# A note on boundedness of singular integral operators on functions spaces
over Local Fields

Salman Ashraf and Qaiser Jahan

###### Abstract.

In this article we consider the classical singular integral operator over a
local field with rough kernels. We study the boundedness of such an operator
on different function spaces by relaxing the smoothness condition on kernels.

###### Key words and phrases:

Local fields, Besov spaces, Triebel–Lizorkin spaces, Hardy spaces, Singular
integral operators.

##  1\. Introduction

In the fundamental paper [3], Singular integral theory were introduced by A.
P. Calderón and A. Zygmund, which plays an important role in Harmonic
analysis. Let $\Omega:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a Lebesgue
measurable function with $\int_{\mathbb{S}^{n-1}}\Omega(y)dy=0.$ We say that a
linear operator $T_{\Omega}:S(\mathbb{R}^{n})\rightarrow
S^{\prime}(\mathbb{R}^{n})$ is a rough singular integral operator if

$\displaystyle T_{\Omega}f(x)=\lim_{\epsilon\to
0}\int_{|y|>\epsilon}\dfrac{\Omega(y/|y|)}{|y|^{n}}f(x-y)dy.$

The rough singular integral operators are one of the main topics in the theory
of singular integral operators. In a series of remarks below we will see some
known results about $L^{p}$ boundedness of $T_{\Omega}.$

  * (i)

In 1956, Calderón and Zygmund [4] proved that if $\Omega\in
L\log^{+}L(\mathbb{S}^{n-1})$ is even or $\Omega\in L^{1}(\mathbb{S}^{n-1})$
is odd then $T_{\Omega}$ is of type $(p,p)$ for $1<p<\infty.$

  * (ii)

In 1979, $L^{p}$ boundedness of $T_{\Omega}$ for $1<p<\infty$ proved
independently by Connett [5] and Ricci-Weiss [21] under less restrictive
condition that $\Omega\in H^{1}(\mathbb{S}^{n-1})$ since

$\displaystyle L\log^{+}L(\mathbb{S}^{n-1})\subset H^{1}(\mathbb{S}^{n-1}).$

Here $H^{1}(\mathbb{S}^{n-1})$ is the Hardy space on the unit sphere.

  * (iii)

In 1998, Grafakos and Stefanov in [10], considered the condition

(1.1)
$\displaystyle\sup_{\xi\in\mathbb{S}^{n-1}}\int_{\mathbb{S}^{n-1}}\big{|}\Omega(\theta)\big{|}\bigg{(}\log\dfrac{1}{|\theta\cdot\xi|}\bigg{)}^{1+\alpha}d\theta\leavevmode\nobreak\
<\leavevmode\nobreak\ +\infty$

for $\alpha>0$ and gave $L^{p}$ boundedness of $T_{\Omega}$ if $\Omega$
satisfies (1.1). Also in [11], Grafakos and Stefanov gave a survey of known
results in the theory of Calderón-Zygmund singular integral operator
$T_{\Omega}.$

The boundedness of $T_{\Omega}$ on homogeneous Besov space $\dot{B}^{s}_{p,q}$
and on homogeneous Triebel-Lizorkin space $\dot{F}^{s}_{p,q}$ were studied by
Chen, Fan and Ying in [6]. The result of [6] is as follows:

######  Theorem 1.1.

(See [6]) Let $1<p,q<\infty$ and $s\in\mathbb{R}.$ If $\Omega\in
L^{r}(\mathbb{S}^{n-1}),\leavevmode\nobreak\ r>1$ with mean value zero, then

  * (i)

$\|T_{\Omega}f\|_{\dot{F}_{p,q}^{s}(\mathbb{R}^{n})}\leq
C\|f\|_{\dot{F}_{p,q}^{s}(\mathbb{R}^{n})}$

  * (ii)

$\|T_{\Omega}f\|_{\dot{B}_{p,q}^{s}(\mathbb{R}^{n})}\leq
C\|f\|_{\dot{B}_{p,q}^{s}(\mathbb{R}^{n})}.$

The result of [6] then improved by Chen and Zhang in [7]:

######  Theorem 1.2.

(See [7]) Let $1<p,q<\infty$ and $s\in\mathbb{R}.$ If $\Omega\in
L^{1}(\mathbb{S}^{n-1})$ with mean value zero and satisfies

$\displaystyle\sup_{\xi\in\mathbb{S}^{n-1}}\int_{\mathbb{S}^{n-1}}\big{|}\Omega(\theta)\big{|}\bigg{(}\log\dfrac{1}{|\theta\cdot\xi|}\bigg{)}^{1+\alpha}d\theta\leavevmode\nobreak\
<\leavevmode\nobreak\ +\infty$

for all $\alpha>1$ then
$\|T_{\Omega}f\|_{\dot{F}_{p,q}^{s}(\mathbb{R}^{n})}\leq
C\|f\|_{\dot{F}_{p,q}^{s}(\mathbb{R}^{n})}.$

Chen and Ding in [8], considered the condition that $\Omega\in
H^{1}(\mathbb{S}^{n-1})$ and proved the following result:

######  Theorem 1.3.

(See [8]) Let $1<p,q<\infty$ and $s\in\mathbb{R}.$ If $\Omega\in
H^{1}(\mathbb{S}^{n-1})$ with mean value zero, then

  * (i)

$\|T_{\Omega}f\|_{\dot{F}_{p,q}^{s}(\mathbb{R}^{n})}\leq
C\|f\|_{\dot{F}_{p,q}^{s}(\mathbb{R}^{n})}$

  * (ii)

$\|T_{\Omega}f\|_{\dot{B}_{p,q}^{s}(\mathbb{R}^{n})}\leq
C\|f\|_{\dot{B}_{p,q}^{s}(\mathbb{R}^{n})}.$

In this paper, our main aim is to establish the boundedness of singular
integral operator with Hardy space kernel on Lebesgue spaces, Triebel–Lizorkin
spaces, and Besov spaces over local fields. In [16], the boundedness of
$p$-adic singular integral operators and their commutators were obtained on
$p$-adic Morrey spaces. In [20], $\delta$-type Calderón–Zygmund operators were
introduced and established the boundedness on weighted Hardy spaces on locally
compact Vilenkin groups. Recently, KP Ho [12] extended the studies of
$\delta$-type Calderón–Zygmund operators on Morrey and Hardy-Morrey spaces in
locally compact Vilenkin groups.

A local field $K$ is a locally compact, totally disconnected, non-Archimedian
norm valued and non-discrete topological field, we refer [25] to basic Fourier
analysis on local fields. The local fields are essentially of two types
(excluding the connected local fields $\mathbb{R}$ and $\mathbb{C}$) namely
characteristic zero and of positive characteristic. The characteristic zero
local fields include the $p$-adic field $\mathbb{Q}_{p}$ and the examples of
positive characteristic are the Cantor dyadic groups, Vilenkin $p$-groups and
$p$-series fields.

Besov spaces $B^{s}_{rt}(K)$ and Triebel-Lizorkin spaces $F^{s}_{rt}(K)$ on
local fields were introduced and studied by Onneweer and Su [17]. In [22],
Weiyi Su, introduced, “$p$-type smoothness” of the functions defined on local
fields. To complete the theoretical base of the Function spaces on local
fields and to broaden the range of its applications, a series of studies have
been carried out such as the construction theory of functions on local fields
[24], the Weierstrass functions, Cantor functions and their $p$-adic
derivatives in local fields, the Lipschitz classes on local fields etc. (see
[23]). Wavelet theory on local fields has also been developed widely. Jiang,
Li and Jin [13] have introduced the concept of multiresolution analysis and
wavelet frames on local fields of positive characteristic [14]. Later, Behera
and Jahan have developed the theory of wavelets on such a field (see [1, 2]).
Recently, Lukomskii and Vodolazov [15] gave the fast algorithm for the
discrete Fourier transform on local fields of characteristic zero.

The classical singular integral operators of Calderón–Zygmund type over local
fields were considered by Phillips and Taibleson (see [18, 19, 25]). Let $K$
be a local field and $\mathfrak{D}^{*}=\\{\xi\in K:|\xi|=1\\}.$ Let $\Omega\in
L^{\infty}(K)$ and satisfy

(1.2) $\displaystyle\Omega(\mathfrak{p}^{j}x)$
$\displaystyle=\Omega(x)\leavevmode\nobreak\ \text{for
all}\leavevmode\nobreak\ j\in\mathbb{Z},$ (1.3)
$\displaystyle\int\limits_{\mathfrak{D^{*}}}\Omega(x)dx$ $\displaystyle=0.$

The singular integral operator is defined by

(1.4) $\displaystyle
Tf(x)=\lim\limits_{k\to-\infty}\int_{|y|>q^{k}}f(x-y)\dfrac{\Omega(y)}{|y|}dy,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\text{for}\leavevmode\nobreak\ k\in\mathbb{Z},$

and the truncated singular integral operator is defined as

(1.5) $\displaystyle
T_{k}f(x)=\int_{|y|>q^{k}}f(x-y)\dfrac{\Omega(y)}{|y|}dy,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\text{for}\leavevmode\nobreak\ k\in\mathbb{Z},$

initially for $f\in S(K).$ In particular, Taibleson [25] proved the following
result:

######  Theorem 1.4.

Suppose that $\Omega\in L^{\infty}(K)$ satisfies the conditions (1.2)-(1.3)
and

(1.6)
$\displaystyle\sup_{|y|=1}\sum\limits_{j=1}^{\infty}\int_{\mathfrak{D}^{*}}\leavevmode\nobreak\
|\Omega(x+\mathfrak{p}^{j}y)-\Omega(x)|\leavevmode\nobreak\ dx<\infty.$

Then $T_{k}$ is bounded in $L^{r}(K)$ for $1<r<\infty.$ Furthermore
$T(f)=\lim_{k\to-\infty}T_{k}(f)$ exists in the $L^{r}(K)$ norm.

In the series of remarks above, we have seen that boundedness of $T_{\Omega}$
is still possible on $L^{p}(\mathbb{R}^{n}),$ homogeneous Besov space
$(\dot{B}^{s}_{p,q})$ and homogenous Triebel-Lizorkin spaces
$(\dot{F}^{s}_{p,q})$ without any smoothness on $\Omega.$ In the setting of
Local field $K$, as in Theorem 1.4, Taibleson considered the smoothness
condition (1.6) on $\Omega,$ which is analogues of Hörmander’s condition and
gave the boundedness of $T_{k}$ on $L^{r}(K)(1<r<\infty)$. Our interest is in
defining the truncated singular integral operator $T_{k}$ with rough kernel
and therefore we replaces the smoothness condition (1.6) with $\Omega\in
H^{1}(\mathfrak{D}^{*})$ and studied the boundedness of $T_{k}$ on the
Lebesgue spaces, Triebel–Lizorkin spaces, and Besov spaces over Local fields.
In particular, we proved the following results:

######  Theorem 1.5.

Let $1<r<\infty,\leavevmode\nobreak\ k\in\mathbb{Z}$ and $T_{k}$ be the
operator defined by (1.5). If $\Omega\in H^{1}(\mathfrak{D}^{*})$ satisfies
(1.2) and (1.3), then

$\displaystyle\|T_{k}f\|_{L^{r}(K)}\leq
Cq^{-k}\|\Omega\|_{H^{1}(\mathfrak{D}^{*})}\|f\|_{L^{r}(K)},$

where $C>0$ is independent of $k.$

######  Theorem 1.6.

Let $1<r,t<\infty,\leavevmode\nobreak\ s>0,\leavevmode\nobreak\
k\in\mathbb{Z}$ and $T_{k}$ be the operator defined by (1.5). If $\Omega\in
H^{1}(\mathfrak{D}^{*})$ satisfies (1.2) and (1.3), then

  * (i)

$\|T_{k}f\|_{F_{rt}^{s}(K)}\leq
Cq^{-k}\|\Omega\|_{H^{1}(\mathfrak{D}^{*})}\|f\|_{F_{rt}^{s}(K)},$

  * (ii)

$\|T_{k}f\|_{B_{rt}^{s}(K)}\leq
Cq^{-k}\|\Omega\|_{H^{1}(\mathfrak{D}^{*})}\|f\|_{B_{rt}^{s}(K)},$

where $C>0$ is independent of $k.$

We remark here that by Theorem 1.5 and Theorem 1.6, it is possible to have
boundedness of $T_{k}$ with $\Omega\in H^{1}(\mathfrak{D}^{*})$ on
$L^{r}(K),B_{rt}^{s}(K)$ and $F_{rt}^{s}(K)$ but the operator norm is
dependent on $k$ by constant $q^{-k}$ and therefore convergence of $T_{k}$ as
$k\rightarrow-\infty$ is not possible in the respective function spaces norm,
which is not the case when $\Omega$ satisfies smoothness condition (1.6) as in
Theorem 1.4.

