# Averaging with the Divisor Function: $\ell^{p}$-improving and Sparse Bounds

Christina Giannitsi  Department of Mathematics, Georgia Institute of
Technology, Atlanta GA 30332, USA<EMAIL_ADDRESS>
###### Abstract.

We study averages along the integers using the divisor function $d(n)$,
defined as

$K_{N}f(x)=\frac{1}{D(N)}\sum_{n\leq N}d(n)\,f(x+n),$

where $D(N)=\sum_{n=1}^{N}d(n)$. We shall show that these averages satisfy a
uniform, scale free $\ell^{p}$-improving estimate for $p\in(1,2)$, that is

$\left(\frac{1}{N}\sum|K_{N}f|^{p^{\prime}}\right)^{1/p^{\prime}}\lesssim\left(\frac{1}{N}\sum|f|^{p}\right)^{1/p}$

as long as $f$ is supported on $[0,N]$.

We will also show that the associated maximal function
$K^{*}f=\sup_{N}|K_{N}f|$ satisfies $(p,p)$ sparse bounds for $p\in(1,2)$,
which implies that $K^{*}$ is bounded on $\ell^{p}(w)$ for $p\in(1,\infty)$,
for all weights $w$ in the Muckenhoupt $A_{p}$ class.

Research supported in part by grant from the US National Science Foundation,
DMS-1949206.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Approximating Multipliers
  4. 4 Fixed Scale
  5. 5 Sparse Bounds

##  1\. Introduction

We establish $\ell^{p}$-improving and sparse bounds for the discrete averages
formed from the divisor function. Let $d(n)=\sum_{d|n}1$ be the usual divisor
function, $D(N)=\sum_{n=1}^{N}d(n)$, and consider the following averaging
operator

$K_{N}f=\frac{1}{D(N)}\sum_{n\leq N}d(n)\,f(x+n)$

and the associated maximal function

$K^{*}f=\sup_{N}|K_{N}f|.$

Let us first establish some notation we shall be using throughout the paper.
For two quantities $a$ and $b$, we shall write $a\lesssim b$ if there exists a
positive constant $C$ such that $a\leq C\,b$. We shall write $a\lesssim_{p}b$
if they implied constant depends on $p$. For a function $f$ on the integers,
and an interval $I\subset\mathbb{Z}$, we write

$\langle f\rangle_{I,p}=\left(\frac{1}{|I|}\,\sum_{x\in
I}|f(x)|^{p}\right)^{1/p}$

Also for an interval $I=[a,b]\cap\mathbb{Z}$ we will write
$2I=[2a-b-1,b]\cap\mathbb{Z}$, $3I=[2a-b-1,2b-a+1]\cap\mathbb{Z}$ for the
doubled and tripled interval respectively. We shall also use $\widehat{f}$ or
$\mathcal{F}f$ for the Fourier transform of $f$, defined as

$\mathcal{F}f(\theta)=\sum_{x\in\mathbb{Z}}f(k)e^{-2\pi ix\theta},$

and $\check{f}$ or $\mathcal{F}^{-1}$ for the inverse Fourier transform,
defined as

$\mathcal{F}^{-1}f(x)=\int_{0}^{1}\widehat{f}(\theta)e^{2\pi
ix\theta}\,d\theta.$

Finally, if $1\leq p\leq\infty$, let $p^{\prime}$ denote its conjugate
exponent, that is $\dfrac{1}{p}+\dfrac{1}{p^{\prime}}=1.$

Our two main results are a scale free, $\ell^{p}$ improving inequality, and a
sparse bound for the maximal function $K^{*}$. More specifically,

######  Theorem 1.1.

For $p\in(1,2)$, there exists $C_{p}>0$ such that for all positive integers
$N$ and functions $f$ supported on an interval $E$ of length $N$, there holds

(1.2) $\langle K_{N}f\rangle_{E,p^{\prime}}\leq C_{p}\,\langle f\rangle_{E,p}$

Now a collection $\mathcal{S}$ of intervals is called sparse if for every
$I\in\mathcal{S}$ there exists a set $E_{I}\subset I$ so that $|E_{I}|>|I|/2$
and $E_{I}$ for $I\in\mathcal{S}$ are pairwise disjoint.

