}}w(F_{i}\cap S(x,r)),$

since for $x\in E_{j}$ and $y\in F_{i}$ we have $d(x,y)=r$ if and only if
$i=j+r-2m$ for some $m\in\\{0,\cdots,r\\}$. The following estimate holds
trivially since for each member of $\mathcal{T}_{i}$ there can be at most
$k^{m}$ elements of $\mathcal{T}_{j}$ with distance $r$.

(3.2) $\displaystyle\sum_{x\in E_{j}}w(F_{i}\cap S(x,r))\leq k^{m}w(F_{i}).$

Now by our assumption (1.3),

(3.3) $\sum_{x\in E_{j}}w(F_{i}\cap S(x,r))\leq\sum_{x\in
E_{j}}k^{\frac{r+i-j}{2}(p-\delta)}k^{r\delta}w(x)^{q/p}\leq
k^{(r-m)(p-\delta)}k^{r\delta}w(E_{j})^{q/p}.$

Hence from (3.2) and (3.3) we obtain

$\displaystyle\sum_{x\in E_{j}}w(F_{i}\cap
S(x,r))\lesssim\min\Big{\\{}k^{(r-m)(p-\delta)}k^{r\delta}w(E_{j})^{q/p},k^{m}w(F_{i})\Big{\\}}.$

Therefore the rest of the proof is devoted to obtain an estimate for the
following quantity:

$M:=\sum_{m=0}^{\infty}\,\sum_{\begin{subarray}{c}i,j\in\mathbb{N}_{0}\\\
i=j+r-2m\end{subarray}}\min\Big{\\{}k^{(r-m)(p-\delta)}k^{r\delta}w(E_{j})^{q/p},k^{m}w(F_{i})\Big{\\}}.$

Denote $A_{j}=\frac{w(E_{j})^{q/p}}{k^{(p-\delta)j}}$ and
$B_{j}=\frac{w(F_{j})}{k^{j}}$ for $j\geq 0$ and $A_{j}=B_{j}=0$ for $j<0$.
Observe that

(3.4) $\sum\limits_{j\geq 0}k^{(p-\delta)j}A_{j}\leq
w(E)^{q/p}~{}~{}\text{and}~{}~{}\sum\limits_{j\geq 0}k^{j}B_{j}=w(F).$

Note that for a real parameter $\rho>0$ to be chosen later we have

(3.5) $\displaystyle M$ $\displaystyle=$
$\displaystyle\sum_{m=0}^{\infty}\,\sum_{\begin{subarray}{c}i,j\in\mathbb{N}_{0}\\\
i=j+r-2m\end{subarray}}\min\Big{\\{}k^{(r-m+j)(p-\delta)}k^{r\delta}A_{j},k^{m+j}B_{i}\Big{\\}}$
$\displaystyle=$
$\displaystyle\sum_{m=0}^{\infty}\,\sum_{\begin{subarray}{c}i,j\in\mathbb{N}_{0}\\\
i=j+r-2m\end{subarray}}\min\Big{\\{}k^{\frac{(i+j+r)(p-\delta)}{2}}k^{r\delta}A_{j},k^{m+j}B_{i}\Big{\\}}$
$\displaystyle\leq$ $\displaystyle
k^{\frac{p+\delta}{2}r}\sum_{j=0}^{\infty}\sum_{i<j+\rho}k^{\frac{(i+j)(p-\delta)}{2}}A_{j}+k^{r/2}\sum_{i=0}^{\infty}\sum_{j\leq
i-\rho}k^{\frac{i+j}{2}}B_{i}$ $\displaystyle\leq$ $\displaystyle
k^{\frac{p+\delta}{2}r}k^{\frac{\rho(p-\delta)}{2}}\sum_{j=0}^{\infty}k^{j(p-\delta)}A_{j}+k^{r/2}k^{-\frac{\rho}{2}}\sum_{i=0}^{\infty}k^{i}B_{i}$
$\displaystyle\leq$ $\displaystyle
k^{\frac{p+\delta}{2}r}k^{\frac{\rho(p-\delta)}{2}}w(E)^{q/p}+k^{r/2}k^{-\frac{\rho}{2}}w(F),$

where in the last inequality we have used (3.4). We need to optimize the
parameter $\rho$ in order to complete the proof. Let

$\phi_{a,b}(\rho):=a\,k^{\frac{\rho(p-\delta)}{2}}+k^{-\frac{\rho}{2}}\,b$

for some positive constants $a$ and $b$. The function $\phi$ attains its
absolute minimum at the point
$\rho=\frac{2}{p+1-\delta}\log_{k}\left(\frac{b}{a(p-\delta)}\right)$.