This paper is organized as follows. In Section 2, we provide a brief
introduction to local fields and test function class and distribution on such
a field. We have also provided the definition of $p$-type derivatives,
Triebel-Lizorkin spaces $F^{s}_{rt}(K)$ and Besov spaces $B^{s}_{rt}(K)$ on
local fields. Further, we have given the definition of atomic Hardy space on
$\mathfrak{D}^{*}$ and some Lemmas for the proof of Theorem 1.5 and Theorem
1.6. Section 3 and section 4 are devoted to the proof of Theorem 1.5 and
Theorem 1.6, respectively.

##  2\. Preliminary definitions and lemmas

###  2.1. Local Fields

Let $K$ be a field and a topological space. If the additive group $K^{+}$ and
multiplicative group $K^{*}$ of $K$ both are locally compact Abelian group,
then $K$ is called locally compact topological field, or local field.

If $K$ is any field and is endowed with the discrete topology, then $K$ is a
local field. Further, if $K$ is connected, then $K$ is either $\mathbb{R}$ or
$\mathbb{C}$. If $K$ is not connected, then it is totally disconnected. So,
here local fields means a field $K$ which is locally compact, non-discrete and
totally disconnected. If $K$ is of characteristic zero, then K is either a
$p$-adic field for some prime number $p$ or a finite algebraic extension of
such a field. If $K$ is of positive characteristic, then $K$ is either a field
of formal Laurent series over a finite field of characteristic $p$ or an
algebraic extension of such a field.

Let $K$ be a local field. Since $K^{+}$ is a locally compact abelian group, we
choose a Haar measure $dx$ for $K^{+}$. If $\alpha\neq 0,\leavevmode\nobreak\
\alpha\in K$, then $d(\alpha x)$ is also a Haar measure and by the uniqueness
of Haar measure we have $d(\alpha x)=|\alpha|dx$. We call $|\alpha|$ the
absolute value or valuation of $\alpha$ which is, in fact, a natural non-
Archimedian norm on $K$. A mapping $|\cdot|\leavevmode\nobreak\
:K\rightarrow\mathbb{R}$ satisfies:

  * (a)

$|x|=0$ if and only if $x=0$;

  * (b)

$|xy|=|x||y|$ for all $x,y\in K$;

  * (c)

$|x+y|\leq\text{max}\\{|x|,|y|\\},$ for all $x,y\in K$,

The property $(c)$ is called the ultrametric inequality. It follows that

$|x+y|=\text{max}\\{|x|,|y|\\}\leavevmode\nobreak\
\text{if}\leavevmode\nobreak\ |x|\neq|y|.$

The set $\mathfrak{D}=\\{x\in K:|x|\leq 1\\}$ is called the ring of integers
in $K$. It is the unique maximal compact subring of $K$. The set
$\mathfrak{P}=\\{x\in K:|x|<1\\}$ is called the prime ideal in $K$. It is the
unique maximal ideal in $\mathfrak{D}$. It can be proved that $\mathfrak{D}$
is compact and open and hence $\mathfrak{P}$ is also compact and open.
Therefore, the residue space $\mathfrak{D}/\mathfrak{P}$ is isomorphic to a
finite field $GF(q)$, where $q=p^{c}$ for some prime number $p$ and
$c\in\mathbb{N}$.

Let $\mathfrak{p}$ be a fixed element of maximum absolute value in,
$\mathfrak{P}$ and it is called a prime element of $K$. For a measurable
subset $E$ of $K$, let $|E|=\int_{K}\mathbb{1}_{E}(x)dx$, where $\mathbb{1}$
is the indicator function of $E$ and $dx$ is the Haar measure of $K$
normalized so that $|\mathfrak{D}|=1$. Then it is easy to prove that
$|\mathfrak{P}|=q^{-1}$ and $|\mathfrak{p}|=q^{-1}$ (see [25]).

Let $\mathcal{U}=\\{a_{i}\\}_{i=0}^{q-1}$ be any fixed full set of coset
representative of $\mathfrak{P}$ in, $\mathfrak{D}$ then each $x\in K$ can be
written uniquely as $x=\sum\limits_{l=k}^{\infty}c_{l}\mathfrak{p}^{k}$, where
$c_{l}\in\mathcal{U}$. If $x(\neq 0)\in K$ then $|x|=q^{k}$ for some
$k\in\mathbb{Z}.$

Let
$\mathfrak{D}^{*}=\mathfrak{D}\setminus\mathfrak{P}=\\{x\in\mathfrak{D}:|x|=1\\}$;
$\mathfrak{D}^{*}$ is the group of units in $K^{*}$. If $x\neq 0$, we can
write as $x=\mathfrak{p}^{k}x^{\prime}$, with $x^{\prime}\in\mathfrak{D}^{*}$.
Let $\mathfrak{P}^{k}=\mathfrak{p}^{k}\mathfrak{D}=\\{x\in K:|x|\leq
q^{-k}\\}\leavevmode\nobreak\ ,k\in\mathbb{Z}$. These are called fractional
ideals.

The set $\\{\mathfrak{P}^{k}\subset K:k\in\mathbb{Z}\\}$ satisfies the
following:

  * (i)

$\\{\mathfrak{P}^{k}\subset K:k\in\mathbb{Z}\\}$ is a base for neighborhood
system of identity in $K$, and
$\mathfrak{P}^{k+1}\subset\mathfrak{P}^{k},\leavevmode\nobreak\
k\in\mathbb{Z};$

  * (ii)

$\mathfrak{P}^{k},\leavevmode\nobreak\ k\in\mathbb{Z},$ is open, closed and
compact in $K$;

  * (iii)

$K=\bigcup\limits_{{k=-\infty}}^{+\infty}\mathfrak{P}^{k}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\\{0\\}=\bigcap\limits_{{k=-\infty}}^{+\infty}\mathfrak{P}^{k}.$

Let $\Gamma$ be the character group of the additive group $K^{+}$ of $K$.
There is a nontrivial character $\chi\in\Gamma$ which is trivial on
$\mathfrak{D}$ but is non-trivial on $\mathfrak{P}^{-1}$. Let $\chi$ be a non-
trivial character on $K^{+}$, then the corresponding relationship $\lambda\in
K\longleftrightarrow\chi_{\lambda}\in\Gamma$ is determined by
$\chi_{\lambda}(x)=\chi(\lambda x),$ and the topological isomorphism is
established for $K$ and $\Gamma$, moreover, we have
$\Gamma=\\{\chi_{\lambda}:\lambda\in K\\}.$

For $k\in\mathbb{Z}$, let $\Gamma^{k}$ be the annihilator of
$\mathfrak{P}^{k},$ that is,

$\displaystyle\Gamma^{k}=\\{\chi\in\Gamma:\forall
x\in\mathfrak{P}^{k}\implies\chi(x)=1\\}.$

The locally compact group $\Gamma=\\{\chi_{\lambda}:\lambda\in K\\}$ can be
endowed with the non-Archimedian norm as
$\Gamma^{k}=\\{\chi_{\lambda}\in\Gamma:|\lambda|\leq
q^{k}\\}=\mathfrak{p}^{-k}\mathfrak{D},\leavevmode\nobreak\ k\in\mathbb{Z}.$

For the Haar measure $dx$ of $K^{+}$, let $d\xi$ be the Haar measure on
$\Gamma$, chosen such that

$|\Gamma^{0}|=|\mathfrak{D}|=1,\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\
|\Gamma^{k}|=\dfrac{1}{|\mathfrak{P}^{k}|}=q^{k}.$

The set $\\{\Gamma^{k}\subset\Gamma:\leavevmode\nobreak\ k\in\mathbb{Z}\\}$
satisfies the following properties:

  * (i)

$\Gamma^{k}\subset\Gamma^{k+1},\leavevmode\nobreak\ k\in\mathbb{Z}$, are
increasing sequence in $\Gamma$ which is open, closed and compact;

  * (ii)

$\Gamma=\bigcup\limits_{{k=-\infty}}^{+\infty}\Gamma^{k}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\\{I\\}=\bigcap\limits_{{k=-\infty}}^{+\infty}\Gamma^{k},$ where $I$ is the
unit element of $\Gamma$;

  * (iii)

$\\{\Gamma^{k}\\}_{k\in\mathbb{Z}}$ is a base of the unit $I$ of character
group $\Gamma$.

###  2.2. Function classes

In this subsection we will introduce some function classes:

Test function class $S(K)$: It is the linear space in which functions have the
form

$\phi(x)=\sum_{j=1}^{n}c_{j}\Phi_{\mathfrak{P}^{j}}(x-h_{j}),\leavevmode\nobreak\
c_{j}\in\mathbb{C},h_{j}\in K,j\in\mathbb{Z},n\in\mathbb{N},$

where $\Phi_{\mathfrak{P}^{j}}$ is the characteristic function of
$\mathfrak{P}^{j}.$ The space $S(K)$ is an algebra of continuous functions
with compact support that separates points. Consequently, $S(K)$ is dense in
$C_{0}(K)$ as well as $L^{r}(K),\leavevmode\nobreak\ 1\leq r<\infty.$
Similarly, the test function class $S(\Gamma)$ on $\Gamma$ can be defined.
However, since $K$ is isomorphic to $\Gamma$, so $S(K)$ and $S(\Gamma)$ can be
regarded as equivalent with respect to absolute value or valuation.

The space $S(K)$ is equipped with a topology as a topological vector space as
follows: Define a null sequence in $S(K)$ as a sequence $\\{\phi_{n}\\}$ of
functions on $S(K)$ in such a way that each $\phi_{n}$ is constant on cosets
of $\mathfrak{P}^{l}$ and is supported on $\mathfrak{P}^{k}$ for a fixed pair
of integers $k$ and $l$ and the sequence converges to zero uniformly. The
space $S(K)$ is complete and separable and is called the _space of testing
function_.

Since $S(K)$ is dense in $L^{1}(K)$, thus the Fourier transformation of
$\phi(x)\in S(K)$ is defined by

$\widehat{\phi}(\xi)\equiv(\mathcal{F}\phi)(\xi)=\int_{K}\phi(x)\bar{\chi}_{\xi}(x)dx,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \xi\in\Gamma,$

and the inverse Fourier transformation of $\phi\in S(K)$ is defined by the
formula

$\widecheck{\phi}(x)\equiv(\mathcal{F}^{-1}\phi)(x)=\int_{\Gamma}\phi(\xi)\chi_{x}(\xi)d\xi,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ x\in K.$

$S^{\prime}(K)$, the space of distributions, is a collection of continuous
linear functional on $S(K)$. $S^{\prime}(K)$ is also a complete topological
linear space. The action of $f$ in $S^{\prime}(K)$ on an element $\phi$ in
$S(K)$ is denoted by $\langle f,\phi\rangle$. The distribution space
$S^{\prime}(K)$ is given the weak$\ast$ topology. Convergence in
$S^{\prime}(K)$ is defined in the following way: $f_{k}$ converges to $f$ in
$S^{\prime}(K)$ if $\langle f_{k},\phi\rangle$ converges to $\langle
f,\phi\rangle$ for any $\phi\in S(K).$

The Fourier transformation $\widehat{f}$ of a distribution $f\in
S^{\prime}(K)$ is defined by

$\langle\widehat{f},\phi\rangle\leavevmode\nobreak\ =\leavevmode\nobreak\
\langle f,\widehat{\phi}\rangle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{for\leavevmode\nobreak\ all}\leavevmode\nobreak\
\phi\in S(K),$

the inverse Fourier transformation $\widecheck{g}$ is defined by

$\langle\widecheck{g},\psi\rangle=\langle
g,\widecheck{\psi}\rangle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{for\leavevmode\nobreak\ all}\leavevmode\nobreak\
\psi\in S(K).$

###  2.3. Derivatives and Integrals

In [22], Weiyi Su introduced $p$-type derivatives and $p$-type integrals for
general locally compact Vilenkin group using pseudo-differential operators,
which also includes derivatives and integrals of fractional orders.

