# Weak type 1-1 bound of multi-parameter maximal function

Hoyoung Song  Hoyoung Song, Department of Mathematics
Yonsei University
Seoul 120-729, Korea<EMAIL_ADDRESS>
###### Abstract.

We define the mulati-parameter maximal function $\mathcal{M}$ as

$\mathcal{M}f(x)=\sup_{0<h_{1},h_{2},\cdots,h_{n}<1}\frac{1}{h_{1}h_{2}\cdots
h_{n}}\left|\int_{0}^{h_{1}}\cdots\int_{0}^{h_{n}}f(x-P(t_{1},\cdots,t_{n}))\mathrm{d}t_{1}\cdots\mathrm{d}t_{n}\right|$

where $P(t_{1},t_{2},\cdots,t_{n})$ is a real-valued multi-parameter
polynomial of real variables $t_{1},t_{2},\cdots,t_{n}$. Then, we prove that
$\mathcal{M}$ is of weak-type 1-1 with a bound that depends only on the
coefficients of $P(t_{1},t_{2},\cdots,t_{n})$.

###### Key words and phrases:

multi parameter maximal function, weak type 1-1, maximal function

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Decomposition And The Calderón-Zygmund Singular Integral Theory
  4. 4 Proof Of Lemma 3.1
  5. 5 Proof Of Lemma 3.2

##  1\. Introduction

If $T$ is a mapping from $L^{1}\left(\mathbb{R}^{n}\right)$ to
$L^{1,\infty}\left(\mathbb{R}^{n}\right)$, then we say that $T$ is a weaktype
1-1 operator if

$|\\{x:Tf(x)>\alpha\\}|\leq\frac{C\|f\|_{L^{1}}}{\alpha}$

where $C$ is a constant independent of $f$ and $\alpha.$ In [1], Carbery,
Ricci and Wright obtained the uniform weak-type 1-1 bounds of

$\displaystyle
M_{0}f(x)=\sup_{0<h<1}\frac{1}{h}\int_{0}^{h}|f(x-P(s))|\mathrm{d}s,$
$\displaystyle H_{0}f(x)=p\cdot v.\int_{|s|<1}f(x-P(s))\frac{\mathrm{d}s}{s}$

where $P$ is real-valued polynomial. In the same spirit of the one-parameter
maximal operator, we shall consider the multi-parameter and one-dimensional
operators

$Mf(x)=\sup_{0<h_{1},h_{2}<1}\frac{1}{h_{1}h_{2}}\left|\int_{0}^{h_{1}}\int_{0}^{h_{2}}f(x-P(t_{1},t_{2}))\mathrm{d}t_{1}\mathrm{d}t_{2}\right|,$
$\mathcal{M}f(x)=\sup_{0<h_{1},h_{2},\cdots,h_{n}<1}\frac{1}{h_{1}h_{2}\cdots
h_{n}}\left|\int_{0}^{h_{1}}\cdots\int_{0}^{h_{n}}f(x-P(t_{1},\cdots,t_{n}))\mathrm{d}t_{1}\cdots\mathrm{d}t_{n}\right|.$

The $L^{p}(\mathbb{R}),p>1$ boundedness of $\mathcal{M}$ is obtained by Ricci
and Stein’s Theorem 7.1 of [10] with de Leeuw’s theorem. In [9], Patel showed
that $\mathfrak{M}$ is of weak-type 1-1 with a bound that depends only on the
coefficients of $P$. There are also other multi-parameter sigular integral
studies. In [2], Carbery, Wainger and Wright obtained the necessary and
sufficient condition for $H$ to be bounded on $L^{p}$, $1<p<\infty$ using the
Newton diagram of $P$ where

$Hf(x,y,z)=p\cdot
v\cdot\int_{|s|<1}\int_{|t|<1}f(x-s,y-t,z-P(s,t))\frac{\mathrm{d}s\mathrm{d}t}{st}.$

In [7], the necessary and sufficient condition for the $L^{p}$ boundedness of
global version of $H$ was obtained by Patel. More recently, the multiple
Hilbert transforms were studied in [3], [4] and [6] where

$\mathcal{H}f(x_{1},x_{2},\cdots,x_{n})=\int
f(x_{1}-P_{1}(t_{1},\cdots,t_{m}),\cdots,x_{n}-P_{n}(t_{1},\cdots,t_{m}))\frac{\mathrm{d}t_{1}\cdots\mathrm{d}t_{m}}{t_{1}\cdots
t_{m}}.$

Unlike the $L^{p}$ boundedness, we do not know much about the weak-type 1-1
estimate. We define $M_{1}f(x,z)$ and $H_{1}f(x,z)$ as

$\displaystyle
M_{1}f(x,z)=\sup_{0<h<1}\frac{1}{h}\int_{0}^{h}|f(x-s,z-P(s))|\mathrm{d}s,$
$\displaystyle
H_{1}f(x,z)=p.v\cdot\int_{|s|<1}f(x-s,z-P(s))\frac{\mathrm{d}s}{s}.$

Although we have $L^{p}$ boundedness of $M_{1}$ and $H_{1}$ (independent of
the coefficients of $P$) for $1<p<\infty$, weak-type 1-1 bounds are not known
for $M_{1}$ and $H_{1}$. We know the necessary and sufficient condition for
$L^{p}$ boundedness of $H_{2}$[8] where

$H_{2}f(x)=p\cdot
v\cdot\int_{|s|<1}\int_{|t|<1}f(x-P(s,t))\frac{\mathrm{d}s\mathrm{d}t}{st}.$

However, there is no result about the exact necessary and sufficient condition
for weak-type 1-1 boundedness of $H_{2}$. In this paper, we obtain the weak-
type 1-1 bound of $\mathcal{M}$ with a bound depending only on the
coefficients of $P$.

######  Theorem 1.1.

$\mathcal{M}$ is of weak-type 1-1 with a bound that is dependent only on the
coefficients of $P$.

###  1.1. Organization

In Section 2, we introduce the Newton diagram of $P$ and its properties and
treat the monomial case. In Section 3, we decompose $\mathcal{M}f(x)$ with
vertices of the Newton diagram of $P$. Then, we treat the non-zero coordinate
case and zero coordinate case of the vertex separately. For the two cases, we
introduce two important Lemmas and show the weak-type 1-1 boundedness of
$\mathcal{M}f(x)$ by the Calderón-Zygmund singular integral theory combined
with the two Lemmas. In Section 4, we prove one of the two Lemmas for the non-
zero coordinate case by the properties of Newton diagram and some techniques
of Harmonic analysis. In Section 5, we also show the other Lemma for the zero
coordinate case. For this, we apply the sublevel set estimate and some
estimates of Harmonic analysis with the properties of Newton diagram.

##  2\. Preliminaries

Let $\mathfrak{t}=(t_{1},t_{2},\cdots,t_{n})$ and
$\mathfrak{m}=(m_{1},m_{2},\cdots,m_{n})$. Then, we can rewrite the polynomial
$P(t_{1},\cdots,t_{n})$ as
$P(\mathfrak{t})=\sum_{\mathfrak{m}\in\Lambda}a_{\mathfrak{m}}\mathfrak{t}^{\mathfrak{m}}$,
where $\Lambda$ is indexing the set of lattice points in $\mathbb{Z}^{n}$ such
that $a_{\mathfrak{m}}\neq 0$. Let $\mathbb{N}_{\star}$ denote the set of non-
negative integers. For the set
$A=\\{a_{1},a_{2}\cdots,a_{n}\\}\subset\\{1,2\cdots,n\\}$, we set $\int
f(t)dt_{i\in A}=\int f(t)dt_{a_{1}}dt_{a_{2}}\cdots dt_{a_{n}}$. $A\sim B$
will be used to indicate that $A$ and $B$ are essentially similar in size,
unimportant error terms are neglected. For the vectors $\bar{v}$ and
$\bar{w}$, let $\bar{v}\geq\bar{w}$ denote all the components of $\bar{v}$ are
greater than or equal to all the components of $\bar{w}$.

###  2.1. Newton Diagram

We shall introduce the Newton Diagram. For the details, see [2], [6] and [9].
For each $\mathfrak{m}\in\Lambda$, set

$Q_{\mathfrak{m}}=\left\\{(x_{1},x_{2},\cdots,x_{n})\in\mathbb{R}^{n}:x_{1}\geq
m_{1},x_{2}\geq m_{2},\cdots,x_{n}\geq m_{n}\right\\}.$

Then the smallest closed convex set containing
$Q=\bigcup_{(m,n)\in\Lambda}Q_{m,n}$ is called the Newton diagram of $P$. We
denote it by $\Pi$ and it is an unbounded polygon with a finite number of
corners. Let $\mathcal{D}$ denote the set of its corner points. Furthermore,
suppose $\mathcal{D}$ consists of $r$ corner points $v_{1},v_{2},\ldots,v_{r}$
with $v_{j}=(m_{1}^{j},\cdots,m_{n}^{j})$ for $1\leq j\leq r$. From now, let
non-negative vector denote the vector whoose all components are not negative
number. By the elementary geometry, one knows that for a corner point $v_{j}$
of $\Pi$, there are exactly $n$’s faces of $\Pi$ intersecting $v_{j}$. Hence,
we can choose $n$’s non-negative vectors
$\bar{n}_{j}^{(1)},\bar{n}_{j}^{(2)},\cdots,\bar{n}_{j}^{(n)}$ that are normal
vectors to the faces intersecting on the corner point $v_{j}$. By the
convexity of the Newton diagram, one has

(2.1) $\displaystyle\bar{n}_{j}^{(1)}\cdot\left(v-v_{j}\right)\geq
0,\bar{n}_{j}^{(2)}\cdot\left(v-v_{j}\right)\geq
0\cdots,\bar{n}_{j}^{(n)}\cdot\left(v-v_{j}\right)\geq 0$

for all $v\in\Lambda$ and $1\leq j\leq r$. Now for $1\leq j\leq r$, define

$\displaystyle T(j)$
$\displaystyle=\left\\{(q_{1},\cdots,q_{n})\in\mathbb{N}^{n}:(q_{1},\cdots,q_{n})\cdot\left(v-v_{j}\right)>0\text{
for all }v\in\Lambda\backslash\left\\{v_{j}\right\\}\right\\}$
$\displaystyle=\bigcap_{v\in\Lambda\backslash\left\\{v_{j}\right\\}}\left\\{(q_{1},\cdots,q_{n})\in\mathbb{N}^{n}:(q_{1},\cdots,q_{n})\cdot\left(v-v_{j}\right)>0\right\\}.$

Then, we have the following Lemma. We omit its proof. For the details, see
[2], [6] and [9].

######  Lemma 2.1.

For $1\leq j\leq r$,
(i)

$\displaystyle
T(j)=\left\\{(q_{1},\cdots,q_{n})\in\mathbb{N}^{n}:(q_{1},\cdots,q_{n})=\alpha_{1}\bar{n}_{j}^{(1)}+\cdots+\alpha_{n}\bar{n}_{j}^{(n)}\text{for
some positive reals}\ \alpha_{1}\cdots,\alpha_{2}\right\\}.$

(ii) For $1\leq j<k\leq r$,

$T(j)\cap T(k)=\varnothing.$

Now, we define $S(j)$ as

$S(j):=\left\\{(q_{1},\cdots,q_{n})\in\mathbb{N}_{\star}^{n}:\exists_{\alpha_{1}\geq
0,\cdots,\alpha_{n}\geq
0}(q_{1},\cdots,q_{n})=\alpha_{1}\bar{n}_{j}^{(1)}+\cdots+\alpha_{n}\bar{n}_{j}^{(n)}\right\\}.$

Then, by the definition of $S(j)$, one can obtain that

$\bigcup_{j=1}^{r}S(j)=\mathbb{N}_{\star}^{n}.$

Moreover, with the elemetary calculation, one can know that there exists
$d_{j}>0$ such that

$S(j):=\left\\{(q_{1},\cdots,q_{n})\in\mathbb{N}_{\star}^{n}:\exists_{(n_{1},\cdots,n_{n})\in\mathbb{N}_{\star}^{n}}(q_{1},\cdots,q_{n})=\frac{n_{1}}{d_{j}}\bar{n}_{j}^{(1)}+\cdots+\frac{n_{n}}{d_{j}}\bar{n}_{j}^{(n)}\right\\}.$

Let $\mathfrak{q}:=(q_{1},\cdots,q_{n})$. For $1<m<n$, we also set
$\bar{k}_{m}:=(k_{1},\cdots k_{m-1},k_{m+1},\cdots,k_{n})$. Especially, let
$\bar{k}_{1}:=(k_{2},\cdots,k_{n})$ and $\bar{k}_{n}:=(k_{1},\cdots,k_{n-1})$.
We define $S_{1}(j),\cdots,S_{n}(j)$ as for $1\leq m\leq n$,

$\displaystyle S_{m}(j):=\\{\mathfrak{q}\in
S(j):\mathfrak{q}=\frac{N+k_{1}}{d_{j}}\bar{n}_{j}^{(1)}+\cdots+\frac{N+k_{m-1}}{d_{j}}$
$\displaystyle\bar{n}_{j}^{(m-1)}+\frac{N}{d_{j}}\bar{n}_{j}^{(m)}+\frac{N+k_{m+1}}{d_{j}}\bar{n}_{j}^{(m+1)}+$
$\displaystyle\cdots+\frac{N+k_{n}}{d_{j}}\bar{n}_{j}^{(n)},\bar{k}_{m}\in\mathbb{N}_{\star}^{n-1},N\in\mathbb{N}_{\star}\\}.$

Note that

$S(j)=\bigcup_{m=1}^{n}S_{m}(j).$

Now, we decompose

$S_{m}(j)=\bigcup_{N\in\mathbb{N}_{\star}}S_{m}^{N}(j)$

where

$\displaystyle S_{m}^{N}(j):=\\{\mathfrak{q}\in
S(j):\mathfrak{q}=\frac{N+k_{1}}{d_{j}}\bar{n}_{j}^{(1)}+\cdots+\frac{N+k_{m-1}}{d_{j}}\bar{n}_{j}^{(m-1)}$
$\displaystyle+\frac{N}{d_{j}}\bar{n}_{j}^{(m)}+\frac{N+k_{m+1}}{d_{j}}\bar{n}_{j}^{(m+1)}+$
$\displaystyle\cdots+\frac{N+k_{n}}{d_{j}}\bar{n}_{j}^{(n)},\bar{k}_{m}\in\mathbb{N}_{\star}^{n-1}\\}.$

######  Lemma 2.2.

For each $1\leq j\leq r$, there exists $\beta_{j}>0$ such that

(2.2) $\displaystyle\mathfrak{q}\cdot\left(v-v_{j}\right)\geq\beta_{j}N$

for every $v\in\Lambda\backslash\left\\{v_{j}\right\\}$ and $\mathfrak{q}\in
S_{m}^{N}(j)$.