######  Theorem 1.3.

For $r,s\in(1,2)$, the maximal operator $K$ is of $(r,s)$-sparse type, that is
there exists $C>0$ such that for all compactly supported functions $f$ and
$g$, there exists a sparse collection $\mathcal{S}$ so that the
$\ell^{2}(\mathbb{Z})$ inner product satisfies

(1.4) $|(K^{*}f,g)|\leq C\sum_{I\in\mathcal{S}}|I|\langle
f\rangle_{2I,r}\langle g\rangle_{I,s}$

The sparse result of Theorem 1.3, implies not only $\ell^{p}$ boundedness for
the maximal operator, but also weighted inequalities for weights $w$ for which
the ordinary maximal function is bounded, which is a new result. Specifically,

######  Corollary 1.5.

For any $p\in(1,\infty)$ and all weights $w$ in the Muckenhoupt class $A_{p}$,
the maximal operator $K^{*}:\ell^{p}(w)\to\ell^{p}(w)$ is a bounded operator.

The study of $\ell^{p}$ improving bounds has been an area of particular
interest in Harmonic Analysis for over five decades, since the work of Littman
[Lit71] and Strichartz [Str70]. Operators, and families of operators can
improve upon the $\ell^{p}$ norm (or the $L^{p}$ one in the continuous case)
by achieving bounds over higher $p$-values. In certain cases those bounds are
scale free, meaning that the implied constant does not depend on the scale of
the operator, however the starting point is usually fixing the scale. Of
course, as mentioned above, the interest in sparse bounds for operators stems
from the fact that they immediately imply weighted $\ell^{p}$ inequalities,
and is an active research topic since it was first introduced by Lerner in
[Ler13]. There has been a plethora of papers focusing on the subject of sparse
bounds and $\ell^{p}$ improving inequalities in the recent years, like [CKL19]
and [Hug20, KL18, Kes18a, Kes18b, KLM20, KLM19, And20] and followed by [HLY19,
HKLY20, HKL+19]. More recent results include [DDL20] and [DHV20].

Our approach is inspired by [HKLY20] and [HLY19]. To prove the estimates, we
use the same two techniques as in the aforementioned papers, adjusted to fit
our particular setting. Particularly, we decompose our operator into a High
pass and a Low pass term [Ion04, Hug17, KLM20, HLY19, HKLY20]. This
decomposition is key to proving both the $\ell^{p}$ improving and the sparse
result, although the sparse decomposition is somewhat different. In both
cases, Ramanujan sums have an important role to play in estimating the low
pass term, and so do results from the number-theoretic literature, involving
exponential sums using the divisor function, [Jut84]. We should also mention
the work in [CW14], where they study ergodic theorems using the divisor
function.

Acknowledgment. The author would like to thank Michael Lacey for his
continuous guidance, support, and encouragement.

##  2\. Preliminaries

Recall the major and minor arcs decomposition. Let $\mathbb{A}_{Q}=\\{1\leq
A<Q\,:(A,Q)=1\\}$ denote the multiplicative group associated with an integer
$Q>1$. For $s\geq 1$ consider the following sets

$\mathcal{R}_{s}=\left\\{\frac{A}{Q}\in[0,1)\,:A\in\mathbb{A}_{Q},\;\;2^{s-1}\leq
Q<2^{s}\right\\}$

For $0<\varepsilon\leq 1/4$ and $\frac{A}{Q}\in\mathcal{R}_{s}$, with $s\leq
j\varepsilon$, we define the j-th major arc at $A/Q$ as

$\mathfrak{M}_{j}\left(\frac{A}{Q}\right)=\left\\{\frac{A}{Q}+\eta:\;|\eta|<2^{(\varepsilon-2)j}\right\\}$

Those are disjoint for $\varepsilon$ small enough, for instance when
$2^{(\varepsilon-2)j}<\frac{1}{10\,Q}$. The $j$-th major arcs are given by
$\mathfrak{M}_{j}=\bigcup_{\frac{A}{Q}\in\mathcal{R}_{s}}\mathfrak{M}_{j}(A/Q)$.
We define the $j$-th minor arcs $\mathfrak{m}_{j}$ as the complement of
$\mathfrak{M}_{j}$.

Also recall the Ramanujan sums, defined as

(2.1) $c_{Q}(n)=\sum_{A\in\mathbb{A}_{Q}}e(An/Q),$

where $e(x):=e^{2\pi ix}$.