######  Definition 2.4.

Let $\alpha>0$, if for a complex Haar measurable function
$f:K\rightarrow\mathbb{C}$, the integral

$T_{\langle\cdot\rangle^{\alpha}}f(x)\equiv\int_{\Gamma}\bigg{\\{}\int_{K}\langle\xi\rangle^{\alpha}f(t)\bar{\chi}_{\xi}(t-x)dt\bigg{\\}}d\xi$

exists at $x\in K$, then $T_{\langle\cdot\rangle^{\alpha}}f(x)$ is said to be
a point-wise $\alpha$-order $p$-type derivative of $f(x)$ at $x$, and it is
denoted by $f^{\langle\alpha\rangle}(x).$

######  Definition 2.5.

Let $\alpha>0$, if for a complex Haar measurable function
$f:K\rightarrow\mathbb{C}$, the integral

$T_{\langle\cdot\rangle^{-\alpha}}f(x)\equiv\int_{\Gamma}\bigg{\\{}\int_{K}\langle\xi\rangle^{-\alpha}f(t)\bar{\chi}_{\xi}(t-x)dt\bigg{\\}}d\xi$

exists at $x\in K$, then $T_{\langle\cdot\rangle^{-\alpha}}f(x)$ is said to be
a point-wise $\alpha$-order $p$-type integral of $f(x)$ at $x$, denoted by
$f_{\langle\alpha\rangle}(x).$

######  Remark 2.6.

We suppose that $\alpha>0$ in the above definitions, thus the order $\alpha$
of $p$-type derivatives and integrals can be any positive real numbers, thus,
fractional order derivatives and fractional order integrals all are contained.
Moreover, for $\alpha=0$,

$f^{\langle 0\rangle}(x)=f(x)=f_{\langle 0\rangle}(x)\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for}\leavevmode\nobreak\ x\in K.$

###  2.7. Function spaces on Local fields

######  Definition 2.8.

Let $S(K)$ and $S(\Gamma)$ be the test function classes on $K$ and on
$\Gamma$, respectively, $S^{\prime}(K)$ and $S^{\prime}(\Gamma)$ are their
distribution spaces, respectively.

Take a sequence $\\{\phi_{j}(\xi)\\}_{j=0}^{+\infty}\subset S(\Gamma)$
satisfying

  * (i)

$\text{supp}\leavevmode\nobreak\ \phi_{0}\subset\\{\xi\in\Gamma:|\xi|<q\\}$,
$\text{supp}\leavevmode\nobreak\
\phi_{j}\subset\\{\xi\in\Gamma:q^{j-1}<|\xi|<q^{j+1}\\},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ j\in\mathbb{N}$

  * (ii)

$\sum_{j=0}^{\infty}\phi_{j}(\xi)=1,\leavevmode\nobreak\ \leavevmode\nobreak\
\xi\in\Gamma.$

  * (iii)

$\bigg{|}(\widecheck{\phi}_{j})^{\langle s\rangle}(x)\bigg{|}\leq
c_{s}q^{-j+js},\leavevmode\nobreak\ s\in(0,+\infty),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
j\in\mathbb{N}_{0},\leavevmode\nobreak\ \leavevmode\nobreak\ x\in K,$

where $(\widecheck{\phi}_{j})^{\langle s\rangle}$ is the point-wise $s$-order
$p$-type derivatives of $\widecheck{\phi}_{j}$ and $c_{s}$ is a constant
depending on $s$ only.

We denote

$A(\Gamma)=\bigg{\\{}\\{\phi_{j}\\}_{j\in\mathbb{N}_{0}}\subset
S(\Gamma)\leavevmode\nobreak\ \text{satisfying}\leavevmode\nobreak\
(i)-(iii)\bigg{\\}}.$

Specially, for a local field $K$, we just need to take
$\\{\phi_{j}\\}_{j=0}^{\infty}\subset A(\Gamma)$ as follows:

$\displaystyle\phi_{0}(\xi)$ $\displaystyle=\Phi_{\Gamma^{0}}(\xi),$
$\displaystyle\phi_{j}(\xi)$
$\displaystyle=\Phi_{\Gamma^{j}\setminus\Gamma^{j-1}}(\xi),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ j\in\mathbb{N},$

where $\Phi_{A}$ is the characteristic function of $A$. It satisfies the
following conditions

  * (i)

$\text{supp}\leavevmode\nobreak\
\Phi_{\Gamma^{0}}\subset\\{\xi\in\Gamma:|\xi|<q\\}$,
$\text{supp}\leavevmode\nobreak\
\Phi_{\Gamma^{j}\setminus\Gamma^{j-1}}\subset\\{\xi\in\Gamma:q^{j-1}<|\xi|<q^{j+1}\\},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
j\in\mathbb{N},$

  * (ii)

$\Phi_{\Gamma^{0}}(\xi)+\sum_{j=1}^{\infty}\Phi_{\Gamma^{j}\setminus\Gamma^{j-1}}(\xi)=1,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \xi\in\Gamma,$

  * (iii)

$\bigg{|}(\widecheck{\Phi}_{\Gamma^{0}})^{\langle s\rangle}(x)\bigg{|}\leq
c_{s},$
$\bigg{|}(\widecheck{\Phi}_{\Gamma^{j}\setminus\Gamma^{j-1}})^{\langle
s\rangle}(x)\bigg{|}\leq c_{s}q^{-j+js},\leavevmode\nobreak\
s\in(0,+\infty),\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
j\in\mathbb{N},\leavevmode\nobreak\ \leavevmode\nobreak\ x\in K,$
where $c_{s}$ is a constant depending on $s$ only.

######  Definition 2.9.

Let $\\{\phi_{j}\\}_{j=0}^{\infty}=\\{\Phi_{\Gamma^{0}},\leavevmode\nobreak\
\Phi_{\Gamma^{j}\setminus\Gamma^{j-1}}\\}_{j=1}^{\infty}\subset A(\Gamma),$ we
define the operators $\Delta_{j}$ as follows

$\displaystyle\Delta_{j}f=\mathcal{F}^{-1}\Big{(}\phi_{j}\mathcal{F}f\Big{)},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
j\in\mathbb{N}_{0},\leavevmode\nobreak\ \leavevmode\nobreak\ f\in
S^{\prime}(K).$

Then we obtain the Littlewood-Paley decomposition,

$f=\sum_{j=0}^{\infty}\Delta_{j}f,$

of all $f\in S^{\prime}(K).$

Note that $\Delta_{j}f\in S^{\prime}(K)$ for any $f\in S^{\prime}(K)$.

######  Proposition 2.10.

Let $u$ be in $S^{\prime}(K).$ Then, we have,
$u=\sum_{j=0}^{\infty}\Delta_{j}u,$ in the sense of the convergence in the
space $S^{\prime}(K).$

######  Definition 2.11.

(B-type space) For $0<r\leq+\infty,\leavevmode\nobreak\
0<t\leq+\infty,\leavevmode\nobreak\ s\in\mathbb{R},$ we define B-type spaces
or Besov spaces on local fields $K$ as

$B_{rt}^{s}(K)=\\{f\in S^{\prime}(K):\leavevmode\nobreak\
\|f\|_{{B_{rt}^{s}}(K)}<\infty\\},$

with norm

$\displaystyle\|f\|_{{B_{rt}^{s}}(K)}$
$\displaystyle=\|q^{sj}\Delta_{j}f\|_{\ell_{t}(L^{r}(K))}$
$\displaystyle=\Bigg{\\{}\sum\limits_{j=0}^{\infty}\|q^{sj}\Delta_{j}f\|^{t}_{L^{r}(K)}\Bigg{\\}}^{\frac{1}{t}}$
$\displaystyle=\Bigg{\\{}\sum_{j=0}^{\infty}q^{sjt}\bigg{\\{}\int_{K}|\Delta_{j}f|^{r}dx\bigg{\\}}^{\frac{t}{r}}\Bigg{\\}}^{\frac{1}{t}}.$

######  Definition 2.12.

(F-type space) For $0<r<\infty,\leavevmode\nobreak\
0<t\leq\infty,\leavevmode\nobreak\ s\in\mathbb{R},$ we define F-type spaces or
Triebel-Lizorkin spaces on local field $K$ as

$F_{rt}^{s}(K)=\\{f\in S^{\prime}(K):\leavevmode\nobreak\
\|f\|_{{F_{rt}^{s}}(K)}<\infty\\},$

with norm

$\displaystyle\|f\|_{{F_{rt}^{s}}(K)}$
$\displaystyle=\|q^{sj}\Delta_{j}f\|_{L^{r}(\ell^{t}(K))}$
$\displaystyle=\Bigg{\|}\Bigg{\\{}\sum_{j=0}^{\infty}|q^{sj}\Delta_{j}f|^{t}\Bigg{\\}}^{\frac{1}{t}}\Bigg{\|}_{L^{r}(K)}$
$\displaystyle=\Bigg{(}\int_{K}\Bigg{\\{}\sum_{j=0}^{\infty}q^{sjt}|\Delta_{j}f|^{t}\Bigg{\\}}^{\frac{r}{t}}dx\Bigg{)}^{\frac{1}{r}}.$

######  Remark 2.13.

The spaces $A_{rt}^{s}(K),\leavevmode\nobreak\ A\in\\{B,F\\}$ are independent
of the selection of sequences $\\{\phi_{j}\\}_{j\in\mathbb{N}_{0}}\subset
A(\Gamma).$ They are quasi-Banach spaces (Banach spaces for $r,t\geq 1$), and
$S(K)\subset A_{rt}^{s}(K)\subset S^{\prime}(K),$ where the first embedding is
dense if $r<\infty$ and $t<\infty.$ The theory of the spaces $A_{rt}^{s}(K)$
has been developed in [23].

Now, we give the definition of atomic Hardy space on $\mathfrak{D}^{*},$ which
can be obtained by the idea in [9].

######  Definition 2.14.

($H^{1}(\mathfrak{D}^{*})$ atoms) A function
$a:\mathfrak{D}^{*}\rightarrow\mathbb{C}$ is a $(1,\infty)$ atom if

  * (i)

$\text{supp}\leavevmode\nobreak\ a\subset\mathfrak{D}^{*},$

  * (ii)

$\|a\|_{L^{\infty}(K)}\leavevmode\nobreak\ \leq\leavevmode\nobreak\
|\mathfrak{D}^{*}|^{-1}=(1-q^{-1})^{-1},$

  * (iii)

$\int_{\mathfrak{D}^{*}}a(x)dx\leavevmode\nobreak\ =\leavevmode\nobreak\ 0.$

######  Definition 2.15.

If $f\in H^{1}(\mathfrak{D}^{*})$ then there exists a sequence
$\\{\lambda_{i}\\}_{i\in\mathbb{N}}\in\ell^{1}$ and a sequence of $(1,\infty)$
atoms $\\{a_{i}\\}_{i\in\mathbb{N}}$ such that
$f=\sum\limits_{i=1}^{\infty}\lambda_{i}a_{i}$ and
$\|f\|_{H^{1}(\mathfrak{D}^{*})}\approx\sum\limits_{i=1}^{\infty}|\lambda_{i}|.$

######  Theorem 2.1.

[2] (Minkowski’s integral inequality) Let $(X,\mu)$ and $(Y,\nu)$ be
$\sigma-$finite measure spaces. Let $1\leq r<\infty$ and $F$ be a function on
the product space $X\times Y.$ Then

(2.1)
$\displaystyle\bigg{(}\int_{X}\bigg{(}\int_{Y}|F(x,y)|d\nu(y)\bigg{)}^{r}d\mu(x)\bigg{)}^{1/r}\leavevmode\nobreak\
\leq\leavevmode\nobreak\
\int_{Y}\bigg{(}\int_{X}|F(x,y)|^{r}d\mu(x)\bigg{)}^{1/r}d\nu(y).$

We will use this inequality for the counting measure
$\mu=\sum_{k=0}^{\infty}\delta_{k}$ on $X=\mathbb{N}_{0}:$

(2.2)
$\displaystyle\bigg{(}\sum_{k=0}^{\infty}\bigg{(}\int_{Y}|F_{k}(y)|d\nu(y)\bigg{)}^{r}\bigg{)}^{1/r}\leavevmode\nobreak\
\leq\leavevmode\nobreak\
\int_{Y}\bigg{(}\sum_{k=0}^{\infty}|F_{k}(y)|^{r}\bigg{)}^{1/r}d\nu(y).$

The following Lemmas are useful for us and given in [25].