###### Proof.

For every $\mathfrak{q}\in S_{m}^{N}(j)$ we can write

$\mathfrak{q}=\frac{N+k_{1}}{d_{j}}\bar{n}_{j}^{(1)}+\cdots+\frac{N+k_{m-1}}{d_{j}}\bar{n}_{j}^{(m-1)}+\frac{N}{d_{j}}\bar{n}_{j}^{(m)}+\frac{N+k_{m+1}}{d_{j}}\bar{n}_{j}^{(m+1)}+\cdots+\frac{N+k_{n}}{d_{j}}\bar{n}_{j}^{(n)}$

for some $\bar{k}_{m}\in\mathbb{N}_{\star}^{n-1}$. By the fact that
$\bar{n}_{j}^{(1)},\cdots,\bar{n}_{j}^{(n)}$ are linearly independent with
$(\ref{1a})$, we have

$\left(v-v_{j}\right)\cdot(\bar{n}_{j}^{(1)}+\cdots+\bar{n}_{j}^{(n)})>0$

for all $v\in\Lambda\backslash\left\\{v_{j}\right\\}$. Taking

(2.3)
$\displaystyle\beta_{j}:=\min_{v\in\Lambda\backslash\left\\{v_{j}\right\\}}\frac{1}{d_{j}}\left(v-v_{j}\right)\cdot(\bar{n}_{j}^{(1)}+\cdots+\bar{n}_{j}^{(n)})>0$

immediately yields (2.2). ∎

###  2.2. The Monomial Case

Note that

$\mathcal{M}f(x)\sim\sup_{\mathfrak{q}\in\mathbb{N}_{\star}^{n}}2^{q_{1}+\cdots
q_{n}}\left|\int_{2^{-q_{1}}}^{2^{-q_{1}+1}}\cdots\int_{2^{-q_{n}}}^{2^{-q_{n}+1}}f(x-P(\mathfrak{t}))\mathrm{d}t_{1}\cdots\mathrm{d}t_{n}\right|.$

######  Lemma 2.3.

If $P(\mathfrak{t})$ is a monomial, then

$\mathcal{M}f(x)\leq 2M_{H}f(x)$

where $M_{H}$ denotes the usual Hardy-Littlewood maximal function.

###### Proof.

Let $P(\mathfrak{t})=a_{\mathfrak{m}}\mathfrak{t}^{\mathfrak{m}}$ and suppose
that the coefficients of $P$ are the positive real number. Then, after a
change of variable
$\left(a_{\mathfrak{m}}\mathfrak{t}^{\mathfrak{m}}=u\right)$, we have

$\displaystyle\mathcal{M}f(x)\sim\sup_{\mathfrak{q}\in\mathbb{Z}_{+}^{n}}\frac{1}{m_{n}\left(a_{\mathfrak{m}}\right)^{1/m_{n}}}2^{q_{1}+\cdots+q_{n-1}}$
$\displaystyle\times\int_{t_{1}=2^{-q_{1}}}^{2^{-q_{1}+1}}\cdots\int_{t_{n-1}=2^{-q_{n-1}}}^{2^{-q_{n-1}+1}}\frac{2^{q_{n}}}{t_{1}^{m_{1}/m_{n}}\cdots
t_{n-1}^{m_{n-1}/m_{n}}}\int_{a_{\mathfrak{m}}t_{1}^{m_{1}}\cdots
t_{n-1}^{m_{n-1}}2^{-m_{n}q_{n}}}^{a_{\mathfrak{m}}t_{1}^{m_{1}}\cdots
t_{n-1}^{m_{n-1}}2^{-m_{n}q_{n}+m_{n}}}f(x-u)\frac{\mathrm{d}u}{u^{1-1/m_{n}}}dt_{1}\cdots
dt_{n-1}.$

So, it suffices to show that

(2.4) $\displaystyle\int_{0}^{a_{\mathfrak{m}}t_{1}^{m_{1}}\cdots
t_{n-1}^{m_{n-1}}2^{-m_{n}q_{n}+m_{n}}}f(x-u)\frac{\mathrm{d}u}{u^{1-1/m_{n}}}\leq
2m_{n}\left(a_{\mathfrak{m}}\right)^{1/m_{n}}2^{-q_{n}}t_{1}^{m_{1}/m_{n}}\cdots
t_{n-1}^{m_{n-1}/m_{n}}M_{H}f(x).$

This is trivial when $n=1$. When $n>1$, note that

$\int_{0}^{a_{\mathfrak{m}}t_{1}^{m_{1}}\cdots
t_{n-1}^{m_{n-1}}2^{-m_{n}q_{n}+q_{n}}}f(x-u)\frac{\mathrm{d}u}{u^{1-1/n}}=\int_{u=0}^{a_{\mathfrak{m}}t_{1}^{m_{1}}\cdots
t_{n-1}^{m_{n-1}}2^{-m_{n}q_{n}+q_{n}}}f(x-u)\left(\int_{z=0}^{1/u^{1-1/n}}\mathrm{~{}d}z\right)\mathrm{d}u.$

By changing the order of integral, one can have (2.4). ∎

##  3\. Decomposition And The Calderón-Zygmund Singular Integral Theory

Our proof is based on the arguments of [1] and [9]. We define
$C^{\infty}$-function $\eta(s)$ supported in $\left[\frac{1}{2},4\right]$ as
$\eta(s)=1$ if $s\in[1,2]$. Then

$\displaystyle\mathcal{M}f(x)$
$\displaystyle\lesssim\sup_{\mathfrak{q}\in\mathbb{N}_{\star}^{n}}2^{q_{1}+\cdots
q_{n}}\left|\int
f(x-P(\mathfrak{t}))\eta(2^{q_{1}}t_{1})\cdots\eta(2^{q_{n}}t_{n})\mathrm{d}t_{1}\cdots\mathrm{d}t_{n}\right|$
$\displaystyle=\sup_{\mathfrak{q}\in\mathbb{N}_{\star}^{n}}\left|\int
f(x-P(2^{-q_{1}}t_{1},\cdots,2^{-q_{n}}t_{n}))\eta(t_{1})\cdots\eta(t_{n})\mathrm{d}t_{1}\cdots\mathrm{d}t_{n}\right|$
$\displaystyle\leq\sum_{j=1}^{r}\sup_{\mathfrak{q}\in S(j)}\left|\int
f(x-P(2^{-q_{1}}t_{1},\cdots,2^{-q_{n}}t_{n}))\eta(t_{1})\cdots\eta(t_{n})\mathrm{d}t_{1}\cdots\mathrm{d}t_{n}\right|$
$\displaystyle:=\sum_{j=1}^{r}\mathcal{M}(j)f(x).$

Hence, the following Theorem 3.1 implies Theorem 1.1

######  Theorem 3.1.

$\mathcal{M}(j)$ is of weak-type 1-1 for $1\leq j\leq r$.

From now, we shall show Theorem 3.1. For this, we consider two cases (non-zero
coordinate case or zero coordinate case).

###  3.1. Non-Zero Coordinate Case

First, we assume that $v_{j}=(m_{1}^{j},\cdots,m_{n}^{j})$ does not have a
vanishing coordinate. This means that $m_{i}^{j}\neq 0$ for all $1\leq i\leq
n$. We can decompose $\mathcal{M}(j)f(x)$ as

$\displaystyle\mathcal{M}(j)f(x)=\sup_{\mathfrak{q}\in S(j)}\left|\int
f(x-P(2^{-q_{1}}t_{1},\cdots,2^{-q_{n}}t_{n}))\eta(t_{1})\cdots\eta(t_{n})\mathrm{d}t_{1}\cdots\mathrm{d}t_{n}\right|$
$\displaystyle\leq\sup_{\mathfrak{q}\in
S(j)}\left|\int\left[f(x-P(2^{-q_{1}}t_{1},\cdots,2^{-q_{n}}t_{n}))-f\left(x-2^{-\mathfrak{q}\cdot
v_{j}}a_{v_{j}}\mathfrak{t}^{v_{j}}\right)\right]\eta(t_{1})\cdots\eta(t_{n})d\mathfrak{t}\right|$
$\displaystyle+\sup_{\mathfrak{q}\in S(j)}\left|\int
f\left(x-2^{-\mathfrak{q}\cdot
v_{j}}a_{v_{j}}\mathfrak{t}^{v_{j}}\right)\eta(t_{1})\cdots\eta(t_{n})d\mathfrak{t}\right|$
$\displaystyle:=M(j)f(x)+Q(j)f(x).$

One can apply Lemma 2.3 to $Q(j)f(x)$. So, it suffices to consider $M(j)f(x)$.
We split $M(j)f(x)$ again as

$\displaystyle M(j)f(x)$ $\displaystyle\leq\sum_{m=1}^{n}\sup_{\mathfrak{q}\in
S_{m}(j)}\left|\int\left[f(x-P(2^{-q_{1}}t_{1},\cdots,2^{-q_{n}}t_{n}))-f\left(x-2^{-\mathfrak{q}\cdot
v_{j}}a_{v_{j}}\mathfrak{t}^{v_{j}}\right)\right]\eta(t_{1})\cdots\eta(t_{n})d\mathfrak{t}\right|$
$\displaystyle:=\sum_{m=1}^{n}M_{m}(j)f(x).$

We can write $P(2^{-q_{1}}t_{1},\cdots,2^{-q_{n}}t_{n})=2^{-\mathfrak{q}\cdot
v_{j}}\tilde{P}_{j}(t_{1},\cdots,t_{n})$ where
$\tilde{P}_{j}(\mathfrak{t}):=\sum_{v=\mathfrak{m}\in\Lambda}2^{-\mathfrak{q}\cdot\left(v-v_{j}\right)}a_{\mathfrak{m}}\mathfrak{t}^{\mathfrak{m}}.$
Then, we have

$\displaystyle M_{m}(j)f(x)=$ $\displaystyle\sup_{\mathfrak{q}\in
S_{m}(j)}\mid\int\left[f(x-2^{-\mathfrak{q}\cdot
v_{j}}\tilde{P}_{j}(\mathfrak{t}))-f(x-2^{-\mathfrak{q}\cdot
v_{j}}a_{v_{j}}\mathfrak{t}^{v_{j}})\right]\eta(t_{1})\cdots\eta(t_{n})d\mathfrak{t}t\mid$
$\displaystyle\leq$ $\displaystyle\sum_{N\geq 0}\sup_{\mathfrak{q}\in
S_{m}^{N}(j)}\left|\int\left[f(x-2^{-\mathfrak{q}\cdot
v_{j}}\tilde{P}_{j}(\mathfrak{t}))-f(x-2^{-\mathfrak{q}\cdot
v_{j}}a_{v_{j}}\mathfrak{t}^{v_{j}})\right]\eta(t_{1})\cdots\eta(t_{n})d\mathfrak{t}\right|$
$\displaystyle=$ $\displaystyle\sum_{N\geq
0}\sup_{\bar{k}_{m}\in\mathbb{N}_{\star}^{n-1}}\left|\int\left[f(x-c_{N}2^{-\sigma_{m}\cdot\bar{k}_{m}}\tilde{P}_{j}(\mathfrak{t}))-f(x-c_{N}2^{-\sigma_{m}\cdot\bar{k}_{m}}a_{v_{j}}\mathfrak{t}^{v_{j}})\right]\eta(t_{1})\cdots\eta(t_{n})d\mathfrak{t}\right|$
$\displaystyle:=$ $\displaystyle\sum_{N\geq 0}M_{j}^{N}f(x)$

where $c_{N}:=2^{-(N/d_{j})[(\bar{n}_{j}^{(1)}+\cdots+\bar{n}_{j}^{(n)})\cdot
v_{j}]}$ and