We present the following lemma, based on an important property of Ramanujan
sums which is due to Bourgain [Bou93]. An alternate proof can be found in
[KLM19, Lemma 3.13]. Our proof is fairly simple and follows the same steps as
[HKLY20, Lemma 3.4].

######  Lemma 2.2.

For any $\varepsilon>0$ and integer $k>1$, uniformly in $N>J^{k}$ for some
integer $J$, there holds

(2.3)
$\left(\frac{1}{N}\sum_{|x|<N}\left|\sum_{Q=1}^{J}\frac{c_{Q}(x)}{Q}\right|^{k}\right)^{1/k}\lesssim_{k,\varepsilon}J^{\varepsilon}.$

###### Proof.

We give an outline of the proof. Bourgain’s estimate [Bou93, (3.44)], tells us
that for any two positive integers $J,\,N$

$\displaystyle\left[\frac{1}{N}\sum_{|x|<N}\left|\sum_{J\leq
Q<2J}|c_{Q}(x)|\right|^{k}\right]^{1/k}$ $\displaystyle\lesssim
J^{1+\frac{\varepsilon}{\log 2}}$

So let $m_{0}$ be an integer such that $2^{m_{0}}\leq J<2^{m_{0}+1}$. Using
this estimate, we have

$\displaystyle\frac{1}{N}\sum_{|x|<N}\left|\sum_{Q=1}^{J}\frac{|c_{Q}(x)|}{Q}\right|^{k}$
$\displaystyle\leq(m_{0}+1)^{k-1}\sum_{m=0}^{m_{0}}\frac{1}{N}\sum_{|x|<N}\left|\sum_{2^{m}\leq
Q<2^{m+1}}\frac{|c_{q}(x)|}{Q}\right|^{k}$ $\displaystyle\lesssim(\log
J)^{k-1}\sum_{m=0}^{m_{0}}2^{\varepsilon km/\log 2}$ $\displaystyle\lesssim
J^{k\varepsilon}$

This completes the proof. ∎

Let us now turn our attention to the divisor function. The asymptotics of the
divisor function are already well known. If $N,n>3$ and $d(n)$ is the divisor
function then

(2.4) $\log d(n)\lesssim\frac{\log n}{\log\log n}=o(\log n)$ (2.5)
$D(N)=N\,\log N+(2\gamma-1)N+O(\sqrt{N})=O(N\log N)$

where $\gamma$ is Euler’s constant.

The kernel of our operator has been extensively studied in the number theory
literature. We shall be using results discussed in [Jut84] and [CW14].
Particularly, the following is Lemma 5.4 from [CW14], for $P_{N}=N^{1/2}$ and
$Q_{N}=N^{1-\varepsilon}$. We have also normalized the estimate by dividing by
$D(N)$.

######  Lemma 2.6.

Suppose $N\geq 1$ and $0<\varepsilon\ll 1$. Let $\theta\in[0,1]$ be such that
for every $1\leq Q\leq\sqrt{N}$ and every $A\in\mathbb{A}_{Q}$,
$|\theta-A/Q|>N^{-1+\varepsilon}$. Then the following is true.

(2.7) $\frac{1}{D(N)}\left|\sum_{1\leq n\leq
N}d(n)e(n\theta)\right|=O(N^{-\varepsilon})$

Now let

$\widehat{\Gamma}_{N}(\theta)=\frac{1}{N}\sum_{n=0}^{N-1}e(n\theta)$

denote the Fourier transform of the usual averages,

$\Gamma_{N}f(x)=\frac{1}{N}\sum_{n=1}^{N}f(x+n).$

It is a well known fact that these averages satisfy

(2.8)
$|\widehat{\Gamma}_{N}(\theta)|\lesssim\min\left\\{1,\frac{1}{N\,|\theta|}\right\\}$

for $|\theta|<1/2$. With that notation we have the following result.