######  Lemma 2.16.

There are constants $A,\leavevmode\nobreak\ B>0$ depending only on $K$ such
that if $f\in L^{1}(K),\leavevmode\nobreak\ f(x)\geq 0,\leavevmode\nobreak\
\lambda>0$ then there is a countable collections of mutually disjoints spheres
$\\{W_{i}\\}_{i\in\mathbb{N}}\leavevmode\nobreak\ ,\leavevmode\nobreak\
D_{\lambda}=\cup_{i}W_{i}$, and a decomposition of $f=f_{1}+f_{2}$ such that:

  * (i)

$|D_{\lambda}|\leavevmode\nobreak\ =\leavevmode\nobreak\
\sum\limits_{i}|W_{i}|<\|f\|_{L^{1}}\lambda^{-1}$

  * (ii)

$|f(x)|\leavevmode\nobreak\ \leq\leavevmode\nobreak\
\lambda\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\text{a.e. for}\leavevmode\nobreak\ x\notin D_{\lambda}$

  * (iii)

$|f_{2}(x)|\leavevmode\nobreak\ \leq\leavevmode\nobreak\
q\lambda\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\text{a.e. for}\leavevmode\nobreak\ x\in D_{\lambda}$

  * (iv)

$f_{2}(x)\leavevmode\nobreak\ =\leavevmode\nobreak\ f(x),\leavevmode\nobreak\
x\notin D_{\lambda}$

  * (v)

$f_{1}(x)\leavevmode\nobreak\ =\leavevmode\nobreak\ 0,\leavevmode\nobreak\
x\notin D_{\lambda}$

  * (vi)

$\int\limits_{W_{i}}f_{1}(x)dx=0\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \forall i.$

######  Remark 2.17.

The conclusions of above lemma yield the following results:

  * (i)

$f_{1}\in L^{1}(K),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \|f_{1}\|_{L^{1}(K)}\leavevmode\nobreak\
\leq\leavevmode\nobreak\ \|f\|_{L^{1}(K)}$

  * (ii)

$f_{2}\in L^{1}(K),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \|f_{1}\|_{L^{1}(K)}\leavevmode\nobreak\
=\leavevmode\nobreak\ \|f\|_{L^{1}(K)}$

  * (iii)

$f_{2}\in L^{\infty}(K),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \|f_{2}\|_{L^{\infty}(K)}\leavevmode\nobreak\
\leq\leavevmode\nobreak\ q\lambda$

  * (iv)

$f_{2}\in L^{2}(K),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \|f_{2}\|^{2}_{L^{2}(K)}\leavevmode\nobreak\
\leq\leavevmode\nobreak\ \lambda q\|f\|_{L^{1}}.$

######  Lemma 2.18.

Suppose $T$ is a linear operator defined on $S(K)$ with values in
$S^{\prime}(K)$. Suppose further that when $f\in S(K)$ then $Tf\in L^{2}(K)$
and $\|Tf\|_{L^{2}(K)}\leq A_{2}\|f\|_{L^{2}(K)},\leavevmode\nobreak\ A_{2}>0$
independent of $f$ and $Tf$ is a function such that for all
$\lambda>0,\leavevmode\nobreak\ \leavevmode\nobreak\ |\\{x\in
K:|Tf(x)|>\lambda\\}|\leq A_{1}\|f\|_{L^{1}(K)}\lambda^{-1},A_{1}>0$
independent of $f$ and $\lambda.$ Then there is a constant $A_{r}>0$
independent of $f,\leavevmode\nobreak\ 1<r\leq 2$ such that for $f\in S(K),$
$Tf\in L^{r}(K)$ and $\|Tf\|_{L^{r}(K)}\leq A_{r}\|f\|_{L^{r}(K)}.$

######  Definition 2.19.

An operator $T$ that is defined on $S(K)$ with values in $S^{\prime}(K)$ is
said to be commute with translations if for all $h\in K,$
$\tau_{h}T=T_{\tau_{h}}.$

######  Lemma 2.20.

Suppose $T$ is a linear operator defined on $S(K)$ with values in
$S^{\prime}(K)$ that commutes with translations. Suppose further that $\phi\in
S(K)$ implies that $T\phi\in L^{r}(K),\leavevmode\nobreak\
\|T\phi\|_{L^{r}(K)}\leq A\|\phi\|_{L^{r}(K)},\leavevmode\nobreak\ A>0$
independent of $\phi,\leavevmode\nobreak\ 1\leq r\leq\infty.$ Let
$\frac{1}{r}+\frac{1}{r^{\prime}}=1.$ Then $T\phi\in L^{r^{\prime}}(K)$ for
all $\phi\in S(K)$ and $\|T\phi\|_{L^{r^{\prime}}(K)}\leq
A\|\phi\|_{L^{r^{\prime}}(K)}.$

##  3\. Proof of Theorem 1.5

We begin by discretising $T_{k}.$ Let $\Omega\in H^{1}(\mathfrak{D}^{*})$ and
for $k\in\mathbb{Z}$ we have

$\displaystyle T_{k}(f)(x)$
$\displaystyle=\int\limits_{|y|>q^{k}}f(x-y)\dfrac{\Omega(y)}{|y|}dy,$
$\displaystyle=\int\limits_{|y|=q^{k+1}}f(x-y)\dfrac{\Omega(y)}{|y|}dy+\int\limits_{|y|=q^{k+2}}f(x-y)\dfrac{\Omega(y)}{|y|}dy+\cdots,$
$\displaystyle=\sum\limits_{j=k}^{\infty}\leavevmode\nobreak\
\int\limits_{|y|=q^{j+1}}f(x-y)\dfrac{\Omega(y)}{|y|}dy,$ (3.1)
$\displaystyle=\sum\limits_{j=k}^{\infty}q^{-(j+1)}\leavevmode\nobreak\
\int\limits_{|y|=q^{j+1}}f(x-y)\Omega(y)dy.$

Since $\Omega\in H^{1}(\mathfrak{D}^{*})$ then we can express $\Omega$ as

(3.2) $\Omega=\sum_{i=1}^{\infty}\lambda_{i}a_{i},$

where $\\{\lambda_{i}\\}_{i\in\mathbb{N}}\in l^{1}$ and $a_{i}\textquotesingle
s$ are $(1,\infty)$ atoms. Putting the value of (3.2) into (3.1), we get

$\displaystyle T_{k}(f)(x)$
$\displaystyle=\sum\limits_{i=1}^{\infty}\sum\limits_{j=k}^{\infty}q^{-(j+1)}\lambda_{i}\int\limits_{|y|=q^{j+1}}f(x-y)a_{i}(y)dy,$
(3.3) $\displaystyle=\sum\limits_{i=1}^{\infty}\lambda_{i}B_{i}f,$

where

$\displaystyle B_{i}f=\sum\limits_{j=k}^{\infty}q^{-(j+1)}\leavevmode\nobreak\
\int\limits_{|y|=q^{j+1}}f(x-y)a_{i}(y)dy.$

For simplicity, we denote $B_{i}$ and $a_{i}$ by $B$ and $a$, respectively.
Define

$\displaystyle g_{j}(y)=a(y)\Phi_{A_{j}}(y),$

where $\Phi_{A_{j}}$ is the characteristic function of
$A_{j}=\\{y\leavevmode\nobreak\ :\leavevmode\nobreak\ |y|=q^{j+1}\\}.$ Then

(3.4) $\displaystyle Bf(x)=\sum\limits_{j=k}^{\infty}q^{-(j+1)}g_{j}\ast
f(x).$

From (3.3) we get

$\displaystyle\|T_{k}f\|_{L^{r}(K)}$
$\displaystyle\leq\sum_{i=1}^{\infty}|\lambda_{i}|\|Bf\|_{L^{r}(K)},$ (3.5)
$\displaystyle\leq
C^{\prime}\|\Omega\|_{H^{1}(\mathfrak{D}^{*})}\|Bf\|_{L^{r}(K)}.$

In order to prove that $T_{k}$ is bounded on $L^{r}(K)\leavevmode\nobreak\
(\text{for}\leavevmode\nobreak\ 1<r<\infty)$, it is enough to show the same
for $B.$ We will do this proof in two steps. In the first step we will show
that $B$ is bounded on $L^{2}(K).$ In second step we will prove that $B$ is
weak type $(1,1)$ and conclude from Lemma 2.18 that $B$ is bounded in
$L^{r}(K)$ for $1<r\leq 2.$ Then we will use Lemma 2.20 to obtain the
boundedness of $B$ in $L^{r}(K)$ for $2<r<\infty.$

###  3.1. $L^{2}(K)$ boundedness of $B$

Let $f\in L^{2}(K)$ then $g_{j}\ast f\in L^{2}(K),$ since $g_{j}\in L^{1}(K).$
Also

$\displaystyle|\widehat{g_{j}}(\xi)|$
$\displaystyle=\bigg{|}\int_{K}a(x)\Phi_{A_{j}}(x)\bar{\chi}_{\xi}(x)dx\bigg{|},$
$\displaystyle=\bigg{|}\int_{\mathfrak{D}^{*}}a(x)\bar{\chi}_{\xi}(x)dx\bigg{|},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
(\because\leavevmode\nobreak\ \text{supp}\leavevmode\nobreak\
a\Phi_{A_{j}}\subset\mathfrak{D}^{*})$
$\displaystyle\leq\int_{\mathfrak{D}^{*}}|a(x)||\bar{\chi}_{\xi}(x)|dx,$
$\displaystyle\leq\|a\|_{L^{\infty}(K)}\leavevmode\nobreak\
\cdot\leavevmode\nobreak\ |\mathfrak{D}^{*}|$
$\displaystyle\leq|\mathfrak{D}^{*}|^{-1}|\mathfrak{D}^{*}|=1.$

By Plancherel, we have that

$\displaystyle\|g_{j}\ast f\|_{L^{2}(K)}$
$\displaystyle\leq\leavevmode\nobreak\
\|\widehat{g_{j}}\|_{L^{\infty}(K)}\|\widehat{f}\|_{L^{2}(K)}$
$\displaystyle\leq\leavevmode\nobreak\ \|\widehat{f}\|_{L^{2}(K)},$

and

$\displaystyle\|Bf\|_{L^{2}(K)}$
$\displaystyle=\bigg{\|}\sum_{j=k}^{\infty}q^{-(j+1)}\leavevmode\nobreak\
g_{j}\ast f\bigg{\|}_{L^{2}(K)}$ $\displaystyle\leq\leavevmode\nobreak\
\sum_{j=k}^{\infty}q^{-(j+1)}\leavevmode\nobreak\ \|g_{j}\ast f\|_{L^{2}(K)}$
$\displaystyle\leq\leavevmode\nobreak\
\sum_{j=k}^{\infty}q^{-(j+1)}\leavevmode\nobreak\ \|f\|_{L^{2}(K)}$ (3.6)
$\displaystyle=\leavevmode\nobreak\ q^{-k}(q-1)^{-1}\|f\|_{L^{2}(K)}.$

###  3.2. Weak $(1,1)$ boundedness of $B$

Let $f\in L^{1}(K)$ and fix $\lambda>0.$ We apply the decomposition in lemma
2.16 to obtain the functions $f_{1},\leavevmode\nobreak\ f_{2}$ so that
$f=f_{1}+f_{2}.$ Now observe that

$\displaystyle\\{x\in K:|Bf(x)|>\lambda\\}\subset\\{x\in
K:|Bf_{1}(x)|>\lambda/2\\}\bigcup\\{x\in K:|Bf_{2}(x)|>\lambda/2\\}\equiv
E_{1}\cup E_{2},$

and so, we have

(3.7) $\displaystyle|\\{x\in K:|Bf(x)|>\lambda\\}|\leavevmode\nobreak\
\leq\leavevmode\nobreak\ |\\{x\in K:|Bf_{1}(x)|>\lambda/2\\}|+|\\{x\in
K:|Bf_{2}(x)|>\lambda/2\\}|$

and need to estimates each of these terms. The estimates of $E_{2}$ is easy
since we have

$\displaystyle|E_{2}|=\int_{K}\Phi_{E_{2}}(x)dx=\int_{E_{2}}1\leavevmode\nobreak\
dx$ $\displaystyle\leavevmode\nobreak\ \leq\leavevmode\nobreak\
\dfrac{4}{{\lambda}^{2}}\|Bf_{2}(x)\|^{2}_{L^{2}(K)}$
$\displaystyle\leavevmode\nobreak\ \leq\leavevmode\nobreak\
\dfrac{4}{{\lambda}^{2}}\|f_{2}\|^{2}_{L^{2}(K)}$
$\displaystyle\leavevmode\nobreak\ \leq\leavevmode\nobreak\
\dfrac{4}{{\lambda}^{2}}\lambda q\|f\|_{L^{1}(K)}$ (3.8)
$\displaystyle\leavevmode\nobreak\ =\leavevmode\nobreak\
\dfrac{4q}{\lambda}\|f\|_{L^{1}(K)}.$

Here, we have used that $B$ is bounded on $L^{2}(K)$ and property (iv) in
Remark 2.17. Now we find estimate on $E_{1}.$ We have

$\displaystyle|E_{1}|\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\ |\\{x\in K:|Bf_{1}(x)|>\lambda/2\\}|$
(3.9) $\displaystyle=\leavevmode\nobreak\
\bigg{|}\bigg{(}\bigcup_{i}W_{i}\bigg{)}\bigcap\\{x\in
K:|Bf_{1}(x)|>\lambda/2\\}\bigg{|}\leavevmode\nobreak\ +\leavevmode\nobreak\
\bigg{|}\bigg{(}\bigcup_{i}W_{i}\bigg{)}^{\complement}\bigcap\\{x\in
K:|Bf_{1}(x)|>\lambda/2\\}\bigg{|}.$