$\sigma_{m}:=(\frac{1}{d}\bar{n}_{j}^{(1)}\cdot
v_{j},\cdots,\frac{1}{d}\bar{n}_{j}^{(m-1)}\cdot
v_{j},\frac{1}{d}\bar{n}_{j}^{(m+1)}\cdot
v_{j},\cdots,\frac{1}{d}\bar{n}_{j}^{(n)}\cdot v_{j}).$

For $f\in S$, we let
$\mu_{\bar{k}_{m}}^{N,(\bar{k}_{m})},\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}$, and
$\nu^{N,(\bar{k}_{m})}$ denote the measures satisfying

$\displaystyle\left\langle f,\mu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}\right\rangle$
$\displaystyle=\int\left[f(c_{N}2^{-\sigma_{m}\cdot\bar{k}_{m}}\tilde{P}_{j}(\mathfrak{t}))-f(c_{N}2^{-\sigma_{m}\cdot\bar{k}_{m}}a_{v_{j}}\mathfrak{t}^{v_{j}})\right]\eta(t_{1})\cdots\eta(t_{n})d\mathfrak{t}$
$\displaystyle\left\langle f,\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}\right\rangle$
$\displaystyle=\int\left[f(2^{-\sigma_{m}\cdot\bar{k}_{m}}\tilde{P}_{j}(\mathfrak{t}))-f(2^{-\sigma_{m}\cdot\bar{k}_{m}}a_{v_{j}}\mathfrak{t}^{v_{j}})\right]\eta(t_{1})\cdots\eta(t_{n})d\mathfrak{t}$
$\displaystyle\left\langle f,\nu^{N,(\bar{k}_{m})}\right\rangle$
$\displaystyle=\int\left[f(\tilde{P}_{j}(\mathfrak{t}))-f(a_{v_{j}}\mathfrak{t}^{v_{j}})\right]\eta(t_{1})\cdots\eta(t_{n})d\mathfrak{t}$

for those $\bar{k}_{m}$ ’s for which

$\mathfrak{q}=\frac{N+k_{1}}{d_{j}}\bar{n}_{j}^{(1)}+\cdots+\frac{N+k_{m-1}}{d_{j}}\bar{n}_{j}^{(m-1)}+\frac{N}{d_{j}}\bar{n}_{j}^{(m)}+\frac{N+k_{m+1}}{d_{j}}\bar{n}_{j}^{(m+1)}+\cdots+\frac{N+k_{n}}{d_{j}}\bar{n}_{j}^{(n)}.$

For all the other $\bar{k}_{m}$ ’s, we define
$\mu_{\bar{k}_{m}}^{N,(\bar{k}_{m})},\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}$, and
$\nu^{N,(\bar{k}_{m})}$ to be zero distributions. Then, we define
$M_{j}^{N}f(x)$ as

$M_{j}^{N}f(x):=\sup_{\bar{k}_{m}\in\mathbb{N}_{\star}^{n-1}}\left|\mu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}*f(x)\right|.$

Also, we define

$L_{j}^{N}f(x):=\sup_{\bar{k}_{m}\in\mathbb{N}_{\star}^{n-1}}\left|\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}*f(x)\right|.$

Note that $(M_{j}^{N}f)(x)=(L_{j}^{N}f_{N})\left(x/c_{N}\right)$ where
$f_{N}(x)=f\left(c_{N}x\right)$. So, we shall consider the weak-type 1-1
estimate for $L_{j}^{N}$.
Now, we introduce the following Lemma whoose proof is postponed in the section
4.

######  Lemma 3.1.

There are are positive real constants $A,B,C,\delta_{1}$ and $\delta_{2}$
independent of $\bar{k}_{m}$ and $N$ such that

(3.1)
$\displaystyle\int\left|\nu^{N,(\bar{k}_{m})}(x-y)-\nu^{N,(\bar{k}_{m})}(x)\right|\mathrm{d}x\leq
A2^{-\delta_{1}N}|y|^{\delta_{2}}\text{ for all }y\in\mathbb{R},$ (3.2)
$\displaystyle\sum_{\bar{k}_{m}\in\mathbb{N}_{\star}^{n-1}}\int_{|x|\geq
2|y|}\left|\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}(x-y)-\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}(x)\right|\mathrm{d}x\leq
C2^{-\delta_{1}N},$ (3.3)
$\displaystyle\left\|\sup_{\bar{k}_{m}\in\mathbb{N}^{n-1}}\left|f*\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}\right|\right\|_{2}\leq
B2^{-\delta_{3}N}\|f\|_{2}.$

We shall apply Lemma 3.1 to show the following Proposition utilizing the
Calderón-Zygmund singular integral theory. The following Proposition implies
that $M_{m}(j)f(x)$ has the weak type 1-1 boundedness.

######  Proposition 3.1.

(3.4)
$\displaystyle\left|\left\\{x:L_{j}^{N}f(x)>\lambda\right\\}\right|\leq\frac{C}{\lambda}2^{-\delta
N}\|f\|_{1}$

where $C$ and $\delta$ are positive real constants independent of $N$ and
$\lambda$.

###### Proof.

First, we shall introduce the the Calderón-Zygmund decomposition briefly. We
choose $\varepsilon>0$ (to be fixed later) and decompose $\mathbb{R}$ as
$\mathbb{R}=F\cup\Omega$ so that:
(1) $F$ is closed and $F\cap\Omega=\varnothing$;
(2) $f(x)\leq 2^{\varepsilon N}\lambda$ a.e. on $F$;
(3) $\Omega$ is the union of intervals, $\Omega=\bigcup_{i=1}^{\infty}Q_{i}$,
whose interiors are mutually disjoint. Moreover,

(3.5) $\displaystyle|\Omega|\leq\frac{C}{2^{\varepsilon
N}\lambda}\int_{\mathbb{R}}f(x)\mathrm{d}x,$ (3.6)
$\displaystyle\frac{1}{\left|Q_{i}\right|}\int_{Q_{i}}f(x)\mathrm{d}x\leq
C2^{\varepsilon N}\lambda.$

The constant $C$ in (3.5) and (3.6) depends only on the dimension of the
space. We define a function $g$ almost everywhere by

(3.7) $\displaystyle g(x)=\begin{cases}f(x)&\text{ for }x\in F\\\
\frac{1}{\left|Q_{i}\right|}\int_{Q_{i}}f(y)\mathrm{d}y,&\text{ for }x\in
Q_{i}^{\circ}\end{cases}.$

If $f(x)=g(x)+b(x)$, then $b(x)=0$ for $x\in F$ and

(3.8) $\displaystyle\int_{Q_{i}}b(x)\mathrm{d}x=0,\text{ for each }Q_{i}.$

Also, one knows that

$|\\{x:L_{j}^{N}f(x)>\lambda\\}|\leq|\\{x:L_{j}^{N}g(x)>\frac{\lambda}{2}\\}|+|\\{x:L_{j}^{N}b(x)>\frac{\lambda}{2}\\}|.$

By (3.5), (3.6) and (3.7), we have

$\displaystyle\|g\|_{2}^{2}$
$\displaystyle=\int_{F}|g(x)|^{2}\mathrm{~{}d}x+\int_{\Omega}|g(x)|^{2}\mathrm{~{}d}x\leq
C2^{\varepsilon N}\lambda\|f\|_{1}.$

Therefore, by Lemma 3.3, we obtain that

$\displaystyle\left|\left\\{x:L_{j}^{N}g(x)>\frac{\lambda}{2}\right\\}\right|$
$\displaystyle\leq\frac{4}{\lambda^{2}}\int\left|L_{j}^{N}g\right|^{2}\mathrm{~{}d}x=\frac{4}{\lambda^{2}}\int\left(\sup_{\bar{k}_{m}\in\mathbb{N}_{\star}^{n-1}}\left|\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}*g(x)\right|\right)^{2}\mathrm{~{}d}x$
$\displaystyle\leq\frac{4}{\lambda^{2}}B^{2}2^{-2\delta_{3}N}\|g\|_{2}^{2}\leq\frac{C}{\lambda}2^{-\left(2\delta_{3}-\varepsilon\right)N}\|f\|_{1}.$

Set

$b_{i}(x)=\begin{cases}b(x)&\text{ if }x\in Q_{i}\\\ 0&\text{ if }x\notin
Q_{i}.\end{cases}$

Then, one has

$\begin{gathered}b(x)=\sum_{i}b_{i}(x)\ \text{and}\
L_{j}^{N}b(x)\leq\sum_{i}L_{j}^{N}b_{i}(x).\end{gathered}$

Let $Q_{i}^{*}$ denote the interval which has the same centre $y^{i}$ as
$Q_{i}$, but which is expanded 2 times. Then $Q_{i}\subseteq Q_{i}^{*}$ and
$\Omega\subseteq\Omega^{*}$ where $\Omega^{*}=\cup Q_{i}^{*}$. Moreover, we
have:
(a) $\left|\Omega^{*}\right|\leq 2|\Omega|$ and $F^{*}\subseteq F$ where
$F^{*}=\left(\Omega^{*}\right)^{c}$,
(b) If $x\notin Q_{i}^{*}$, then $\left|x-y^{i}\right|\geq
2\left|y-y^{i}\right|$ for all $y\in Q_{i}$.
Now, by (3.8) we have

$\displaystyle L_{j}^{N}b_{i}(x)$
$\displaystyle=\sup_{\bar{k}_{m}\in\mathbb{N}_{\star}^{n-1}}\left|\int\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}(x-y)b_{i}(y)\mathrm{d}y\right|$
$\displaystyle=\sup_{\bar{k}_{m}\in\mathbb{N}_{\star}^{n-1}}\left|\int_{Q_{i}}\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}(x-y)b(y)\mathrm{d}y\right|$
$\displaystyle=\sup_{\bar{k}_{m}\in\mathbb{N}_{\star}^{n-1}}\left|\int_{Q_{i}}\left[\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}(x-y)-\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}\left(x-y^{i}\right)\right]b(y)\mathrm{d}y\right|.$

So, we know that

$\displaystyle\int_{F^{*}}$ $\displaystyle L_{j}^{N}b(x)\mathrm{d}x$
$\displaystyle\leq\sum_{i}\int_{x\notin
Q_{i}^{*}}L_{j}^{N}b_{i}(x)\mathrm{d}x$
$\displaystyle\leq\sum_{i}\int_{x\notin Q_{i}^{*}}\int_{y\in
Q_{i}}\sup_{\bar{k}_{m}\in\mathbb{N}_{\star}^{n-1}}\left|\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}(x-y)-\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}\left(x-y^{i}\right)\right||b(y)|\mathrm{d}y\mathrm{~{}d}x$
$\displaystyle\leq\sum_{i}\int_{y\in Q_{i}}\left\\{\int_{x\notin
Q_{i}^{*}}\sum_{\bar{k}_{m}\in\mathbb{N}_{\star}^{n-1}}\left|\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}(x-y)-\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}\left(x-y^{i}\right)\right|\mathrm{d}x\right\\}|b(y)|\mathrm{d}y$
$\displaystyle\leq\sum_{i}\int_{y\in
Q_{i}}\left\\{\int_{\left|x^{\prime}\right|\geq
2\left|y^{\prime}\right|}\sum_{\bar{k}_{m}\in\mathbb{N}_{\star}^{n-1}}\left|\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}\left(x^{\prime}-y^{\prime}\right)-\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}\left(x^{\prime}\right)\right|\mathrm{d}x^{\prime}\right\\}|b(y)|\mathrm{d}y$

using (b) for $y\in Q_{i}$ if $x^{\prime}=x-y^{i}$ and $y^{\prime}=y-y^{i}$.
Then, by (3.2) we obtain that

$\displaystyle\int_{F^{*}}L_{j}^{N}b(x)\mathrm{d}x$ $\displaystyle\leq
C2^{-\delta_{1}N}\sum_{i}\int_{y\in Q_{i}}|b(y)|\mathrm{d}y$
$\displaystyle\leq C2^{-\delta_{1}N}\left\\{\sum_{i}\int_{y\in
Q_{i}}|f(y)|\mathrm{d}y+\sum_{i}\int_{y\in Q_{i}}|g(y)|\mathrm{d}y\right\\}$
$\displaystyle\leq
C2^{-\delta_{1}N}\left\\{\int_{\Omega}|f(y)|\mathrm{d}y+\sum_{i}\int_{y\in
Q_{i}}\left\\{\frac{1}{\left|Q_{i}\right|}\int_{Q_{i}}|f(z)|\mathrm{d}z\right\\}\mathrm{d}y\right\\}$
$\displaystyle\leq
C2^{-\delta_{1}N}\left\\{\|f\|_{1}+\int_{\Omega}|f(z)|\mathrm{d}z\right\\}$
$\displaystyle\leq C2^{-\delta_{1}N}\|f\|_{1}.$

Thus, we have

$\displaystyle\left|\left\\{x:L_{j}^{N}b(x)>\frac{\lambda}{2}\right\\}\right|$
$\displaystyle\leq\left|\left\\{x\in
F^{*}:L_{j}^{N}b(x)>\frac{\lambda}{2}\right\\}\right|+\left|\left\\{x\in\Omega^{*}:L_{j}^{N}b(x)>\frac{\lambda}{2}\right\\}\right|$
$\displaystyle\leq\frac{2}{\lambda}\int_{F^{*}}L_{j}^{N}b(x)\mathrm{d}x+\left|\Omega^{*}\right|$
$\displaystyle\leq
C\left(\frac{1}{\lambda}2^{-\delta_{1}N}\|f\|_{1}+\frac{2^{-\varepsilon
N}}{\lambda}\|f\|_{1}\right)$
$\displaystyle\leq\frac{C}{\lambda}2^{-\delta_{4}N}\|f\|_{1}$