######  Lemma 2.9.

Suppose $N\geq 2$, $Q\leq\sqrt{N}$ and $0<\varepsilon\ll 1$. Let $\gamma$
denote Euler’s constant.
There exists a constant $C$ such that for all $A\in\mathbb{A}_{Q}$,

(2.10 a) $\left|\sum_{n\leq N}d(n)e\left(nA/Q\right)-2\frac{N\,(\log N-2\log
Q+2\gamma-1)}{Q}\right|\,\leq\,C\,N^{\frac{2}{3}+\varepsilon}$

There exists a constant $C$, such that if
$0<\left|\eta\right|<\frac{1}{Q\sqrt{N}}$, then for all $A\in\mathbb{A}_{Q}$,

(2.10 b) $\displaystyle\left|\sum_{n\leq
N}d(n)e\Big{(}n\big{(}\frac{A}{Q}+\eta\big{)}\Big{)}-\frac{e(\eta)-1}{\pi
i\eta}\cdot\frac{N\,(\log N-2\log
Q+2\gamma-1)\,\widehat{\Gamma}_{N}(\eta)}{Q}\right|\,\leq\,C\,N^{\frac{2}{3}+\varepsilon}$

There exists a constant $C$, such that if
$0<\left|\theta\right|<\frac{1}{Q\sqrt{N}}$, then

(2.10 c) $\left|\sum_{n\leq N}d(n)e\left(n\theta\right)-\frac{e(\theta)-1}{\pi
i\theta}\cdot\frac{N}{Q}\,(\log
N+2\gamma-1)\,\widehat{\Gamma}_{N}(\theta)\right|\,\leq
C\,N^{\frac{2}{3}+\varepsilon}$

Lemma 2.9 is a direct result of Theorem 5 of [Jut84], along with Remarks 1, 2,
and equation (1.4) of the same paper, as well as the fact that

$\displaystyle\sum_{n=0}^{N-1}e(-n\eta)=\frac{e(N\eta)-1}{e(\eta)-1}.$

######  Remark 2.11.

We point out that Lemma 2.9 is an estimate of the sum before normalization.
Therefore, using (2.5), the implied estimate for the differences is of the
order $N^{-\frac{1}{3}+\varepsilon}$.

######  Remark 2.12.

If we look at the Taylor expansion of $\frac{e(\theta)-1}{\pi i\theta}$, we
can easily see that

$\left|\frac{e(\theta)-1}{\pi i\theta}-2\right|\lesssim|\theta|.\\\ $

######  Remark 2.13.

Note that by (2.5) we have that $|N(\log N+2\gamma-1)-D(N)|\lesssim\sqrt{N}$.

##  3\. Approximating Multipliers

We consider a Schwartz function $\chi$ that satisfies
$\mathbbm{1}_{[-\frac{1}{8},\frac{1}{8}]}\leq\chi\leq\mathbbm{1}_{[-\frac{1}{4},\frac{1}{4}]}$,
and its dilations $\chi_{s}(\theta)=\chi(2^{3s}\theta)$. Now consider the
following multipliers, for $s\geq 1$ and $2^{s}\leq Q<2^{s+1}$.

(3.1a) $\displaystyle\widehat{L}_{1,N}(\theta)$
$\displaystyle=2\,\widehat{\Gamma}_{N}(\theta)\chi_{1}(\theta)$ (3.1b)
$\displaystyle\widehat{L}_{Q,N}(\theta)$
$\displaystyle=\frac{2}{Q}\sum_{A\in\mathbb{A}_{Q}}\,\widehat{\Gamma}_{N}(\theta-A/Q)\chi_{s}(\theta-A/Q)$

Using the definition of Ramanujan sums $c_{Q}(x)$ in (2.1), we calculate the
inverse Fourier transform of our multipliers.

######  Lemma 3.2.

With the definition of (3.1) we have

(3.3a) $\displaystyle L_{1,N}(x)$
$\displaystyle=2\,\Gamma_{N}*\check{\chi}_{1}(x)$ (3.3b) $\displaystyle
L_{Q,N}(x)$
$\displaystyle=\dfrac{2}{Q}\,c_{Q}(x)\,\big{(}\Gamma_{N}*\check{\chi}_{s}\big{)}(x)$