Consider the first term of (3.9),

$\displaystyle\bigg{|}\bigg{(}\bigcup_{i}W_{i}\bigg{)}\bigcap\\{x\in
K:|Bf_{1}(x)|>\lambda/2\\}\bigg{|}$ $\displaystyle\leavevmode\nobreak\
=\leavevmode\nobreak\ \bigg{|}\bigcup_{i}W_{i}\cap\\{x\in
K:|Bf_{1}(x)|>\lambda/2\\}\bigg{|}$ $\displaystyle\leavevmode\nobreak\
\leq\leavevmode\nobreak\ \bigg{|}\bigcup_{i}W_{i}\bigg{|}$
$\displaystyle\leavevmode\nobreak\ \leq\leavevmode\nobreak\
\sum\limits_{i}|W_{i}|$ (3.10) $\displaystyle\leavevmode\nobreak\
\leq\leavevmode\nobreak\ \|f\|_{L^{1}(K)}\leavevmode\nobreak\ \lambda^{-1}.$

Now we just have to handle the term

$\displaystyle\bigg{|}\bigg{(}\bigcup_{i}W_{i}\bigg{)}^{\complement}\bigcap\\{x\in
K:|Bf_{1}(x)|>\lambda/2\\}\bigg{|},$

for this one we prove that

(3.11) $\displaystyle\text{if}\leavevmode\nobreak\
x\notin\cup_{i}W_{i}\leavevmode\nobreak\ \text{then}\leavevmode\nobreak\
Bf_{1}(x)=0,$

and which gives that the above term is zero and reduces (3.9) to one term
only. We turn to proving (3.11),

$\displaystyle g_{j}\ast f_{1}(x)$
$\displaystyle=\int\limits_{K}a(y)\Phi_{A_{j}}(y)f_{1}(x-y)dy$
$\displaystyle=\int\limits_{\mathfrak{D}^{*}}a(y)f_{1}(x-y)dy,$

since $\text{supp}\leavevmode\nobreak\
a\Phi_{A_{j}}\subset\mathfrak{D}^{*}\cap A_{j}=\mathfrak{D}^{*}.$ Thus

$\displaystyle g_{j}\ast f_{1}(x)$
$\displaystyle=\int\limits_{y\in(x-W_{i})\cap\mathfrak{D}^{*}}a(y)f_{1}(x-y)dy.$

Let us note that $0\notin x-W_{i}.$ If $0\in x-W_{i}$ then $x\in W_{i},$ a
contradiction since $x\in(\cup_{i}W_{i})^{\complement}.$ Then
$(x-W_{i})\cap\mathfrak{D}^{*}=\emptyset$ and $g_{j}\ast f_{1}(x)$ is zero.
Hence

$\displaystyle Bf_{1}(x)$
$\displaystyle=\sum\limits_{j=k}^{\infty}q^{-(j+1)}g_{j}\ast f_{1}(x)$
$\displaystyle=0.$

By (3.7), (3.8) and (3.10), we obtain

$\displaystyle|\\{x\in K:|Bf(x)|>\lambda\\}|$
$\displaystyle\leq\lambda^{-1}\leavevmode\nobreak\
\|f\|_{L^{1}(K)}+4q\lambda^{-1}\leavevmode\nobreak\ \|f\|_{L^{1}(K)}$ (3.12)
$\displaystyle=(1+4q)\lambda^{-1}\leavevmode\nobreak\ \|f\|_{L^{1}(K)}.$

This concludes the proof of the fact that $B$ is weak type $(1,1).$ From (3.6)
and (3.12) we have

$\displaystyle\|Bf\|_{L^{2}(K)}\leq q^{-k}(q-1)^{-1}\|f\|_{L^{2}(K)},$

and

$\displaystyle|\\{x\in
K:|Bf(x)|>\lambda\\}|\leq(1+4q)\lambda^{-1}\|f\|_{L^{1}(K)}.$

Applying Lemma 2.18 we obtain that $T_{k}$ is of type $(r,r)(1<r\leq 2),$ i.e,
for $f\in S(K),$ $Bf\in L^{r}(K)(1<r\leq 2)$ and

(3.13) $\displaystyle\|Bf\|_{L^{r}(K)}\leq Aq^{-k}\|f\|_{L^{r}(K)},$

where $A$ is independent of $k.$

###  3.3. $B$ commutes with translations

Let $h\in K$ and $f\in S(K),$

$\displaystyle\widehat{\tau_{h}(Bf)}$
$\displaystyle=\sum\limits_{j=k}^{\infty}q^{-(j+1)}\widehat{(\tau_{h}(g_{j}\ast
f))}$
$\displaystyle=\sum\limits_{j=k}^{\infty}q^{-(j+1)}\bar{\chi_{h}}\widehat{(g_{j}\ast
f)}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
(\because\leavevmode\nobreak\ \widehat{\tau_{h}f}=\bar{\chi_{h}}\widehat{f})$
$\displaystyle=\sum\limits_{j=k}^{\infty}q^{-(j+1)}\widehat{g_{j}}(\bar{\chi_{h}}\widehat{f})$
$\displaystyle=\sum\limits_{j=k}^{\infty}q^{-(j+1)}\widehat{g_{j}}(\widehat{\tau_{h}f})$
$\displaystyle=\sum\limits_{j=k}^{\infty}q^{-(j+1)}\widehat{(g_{j}\ast\tau_{h}f)}$
$\displaystyle=\widehat{(B\tau_{h}f)},$

which implies that $B$ commutes with translations. From (3.13) and Lemma 2.20,
we obtain that for $f\in S(K),$ $Bf\in L^{r}(K)(2<r<\infty)$ and

(3.14) $\displaystyle\|Bf\|_{L^{r}(K)}\leq Aq^{-k}\|f\|_{L^{r}(K)}.$

Combining (3.13) and (3.14), we get whenever $f\in S(K),$ $Bf\in L^{r}(K)$ and

$\displaystyle\|Bf\|_{L^{r}(K)}\leq
Aq^{-k}\|f\|_{L^{r}(K)},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ 1<r<\infty.$

Therefore by (3.3), for $f\in S(K),$ $T_{k}f\in L^{r}(K)$ and

(3.15) $\displaystyle\|T_{k}f\|_{L^{r}(K)}\leq
Cq^{-k}\|\Omega\|_{H^{1}(\mathfrak{D}^{*})}\|f\|_{L^{r}(K)},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ 1<r<\infty.$

Let $f\in L^{r}(K),$ since $S(K)$ is dense in $L^{r}(K)\leavevmode\nobreak\
1\leq r<\infty,$ i.e. there is a sequence $(f_{j})_{j\in\mathbb{N}}\subseteq
S(K)$ with $f_{j}\rightarrow f$ in $L^{r}(K).$ From (3.15), we see that

$\displaystyle\|T_{k}f_{j}-T_{k}f_{j^{\prime}}\|_{L^{r}(K)}$
$\displaystyle=\|T_{k}(f_{j}-f_{j^{\prime}})\|_{L^{r}(K)}$ $\displaystyle\leq
Cq^{-k}\|\Omega\|_{H^{1}(\mathfrak{D}^{*})}\|f_{j}-f_{j^{\prime}}\|_{L^{r}(K)},$

and so $(T_{k}f_{j})_{j\in\mathbb{N}}$ is a Cauchy sequence in $L^{r}(K).$ By
completeness, the limit $T_{k}f_{j}\rightarrow T_{k}f$ exists in $L^{r}(K)$
and by (3.15),

$\displaystyle\|T_{k}f\|_{L^{r}(K)}$
$\displaystyle=\lim_{j\to\infty}\|T_{k}f_{j}\|_{L^{r}(K)}$ $\displaystyle\leq
Cq^{-k}\|\Omega\|_{H^{1}(\mathfrak{D}^{*})}\lim_{j\to\infty}\|f_{j}\|_{L^{r}(K)}$
$\displaystyle\leq
Cq^{-k}\|\Omega\|_{H^{1}(\mathfrak{D}^{*})}\|f\|_{L^{r}(K)},$

which completes the proof of Theorem 1.5.

##  4\. Proof of Theorem 1.6

Let $1<r,t<\infty$ and $s>0.$ Let $\Omega\in H^{1}(\mathfrak{D}^{*})$ then it
follows from (3.3) that

$\displaystyle T_{k}(f)(x)$
$\displaystyle=\sum\limits_{i=1}^{\infty}\lambda_{i}Bf,$

and

$\displaystyle\|T_{k}f\|_{F_{rt}^{s}(K)}$
$\displaystyle\leq\sum\limits_{i=1}^{\infty}|\lambda_{i}|\|Bf\|_{F_{rt}^{s}(K)}$
(4.1) $\displaystyle\leq
C\|\Omega\|_{H^{1}(\mathfrak{D}^{*})}\|Bf\|_{F_{rt}^{s}(K)},$

where

$\displaystyle Bf(x)=\sum\limits_{j=k}^{\infty}q^{-(j+1)}g_{j}\ast f(x).$

Then

$\displaystyle\|Bf\|_{F_{rt}^{s}(K)}\leq\sum_{j=k}^{\infty}q^{-(j+1)}\leavevmode\nobreak\
\|g_{j}\ast f(x)\|_{F_{rt}^{s}(K)}.$

So, to estimate the $F_{rt}^{s}(K)$ norm of $T_{k}$ it is enough to find
$\|g_{j}\ast f(x)\|_{F_{rt}^{s}(K)}.$ We begin with the observations:

$\displaystyle\Delta_{l}(g_{j}\ast f)$ $\displaystyle=\Delta_{l}(f\ast g_{j})$
$\displaystyle=\mathcal{F}^{-1}\phi_{l}\mathcal{F}(f\ast g_{j})$ (4.2)
$\displaystyle=(\mathcal{F}^{-1}\phi_{l}\mathcal{F}f)\ast g_{j},$

where $\phi_{l}$ is decomposition of unity see Definition 2.8 and

(4.3) $\displaystyle\|(\mathcal{F}^{-1}\phi_{l}\mathcal{F}f)\ast
g_{j}\|_{\ell^{t}}\leq(\|\mathcal{F}^{-1}\phi_{l}\mathcal{F}f\|_{\ell^{t}})\ast
g_{j},$

which follows from (2.2) with
$F_{l}=(\mathcal{F}^{-1}\phi_{l}\mathcal{F}f)\ast g_{j}.$ Now

$\displaystyle\|g_{j}\ast f(x)\|_{F_{rt}^{s}(K)}$
$\displaystyle=\Bigg{\|}\Bigg{\\{}\sum_{l=0}^{\infty}|q^{sl}\Delta_{l}(g_{j}\ast
f)|^{t}\Bigg{\\}}^{\frac{1}{t}}\Bigg{\|}_{L^{r}(K)}$
$\displaystyle=\Bigg{\|}\Bigg{\\{}\sum_{l=0}^{\infty}q^{slt}|(\mathcal{F}^{-1}\phi_{l}\mathcal{F}f)\ast
g_{j}|^{t}\Bigg{\\}}^{\frac{1}{t}}\Bigg{\|}_{L^{r}(K)}$
$\displaystyle\leq\Bigg{\|}\Bigg{\\{}\sum_{l=0}^{\infty}q^{slt}|\mathcal{F}^{-1}\phi_{l}\mathcal{F}f|^{t}\Bigg{\\}}^{\frac{1}{t}}\ast
g_{j}\Bigg{\|}_{L^{r}(K)}$
$\displaystyle\leq\Bigg{\|}\Bigg{\\{}\sum_{l=0}^{\infty}q^{slt}|\mathcal{F}^{-1}\phi_{l}\mathcal{F}f|^{t}\Bigg{\\}}^{\frac{1}{t}}\Bigg{\|}_{L^{r}(K)}\|g_{j}\|_{L^{1}(K)}$
$\displaystyle\leq\|f\|_{F_{rt}^{s}(K)}.$

Therefore

$\displaystyle\|Bf\|_{F_{rt}^{s}(K)}$
$\displaystyle\leq\sum_{j=k}^{\infty}q^{-(j+1)}\leavevmode\nobreak\
\|f\|_{F_{rt}^{s}(K)}$ (4.4)
$\displaystyle=q^{-k}(q-1)^{-1}\|f\|_{F_{rt}^{s}(K)}.$

By (4.1) and (4.4), we have

$\displaystyle\|T_{k}f\|_{F_{rt}^{s}(K)}$ $\displaystyle\leq
C\|\Omega\|_{H^{1}(\mathfrak{D}^{*})}\|Bf\|_{F_{rt}^{s}(K)}$
$\displaystyle\leq
Cq^{-k}\|\Omega\|_{H^{1}(\mathfrak{D}^{*})}\|f\|_{F_{rt}^{s}(K)}.$

Since $S(K)$ is dense in $F_{rt}^{s}(K)\leavevmode\nobreak\ 1\leq r,t<\infty,$
therefore for $f\in F_{rt}^{s}(K)$ we have

$\displaystyle\|T_{k}f\|_{F_{rt}^{s}(K)}$ $\displaystyle\leq
Cq^{-k}\|\Omega\|_{H^{1}(\mathfrak{D}^{*})}\|f\|_{F_{rt}^{s}(K)}.$