∎

###  3.2. Zero Coordinate Case

Now, we shall treat the case where $v_{j}=(m_{1}^{j},\cdots,m_{n}^{j})$ has a
vanishing coordinate. Assume that

$\displaystyle\begin{cases}&m_{i}^{j}=0\text{ for }i\in A,\\\ &m_{i}^{j}\neq
0\text{ for }i\in B\end{cases}$

where $v_{j}=(m_{1}^{j},\cdots,m_{n}^{j})$. Also, suppose that

(3.9) $\displaystyle A=\\{a_{1},a_{2},\cdots,a_{\alpha}\\}\ \text{and}\
B=\\{b_{1},b_{2},\cdots,b_{\beta}\\}\quad(A\cup B=\\{1,2\cdots,n\\}).$

Now, we set

$\Lambda_{0}:=\\{v=(m_{1},\cdots,m_{n}):m_{i}=0\quad\text{if}\quad i\in A\\}.$

Then, we split $\mathcal{M}(j)f(x)$ as

$\displaystyle\mathcal{M}(j)f(x)=\sup_{\mathfrak{q}\in S(j)}\left|\int
f(x-P(2^{-q_{1}}t_{1},\cdots,2^{-q_{n}}t_{n}))\eta(t_{1})\cdots\eta(t_{n})\mathrm{d}t_{1}\cdots\mathrm{d}t_{n}\right|$
$\displaystyle\leq\sup_{\mathfrak{q}\in
S(j)}\left|\int[f(x-P(2^{-q_{1}}t_{1},\cdots,2^{-q_{n}}t_{n}))-f(x-2^{-\mathfrak{q}\cdot
v_{j}}\sum_{v\in\Lambda_{0}}2^{-\mathfrak{q}\cdot(v-v_{j})}a_{v}\mathfrak{t}^{v})]\eta(t_{1})\cdots\eta(t_{n})d\mathfrak{t}\right|$
$\displaystyle+\sup_{\mathfrak{q}\in S(j)}\left|\int f(x-2^{-\mathfrak{q}\cdot
v_{j}}\sum_{v\in\Lambda_{0}}2^{-\mathfrak{q}\cdot(v-v_{j})}a_{v}\mathfrak{t}^{v})\eta(t_{1})\cdots\eta(t_{n})d\mathfrak{t}\right|$
$\displaystyle:=M(j)f(x)+Q(j)f(x).$

First, we shall consider $Q(j)f(x)$. In order to show that $Q(j)f(x)$ is weak-
type 1-1 with a bound dependent on the coefficients, we shall use the
induction argument. Assume that $n_{0}$-parameter maximal function operator
$\mathcal{M}^{\prime}$ is weak-type 1-1 with a bound dependent on the
coefficients of $P_{0}(t_{1},\cdots,t_{n_{0}})$ where $n_{0}<n$ and

(3.10)
$\displaystyle\mathcal{M}^{\prime}f(x)=\sup_{0<h_{1},h_{2},\cdots,h_{n_{0}}<1}\frac{1}{h_{1}h_{2}\cdots
h_{n_{0}}}\left|\int_{0}^{h_{1}}\cdots\int_{0}^{h_{n_{0}}}f(x-P_{0}(t_{1},\cdots,t_{n_{0}}))\mathrm{d}t_{1}\cdots\mathrm{d}t_{n_{0}}\right|.$

Then, by the following inequality:

$\displaystyle Q(j)f(x)$ $\displaystyle=\sup_{\mathfrak{q}\in S(j)}\left|\int
f(x-2^{-\mathfrak{q}\cdot
v_{j}}\sum_{v\in\Lambda_{0}}2^{-\mathfrak{q}\cdot(v-v_{j})}a_{v}\mathfrak{t}^{v})\eta(t_{1})\cdots\eta(t_{n})d\mathfrak{t}\right|$
$\displaystyle=\sup_{\mathfrak{q}\in S(j)}\left|\int
f(x-\sum_{v\in\Lambda_{0}}2^{-\mathfrak{q}\cdot
v}a_{v}\mathfrak{t}^{v})\eta(t_{1})\cdots\eta(t_{n})d\mathfrak{t}\right|$
(3.11) $\displaystyle\leq\int\left\\{\sup_{\mathfrak{q}\in S(j)}\left|\int
f(x-\sum_{v\in\Lambda_{0}}2^{-\mathfrak{q}\cdot
v}a_{v}\mathfrak{t}^{v})\prod_{i\in B}\eta(t_{i})dt_{i\in
B}\right|\right\\}\prod_{i\in A}\eta(t_{i})dt_{i\in A},$

we can obtain that $Q(j)f(x)$ is also weak-type 1-1 with a bound dependent on
the coefficients of $P$ by the assumption. This is because (3.11) can be
controlled by the $n_{0}$-parameter maximal function that is mentioned in
(3.10).
Now, we shall treat $M(j)f(x)$. We write
$P\left(2^{-q_{1}}t_{1},\cdots,2^{-q_{n}}t_{n}\right)=2^{-\mathfrak{q}\cdot
v_{j}}\tilde{P}(\mathfrak{t})$ where

$\tilde{P}(\mathfrak{t})=\sum_{v=\mathfrak{m}\in\Lambda}2^{-\mathfrak{q}\cdot\left(v-v_{j}\right)}a_{\mathfrak{m}}\mathfrak{t}^{\mathfrak{m}}$

and set

$\tilde{P}_{0}(\mathfrak{t})=\sum_{v=\mathfrak{m}\in\Lambda_{0}}2^{-\mathfrak{q}\cdot\left(v-v_{j}\right)}a_{\mathfrak{m}}t^{\mathfrak{m}}.$

Hence, we rewrite $M(j)f(x)$ as

$\displaystyle M(j)f(x)$ (3.12) $\displaystyle=\sup_{\mathfrak{q}\in
S(j)}|\int\left[f(x-2^{-\mathfrak{q}\cdot
v_{j}}\tilde{P}(\mathfrak{t}))-f(x-2^{-\mathfrak{q}\cdot
v_{j}}\tilde{P}_{0}(\mathfrak{t}))\right]\eta(t_{1})\cdots\eta(t_{n})d\mathfrak{t}.$

Note that for $v_{j}=(m_{1}^{j},\cdots,m_{n}^{j})$ and each $i\in A$ in (3.9),
there exists non-negative vector $n_{j}^{(a_{i})}$ which is parallel with the
vector $i$-th axis. Then, let $n_{j}^{(b_{1})},\cdots,n_{j}^{(b_{\beta})}$
denote the other normal non-negative vectors generated by the corner point
$v_{j}$.
For $\bar{k}=(k_{1},k_{2},\cdots,k_{\alpha})\in\mathbb{N}_{\star}^{\alpha}$,
we define $S^{\bar{k}}(j)$ as

$S^{\bar{k}}(j)=\left\\{\mathfrak{q}\in
S(j):\mathfrak{q}=\frac{k_{1}}{d}n_{j}^{(a_{1})}+\cdots+\frac{k_{\alpha}}{d}n_{j}^{(a_{\alpha})}+\frac{\ell_{1}}{d}n_{j}^{(b_{1})}+\cdots+\frac{\ell_{\beta}}{d}n_{j}^{(b_{\beta})};\bar{\ell}=(\ell_{1},\cdots,\ell_{\beta})\in\mathbb{N}_{\star}^{\beta}\right\\}.$

Then, (3.2) is bounded by

$\displaystyle\sum_{\bar{k}\in\mathbb{N}_{\star}^{\alpha}}\sup_{\mathfrak{q}\in
S^{\bar{k}}(j)}\left|\int\left[f(x-2^{-\mathfrak{q}\cdot
v_{j}}\tilde{P}(\mathfrak{t}))-f(x-2^{-\mathfrak{q}\cdot
v_{j}}\tilde{P}_{0}(\mathfrak{t}))\right]\eta(t_{1})\cdots\eta(t_{n})d\mathfrak{t}\right|$
$\displaystyle:=\sum_{\bar{k}\in\mathbb{N}_{\star}^{\alpha}}M^{\bar{k}}(j)f(x).$

So, if we have the following Proposition, then one can obtain the weak type
1-1 bound of the operator $M(j)$.

######  Proposition 3.2.

There exists $\gamma^{\prime}\in\mathbb{Q}_{+}^{\beta}$ such that

(3.13)
$\displaystyle\left|\left\\{x:M^{\bar{k}}(j)f(x)>\lambda\right\\}\right|\leq\frac{C}{\lambda}2^{-\gamma^{\prime}\cdot\bar{k}}\|f\|_{1}$

where $\gamma^{\prime}$ and $C>0$ are independent of $\bar{k}$ and $\lambda$.

###### Proof.

Observe that for $\mathfrak{q}\in S^{k}(j)$,

$\displaystyle\mathfrak{q}\cdot v_{j}$
$\displaystyle=(\frac{\ell_{1}}{d}n_{j}^{(b_{1})}+\cdots+\frac{\ell_{\beta}}{d}n_{j}^{(b_{\beta})})\cdot
v_{j}$

since $\bar{n}_{j}^{(a_{i})}\cdot v_{j}=0$ for $i\in A$. We set

$\sigma=\left(\frac{1}{d}(\bar{n}_{j}^{(b_{1})}\cdot
v_{j}),\cdots,\frac{1}{d}(\bar{n}_{j}^{(b_{\beta})}\cdot v_{j})\right).$

One can easily check that $\sigma$ is not a zero vector. Then, we rewrite
$M^{\bar{k}}(j)f(x)$ with $\bar{\ell}=(\ell_{1},\cdots,\ell_{\beta})$ as

$\displaystyle M^{\bar{k}}(j)f(x)=$
$\displaystyle\sup_{\bar{\ell}\in\mathbb{N}_{\star}^{\beta}}\left|\int\left[f(x-2^{-\sigma\cdot\bar{\ell}}\tilde{P}(\mathfrak{t}))-f(x-2^{-\sigma\cdot\bar{\ell}}\tilde{P}_{0}(\mathfrak{t}))\right]\eta(t_{1})\cdots\eta(t_{n})d\mathfrak{t}\right|$
$\displaystyle:=\sup_{\bar{\ell}\in\mathbb{N}_{\star}^{\beta}}\left|\nu_{\bar{\ell}}^{\bar{k},(\bar{\ell})}*f(x)\right|$

where $v_{\bar{\ell}}^{\bar{k},(\bar{\ell})}$ and $v^{\bar{k},(\bar{\ell})}$
are the measures defined by

$\displaystyle\left\langle
f,v_{\bar{\ell}}^{\bar{k},(\bar{\ell})}\right\rangle$
$\displaystyle=\int\left[f(2^{-\sigma\cdot\bar{\ell}}\tilde{P}(\mathfrak{t}))-f(2^{-\sigma\cdot\bar{\ell}}\tilde{P}_{0}(\mathfrak{t}))\right]\eta(t_{1})\cdots\eta(t_{n})d\mathfrak{t}$
$\displaystyle\left\langle f,v^{\bar{k},(\bar{\ell})}\right\rangle$
$\displaystyle=\int\left[f(\tilde{P}(\mathfrak{t}))-f(\tilde{P}_{0}(\mathfrak{t}))\right]\eta(t_{1})\cdots\eta(t_{n})d\mathfrak{t}$

for those $\bar{\ell}$ ’s for which

$\frac{k_{1}}{d}n_{j}^{(a_{1})}+\cdots+\frac{k_{\alpha}}{d}n_{j}^{(a_{\alpha})}+\frac{\ell_{1}}{d}n_{j}^{(b_{1})}+\cdots+\frac{\ell_{\beta}}{d}n_{j}^{(b_{\beta})}=\mathfrak{q}\in
S^{\bar{k}}(j)$

and for all the other $\bar{\ell}$ ’s we define them to be zero distributions.
So, the proof is same as proof of Proposition 3.1 if we have the following
lemmas. ∎

######  Lemma 3.2.

There are positive real constant $C,\delta$ and vector
$\gamma\in\mathbb{Q}_{+}^{\beta}$ independent of $\bar{k}$ and $\bar{\ell}$
such that

(3.14)
$\displaystyle\int\left|v^{\bar{k},(\bar{\ell})}(x-y)-v^{\bar{k},(\bar{\ell})}(x)\right|\mathrm{d}x\leq
C2^{-\gamma\cdot\bar{k}}|y|^{\delta}\text{ for all }y\in\mathbb{R},$ (3.15)
$\displaystyle\sum_{\bar{\ell}\in\mathbb{N}_{\star}^{\beta}}\int_{|x|\geq
2|y|}\left|v_{\bar{\ell}}^{\bar{k},(\bar{\ell})}(x-y)-v_{\bar{\ell}}^{\bar{k},(\bar{\ell})}(x)\right|\mathrm{d}x\leq
C2^{-\gamma\cdot k},$ (3.16)
$\displaystyle\left\|\sup_{\bar{\ell}\in\mathbb{N}_{\star}^{\beta}}\left|f*v_{\bar{\ell}}^{\bar{k},(\bar{\ell})}\right|\right\|_{2}\leq
C2^{-\gamma\cdot k}\|f\|_{2}.$

##  4\. Proof Of Lemma 3.1

In this section, we shall show Lemma 3.1.