###### Proof.

We compute

$\displaystyle L_{1,N}(x)$
$\displaystyle=2\int_{\mathbb{T}}\widehat{\Gamma}_{N}(\theta)\chi_{1}(\theta)\,e(x\theta)\,d\theta=2\Gamma_{N}*\check{\chi}_{1}$
$\displaystyle L_{Q,N}(x)$
$\displaystyle=\dfrac{2}{Q}\,\sum_{A\in\mathbb{A}_{Q}}\int_{\mathbb{T}}\,\widehat{\Gamma}_{N}(\theta-A/Q)\,\chi_{s}(\theta-A/Q)\,e(\chi\theta)\,d\theta$
$\displaystyle=\dfrac{2}{Q}\,\sum_{A\in\mathbb{A}_{Q}}e(xA/Q)\int_{\mathbb{T}}\,\,\widehat{\Gamma}_{N}(\theta)\,\chi_{s}(\theta)\,e(\chi\theta)\,d\theta$
$\displaystyle=\dfrac{2}{Q}\,\sum_{A\in\mathbb{A}_{Q}}e(xA/Q)\,\Gamma_{N}*\check{\chi}_{s}$
$\displaystyle=\dfrac{2}{Q}\,c_{Q}(x)\,\Gamma_{N}*\check{\chi}_{s}$

∎

######  Theorem 3.4.

Let $N>3$ and $0<\varepsilon<1/2$. If $1<P\leq\sqrt{N}$ then

(3.5) $\widehat{K}_{N}=\sum_{Q=1}^{P}\widehat{L}_{Q,N}\,+\widehat{r}_{N,P},$

where $\|\widehat{r}_{N,P}\|_{\infty}\lesssim P^{-\varepsilon}$.

###### Proof.

The first thing we need to note is that, by construction, our multipliers
$\widehat{L}_{Q,N}$ are supported on disjoint intervals, centered at rationals
$A/Q$, for $2^{s}\leq Q<2^{s+1}$, $s\geq 1$, and $A\in\mathbb{A}_{Q}.$ Using
(2.8), we have that

(3.6) $\left\|\sum_{2^{s}\leq
Q<2^{s+1}}\widehat{L}_{Q,N}\right\|_{\infty}\lesssim 2^{-s}$

We first show the theorem holds when $P=\sqrt{N}$. Let us fix $\varepsilon>0$
and $\theta\in\mathbb{T}$. We distinguish three cases.

First Case. Suppose there exists $Q\leq\sqrt{N}$, such that for some
$A\in\mathbb{A}_{Q}$, $\theta$ satisfies $|\theta-A/Q|<N^{-1+\varepsilon}$.
Then for any other $B\in\mathbb{A}_{Q}$ with $B\neq A$, we have
$\chi_{s}(\theta-B/Q)=0$, while $\chi_{s}(\theta-A/Q)=1$. This means that all
the other terms in the sum of our approximating multipliers vanish, and we are
only left with

$\widehat{L}_{Q,N}(\theta)=\dfrac{2}{Q}\,\widehat{\Gamma}_{N}(\theta-A/Q)$

Notice that using (2.8), and Remarks 2.11, 2.12 and 2.13, we see that

$\displaystyle\Big{|}\widehat{L}_{Q,N}(\theta)-\frac{e(\theta-A/Q)-1}{\pi
i(\theta-A/Q)}\cdot$ $\displaystyle\frac{N\,(\log N-2\log Q+2\gamma-1)}{Q\cdot
D(N)}\,\widehat{\Gamma}_{N}(\theta-A/Q)\Big{|}$
$\displaystyle\lesssim|\theta-A/Q|+2\frac{2\log Q+\sqrt{N}}{Q\,D(N)}$
$\displaystyle\lesssim N^{-1/2}$

Using this estimate, part (b) of Lemma 2.9 and the triangle inequality,

(3.7) $|\widehat{K}_{N}(\theta)-\widehat{L}_{Q,N}(\theta)|\lesssim
N^{-\frac{1}{3}+\varepsilon}.$

For the rest of the $Q^{\prime}\leq\sqrt{N}$, $Q^{\prime}\neq Q$, we have that
for all $A^{\prime}\in\mathbb{A}_{Q^{\prime}}$,
$|\theta-A^{\prime}/Q^{\prime}|>N^{-1+\varepsilon}$, so using (2.8) we get

$\left|\widehat{L}_{Q^{\prime},N}(\theta)\right|\lesssim\frac{N^{1-\varepsilon}}{Q^{\prime}\,N}$

Summing over all such $Q^{\prime}$, we get

(3.8) $\sum_{\begin{subarray}{c}1\leq Q^{\prime}\leq\sqrt{N}\\\ Q^{\prime}\neq
Q\end{subarray}}\left|\widehat{L}_{Q^{\prime},N}(\theta)\right|\lesssim
N^{-\varepsilon}$

Using (3.6), (3.7), and (3.8) we can get the desired bound for this choice of
$\theta$. It is worth mentioning that the value of $\varepsilon$ might change
from one line to the next, but as always it denotes a small positive number.