This is the conclusion ($i$) of Theorem 1.6. By Theorem 1.5, we have for
$1<r,t<\infty$

$\displaystyle\bigg{(}\sum_{l=0}^{\infty}\|T_{k}f_{j}\|^{t}_{L^{r}(K)}\bigg{)}^{1/t}\leq
Cq^{-k}\|\Omega\|_{H^{1}(\mathfrak{D}^{*})}\bigg{(}\sum_{l=0}^{\infty}\|f_{j}\|^{t}_{L^{r}(K)}\bigg{)}^{1/t}.$

Then we get for any $1<r,t<\infty$ and $s>0,$

$\displaystyle\|T_{k}f\|_{B_{rt}^{s}(K)}$
$\displaystyle=\Bigg{(}\sum_{l=0}^{\infty}q^{slt}\|\Delta_{l}(T_{k}f)\|_{L^{r}(K)}^{t}\Bigg{)}^{\frac{1}{t}}$
$\displaystyle=\Bigg{(}\sum_{l=0}^{\infty}q^{slt}\|T_{k}(\Delta_{l}f)\|_{L^{r}(K)}^{t}\Bigg{)}^{\frac{1}{t}}$
$\displaystyle\leq
Cq^{-k}\|\Omega\|_{H^{1}(\mathfrak{D}^{*})}\Bigg{(}\sum_{l=0}^{\infty}q^{slt}\|\Delta_{l}f\|_{L^{r}(K)}^{t}\Bigg{)}^{\frac{1}{t}}$
$\displaystyle=Cq^{-k}\|\Omega\|_{H^{1}(\mathfrak{D}^{*})}\|f\|_{B_{rt}^{s}(K)}.$

Again using the fact that $S(K)$ is dense in
$B_{rt}^{s}(K)\leavevmode\nobreak\ 1\leq r,t<\infty,$ therefore for $f\in
B_{rt}^{s}(K)$ we have

$\displaystyle\|T_{k}f\|_{B_{rt}^{s}(K)}$ $\displaystyle\leq
Cq^{-k}\|\Omega\|_{H^{1}(\mathfrak{D}^{*})}\|f\|_{B_{rt}^{s}(K)}.$

This concludes the proof of Theorem 1.6.

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# Cosmological constraints from the density gradient weighted correlation
function

Xiaoyuan Xiao1,2, Yizhao Yang1,2, Xiaolin Luo1,2,†, Jiacheng Ding1,2,†, Zhiqi
Huang1,2, Xin Wang1,2, Yi Zheng1,2,‡, Cristiano G. Sabiu3, Jaime Forero-
Romero4, Haitao Miao5, Xiao-Dong Li1,2,⋆
1 School of Physics and Astronomy, Sun Yat-Sen University, Guangzhou 510297,
P.R.China
2 CSST Science Center for the Guangdong-Hong kong-Macau Greater Bay Area, SYSU
3 Natural Science Research Institute, University of Seoul, 163
Seoulsiripdaero, Dongdaemun-gu, Seoul, 02504, Republic of Korea
4 Departamento de Física, Universidad de los Andes, Cra. 1 No. 18A-10 Edificio
Ip, CP 111711, Bogotá, Colombia
5 Key Laboratory for Computational Astrophysics, National Astronomical
Observatories, Chinese Academy of Sciences, 20A Datun Road, Beijing 100012,
People’s Republic of China
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The mark weighted correlation function (MCF) $W(s,\mu)$ is a computationally
efficient statistical measure which can probe clustering information beyond
that of the conventional 2-point statistics. In this work, we extend the
traditional mark weighted statistics by using powers of the density field
gradient $|\nabla\rho/\rho|^{\alpha}$ as the weight, and use the angular
dependence of the scale-averaged MCFs to constrain cosmological parameters.
The analysis shows that the gradient based weighting scheme is statistically
more powerful than the density based weighting scheme, while combining the two
schemes together is more powerful than separately using either of them.
Utilising the density weighted or the gradient weighted MCFs with
$\alpha=0.5,\ 1$, we can strengthen the constraint on $\Omega_{m}$ by factors
of 2 or 4, respectively, compared with the standard 2-point correlation
function, while simultaneously using the MCFs of the two weighting schemes
together can be $1.25$ times more statistically powerful than using the
gradient weighting scheme alone. The mark weighted statistics may play an
important role in cosmological analysis of future large-scale surveys. Many
issues, including the possibility of using other types of weights, the
influence of the bias on this statistics, as well as the usage of MCFs in the
tomographic Alcock-Paczynski method, are worth further investigations.

###### keywords:

Cosmology: cosmological parameters – large-scale structure of universe;
Methods: statistical

††pubyear: 2021††pagerange: Cosmological constraints from the density gradient
weighted correlation function–LABEL:lastpage

##  1 Introduction

The cosmic acceleration (Riess et al., 1998; Perlmutter et al., 1999)
indicates either the presence of a “dark energy” component in the Universe or
the breakdown of general relativity on cosmological scales. In the last two
decades, tremendous effort has been put on the theoretical explanation and
observational probes of acceleratting universe, yet they are still not well
comprehended or accurately measured (Weinberg, 1989; Li et al., 2011; Yoo &
Watanabe, 2012; Weinberg et al., 2013).

On large scales, the spatial distribution of galaxies forms a unique, and
complex filamentary motif known as the ’cosmic web’ (Bardeen et al., 1986; de
Lapparent et al., 1986; Huchra et al., 2012; Tegmark et al., 2004; Guzzo et
al., 2014), which encodes a significant amount of information about the
expansion and structure growth history of the Universe. Over the next few
years, surveys of large-scale structure (LSS) like
DESI111https://desi.lbl.gov/, EUCLID222http://sci.esa.int/euclid/,
Rubin333https://www.lsst.org/, Roman444https://wfirst.gsfc.nasa.gov/, and CSST
(Gong et al., 2019), will explore an unprecedented large volume of the
Universe with extraordinary precision. So the development of powerful
statistics is of great significance to comprehensively and reliably derive the
cosmological parameters from LSS.

The most widely-adopted LSS statistics are the 2-point correlation function
(2pCF) or power spectrum measurements (Kaiser, 1987; Ballinger et al., 1996;
Eisenstein et al., 1998; Blake & Glazebrook, 2003; Seo & Eisenstein, 2003).
These methods have had excellent success when applied to galaxy redshift
surveys such as the 2-degree Field Galaxy Redshift Survey (2dFGRS; Colless et
al. (2003)), the 6-degree Field Galaxy Survey (6dFGS; Beutler et al. (2012)),
the WiggleZ survey Blake et al. (2011b, a), and the Sloan Digital Sky Survey
(SDSS; York et al. (2000); Eisenstein et al. (2005); Percival et al. (2007);
Anderson et al. (2012); Sánchez et al. (2012, 2013); Anderson et al. (2014);
Samushia et al. (2014); Ross et al. (2015); Beutler et al. (2017); Sánchez et
al. (2017); Alam et al. (2017); Chuang et al. (2017)). However, the main
defect of these methods is that they are only sensitive to the Gaussian
statistical properties of the density field, while both structure formation
processes and the primordial conditions can introduce significant amounts of
non-Gaussianity into the LSS.

Ongoing studies seek to go beyond the 2-point statistics explored methods like
the 3-point statistics (Sabiu et al., 2016; Slepian et al., 2017), the 4-point
statistics (Sabiu et al., 2019), the density field or voids (Ryden, 1995;
Lavaux & Wandelt, 2012; Sutter et al., 2012; Bos et al., 2012; Chan et al.,
2014; Cai et al., 2015; Sutter et al., 2014; Mao et al., 2017; Ramanah et al.,
2019; Hamaus et al., 2020; Lavaux & Jasche, 2021), the Minkowski functionals
(Minkowski, 1903; Mecke et al., 1994; Schmalzing & Gorski, 1998; Kerscher et
al., 1998; Park & Kim, 2010; Appleby et al., 2020, 2021a, 2021b), deep
learning (Ravanbakhsh et al., 2017; Mathuriya et al., 2018; He et al., 2019;
Ntampaka et al., 2019; Pan et al., 2020; Li et al., 2020; Mao et al., 2021;
Villaescusa-Navarro et al., 2021; Ni et al., 2021; Wu et al., 2021), and so
on. While all of them can explore the non-Gaussian clustering information
encoded in the LSS, in this analysis we investigate a statistical tool namely
the mark weighted correlation function (MCF; Beisbart & Kerscher, 2000;
Beisbart et al., 2002; Gottlöber et al., 2002; Sheth & Tormen, 2004; Sheth et
al., 2005; Skibba et al., 2006; White & Padmanabhan, 2009; White, 2016;
Satpathy et al., 2019; Alam et al., 2021; Philcox et al., 2020; Massara et
al., 2021) which is simple and computationally efficient compared with the
aforementioned statistics.

The MCF is simply the 2-point statistics but with each object weighted by a
“mark” related with its local environment. The weights are generally
proportional to the local density to a positive or negative power. In this
way, the statistics give more emphasis on the dense or the undense regions.
This leads to better discrimination of the different patterns of galaxy
clustering and redshift space distortions (RSDs) in the dense or under dense
areas, and thus can lead to tighter cosmological constraints compared with the
standard 2pCF (Yang et al., 2020).

Figure 1:  A slice of halos selected from the BigMDPL simulation. Color of the
points represents their weights in three weighting schemes. Left panel: In the
density weighting scheme, weight is solely determined by the local density.
Objects in the regions of clusters (voids) are assigned with very large
(small) weights. Middle panel: A scheme wherein the weights are set as
$\nabla\rho$. In this case the weights are still crucially determined by the
values of $\rho$. Right panel: The gradient weighting scheme we used in this
analysis. To eliminate the dependence on $\rho$ we set ${\rm
weight}\propto|\nabla\rho|/\rho$. This weighting is thus sensitive to the
change of density in the local environment, while not being overly sensitive
to its absolute value. In this plot, values of the object weights in the voids
and clusters have the same order of magnitude.

While using the local density is the simplest way to include the local
environment information into the analysis, this approach is certainly far from
capturing the full information of the environment. In previous works, many
statistical quantities related with the local environment have been proposed
and studied (Gott et al., 1986; Mecke et al., 1994; Hahn et al., 2007; Gott et
al., 2009; Forero-Romero et al., 2009; Sousbie, 2011; Hoffman et al., 2012;
Park et al., 2013; Wang et al., 2014; Forero-Romero et al., 2014; Pranav et
al., 2019; Fang et al., 2019; Hong et al., 2020; García-Alvarado et al., 2020;
Shim et al., 2021a, b; Suárez-Pérez et al., 2021). So it may be worth
considering other statistical quantities in the construction of the MCFs, so
as to extract more information from the LSS.

In this paper, we extend the scope of the marked statistics by considering
more possible weighting in the correlation functions. Tentatively, in this
analysis we try using the density gradient as the mark, based on three
considerations. 1) The first order derivative is the simplest extension to the
value itself. It is natural to consider the spatial variation of the field
after its value being considered. 2) The density can be used to classify the
universe into topological regions of cluster, filament, sheet and void. In
contrast, the gradient traces regions where the density changes rapidly, and
thus marks the boundaries among the different topological regions. From this
viewpoint, the density and the density gradient are complementary descriptions
of the LSS. So they could also be complementary in a certain statistics. 3)
Phenomenologically, regions having large density gradient are expected to have
large gravitational acceleration. Since velocity is not directly observable in
surveys, density gradient could be the simplest and most convenient quantity
if one were to study the dynamical property of the LSS.

This paper is organized as follows. Section 2 describes the simulation data
used in this analysis. In Section 3, we explained the concept and the
mathematical definition of the marked statistics we designed. In Section 4, we
applied the marked statistics to the simulation data, and presents the
obtained cosmological results. We discuss and conclude in Section 5.

Figure 2:  Histograms of the object weights defined as $\rho$, $|\nabla\rho|$
and $|\nabla\rho|$/$\rho$, respectively. The distributions of the former two
schemes are quite similar to each other. The last weighting scheme exhibits a
clearly different distribution, and the dispersion of the values is
significantly smaller compared with the other two schemes.

##  2 DATA

To conduct a cosmological analysis using the MCFs of the two weighting
schemes, we use the BigMulti-Dark555webpage: https://www.cosmosim.org  (BigMD)
N-body simulation, and also a set of samples produced using the COLA (COmoving
Lagrangian Acceleration) algorithm (Tassev et al., 2013). In all simulations,
we identify gravitationally bound structures using ROCKSTAR (Behroozi et al.,
2013), which is a halo finder based on adaptive hierarchical refinement of
friends-of-friends groups in six phase-space dimensions and one time dimension
to guarantee robust tracking of substructure. We fixed a halo number density
$\bar{n}=$ 0.001 $(h^{-1}\rm Mpc)^{-3}$ in all simulations to ensure the
comparability. Both halos and subhalos are used in our analysis.