Now choosing $a_{0}=k^{\frac{p+\delta}{2}r}w(E)^{q/p}$ and
$b_{0}=k^{r/2}w(F)$, we obtain that (3.5) attains its minimum value if we
choose
$\rho=\rho_{0}:=\frac{2}{p+1-\delta}\log_{k}\left(\frac{k^{r/2}w(F)}{k^{\frac{p+\delta}{2}r}w(E)^{q/p}(p-\delta)}\right)$
and the value is given by the following

$\displaystyle\phi_{a_{0},b_{0}}(\rho_{0})$ $\displaystyle=$ $\displaystyle
k^{\frac{p+\delta}{2}r}w(E)^{\frac{q}{p}}k^{\frac{p-\delta}{p+1-\delta}\log_{k}\left(\frac{k^{\frac{r}{2}}w(F)}{(p-\delta)k^{\frac{p+\delta}{2}r}w(E)^{\frac{q}{p}}}\right)}+k^{\frac{r}{2}}w(F)k^{\frac{-1}{p+1-\delta}\log_{k}\left(\frac{k^{\frac{r}{2}}w(F)}{(p-\delta)k^{\frac{p+\delta}{2}r}w(E)^{\frac{q}{p}}}\right)}$
$\displaystyle=$ $\displaystyle
k^{\left(\frac{p+\delta}{2}-\frac{(p-\delta)(p+\delta-1)}{p+1-\delta}\right)r}\,w(E)^{\frac{q}{p}}\left(\frac{w(F)}{(p-\delta)w(E)^{\frac{q}{p}}}\right)^{\frac{p-\delta}{p+1-\delta}}$
$\displaystyle\qquad+\
k^{\frac{r}{2}\left(1-\frac{1-p-\delta}{p+1-\delta}\right)}w(F)\left(\frac{(p-\delta)w(E)^{\frac{q}{p}}}{w(F)}\right)^{\frac{1}{p+1-\delta}}$

We conclude that

(3.6) $\phi_{a_{0},b_{0}}(\rho_{0})\lesssim
k^{\frac{p}{p-\delta+1}r}w(F)^{\frac{p-\delta}{p+1-\delta}}w(E)^{\frac{q}{p(p+1-\delta)}}.$

For the choice of the $\delta=\frac{1-\alpha p-\alpha p^{2}}{1-\alpha p}$ and
$\frac{1}{q}=\frac{1}{p}-\alpha$, we have

$\displaystyle\frac{p-\delta}{p+1-\delta}$
$\displaystyle=\frac{p-\frac{1-\alpha p-\alpha p^{2}}{1-\alpha
p}}{p+1-\frac{1-\alpha p-\alpha p^{2}}{1-\alpha p}}$
$\displaystyle=\frac{p-1+\alpha p}{p}$
$\displaystyle=\frac{1}{p^{\prime}}+\alpha=1-\frac{1}{q}.$

Similarly $\frac{1}{p-\delta+1}=\frac{1-\alpha p}{p}=\frac{1}{q}$. Combining
these facts with (3.6) we conclude

$\displaystyle\sum_{x\in E}w(F\cap S(x,r))$ $\displaystyle\leq$ $\displaystyle
C_{p,\alpha}k^{\frac{p}{p-\delta+1}r}w(F)^{\frac{p-\delta}{p+1-\delta}}w(E)^{\frac{q}{p(p+1-\delta)}}$
$\displaystyle=$ $\displaystyle C_{p,\alpha}k^{(1-\alpha
p)r}w(E)^{\frac{1}{p}}w(F)^{1-\frac{1}{q}}$ $\displaystyle=$ $\displaystyle
C_{p,\alpha}k^{(1-\alpha)\epsilon r}w(E)^{\frac{1}{p}}w(F)^{1-\frac{1}{q}},$

provided we choose $\epsilon=\frac{1-p\alpha}{1-\alpha}\in(0,1)$. This
completes the proof. ∎

###### Proof of Theorem 1.5.

For $\beta=0$, the weight is the constant weight and the result trivially
follows. Hence we assume $0<\beta\leq\frac{p(p-1)}{q}$. We have
$1<p<\frac{1}{\alpha}$ and $\frac{1}{q}=\frac{1}{p}-\alpha$.