###  4.1. Proof of (3.1)

Recall that we write
$P(2^{-q_{1}}t_{1},\cdots,2^{-q_{n}}t_{n})=2^{-\mathfrak{q}\cdot
v_{j}}\tilde{P}_{j}(t_{1},\cdots,t_{n})$ where

$\tilde{P}_{j}(\mathfrak{t}):=\sum_{v=\mathfrak{m}\in\Lambda}2^{-\mathfrak{q}\cdot\left(v-v_{j}\right)}a_{\mathfrak{m}}\mathfrak{t}^{\mathfrak{m}}=a_{v_{j}}\mathfrak{t}^{v_{j}}+\sum_{v=\mathfrak{m}\in\Lambda\setminus{v_{j}}}2^{-\mathfrak{q}\cdot\left(v-v_{j}\right)}a_{\mathfrak{m}}\mathfrak{t}^{\mathfrak{m}}.$

Since $\mathfrak{q}\cdot(v-v_{j})\geq 0$ for any $\mathfrak{q}\in S(j)$,
$\tilde{P}_{j}(\mathfrak{t})$ has uniformly bounded $\mathcal{C}_{m}$ norm for
all $\mathfrak{q}\in S(j)$. Moreover, by the term
$a_{v_{j}}\mathfrak{t}^{v_{j}}$ of $\tilde{P}_{j}(\mathfrak{t})$, one can
knows that $\partial^{\alpha}\tilde{P}_{j}(\mathfrak{t})$ is uniformly bounded
below for some $\alpha$ with
$1\leq|\alpha|\leq\operatorname{deg}\tilde{P}_{j}$.
$\tilde{P}_{0}(\mathfrak{t})$ also have two same properties. Hence, by
Proposition 7.2 of [5] combined with the above two properties of
$\tilde{P}_{j}(\mathfrak{t})$ and $\tilde{P}_{0}(\mathfrak{t})$, we have

(4.1)
$\displaystyle\int\left|\nu^{N,(\bar{k}_{m})}(x-y)-\nu^{N,(\bar{k}_{m})}(x)\right|\mathrm{d}x\leq
C|y|^{\epsilon_{0}}$

for all $y\in\mathbb{R}$ with $C,\epsilon_{0}>0$ independent of $\bar{k}_{m}$
and $N$.

Now, we consider the case where $N$ is sufficiently large. By Lemma 2.2, when
$N$ is sufficiently large, $\nabla\tilde{P}_{j}(\mathfrak{t})$ is uniformly
bounded below within the support of $\eta(t_{1})\cdots\eta(t_{n})$ . Hence,
one has

(4.2)
$\displaystyle\left|\widehat{\nu^{N,(\bar{k}_{m})}}(\xi)\right|=\left|\int\left[\exp(i\xi\tilde{P}_{j}(\mathfrak{t}))-\exp\left(i\xi
a_{v_{j}}\mathfrak{t}^{v_{j}}\right)\right]\eta(t_{1})\cdots\eta(t_{n})d\mathfrak{t}\right|\leq\frac{C_{m}}{|\xi|^{m}}$

for all $m\in\mathbb{N}$ and $N\geq N_{0}$. Here, $N_{0}$ depends only on the
coefficients of polynomial $P$. Also, by the mean value theorem, we have

(4.3) $\displaystyle\left|\widehat{\nu^{N,(\bar{k}_{m})}}(\xi)\right|\leq
C2^{-\beta_{j}N}|\xi|$

where $\beta_{j}$ is defined in (2.3). Thus, by (4.2) and (4.3) combined with
the convexity, there exists $\sigma_{j}>0$ such that

$\displaystyle\int\left|\nu^{N,(\bar{k}_{m})}(x-y)-\nu^{N,(\bar{k}_{m})}(x)\right|\mathrm{d}x$
(4.4) $\displaystyle\leq C\left\\{\int\left|(e^{-2\pi
iy\xi}-1)\cdot\widehat{\nu^{N,(\bar{k}_{m})}}(\xi)\right|^{2}\mathrm{~{}d}\xi\right\\}^{1/2}$
$\displaystyle=C\left\\{\int_{|\xi|\leq 1}\left|(e^{-2\pi
iy\xi}-1)\cdot\widehat{\nu^{N,(\bar{k}_{m})}}(\xi)\right|^{2}\mathrm{~{}d}\xi+\int_{|\xi|\geq
1}\left|(e^{-2\pi
iy\xi}-1)\cdot\widehat{\nu^{N,(\bar{k}_{m})}}(\xi)\right|^{2}\mathrm{~{}d}\xi\right\\}^{1/2}$
(4.5) $\displaystyle\leq C2^{-\sigma_{j}N}|y|.$

Here, we have used the Cauchy-Schwarz inequality with the fact that
$v^{N,(\bar{k}_{m})}$ has a compact support and Plancherel’s theorem to have
(4.4).

###  4.2. Proof of (3.2)

Suppose that the uniform compact support of the $\nu^{N,(\bar{k}_{m})}$ (for
all $N$ and $\bar{k}_{m}$) is contained in a ball $B(0,R)$. Since

$\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}(x)=2^{\sigma_{m}\cdot\bar{k}_{m}}\nu^{N,(\bar{k}_{m})}\left(2^{\sigma_{m}\cdot\bar{k}_{m}}x\right),$

the support of $v_{k}^{N,(k)}$ is contained in
$B(0,2^{-\sigma_{m}\cdot\bar{k}_{m}}R)$. Let $2^{k_{0}-1}<|y|\leq 2^{k_{0}}$
for some $k_{0}\in\mathbb{Z}$. Then, $|x|\geq 2|y|$ means that

$|x|>2^{k_{0}}\text{ and }|x-y|\geq|y|>2^{k_{0}-1}.$

Hence, if $2^{-\sigma_{m}\cdot\bar{k}_{m}}R<2^{k_{0}-1}$,

$\int_{|x|\geq
2|y|}\left|\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}(x-y)-\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}(x)\right|\mathrm{d}x\equiv
0.$

Therefore, by (3.1) we have

$\displaystyle\sum_{\bar{k}_{m}}\int_{|x|\geq
2|y|}\left|\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}(x-y)-\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}(x)\right|\mathrm{d}x$
$\displaystyle\leq\sum_{\bar{k}_{m}:\sigma_{m}\cdot\bar{k}_{m}+k_{0}\leq\log
R+1}\int\left|\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}(x-y)-\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}(x)\right|\mathrm{d}x$
$\displaystyle=\sum_{\bar{k}_{m}:\sigma_{m}\cdot\bar{k}_{m}+k_{0}\leq\log
R+1}\int\left|\nu^{N,(\bar{k}_{m})}\left(z-2^{\sigma_{m}\cdot\bar{k}_{m}}y\right)-\nu^{N,(\bar{k}_{m})}(z)\right|\mathrm{d}z$
$\displaystyle\leq\sum_{\bar{k}_{m}:\sigma_{m}\cdot\bar{k}_{m}+k_{0}\leq\log
R+1}A2^{-\delta_{1}N}\left(2^{\sigma_{m}\cdot\bar{k}_{m}}|y|\right)^{\delta_{2}}$
$\displaystyle\leq
A2^{-\delta_{1}N}\sum_{\bar{k}_{m}:\sigma_{m}\cdot\bar{k}_{m}+k_{0}\leq\log
R+1}\left(2^{\sigma_{m}\cdot\bar{k}_{m}}2^{k_{0}}\right)^{\delta_{2}}$
$\displaystyle\leq A2^{-\delta_{1}N}C\left(R,\sigma_{1},\delta_{2}\right).$

###  4.3. Proof of (3.3)

By the mean value theorem, one has

(4.6)
$\displaystyle\left|\widehat{\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}}(\xi)\right|\leq
C|\xi|2^{-\sigma_{m}\cdot\bar{k}_{m}}2^{-\beta_{j}N}.$

Furthermore, by the van der Corput’s lemma, one knows that

(4.7)
$\displaystyle\left|\widehat{\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}}(\xi)\right|\leq\frac{C}{\left(|\xi|2^{-\sigma_{m}\cdot\bar{k}_{m}}\right)^{\epsilon}}.$

Therefore, by the convexity, there exists $\delta_{3}>0$ such that

(4.8)
$\displaystyle\left|\widehat{\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}}(\xi)\right|\leq
C2^{-\delta_{3}N}\min\left(|\xi|2^{-\sigma_{m}\cdot\bar{k}_{m}},\frac{1}{\left(|\xi|2^{-\sigma_{m}\cdot\bar{k}_{m}}\right)^{\epsilon/2}}\right).$

Thus, by the standard Littlewood-Paley theory, we have

(4.9)
$\displaystyle\left\|\sup_{\bar{k}_{m}\in\mathbb{N}_{\star}^{n-1}}\left|f*\nu_{\bar{k}_{m}}^{N,(\bar{k}_{m})}\right|\right\|_{2}\leq
B2^{-\delta_{3}N}\|f\|_{2}.$

For the details, see [10].

##  5\. Proof Of Lemma 3.2

The proof of (3.15) and (3.16) in Lemma 3.2 are same as the proof of (3.2) and
(3.3) respectively. Hence, we shall show the inequality (3.14).

###  5.1. Proof of (3.14)

Recall that we write
$P\left(2^{-q_{1}}t_{1},\cdots,2^{-q_{n}}t_{n}\right)=2^{-\mathfrak{q}\cdot
v_{j}}\tilde{P}(\mathfrak{t})$ where

$\tilde{P}(\mathfrak{t})=\sum_{v=\mathfrak{m}\in\Lambda}2^{-\mathfrak{q}\cdot\left(v-v_{j}\right)}a_{\mathfrak{m}}\mathfrak{t}^{\mathfrak{m}}$

and

$\tilde{P}_{0}(\mathfrak{t})=\sum_{v=\mathfrak{m}\in\Lambda_{0}}2^{-\mathfrak{q}\cdot\left(v-v_{j}\right)}a_{\mathfrak{m}}t^{\mathfrak{m}}.$

Observe that for $v\in\Lambda\backslash\Lambda_{0}$ and $i\in A$,
$\bar{n}_{j}^{(a_{i})}\cdot\left(v-v_{j}\right)>0$ from the definition of
$\Lambda_{0}$. So, we set
$\gamma_{j}=(\gamma_{j}^{1},\cdots,\gamma_{j}^{\alpha})$ where

$\gamma_{j}^{i}=\frac{1}{d}\min_{v\in\Lambda\backslash\Lambda_{0}}\left\\{\bar{n}_{j}^{(a_{i})}\cdot\left(v-v_{j}\right)\right\\}>0,\
(i\in\\{1,2,\cdots,\alpha\\}).$

Now, we rewrite
$\tilde{P}(\mathfrak{t})=\tilde{P}_{0}(\mathfrak{t})+2^{-\gamma_{j}\cdot\bar{k}}\tilde{Q}(\mathfrak{t})$
where $\bar{k}=(k_{1},k_{2},\cdots,k_{\alpha})\in\mathbb{N}_{\star}^{\alpha}$.
By the fact that for $v\in\Lambda\backslash\Lambda_{0}$ and $\mathfrak{q}\in
S(j)$,

$\displaystyle\mathfrak{q}\cdot\left(v-v_{1}\right)$
$\displaystyle\geq\frac{k_{1}}{d}\bar{n}_{j}^{(a_{1})}\cdot\left(v-v_{j}\right)+\cdots+\frac{k_{\alpha}}{d}\bar{n}_{j}^{(a_{\alpha})}\cdot\left(v-v_{j}\right)$
(5.1) $\displaystyle\geq\gamma_{j}\cdot\bar{k},$

one knows that there exists a constant $M>0$(depends only on the coefficients
of P) such that

(5.2)
$\displaystyle\left\|\nabla\tilde{Q}(\mathfrak{t})\right\|_{L^{\infty}([1/2,4]^{n})}\leq
M.$

Let

$I_{\theta}:=\\{\mathfrak{t}\in[\frac{1}{2},4]^{n}:|\nabla\tilde{P}_{0}(t)|>2^{-\theta(\gamma_{j}\cdot\bar{k})}\\}.$

for some sufficiently small $\theta\in\mathbb{Q}_{+}$. For $I_{\theta}$, we
set

$\widehat{v_{I_{\theta}}^{\bar{k},(\bar{\ell})}}(\xi):=\int_{I_{\theta}}\left[e^{-2\pi
i\xi\cdot\tilde{P}(\mathfrak{t})}-e^{-2\pi
i\xi\cdot\tilde{P}_{0}(\mathfrak{t})}\right]\eta(t_{1})\cdots\eta(t_{n})d\mathfrak{t}$

for those $\bar{\ell}$ ’s for which

$\frac{k_{1}}{d}n_{j}^{(a_{1})}+\cdots+\frac{k_{\alpha}}{d}n_{j}^{(a_{\alpha})}+\frac{\ell_{1}}{d}n_{j}^{(b_{1})}+\cdots+\frac{\ell_{\beta}}{d}n_{j}^{(b_{\beta})}=\mathfrak{q}\in
S^{\bar{k}}(j)$

and for all the other $\bar{\ell}$ ’s we define the Fourier multiplier to be
zero. Then, by the inequality (5.2) combined with
$\tilde{P}(\mathfrak{t})=\tilde{P}_{0}(\mathfrak{t})+2^{-\gamma_{j}\cdot\bar{k}}\tilde{Q}(\mathfrak{t})$,
one can obtain that