###  2.1 BigMD simulation

The Big MultiDark (BigMD) Planck simulation belongs to the series of MultiDark
simulations. It is generated using $3\,840^{3}$ particles in a volume of
$(2.5h^{-1}\rm Gpc)^{3}$, based on a flat $\Lambda$CDM cosmology with Planck
parameters $H_{0}=67.77\ {\rm km}\ s^{-1}{\rm Mpc}^{-1}$,
$\Omega_{m}=0.307115$, $\Omega_{b}=0.048206$, $n_{s}=0.9611$, and
$\sigma_{8}=0.8288$ (Klypin et al., 2016). The initial condition was set by
using the Zeldovich approximation at redshift $z_{\rm init}=100$. Both the
volume and the number of particles are very huge, so this simulation is among
the best choices for testing our methodology.

###  2.2 COLA MultiCosmo Simulations

In order to study the sensitivity of the statistics to cosmological
parameters, we also generate five COLA simulations using different
cosmological parameters of $\Omega_{m}=0.2671,\ 0.2871,\ 0.3071,\ 0.3271$ and
$0.3471$, respectively. Values of the other parameters, i.e. $\Omega_{b}$,
$\sigma_{8}$, $n_{s}$ and $H_{0}$, are fixed to the values adopted in the
BigMD simulation. To minimize the effect of cosmic variance, we maintain the
same random seed when generating their initial conditions. These simulations
are produced using $1024^{3}$ particles in a box of $(680h^{-1}\rm Mpc)^{3}$.
For convenience, in what follows we refer to them as COLA MultiCosmo
Simulations.

COLA combines 2LPT (the second order Lagrangian perturbation theory), for time
integration for large scale dynamical evolution, with a full-blown N-body
Particle-Mesh (PM) algorithm to calculate the small scale dynamics (Tassev et
al., 2013). In this work, COLA was chosen because it is hundreds of times
faster than N-body simulation and can still maintain a good accuracy on non-
linear scales.

###  2.3 Simulations for Covariance Estimation

To compute the covariance matrices we generate 180 simulations using COLA. All
simulations are produced using $600^{3}$ particles in a $(512h^{-1}\rm
Mpc)^{3}$ box, and their cosmological parameters are the same as those adopted
in the BigMD simulation. The only difference of these simulations is that we
use different random seeds in creating their initial conditions.

##  3 Methodology

As a simple generalization of the standard 2-point correlation function, MCF
assign each object an environment dependent weight, which is commonly chosen
as a form proportional to the local density White (2016); Yang et al. (2020)

${\rm weight}=\rho_{n_{\rm NB}}^{\alpha},$ (1)

Here we estimate the local density $\rho_{n_{\rm NB}}$ using its $n_{\rm NB}$
nearest neighbors, i.e.

$\rho_{n_{\rm NB}}({\bf r})=\sum_{i=1}^{n_{\rm NB}}W_{k}({\bf
r-r_{i}},h_{W}),$ (2)

where $\rho_{n_{\rm NB}}(\bf r)$ is the environmental number density around a
specific galaxy located at $r$, and $W_{k}$ is the smoothing kernel, for which
we choose the 3rd order B-spline functions having non-zero value within a
sphere of radius $2h_{W}$ $h^{-1}$ Mpc (Gingold & Monaghan, 1977; Lucy, 1977).
In order to ensure that the kernel always contains $n_{\rm NB}$ nearest
neighbor halos within a radius of $2h_{W}$, We use an adjustable radius for
the smoothing kernel.

As the simplest generalization of Equation 1, in this analysis we use the
gradient of the local density as the weight, and define

${\rm weight}=(|\nabla\rho_{n_{\rm NB}}|/\rho_{n_{\rm NB}})^{\alpha},$ (3)

where the gradient is evaluated via the derivatives of the kernel,

$\nabla\rho_{n_{\rm NB}}({\bf r})=\sum_{i=1}^{n_{\rm NB}}\nabla W_{k}({\bf
r-r_{i}},h_{W}).$ (4)

Equation 3 describes the variation of the local density around a certain
position. In order to focus on the variation of $\rho$, rather than being
sensitive to its value, we divide the value of the gradient by the value of
$\rho$. Throughout this analysis, we choose $n_{\rm NB}=30$, which leads to a
smoothing radius $h_{W}=8.3\pm 3.7h^{-1}\rm Mpc$.

For simplicity, we will refer the above two weighting schemes as density
weighting scheme and gradient weighting scheme, respectively. We will apply
both of them to the simulation data and compare the cosmological results
obtained.

As an illustration of the weighting schemes, Figure 1 shows a slice of objects
selected from the BigMD simulation, with weights of the objects represented by
their colors. In the density weighting scheme, objects in the most dense
regions (clusters) are assigned with very large weights, while objects in void
regions are assigned with negligible weights. In case that we use the
gradient, and assign each object a ${\rm weight}$ defined by $\nabla\rho$, the
results are still quite similar to the results of the density weighting
scheme, The reason is that the value of $|\nabla\rho|$ is mostly determined by
the value of $\rho$ itself. Finally, the right panel represents the weighting
scheme used in this analysis, where we use ${\rm
weight}\propto|\nabla\rho|/\rho$, which is not too dependent on $\rho$ while
still sensitive to the variation of the environment.

The distribution of the weights are illustrated in Figure 2. The distribution
of the $\rho$ and $|\nabla\rho|$ weighting schemes are similar. For both of
them, we detect 4-5 orders of magnitude variation among the majority of
objects. In contrast, the $|\nabla\rho/\rho|$ weighting scheme displays a
clearly different distribution, and the dispersion of the objects’ weights is
only $\sim$1-2 magnitude. We consider the low dispersion to be a particularly
advantageous property of this weighting scheme.

Except the difference in the weighting, the other process of the statistics
are exactly the same as those in general MCFs. In the measurement of the 2pCF,
we adopt the most commonly used Landy-Szalay estimator

$W(s,\mu)=\frac{WW(s,\mu)-2WR(s,\mu)+RR(s,\mu)}{RR(s,\mu)}.$ (5)

Here $WW(s,\mu)$ denotes the weighted number of galaxy-galaxy pairs,
$WR(s,\mu)$ presents the number of galaxy-random pairs, and $RR(s,\mu)$
corresponds to the count of random-random pairs. The pairs separation of
objects are defined by $s\pm\Delta s$ and $\mu\pm\Delta\mu$, where the letter
$s$ is the distance between the pair of objects in the redshift space and
$\mu=\cos(\theta)$, where $\theta$ denotes the angle between the line joining
the pair and the line of sight direction 666We use $s$ instead of $r$ here,
because the statistics is usually computed using the distance in redshift
space, and due to the RSDs they are correlated with each other via
$s=r+v/(aH)$.. We always use 10 times more particles in a random samples than
in the data samples.

Figure 3:  The monopole measurements $s^{2}W(s)$ in the density and gradient
weighting schemes. Values of $\alpha$ are set as $-1,\ -0.5,\ 0,\ 0.5$ and
$1$, respectively. The uppermost panels show the results at $z=0$, while
panels in the other rows compare the $z=0.51$, 1.0, 1.45 results to the
redshift zero results. The two schemes show largest difference in the case of
$\alpha=-1$, In the density weighting scheme, the amplitude of the correlation
function increases faster as we increase $\alpha$.  Figure 4:  The same to
Figure 3, but for the $s^{2}W_{2}(s)$ measurements at $z=0$.

In the counting of pairs, the weighting scheme is applied to both the data and
the random samples (except that we use a 10x larger $n_{\rm NB}$ in the
random). This treatment is different from what adopted in (Yang et al., 2020)
(where the authors fixed the weights of the randoms as 1). Here we do this to
take into consideration the possible correlations between nearby objects
induced by the smoothing, although we tested and found that there are no
significant changes to the measured 2pCFs.

##  4 Results

In what follows we present our results. Sec. 4.1 presents the monopole $W(s)$
and quadrupole $W_{2}(s)$ of the two weighting schemes. Sec. 4.2 presents the
anisotropic $W_{\Delta s}(\mu)$ measurements. Their covariance matrices are
presented in Sec. 4.3. We derive the cosmological constraints in Sec. 4.4.

###  4.1 The angular-averaged measurements

Figure 5:  The scale-averaged statistics $W_{\Delta s}(\mu)$ measured in the
two weighting schemes. The slope decreases as we go to higher redshifts where
the RSD effect becomes weaker.

In this subsection we present the angular-averaged MCFs (i.e. the multipoles),
which are defined as

$W_{l}(s)\equiv\frac{2l+1}{2}\int_{-1}^{1}L_{l}(\mu)W(\mu,s)d\mu.$ (6)

In what follows, we present the monopole and quadrupole measurements. For
simplicity, we will use $W(s)$ rather than $W_{0}(s)$ to denote the monopoles.

Figure 3 shows the monopole measurements $s^{2}W(s)$ obtained using the
density and gradient weighting schemes, with $\alpha$ taken as -1, -0.5, 0,
0.5 and 1, respectively. Each measurement uses a halo/subhalo sample
identified from one BigMDPL snapshot, with a fixed number density
$10^{-3}(h^{-1}\rm Mpc)^{-3}$ produced by applying a minimal cut of
halo/subhalo mass. As described in the previous section, values of $\rho$ and
$\nabla\rho$ are estimated using 30 nearest neighbors, corresponding to a
smoothing radius of $h_{W}=8.3\pm 3.7h^{-1}\rm Mpc$.

The first row shows the results measured at $z=0$. From left to right, we see
a clear trend of increasing clustering strength as we increase the value of
$\alpha$, meaning that in regions with larger $\rho$ or $|\nabla\rho|/\rho$
objects are more clustered. But the trend is less significant in the gradient
weighting scheme. The shapes and amplitudes of the $s^{2}W(s)$s are affected
by the choice of the weighting schemes.

The largest difference between the two schemes is detected at the $\alpha=-1$
case. Effectively, weighting the objects by $\rho^{-1}$ is equivalent to
transforming the distribution to a spatially uniform distribution. So in the
density weighting scheme we find $W(s)\approx 0$ at $s\gtrsim 20{h^{-1}}\rm
Mpc$ 777Considering that the radii of the smoothing sphere is $2h_{W}=16.6\pm
7.4{h^{-1}}\rm Mpc$, clustering signal above this scale is is erased. Below
this scale, the smoothing kernel can induce some “weird” clustering features
(the sharp peaks and valleys in $\alpha=-0.5$ and $\alpha=-1$ cases), which is
discussed in Yang et al. (2020) and is found to be related with the scale of
the smoothing.. Of course, this phenomenon is not detected if weighting is
$|\nabla\rho/\rho|^{-1}$.

Figure 3 also compares the $W(s)$ measurements at different redshifts. In all
cases, we observe a similar amplitude at different redshifts, and a more
prominent BAO peak at higher redshifts.

Finally, Figure 4 shows the quadrupole measurements at $z=0$. Similar to the
monopole results, here we see the two weighting schemes lead to measurements
having significantly different shapes and amplitudes, and the density
weighting scheme measurement varies more dramatically when we change the value
of $\alpha$.

###  4.2 The scale averaged measurements $W_{\Delta s}(\mu)$

The dynamics of objects crucially depends on their environment. This results
in an environmental dependent RSD, and motivates us to study the anisotropic
clustering using the mark weighted correlation functions. To do this, we
measured the scale-averaged correlation function $W(s,\mu)$, which quantifies
the anisotropy via

$W_{\Delta s}(\mu)\equiv\int_{s_{\rm min}}^{s_{\rm max}}W(s,\mu)ds.$ (7)

and its normalized version

$\hat{W}_{\Delta s}(\mu)\equiv\frac{W_{\Delta s}(\mu)}{\int_{0}^{\mu_{\rm
max}}W_{\Delta s}(\mu)\ d\mu}.$ (8)

These two quantities were used to quantify the RSDs and the Alcock-Paczynski
(AP) distortions in the tomographic AP method (Li et al., 2015; Park et al.,
2019; Ma et al., 2020), and have been proved useful in constraining the dark
matter ratio $\Omega_{m}$ (Li et al., 2016), the dark-energy equation of state
parameter $w$ (Li et al., 2016) and its time evolution (Li et al., 2018; Zhang
et al., 2019), the Hubble constant (Zhang et al., 2019), the curvature and the
properties of neutrinos (Li et al., 2019).

Figure 5 present the measured $W_{\Delta s}(\mu)$s when $s_{\rm min},\ s_{\rm
max}$ are set as 6, 40 $h^{-1}\rm Mpc$, respectively. In all cases, we detect
a sharp peak at $0\lesssim 1-\mu\lesssim 0.1$, and a slope in the range of
$0.1\lesssim 1-\mu$ The former is produced by the finger-of-god (FOG) effect
(Jackson, 1972), while the latter is the result of the Kaiser effect (Kaiser,
1987).