$w(x)=\sum_{j\geq 0}k^{\beta j}\chi_{\mathcal{T}_{j}}.$

It is sufficient to show that $w$ satisfies condition (1.3). Let
$x\in\mathcal{T}_{j}$ and $|i-j|\leq r$ and $m\in\\{0,\cdots,r\\}$ be the
unique integer such that $i=j+r-2m$. Then

$\displaystyle w(\mathcal{T}_{i}\cap S(x,r))$ $\displaystyle\leq$
$\displaystyle|\mathcal{T}_{i}\cap S(x,r)|k^{\beta i}$ $\displaystyle\leq$
$\displaystyle k^{r-m}k^{(i-j)\frac{\beta q}{p}}k^{i(\beta-\frac{\beta
q}{p})}k^{\frac{j\beta q}{p}}$ $\displaystyle=$ $\displaystyle
k^{r-m}k^{(r-2m)\frac{\beta q}{p}}k^{i(\beta-\frac{\beta q}{p})}w(x)^{q/p}$
$\displaystyle=$ $\displaystyle k^{(r-m)(1+2\frac{\beta
q}{p})}k^{-r\frac{\beta q}{p}}k^{i(\beta-\frac{\beta q}{p})}w(x)^{q/p}.$

In order to complete the proof it is enough to show that the following
inequality holds

$k^{(r-m)(1+2\frac{\beta q}{p})}k^{-r\frac{\beta q}{p}}k^{i(\beta-\frac{\beta
q}{p})}\leq k^{(r-m)(p-\delta)}k^{r\delta}.$

But this is equivalent to

(3.7) $(r-m)\left(1+2\frac{\beta q}{p}-p+\delta\right)\leq
r\delta+r\frac{\beta q}{p}-i(\beta-\frac{\beta q}{p}).$

Since $(\beta-\frac{\beta q}{p})\leq 0$ and we need to prove the above for all
$i\geq 0$, thus (3.7) will follow if we ensure the following

$\displaystyle r\left(1+2\frac{\beta q}{p}-p+\delta\right)\leq
r\delta+r\frac{\beta q}{p}\iff\beta\leq\frac{p(p-1)}{q}.$

This completes the proof of Theorem 1.5. ∎

######  Remark 3.1 (Two weight estimates).

We would like to highlight that approach is also applicable in the two weight
setting. Let $u,v$ be two weights on $\mathcal{T}^{k}$ satisfying the
following condition: Let $1<p\leq q<\infty$. There exists $\epsilon\in(0,1)$
such that for all $E,F\subset\mathcal{T}^{k}$ and $r\in\mathbb{N}_{0}$

(3.8) $\displaystyle\sum_{x\in E}u(F\cap S(x,r))\leq C\ k^{\epsilon
r(1-\alpha)}v(E)^{\frac{1}{p}}u(F)^{1-\frac{1}{q}}.$

Then one can prove $\mathcal{S}_{\alpha}$ maps $L^{p}(v)$ to $L^{q}(u)$. We
just point out that arguing as in Lemma 2.1, under the condition (3.8) the
following holds

$u(\\{A_{r,\alpha}f>\lambda\\})\leq\frac{1}{(2^{\beta}-1)^{q}}\sum\limits_{n\in\mathbb{N}_{0};1\leq
2^{n}\leq
k^{r}}2^{nq}\left(\frac{k^{r}}{2^{n}}\right)^{q\beta}\frac{k^{rq\epsilon(1-\alpha)}}{k^{rq}}v\left(\Big{\\{}x\in\mathcal{T}^{k}:\frac{|f|k^{r\alpha}}{\lambda}\geq
2^{n-1}\Big{\\}}\right)^{q/p},$

for all $\lambda>0,\ r\in\mathbb{N}_{0}$ and $\beta\in(0,1)$. After this the
proof follows the exact arguments as in Theorem 1.3 with necessary
modifications. It is also possible to provide an analogous condition of (1.3)
in the two weight setting but for brevity we do not include it here.

## Acknowledgements

The first author is supported by the institute postdoctoral fellowship from
Tata Institute of Fundamental Research, Centre for Applicable Mathematics.

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