(5.3)
$\displaystyle\left|\widehat{v_{I_{\theta}}^{\bar{k},(\bar{\ell})}}(\xi)\right|\leq\frac{C_{m}}{|\xi|^{m}}$

for all $m\in\mathbb{N}$ and $\bar{k}\geq(k_{0},k_{0}\cdots,k_{0})$ when we
fix $\theta$ such that $\theta\leq\theta_{0}$. Here, $\theta_{0}$ is
determined by the $M$ in (5.2) and $k_{0}$ is choosen by the $M$ in (5.2) and
the sufficiently small $\theta$ satisfying $\theta\leq\theta_{0}$.
Furthermore, by the mean-value theorem with (5.1), we have

(5.4)
$\displaystyle\left|\widehat{v_{I_{\theta}}^{\bar{k},(\bar{\ell})}}(\xi)\right|\leq
C|\xi|2^{-\gamma_{j}\cdot\bar{k}}.$

Therefore, by the same argument in the proof of (3.1) combined with (5.3) and
(5.4), we have

(5.5)
$\displaystyle\int\left|v_{I_{\theta}}^{\bar{k},(\bar{\ell})}(x-y)-v_{I_{\theta}}^{\bar{k},(\bar{\ell})}(x)\right|\mathrm{d}x\leq
C2^{-\gamma\cdot\bar{k}}|y|^{\delta}$

for all $y\in\mathbb{R}$ and $\bar{k}\geq(k_{0},k_{0}\cdots,k_{0})$. On the
other hand, using Proposition 7.2 of [5], we also know that

(5.6)
$\displaystyle\int\left|v_{I_{\theta}}^{\bar{k},(\bar{\ell})}(x-y)-v_{I_{\theta}}^{\bar{k},(\bar{\ell})}(x)\right|\mathrm{d}x\leq
C|y|^{\epsilon_{0}}$

for all $y\in\mathbb{R}$ with $C,\epsilon_{0}>0$ independent of $\bar{k}$,
$\bar{\ell}$ and $I_{\theta}$. For the details, see the arguments to have
(4.1).

Now, we shall consider
$\widehat{v_{I_{\theta}^{c}}^{\bar{k},(\bar{\ell})}}(\xi)$ defined by

$\widehat{v_{I_{\theta}^{c}}^{\bar{k},(\bar{\ell})}}(\xi):=\widehat{v^{\bar{k},(\bar{\ell})}}(\xi)-\widehat{v_{I_{\theta}}^{\bar{k},(\bar{\ell})}}(\xi)$

for those $\bar{\ell}$ ’s for which

$\frac{k_{1}}{d}n_{j}^{(a_{1})}+\cdots+\frac{k_{\alpha}}{d}n_{j}^{(a_{\alpha})}+\frac{\ell_{1}}{d}n_{j}^{(b_{1})}+\cdots+\frac{\ell_{\beta}}{d}n_{j}^{(b_{\beta})}=\mathfrak{q}\in
S^{\bar{k}}(j)$

For all the other $\bar{\ell}$ ’s, we define the Fourier multiplier to be
zero. By the sublevel set estiamte, there exists $\delta^{\prime}>0$ such that

(5.7) $\displaystyle
I_{\theta}^{c}:=\left|\\{\mathfrak{t}\in[\frac{1}{2},4]^{n}:|\nabla\tilde{P}_{0}(t)|\leq
2^{-\theta(\gamma_{j}\cdot\bar{k})}\\}\right|\leq
C(2^{-\theta(\gamma_{j}\cdot\bar{k})})^{\delta^{\prime}}.$

So, due to (5.7), we have

(5.8)
$\displaystyle\int\left|v_{I_{\theta}^{c}}^{\bar{k},(\bar{\ell})}(x)\right|\mathrm{d}x$
$\displaystyle\leq\int|\int_{I_{\theta}^{c}}\int
e^{2\pi(x-\tilde{P}(\mathfrak{t}))\xi}d\xi
d\mathfrak{t}|dx+\int|\int_{I_{\theta}^{c}}\int
e^{2\pi(x-\tilde{P}_{0}(\mathfrak{t}))\xi}d\xi d\mathfrak{t}|dx$
$\displaystyle=|I_{\theta}^{c}|(\int|\delta(x)|dx+\int|\delta(x)|dx)$
$\displaystyle\leq C(2^{-\theta(\gamma_{j}\cdot k)})^{\delta^{\prime}}$

where $\delta(x)$ is the Dirac delta distribution. On the other hand, by
Proposition 7.2 of [5] again, one can obtain that

(5.9)
$\displaystyle\int\left|v_{I_{\theta}^{c}}^{\bar{k},(\bar{\ell})}(x-y)-v_{I_{\theta}^{c}}^{\bar{k},(\bar{\ell})}(x)\right|\mathrm{d}x\leq
C|y|^{\epsilon_{0}}$

for all $y\in\mathbb{R}$ with $C,\epsilon_{0}>0$ independent of $\bar{k}$ and
$\bar{\ell}$.
Therefore, by (5.8) and (5.9) we have

(5.10)
$\displaystyle\int\left|v_{I_{\theta}^{c}}^{\bar{k},(\bar{\ell})}(x-y)-v_{I_{\theta}^{c}}^{\bar{k},(\bar{\ell})}(x)\right|\mathrm{d}x\leq
C2^{-\gamma\cdot\bar{k}}|y|^{\delta}\text{ for all }y\in\mathbb{R}.$

(5.5), (5.6) and (5.10) implies the desired results.

## References

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11institutetext:  Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126
Pisa, Italy
11email<EMAIL_ADDRESS>22institutetext: Dipartimento di Fisica,
Sapienza, Università di Roma, Piazzale Aldo Moro 5, 00185, Roma, Italy
33institutetext: INAF - Osservatorio Astronomico di Bologna, via Piero Gobetti
93/3, I-40129 Bologna, Italy 44institutetext: INAF-Osservatorio astronomico di
Capodimonte, Via Moiariello 16, I-30131 Napoli, Italy

# Are there more galaxies than we see around high-$z$ quasars?

Tommaso Zana 1122 Stefano Carniani 11 David Prelogović 11 Fabio Vito 33 Viola
Allevato 1133 Andrea Ferrara 11 Simona Gallerani 11 Eleonora Parlanti 11

(Received June 13, 2023;)

###### Abstract

Context. Whether or not $z\gtrsim 6$ quasars lie in the most massive dark-
matter halos of the Universe is still a subject of dispute. While most
theoretical studies support this scenario, current observations yield
discordant results when they probe the halo mass through the detection rate of
quasar companion galaxies. Feedback processes from supermassive black holes
and dust obscuration have been blamed for this discrepancy, but the impact of
these effects is complex and far from being clearly understood.

Aims. This paper aims to improve the interpretation of current far-infrared
observations by taking into account the cosmological volume probed by the
Atacama Large Millimeter/submillimeter Array Telescope and to explain the
observational discrepancies.

Methods. We statistically investigate the detection rate of quasar companions
in current observations and verify if they match the expected distribution
from various theoretical models, once convolved with the ALMA field-of-view,
through the use of Monte Carlo simulations.

Results. We demonstrate that the telescope geometrical bias is fundamental and
can alone explain the scatter in the number of detected satellite galaxies in
different observations. We conclude that the resulting companion densities
depend on the chosen galaxy distributions. According to our fiducial models,
current data favour a density scenario where quasars lie in dark-matter halos
of viral mass $M_{\rm vir}\gtrsim 10^{12}\leavevmode\nobreak\ {\rm
M_{\odot}}$, in agreement with most theoretical studies. According to our
analysis, each quasar has about 2 companion galaxies, with a [CII] luminosity
$L_{\rm[CII]}\gtrsim 10^{8}\leavevmode\nobreak\ \\!{\rm L}_{\odot}$, within a
distance of about 1 Mpc from the quasar.

###### Key Words.:

Methods: statistical – Methods: numerical – quasars: general – Galaxies: halos
– Galaxies: high-redshift – Infrared: galaxies

##  1 Introduction

Most luminous ($L_{\rm bol}>10^{46}$ erg s-1) quasars at redshift $z\gtrsim 6$
are powered by the accretion process of gas onto the most massive supermassive
black-holes ($10^{8}-10^{10}\leavevmode\nobreak\ \\!{\rm M}_{\odot}$;
Ferrarese & Ford 2005; Meyer et al. 2022a; Wang et al. 2018). According to
theoretical models, including numerical simulations (Sijacki et al. 2009; Di
Matteo et al. 2012; Costa et al. 2014; Weinberger et al. 2018; Barai et al.
2018; Habouzit et al. 2019; Ni et al. 2020; Valentini et al. 2021), and
clustering studies (Hickox et al. 2009; Ross et al. 2009; Cappelluti et al.
2010; Allevato et al. 2011, 2012), supermassive black-holes reside in the
densest regions of the Universe, such as the centre of most massive dark-
matter halos, with virial masses ranging from $10^{12}$ to
$10^{13}\leavevmode\nobreak\ \\!{\rm M}_{\odot}$ at $z\sim 6$.

However, observations do not always agree with such studies at nearly any
redshift. Numerous observational works have exploited current astronomical
facilities to investigate the properties of the environments where quasars
reside, yielding discordant results. On the one hand, some studies suggest
that quasars are located in regions characterized by large densities of
galaxies, such as Steidel et al. (2005) at $z=2.3$, Hall et al. (2018) at
$0.5\lesssim z\lesssim 3.5$, Swinbank et al. (2012) at $2.2\lesssim z\lesssim
4.5$, García-Vergara et al. (2019) and Uchiyama et al. (2020) at $z\sim 4$,
Husband et al. (2013) and Capak et al. (2011) at $z\sim 5$, and Kim et al.
(2009), Morselli et al. (2014), Balmaverde et al. (2017), Decarli et al.
(2017), Decarli et al. (2018), Mignoli et al. (2020), and Venemans et al.
(2020) at $z\gtrsim 6$. On the other hand, some works, such as Francis &
Bland-Hawthorn (2004) and Simpson et al. (2014) at $z\gtrsim 2$, Uchiyama et
al. (2018) at $z\sim 4$, Kashikawa et al. (2007) and Kikuta et al. (2017) at
$z\sim 5$, and Bañados et al. (2013), Mazzucchelli et al. (2017), Champagne et
al. (2018), Meyer et al. (2022b) at $z\gtrsim 6$, find that the number of
companion galaxies is, at most, similar to the galaxy density estimated in
blank fields. Overall, the question of whether quasars tend to live in over-
dense regions of the Universe remains an active area of research.

The detection of Lyman-break galaxies (LBGs) and Lyman-alpha emitting galaxies
(LAEs) in the neighbourhoods of quasars and active galactic nuclei is often
used in literature to probe the underlying dark matter distribution of high
redshift structures (see, for instance, Intema et al. 2006; Venemans et al.
2007; Bañados et al. 2013; Hennawi et al. 2015; Mazzucchelli et al. 2017,
investigating the satellite number counts, or García-Vergara et al. 2019,
studying the cross-correlation between galaxies and quasars). The
contradictory conclusions of these studies may also stem from the intrinsic
challenges in constraining the redshift of LBGs and the significant influence
of dust extinction and obscuration on both LBGs and LAEs. For this reason, the
best strategy to detect high-$z$ quasar companions relies on the observation
of their [CII]158$\mu$m emission, as this gas tracer is not affected by dust-
attenuation and intergalactic medium absorption (e.g., Maiolino et al. 2005;
Walter et al. 2009). In this context, the Atacama Large Millimeter Array
(ALMA) has provided the most numerous spectroscopically confirmed detections
of $z\gtrsim 6$ quasar companions (Decarli et al. 2017, 2018; Venemans et al.
2020). In particular, Venemans et al. (2020, hereafter V20) found 27 line-
emitter candidates within 27 quasar fields. Out of this initial pool, 10 line
emitters were already recognised as quasar companions in previous works
(Decarli et al. 2017; Willott et al. 2017; Neeleman et al. 2019; Venemans et
al. 2019). The remaining candidates have been identified as companions by
assuming that the detected line does correspond to the carbon [CII]
transition, emitted within $\pm\Delta v$ from the quasar.111In this work, we
assume the validity of V20 hypothesis and therefore consider all the
additional 17 line emitters as proper quasar companions. Two thresholds have
been adopted in V20, leading to slightly different results: 7 additional
satellites are detected with $\Delta v=1000$ km s-1, and 9 with $\Delta
v=2000$ km s-1. The V20 sample is nowadays the largest and most complete
catalogue of high-$z$ quasar companions with constrained redshift.

One debated possibility to explain the differences among these observational
results is associated with the quenching effect of star formation in galactic
satellites driven by quasar feedback (Efstathiou 1992; Thoul & Weinberg 1996;
Okamoto et al. 2008; Dashyan et al. 2019; Martín-Navarro et al. 2019).
According to this scenario, quasars would still reside in the most massive
halos as predicted by theoretical models, but their companions would be gas-
poor and would exhibit low star-formation activity, making them impossible to
detect with current observational facilities. This argument has been put forth
to account for the absence of galaxy over-densities within $\sim 1$ Mpc from
certain high-z quasar objects.