Yang et al. (2020) showed that $\hat{W}_{\Delta s}(\mu)$ has a statistically
stronger constraint on cosmological parameters than $W(s)$. Also, in the
construction of $\hat{W}_{\Delta s}$ we integrate the scale-dependence and
normalize the amplitude, so compared with $W(s)$ it is less likely to be
affected by scale-dependent bias. So in the following analysis we only use
$\hat{W}_{\Delta s}(\mu)$.

###  4.3 Covariance measurements

Figure 6:  Correlation coefficient of $W_{\Delta s}(\mu)$. In both of the two
weighting schemes there exist positive correlation among nearby $\mu$-bins.
Compared with the density weighting scheme, in the gradient weighting scheme
the correlation among nearby $\mu$-bins is weaker. Strong cross-correlation
between the two schemes is detected.  Figure 7: Using the MCFs to distinguish
two different cosmologies with different values of $\Omega_{m}$. Upper panels
show the measured $W_{\Delta s}(\mu)$, while lower panels show the difference
between the two cosmologies. In the left and right panels, we show the results
obtained using $\alpha=0.5$ and 1, respectively. The gradient weighting scheme
has smaller statistical errors compared with the density weighting scheme.

Figure 6 shows the correlation coefficients of the $W_{\Delta s}(\mu)$ in 120
$\mu$-bins. Without loss of generality we only present the results using
$\alpha=0.5$.

In both of the weighting schemes there exist positive correlation among the
nearby $\mu$-bins, although in the gradient weighting scheme this correlation
is slightly weaker. Although measures have been taken to eliminate the
similarity between the two weighting schemes, their $W_{\Delta s}(\mu)$
measurements are still far away from being independent. We find the cross-
correlation coefficient $\gtrsim 0.8$ for the near diagonal terms. So we must
to take the cross-correlation into consideration when simultaneously using the
two weight schemes.

###  4.4 Cosmological constraints

In what follows, we study whether the new weighting scheme is helpful in
improving the cosmological constraints. To do this, we try to distinguish the
COLA MultiCosmo Simulations using MCFs with different weighting schemes, and
evaluate their statistical power. The COLA MultiCosmo Simulations are created
using values of $\Omega_{m}=0.2671,0.2871,0.3071,0.3271$ and $0.3471$,
respectively. A varying mass-cut is applied to create a series of
$\bar{n}=10^{-3}\ (h^{-1}\rm Mpc)^{-3}$ halo/subhalo samples. For simplicity,
the $\Omega_{m}=0.3071$ simulation is chosen as the baseline, and all
simulations are compared with it. The difference is characterized by

$\chi^{2}=\Delta{\bf p}^{\rm T}\cdot{\bf Cov}^{-1}\cdot{\bf\Delta p},$ (9)

where $\Delta\bf p$ is defined as

$\Delta{\bf p}\equiv{\bf p}(\Omega_{m}=0.3071)-{\bf p}(\Omega_{m})$ (10)

and $\bf p$ refer to one or several MCFs,whose weighting scheme and weighting
power depends on our choice. Throughout the analysis, we use $n_{\mu}=6$ bins
for a single $\hat{W}_{\Delta s}(\mu)$ measurement. This means that if
combining together 2 or 4 MCFs we will have 12 or 24 statistical variables.
These numbers are safely small compared with the number of mocks (180) used to
compute the covariance matrix (Hartlap et al., 2007).

Figure 8: Using different MCFs to distinguish simulations with different
cosmologies. The baseline cosmology has $\Omega_{m}=0.3071$, and the other
cosmologies are compared to it. The $\chi^{2}$ and likelihood distributions of
the cosmologies are plotted in the left and middle panels, respectively.
Clearly, the gradient weighting scheme leads to tighter constraint than the
density weighting scheme, while combining the two schemes together leads to a
constraint tighter than solely using any of them. From left to right, we show
the $\chi^{2}$, likelihood and reduced $\chi^{2}$, respectively.

Figure 7 presents the $W_{\Delta s}(\mu)$ measured in two different
cosmologies. In all plots, we impose a cut $\mu>0.97$ to eliminate the
clustering regimes severely affected by the FOG 888This eliminates the sharp
peak in $W_{\Delta s}(\mu)$ near the $1-\mu=0$ side; see (Li et al., 2015,
2016) for details.. In both the density and gradient weighting schemes, we are
able to distinguish the two cosmologies using the MCFs. In the lower panels
(where we show the different between the MCFs in the two cosmologies),
$\Delta{\bf p}$ of two weighting schemes does not coincide. In what follows,
we will consider simultaneously using them in the analysis and check whether
combining them together is helpful in tightening the cosmological constraints.

The results are illustrated in Figure 8. The left and middle panels show the
distributions of the $\chi^{2}$s and the likelihoods $\mathcal{L}\propto
e^{-\frac{1}{2}\chi^{2}}$, respectively. The plots include the results
obtained using the standard 2pcf ($\alpha=0$), two density weighting scheme
MCFs using $\alpha=0.5,\ 1$, two gradient weighting scheme MCFs using
$\alpha=0.5,\ 1$, and the combination of the above four MCFs, respectively
999We do not use the $\alpha=-0.5$ and -1 results which are found to be
statistically weaker.. Since we are using the covariance estimated using the
180 COLA simulations, all results represent the statistical power of a
$\bar{n}=10^{-3}(h^{-1}{\rm Mpc})^{-3}$ halo/subhalo sample in a
$(512h^{-1}{\rm Mpc})^{3}$ volume.

Amazingly, we find that the gradient weighting scheme is statistically more
powerful than the density weighting scheme. Taking the $\Omega_{m}=0.3271$
cosmology as an example. The standard 2pCF yields $\chi^{2}\approx 1.3$, which
means that we can hardly detect the difference between this cosmology and the
baseline cosmology. If using the two MCFs with ${\rm
weight}\propto\rho^{0.5},\ \rho$, the statistical significance is increased
and we get $\chi^{2}\approx 2.9$. In contrast, in the gradient weighting
scheme, we get $\chi^{2}\approx 6$, which is almost twice as large as the
former results. Finally, combining the two weighting schemes yields
$\chi^{2}\approx 8$, which is better than using either of the schemes alone,
and is almost 5 times larger than the $\chi^{2}$ obtained using the standard
2pCF.

To take into consideration the degree of freedom (dof), the rightmost column
shows the reduced $\chi^{2}$, which is defined as the $\chi^{2}$ divided by
the dof 101010As a not too bad approximation, here we define the dof as the
number of points in the curve minus one (the number of parameters), although
the dof should be smaller due to the non-gaussian correlation among the points
in the curve.. When looking at the values of the reduced $\chi^{2}$, the
results of the density weighting, gradient weighting, or their combination are
not too different from each other. This means that, the enlarged $\chi^{2}$ in
the combined scheme is mainly due to the enlarged size of the data vector.
Also, compared with the results of the baseline cosmology, the reduced
$\chi^{2}$ in the wrong cosmologies are enlarged by 0.5-1.5, meaning that they
are strongly disfavored by the fitting 111111Since the $\Omega_{m}=0.3071$ is
chosen as the baseline here it perfectly “fits” the noise and leads to a zero
reduced $\chi^{2}$. This will not happen when we were analyzing the real
observational data..

In summary, compared with the standard correlation function, the density
weighting or gradient weighting MCFs (with $\alpha=0.5$ and $1$) are roughly 2
or 4 times more powerful in constraining $\Omega_{m}$, while simultaneously
using the two weighting scheme is $\approx 1.25$ times more powerful than
using only the gradient weighting scheme. The gradient weighting scheme is not
only more powerful than the density weighting scheme, but also is
statistically complementary to it.

Finally, we mention three caveats to the above analysis. 1) We are limited by
computational resources, in comparing cosmologies we use five COLA simulations
created by $1024^{3}$ particles in a $(680h^{-1}{\rm Mpc})^{3}$ box.
Considering that we were using clustering at scales of $6-40{h^{-1}}\rm Mpc$,
the volume of the simulations should be large enough for the statistics.
However larger, and more accurate simulations are necessary if we are to
achieve precise modelling of the cosmological dependence of $W_{\Delta
s}(\mu)$. 2) $\hat{W}_{\Delta s}(\mu)$ may also not be the optimal choice of
statistical quantity. This issue is worth further investigation. 3) Our
analysis only considers $\alpha=0.5$ and 1, so the conclusion is only valid if
one were using these values of power.

##  5 Conclusions and Discussions

The MCFs (or the mark weighted power spectrum) can explore the clustering
information not contained in the standard 2-point statistics, while also being
much more computationally efficient than three point or higher order
statistics. It is of ultra importance for us to studying them, since they may
play an important role in the analysis of future surveys.

In this work we consider using the power of the density gradient
$(|\nabla\rho|/\rho)^{\alpha}$ as the weight in the marked two point
statistics. We use the anisotropic $\hat{W}_{\Delta s}(\mu)$ to distinguish
simulations with different values of $\Omega_{m}$. The $\chi^{2}$ and reduced
$\chi^{2}$ shows that both weighting schemes are able to distinguish the
different simulations. The $\chi^{2}$ values indicate that, the gradient
weighting scheme is found to be statistically more powerful than the density
weighted scheme, while combining the two schemes together can further improve
the statistical power.

The work can be considered as an extension of Yang et al. (2020) and Massara
et al. (2021), where the authors found that the density weighting scheme can
significantly increase the amount of statistical information of the standard
two point analysis. We show that the gradient weighted scheme can be
complementary and even more powerful.

Phenomenologically it is not difficult to understand why the MCF is more
powerful than the standard 2pCF. The clustering in different environments is
different. While the information from these regions are mixed together in the
standard 2pCF, they are separated if a weighting is applied, enabling us to
extract more information. However, it is much more difficult to explain 1) why
the gradient weighting scheme is statistically more powerful than the density
weighting scheme and 2) why the two weighting schemes are statistically
complementary to each other. Answering these two questions require accurate
theoretical modeling of the statistical quantities (e.g. the work of Philcox
et al. (2020)), and is beyond the scope of this work.

Although this proof-of-concept work shows that the density weighted
correlation functions are helpful in the LSS analysis, there is still a long
way to go before we apply them to real observational data. Here we list two
caveats. 1) Weighted by the density or the density gradient, the mark weighted
statistics could be more affected by the bias or the velocity bias of the
tracers. 2) If one were to combine many MCFs together, a large amount of mocks
are required to compute the covariance matrix. In this case, one may have to
use methods such like the principle component analysis to reduce the degrees
of freedom.

It is difficult to accurately calculate the MCFs from fundamental theory,
hindering the comparison between theory and observations. One possibility
would be to run a large number of simulations and build up emulators for the
MCFs. An alternative option is to use the MCFs in the tomographic AP method
(Li et al., 2014, 2015, 2016). The $W_{\Delta s}(\mu)$ with $\alpha=0$ is
exactly the statistical quantity used in that method, and so it is very likely
to improve the power of that method via using more marked statistics. We will
explore this issue in future works.

There are still many issues related with the MCFs which are important but not
addressed in this simple work. As a preliminary study, here we only focus on
the constraints of $\Omega_{m}$, though it is worth exploring other physics
such as modified gravity theories, neutrino mass and species, and the dark
energy equation of state. Also, while this work shows that one can improve the
statistical power of the MCFs by using the density gradient, it is known that
this gradient encodes geometric information of the local cosmic web
environment (Novikov et al., 2006). This could motivate further study of other
cosmic web scalars as markers. Some possibilities include the hessian of the
potential field Forero-Romero et al. (2009), the hessian of the density field
(Bond et al., 2010), the tidal anisotropy (Bustamante & Forero-Romero, 2015)
or even graph metrics (Suárez-Pérez et al., 2021),

## Acknowledgements

The CosmoSim database used in this paper is a service by the Leibniz-Institute
for Astrophysics Potsdam (AIP). The MultiDark database was developed in
cooperation with the Spanish MultiDark Consolider Project CSD2009-00064.

This work is supported by National SKA Program of China No. 2020SKA0110401,
and the science research grants from the China Manned Space Project with No.
CMS-CSST-2021-A03, No. CMS-CSST-2021-B01. XDL acknowledges the support from
the NSFC grant (No. 11803094), the Science and Technology Program of
Guangzhou, China (No. 202002030360). ZQH acknowledges the support from the
NSFC grant (No. 12073088). CGS acknowledges support via the Basic Science
Research Program from the National Research Foundation of South Korea (NRF)
funded by the Ministry of Education (2018R1A6A1A06024977 and
2020R1I1A1A01073494).

## Data availability

The BigMD simulation used in this paper is availalbe via the CosmoSim database
(https://www.cosmosim.org/). The COLA simulation data used in this paper is
available upon request.

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