Nevertheless, this alleged negative feedback effect is currently far from
established. Some works proposed even a possible positive interaction between
quasar feedback and star formation in companion galaxies (Fragile et al. 2017;
Martín-Navarro et al. 2021; Zana et al. 2022; Ferrara et al. 2023). In
particular, Zana et al. (2022), based on cosmological simulations, and Ferrara
et al. (2023), using semi-analytic models, claimed that quasar feedback always
enhances the process of star-formation in satellites, provided that the
satellites are not disrupted by the strong quasar outflows. These recent works
suggest that the discrepant observational results on the environments of
high-$z$ quasars might not be due to feedback and that additional effects and
possible observational biases might be in place.

Zana et al. (2022) analysed the cosmological simulation by Barai et al. (2018)
and focussed on the environment of a massive quasar at $z\gtrsim 6$, finding
2-3 satellites with a [CII] luminosity $L_{\rm[CII]}\gtrsim
10^{8}\leavevmode\nobreak\ \\!{\rm L}_{\odot}$. They suggested that their
findings match only the most densely populated field in V20, due to the
geometrical effects introduced by the ALMA telescope. In fact, the relatively
narrow ALMA field of view (FoV) compared with the much larger spatial range
probed by the line of sight (LoS) where satellites can be identified, could
inherently hinder the detection of a potentially significant fraction of the
actual population of satellites, even if they are above the instrumental
sensitivity threshold. Other studies, such as Champagne et al. (2018) and
Meyer et al. (2022b) are in agreement with this hypothesis and conclude that
observational campaigns adopting large FoV are fundamental to recover the
missing satellite populations.

In this work, we investigate the geometrical bias introduced by ALMA on the
satellite detection rate. In particular, we use different mock distributions
of galactic satellite populations, based on theoretically motivated models, to
measure the geometrical response of the ALMA limited observational volume. Our
ultimate goal is to derive a rigorous statistical framework to interpret the
most recent observational survey.

We assume a flat $\Lambda$CDM model, with the cosmological parameters
$\Omega_{\rm M,0}=0.3089$, $\Omega_{\Lambda,0}=1-\Omega_{\rm M,0}=0.6911$, and
$H_{0}=67.74$ Mpc s-1 (Planck Collaboration et al. 2016, results XIII). All
lengths and volumes in the text are assumed to be physical unless otherwise
specified.

The paper is structured as follows: we describe the method adopted to
statistically investigate the geometrical bias in Sec. 2, in Sec. 3 we present
the outcomes, and in Sec. 4 we discuss the results in the context of current
literature. We finally summarise our findings in Sec. 5.

##  2 Method

Figure 1 shows the incidence of [CII]-emitting companions in the sample of 27
quasars reported by V20, in the range $\Delta v=1000$ km s-1. Approximately
half of quasars do not exhibit any detected companions, whereas in other
cases, up to three galaxies are observed. Since all datasets analysed by V20
have comparable exposure times, the frequency of detections in this sample
cannot be solely attributed to possible different sensitivities of the
observations. Under the assumption that all the observed quasars live in
environments with similar properties and that the quasar feedback has the same
effect on all the companions, our objective is to assess if the limited cosmic
volume probed by ALMA can explain the large detection range shown in Figure 1.

Figure 1: Distribution of the number of satellites detected as [CII] emitters
from V20, during an ALMA survey of 27 $z\gtrsim 6$ quasars. All these sources
reside within $\pm 1000$ km s-1, i.e. about $\pm 1.3$ Mpc at $z=6.5$ from the
central quasar.

ALMA observations of $z\sim 6.5$ quasars are characterised by a circular FoV
of radius $\simeq 90$ kpc and cover a wavelength range of $\Delta
v=1000-2000\leavevmode\nobreak\ {\rm km\leavevmode\nobreak\ s^{-1}}$ with
respect to the systematic redshift of each quasar (V20). The explored
cosmological volume is thus a cylinder with a base of about $2.6\times 10^{4}$
kpc2 and a height of about 2.5-5.0 Mpc at $z=6.5$, depending on the spectral
setting of the observations, as it is shown in Figure 2. Here, we focus on the
fixed LoS range $\Delta v=1000$ km s-1 to be more conservative, given the
large distances involved, and we compare with V20 equivalent sample.
Additionally, we consider the median redshift $z=6.5$ as representative of the
range $6\leq z\leq 7$ for the whole V20 sample.222The ratios amongst the
comoving quantities (the position of the satellites and the ALMA observational
range) are conserved in redshift. We have nonetheless run our analysis also at
different mean redshift, confirming no significant variation.

Given the geometry of the probed volume, the detection rate of companions
would depend on both the distribution of galaxies in the quasar environment
and on the orientation of the cylinder axis. Therefore, a fair comparison
between theoretical studies and observations must take into account this
effect.

Figure 2: Schematic representation of an ALMA observation of a quasar (yellow
star). Here, two galaxy satellites (red stars) are detected, whereas a third
one remains outside the observer’s cylinder and would be detected only through
a different line of sight.

We create different mock distributions of quasar satellites and study the
frequency of detections when the companions lie within a random observational
cylinder. Ideally, we would run numerous, cosmologically motivated, N-body
simulations of high-$z$ quasar companions and select a single observational
cylinder in each simulation. This approach would be prohibitive in terms of
time and computational costs. Alternatively, we run various sets of Monte
Carlo simulations where we spawn, for each set, an increasing number of
satellites $N$ in the quasar-centred sphere with radius $D_{z=6.5}$, where
$D_{z=6.5}$ is the distance corresponding to the recession velocity $\Delta
v=1000$ km s-1 at $z=6.5$, i.e. about 1.3 Mpc. The $N$ satellites represent
massive galaxies bright enough to be detected by ALMA as [CII] emitters,
provided that they fall within the volume probed by the telescope. They indeed
represent only the “detectable” fraction of the whole satellite population
and, in principle, could be connected to the total dark matter density where
the quasar resides. For each set of simulations (at fixed $N$), we intersect
the quasar-centred sphere with the ALMA cylinder with a random orientation, as
exemplified in Figure 2. In particular, for each given $N$, we perform
$10^{5}$ simulations, varying the position of the satellites, along with the
orientation of the observational cylinder. Hence, we compute the probability
$P(N_{\rm o}|N)$, that $N_{\rm o}$ satellites are included in the cylinder,
and thus detectable by ALMA, as a function of $N$. We adopt 9 different dark-
matter distributions to generate the satellite populations:

  * •

Homogeneous distribution: satellites are generated by following a homogeneous
distribution within the spherical volume, with fixed radius $D_{z=6.5}$. At
$z=6.5$, the volume measures about $9$ Mpc3. This model is dubbed as
Homogeneous and represents the simplest case, with no assumption on the radial
distribution.

  * •

Plummer model: satellite galaxies are spawned by following a Plummer density
profile (Plummer 1911), with cumulative distribution function (CDF)

$P(r)=\frac{r^{3}}{\left(r^{2}+a^{2}\right)^{3/2}},$ (1)

where $r$ is the radius from the central quasar and $a$ is the scale radius.
We adopt two instances for this distribution by choosing $a=25$ and 100 kpc.
These models are dubbed as Plummer25 and Plummer100.

  * •

Hernquist model: analogously to the Plummer model case, here satellites are
generated by mimicking a Hernquist density profile (Hernquist 1990), through
the CDF

$P(r)=\frac{r^{2}}{(a+r)^{2}}.$ (2)

This model, where we fix the scale radius $a=5$ kpc, is dubbed as Hernquist5.

  * •

NFW model: we sample 2 Navarro-Frenk-White density profiles (Navarro et al.
1996), dubbed as NFW25 and NFW100, via the CDF

$\displaystyle P(r)$
$\displaystyle=\mathcal{N}(a)\left[\ln\left(\frac{a+r}{a}\right)-\frac{r}{a+r}\right]$
(3) $\displaystyle\mathcal{N}(a)$ $\displaystyle=\left[\ln\left(\frac{a+r_{\rm
cut}}{a}\right)-\frac{r_{\rm cut}}{a+r_{\rm cut}}\right]^{-1},$

with scale radii $a$ equal to 25 kpc and 100 kpc, respectively.
$\mathcal{N}(a)$ is a normalization constant with $r_{\rm cut}=D_{z=6.5}$.

  * •

CC function: companion galaxies are randomly spawned by following the cross-
correlation function quasar-galaxies
$\xi(r)=\left(\frac{r}{a}\right)^{\gamma}$, with $a=4.1$ comoving Mpc and
$\gamma=1.8$ from García-Vergara et al. (2019). In particular, we adopt the
CDF

$\displaystyle P(r)$
$\displaystyle=\mathcal{N}r^{3}\left[\frac{1}{3}+\frac{1}{3-\gamma}\left(\frac{a}{r}\right)^{\gamma}\right]$
(4) $\displaystyle\mathcal{N}$ $\displaystyle=r_{\rm
cut}^{3}\left[\frac{1}{3}+\frac{1}{3-\gamma}\left(\frac{a}{r_{\rm
cut}}\right)^{\gamma}\right]^{-1},$

where $\mathcal{N}$ is a normalization constant. This model – dubbed as CCF –
wants to mimic only the shape of the galaxy distribution function around
quasars. The density is not fixed and varies with $N$ in the Monte Carlo
simulations.

  * •

Cosmological simulations: finally, satellite galaxies are spawned by following
the dark-matter distribution of cosmological zoom-in simulations of high-
redshift quasars. In this work, we employ two different cosmological
simulations, dubbed here with CosmoSim1 (Barai et al. 2018) and CosmoSim2
(Valentini et al. 2021). In particular, satellite locations are randomly
selected amongst the dark-matter potential wells in a $z=6.5$ snapshot of each
simulation, after running AMIGA halo finder code (Knollmann & Knebe 2009) with
a minimum of 20 bound particles to define a halo. The dark-matter halos are
considered only if their distance from the centre of the main halo is, at
most, $D_{z=6.5}$. CosmoSim1 and CosmoSim2 represent our fiducial models, for
they have been derived from a complex framework specially designed to describe
the environment of high-$z$ quasars.333We note also that only cosmological
simulations, amongst our models, succeed in reproducing the complex
filamentary structures expected to form during the collapse of primordial
dark-matter fluctuations. Although both CosmoSim1 and CosmoSim2 describe the
evolution of two very massive dark-matter halos, their virial masses and radii
are quite diverse, being $M_{\rm vir}=3.3\times 10^{12}\leavevmode\nobreak\
\\!{\rm M}_{\odot}$, $R_{\rm vir}=66$ kpc and $M_{\rm vir}=1.2\times
10^{12}\leavevmode\nobreak\ \\!{\rm M}_{\odot}$, $R_{\rm vir}=47$ kpc at
$z=6$, for CosmoSim1 and CosmoSim2, respectively,444We define the virial mass
as $M_{\rm vir}=M_{200}=\frac{4\pi}{3}200\rho_{c}R^{3}_{200}$, where
$\rho_{c}$ is the critical density of the Universe and $R_{200}$ is the radius
enclosing 200 times $\rho_{c}$. and this increases the generality of the
investigation. We also note that the feedback prescriptions for the active
galactic nuclei are significantly different in the two simulations, and this
has a non-negligible impact on the dark-matter distribution (see the effect on
the merger history in Zana et al. 2022).

The models based on the Plummer, Hernquist, and NFW profiles adopt a set of
parameters ($a$ and $\mathcal{N}$) to force their CDFs to have $P(r)\gtrsim
0.99$ for $r=D_{z=6.5}$. In the rare occasion where a galaxy is spawned at
$r>D_{z=6.5}$, the extraction process is repeated. Thanks to this expedient,
the number density of companion galaxies within the spherical volume of radius
$D_{z=6.5}$ is directly comparable amongst all the models.

We note that these models do not claim to completely cover all the possible
satellite distributions, but rather aim to probe the implications of highly
diverse environments.

##  3 Results

Figure 3: Monte Carlo simulations for all the matter distribution models
adopted, i.e. Homogeneous, Plummer25, Plummer100, Hernquist5, NFW5, NFW15,
CCF, CosmoSim1, and CosmoSim2. Each panel shows the probability $P(N_{\rm
o}|N)$ of detection of $N_{\rm o}$ satellites, given $N$ satellites spawned.

Figure 3 shows the outcome of the Monte Carlo simulations for the
aforementioned nine satellite distributions. In most cases, the number of
detected companions is significantly smaller than $N$. For example, in the
fiducial distribution model CosmoSim1, with three seeded satellites, we
estimate $P(N_{\rm o}=0|N=3)=0.29$. This indicates that the probability of no
detection due to the ALMA FoV is $\sim 30\%$, despite the presence of three
satellites in the nearby environment. We also observe that satellite
distributions based on simulations CosmoSim1 and CosmoSim2 produce almost
identical results, although they refer to two distinct quasar hosts.

This hints at a major bias produced by the telescope observational volume that
could explain why observations of quasar satellites return so different
numbers in detecting serendipitous [CII] emitters.

The spherical geometry of NFW25 produces a very similar trace to the
simulation-oriented distributions and, therefore, could describe a similar
distribution of galaxies, on average. On the other hand, those distributions
where satellites are almost always spawned in close proximity to the quasar,
(i.e.Plummer25, or Hernquist5), result in a nearly one-to-one correspondence
between seeded and detected satellites. Interestingly, if we distribute a
variable number of companion galaxies by following an empirical cross-
correlation function quasar-LAEs (CCF), galaxies are generated farther from
the quasar and the detection efficiency drops at higher $N$. Finally,
Homogeneous and NFW100 yield similar outcomes to each other, even if they are
in principle based on very different distribution laws. It is worth mentioning
that Homogeneous represents a control scenario, with no additional conjecture
on the galaxy distribution.

The left panel of Figure 4 shows $P(N_{\rm o}\\!=\\!N)$ for all the probed
satellite populations. As the seeded population grows, it becomes increasingly
challenging to detect all the quasar satellites. $P(N_{\rm o}\\!=\\!N)$ is
always lower than 1 for $N>0$ and it decreases more rapidly the less compact
the distribution is. In Sec. 4, we discuss the possibility of sampling a
cylinder with a radius three times larger using ALMA. The right panel of
Figure 4 presents the result for that case, indicating a much higher detection
rate.

Figure 4: Probability of total detection, $P(N_{\rm o}=N)$, for all the
density profiles adopted in this work. Left: standard case of Figure 3, where
$R_{\rm FoV}\simeq 90\leavevmode\nobreak\ {\rm kpc}$. Right: mosaic-case with
$R_{\rm FoV}\simeq 270\leavevmode\nobreak\ {\rm kpc}$.

A general quasar environment is better quantified with a mean companion
number, rather than their precise count, given the natural variance of a
realistic system. Therefore, we introduce a further step to take into account
a scatter in the number of satellites that can be seeded. In particular, we
connect the number $N$ of seeded satellites to the average number $\langle
N\rangle$ of a Poissonian distribution555The use of a Poissonian distribution
in this context is a commonly made choice. We note that other scatter-laws
could be adopted with very similar results.

$P(N|\langle N\rangle)=\frac{\langle N\rangle^{N}}{N!}e^{-\langle N\rangle}.$
(5)

Hence, we can build the conditional probability

$P(N_{\rm o},N|\langle N\rangle)=P(N_{\rm o}|N)\cdot P(N|\langle N\rangle),$
(6)

and the likelihood function for the average seeded number $\langle N\rangle$,
given a single observation $N_{\rm o}$, $P(N_{\rm o}|\langle N\rangle)$:

$P(N_{\rm o}|\langle N\rangle)=\sum_{N}P(N_{\rm o},N|\langle N\rangle).$ (7)

Finally, we compute the distribution $\mathcal{P}(\langle N\rangle)$ by
including all the detections from V20 and considering them to be equiprobable.
Operatively, we multiply together the likelihood associated with every
detection and assume a flat prior over $\langle N\rangle$:

$\mathcal{P}(\langle N\rangle)=\frac{\prod_{N_{\rm o}}P(N_{\rm o}|\langle
N\rangle)^{f(N_{\rm o})}}{\sum_{N_{\rm o}}\left[\prod_{N_{\rm o}}P(N_{\rm
o}|\langle N\rangle)^{f(N_{\rm o})}\right]},$ (8)

where $f(N_{\rm o})$ is the frequency of detection reported in Figure 1. The
final likelihood functions, reproduced in Figure 5 for all the distribution
models, show that each model has a characteristic peak.

Figure 5: Posterior functions for the different galaxy distributions adopted
in this work. The curves show the probability that an average number of
massive satellites $\langle N\rangle$ is actually orbiting around a $z\gtrsim
6$ quasar, given the observed data from V20. The colour coding is the same as
Figure 4

As mentioned before, the CCF model predicts the highest number of average
intrinsic satellites in order to justify current observations. On the other
hand, compact distributions can describe observations with the smallest
possible number of satellites. In the specific cases of Plummer20 and
Hernquist20 – both sharply peaking around $\langle N\rangle=1$ – V20
observations with $N_{\rm o}=2$ or 3 would be explainable entirely through
Poissonian fluctuations.

##  4 Discussion

There is currently not enough observational data to constrain the spatial
distribution of galactic companions around high redshift quasars. For this
reason, we have explored various possibilities, each with different degrees of
realism and assumptions. While numerical simulations describe the most
realistic scenarios, the other radial profiles also offer a reasonable
representation of reality, especially due to the limited number of objects
involved. Apart from the CCF model, which, nevertheless, has its roots in
dedicated observational campaigns, all the spherically symmetric distributions
adopted in this work are the most commonly used radial laws in astrophysics to
model a gravitationally bound system, even if they are not directly connected
to the scales and masses studied here. Homogeneous is the model with the
fewest assumptions, where galaxies are randomly seeded in the surrounding
volume. The other spherically symmetrical models tend to refine the guess and
distribute the galaxies preferentially closer to the central quasar.
Plummer25, Plummer100, Hernquist5, NFW25, and NFW25 use different radial laws,
with different scale radii. Most compact distributions, such as Plummer25,
Plummer100, and Hernquist5 are less observationally motivated, especially at
high $N$, where numerous companions are spawned within the virial radius of
the quasar host, whereas the vast majority of V20 satellites are detected at
$r>100$ kpc. On the other hand, the spherical model CCF describes a relation
that is observed to hold for scales larger than $D_{z=6.5}$, beyond the limit
of $1000$ km s-1.

More consistent models, such as CosmoSim1, CosmoSim2, NFW25, and NFW100,
predict $\langle N\rangle\gtrsim 2$ on the basis of current observations. This
finding has numerous fundamental implications: the geometrical bias of ALMA
can explain the scatter in recent observations.666This is valid even before
the addition of the Poissonian scatter. In other words, we support the
scenario in which high-$z$ quasars have an average of two massive companions,
with [CII] luminosities $L_{\rm[CII]}\gtrsim 10^{8}\leavevmode\nobreak\
\\!{\rm L}_{\odot}$, but their detection depends on the volume probed by the
telescope and thus on the specific LoS of the observation.

This result suggests that, for those quasar fields in which no satellite has
been observed, mosaic observations with ALMA would result in the detection of
2 new galactic companions per quasar field, on average (in the cases described
by our fiducial models, $\langle N\rangle\simeq 2$). In the right panel of
Figure 4, we report the probability $P(N_{\rm o}\\!=\\!N)$, as a function of
the number of seeded satellites if the FoV is expanded to $R_{\rm FoV}\simeq
270\leavevmode\nobreak\ {\rm kpc}$, mimicking an observation with an
additional layer of 9 adjacent “standard” FoV, around the original one. For
CosmoSim1 and CosmoSim2, in particular, the probability of detecting the whole
population of 2 seeded satellites is 0.72, which is almost six times larger
than the probability estimated for a single ALMA pointing ($R_{\rm FoV}\simeq
90\leavevmode\nobreak\ {\rm kpc}$). If the intrinsic number of satellites $N$
were larger instead, $P(N_{\rm o}\\!=\\!N)\gtrsim 0.5$, up to $N\sim 6$.
Accordingly, follow-up priority should be given to those quasar fields which,
so far, have shown the least amount of companions, given the high probability
of detecting the whole missing population with an individual mosaic
observation.

A further consideration can be drawn if we examine our fiducial model which
predicts $\langle N\rangle\simeq 2$, being in very close agreement with the
twin model CosmoSim2. In Zana et al. (2022), we found that a total of 2-3
satellites were detectable, having a [CII] luminosity higher than a current
sensitivity threshold of about $10^{8}\leavevmode\nobreak\ \\!{\rm
L}_{\odot}$. These results were based on a post-processing study of a suite of
cosmological zoom-in simulations, including the evolution of baryons and
numerous state-of-the-art sub-grid prescriptions. In the present
investigation, we have generalized the dark-matter distribution of satellites
of such simulations (with no assumption on the mass of the halo) and
demonstrated that current observations (V20) agree remarkably well with our
previous prediction. As a consequence, the dense quasar-host environment
studied in Zana et al. (2022), and now confirmed by observations via our
geometrical interpretation, likely describes real-quasar systems and this
implies that most powerful high-$z$ quasars would live in the densest regions
of the Universe.

Finally, we provide an additional interpretation by comparing our predicted
number of satellites in the most realistic cases, with recent number-density
measurements in a field galaxy population. In the context of the large program
ASPECS, Decarli et al. (2020) and Uzgil et al. (2021) have reported no
detection of [CII]-emitters with $L_{\rm[CII]}>1.89\times
10^{8}\leavevmode\nobreak\ \\!{\rm L}_{\odot}$, in the redshift range $z\sim
6-8$, within a comoving volume of about $12500$ Mpc3, resulting in an upper
limit on the galaxy density of $3.4\times 10^{-4}$ Mpc-3. If we consider our
predictions to take place within the quasar-centered sphere with radius
$D_{z=6.5}$ (the volume enclosing an ALMA cylinder with every possible
orientation), our fiducial models lead to a density $\gtrsim 0.2$ Mpc-3, i.e.,
larger by almost a factor 600 with respect to the field environment probed by
ASPECS. Given that no [CII]-emitters have been found in the Hubble Ultra Deep
Field via ASPECS, this result still holds true even if we compute our detected
companions within a larger sphere, where the density could be reduced.

We conclude by observing that, the analysis conducted and its findings extend
beyond ALMA, as they can be applied to any instrument with a limited FoV, due
to the straightforward geometric nature of our study.

##  5 Summary and conclusions

Through various Monte Carlo simulations, we evaluated the geometrical bias of
the ALMA telescope on the detection rate of high-$z$ quasar satellites (see
Figure 3). We convolved the resulting probability with a set of Poissonian
distributions in order to estimate the intrinsic number of detectable
companions, which are orbiting around a given quasar potential well. We remark
that each detectable satellite represents a galaxy massive enough to be
detected as a [CII] emitter if it lies within the telescope geometry (see
Figure 2). We produced different likelihood functions for the average number
of orbiting satellites, in order to explain the most recent ALMA observations
(see Figure 1).

We have demonstrated that:

  1. (i)

The telescope bias can entirely explain the differences amongst the number of
detected high-$z$ quasar companions. Hence, every quasar could conceivably
reside within almost the same environment, but we would detect only a part of
the orbiting galaxies, depending on our line of sight.

  2. (ii)

If we consider the most physically motivated distribution profiles, e.g.,
CosmoSim1 and CosmoSim2, we can infer an intrinsic number of $\sim 2$
satellites, massive enough to be detected by ALMA as [CII] emitters with
$L_{\rm[CII]}\gtrsim 10^{8}\leavevmode\nobreak\ \\!{\rm L}_{\odot}$ and
orbiting within $\sim 1.3$ Mpc from the quasar.

  3. (iii)

Interestingly, we expect to discover more satellites, in the case of CosmoSim1
and CosmoSim2, via a simple mosaic observing campaign which targets those
quasars where no companion galaxies have been observed so far. We predict that
a single additional layer of ALMA FoVs around those objects would be enough to
observe the total population of $N\sim 2$ satellites with a probability of
0.72, compared to 0.13 in the single-observation case. In general, we expect
such a campaign to detect, half of the time, the total population of
companions, up to 6 satellites, i.e. $P(N_{\rm o}=N|N)\gtrsim 0.5$ for
$N\lesssim 6$.

  4. (iv)

Our predicted number of satellites ($ii$) in the case CosmoSim1 is compatible
with the analysis performed in Zana et al. (2022). In other words, ALMA
observations are consistent with cosmological hydro-dynamical simulations
evolving $z\sim 6$ quasars in the most massive dark-matter halos with $M_{\rm
vir}>10^{12}-10^{13}\leavevmode\nobreak\ \\!{\rm M}_{\odot}$, corresponding to
fluctuations of about $3-4\sigma$ in the density field.

  5. (v)

We compared our fiducial predictions with a recent survey from the ASPECS
program, where no [CII] emitters have been confirmed in the field above
$L_{\rm[CII]}=1.89\times 10^{8}\leavevmode\nobreak\ \\!{\rm L}_{\odot}$ in our
same redshift range, attesting once more the over-dense nature of high-$z$
quasar environments.

Future additional ALMA detections of quasar companions will increase our
current sample, allowing us to tune this model further and better constrain
our predictions. A contribution may also come from the AtLAST telescope
(Klaassen et al. 2020) which, despite its low angular resolution, can detect
high-$z$ satellites at a greater distance compared to ALMA. Follow-up
observations of V20 galaxies and the possible detection of supplementary lines
could confirm the status of companion or update the catalogues of quasar
satellites. We note, however, that the Poissonian scatter included in our
analysis takes into account such potential small variations. Moreover, the
next deep JWST (Gardner et al. 2023) observational campaigns of quasar
environments may also help to shed light on the spatial distribution of
satellite galaxies, given the much larger FoV. However, the galactic tracers
probed will be completely different with respect to ALMA and will be subjected
to dust extinction. For this reason, a new and appropriate set of predictions
is required.

###### Acknowledgements.

Funded by the European Union (ERC, WINGS, 101040227). Views and opinions
expressed are however those of the authors only and do not necessarily reflect
those of the European Union or the European Research Council Executive Agency.
Neither the European Union nor the granting authority can be held responsible
for them. TZ and VA acknowledge support from INAF-PRIN 1.05.01.85.08. The
authors greatly thank the anonymous referee for useful comments which improved
the quality of this manuscript.

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