Second Case. This is the case where $|\theta|\leq N^{-1+\varepsilon}$. This
case works exactly like the previous one, except we use part (a) of the
definition of the multipliers, equation (3.1), and parts (a) and (c) of the
estimate in Lemma 2.9.

Third Case. Our last case is when $\theta$ does not meet any of the criteria
of the previous two cases. This estimate is similar to our estimate of (3.8).
Indeed, using (2.8), and the fact that now $|\theta-A/Q|>N^{-1+\varepsilon}$
for all choices of $Q\leq\sqrt{N}$ and $A\in\mathbb{A}_{Q}$, we get that

(3.9) $\sum_{1\leq
Q\leq\sqrt{N}}\left|\widehat{L}_{Q,N}(\theta)\right|\lesssim\sum_{1\leq
Q\leq\sqrt{N}}\frac{N^{-\varepsilon}}{Q}\lesssim N^{-\varepsilon}$

This, combined with (2.7) concludes this case. This implies that
$\widehat{K}_{N}=\sum_{Q=1}^{\sqrt{N}}\widehat{L}_{Q,N}\,+\widehat{r}_{N,\sqrt{N}},$
where
$\|\widehat{r}_{N,\sqrt{N}}\|_{\infty}\lesssim(\sqrt{N})^{-\varepsilon}$.

Now suppose that $P<\sqrt{N}$. Then we see that
$\widehat{K}_{N}=\sum_{Q=1}^{P}\widehat{L}_{Q,N}\,+\widehat{r}_{N,P},$ where

$\widehat{r}_{N,P}=\sum_{Q=P+1}^{\sqrt{N}}\widehat{L}_{Q,N}+\widehat{r}_{N,\sqrt{N}}$

Using the same argument as in (3.6), we see that

$\displaystyle\left\|\sum_{Q=P+1}^{\sqrt{N}}\widehat{L}_{Q,N}\right\|_{\infty}\leq\max_{P+1\leq
Q\leq N}|\widehat{L}_{Q,N}|\leq\frac{1}{P+1}\lesssim P^{-\varepsilon}.$

This, combined with the fact that
$\|\widehat{r}_{N,\sqrt{N}}\|_{\infty}\leq(\sqrt{N})^{-\varepsilon}\lesssim
P^{-\varepsilon}$ completes the proof. ∎

##  4\. Fixed Scale

We will now focus on a fixed scale estimate, so let us fix an $N$, let $E$ be
an interval such that $|E|=N$ and fix $p\in(1,2)$. Our goal is to show that if
$f=\mathbbm{1}_{F}$ is supported on $E$ and $g=\mathbbm{1}_{G}$ is supported
on $E$ then

(4.1) $\frac{1}{N}(K_{N}f,g)\leq C_{p}\,\langle f\rangle_{E,p}\,\langle
g\rangle_{E,p},$

where $C_{p}$ is positive constant independent of $N$. The proof of Theorem
1.1 follows immediately by an additional elementary argument.

Notice that it is trivial to obtain the following bound using (2.4):

(4.2) $\frac{1}{N}(K_{N}f,g)\lesssim\,2N^{\delta}\,\langle
f\rangle_{E,1}\,\langle g\rangle_{E,1},$

for a fixed $0<\delta<\frac{(p-1)^{2}}{p}$. While this seems an arbitrary
choice, it is a bound that is required in the proof of Lemma 4.4. Note that
for $p\in(1,2)$ this is a reasonable bound for $\delta$, and highlights how
the endpoint $p=1$ would be problematic, as it would require $\delta=0$. This
immediately implies that if

$2\,N^{\delta}\,\langle f\rangle_{E,1}^{1/p^{\prime}}\,\langle
g\rangle_{E,1}^{1/p^{\prime}}\leq 1$

then our fixed scale estimate holds, in which case we are done. So suppose the
result fails. That means both of the following conditions are true

(4.3) $\displaystyle\langle f\rangle_{E,1}\geq
2^{-p^{\prime}}N^{-\delta\,p^{\prime}}$ $\displaystyle\langle
g\rangle_{E,1}\geq 2^{-p^{\prime}}N^{-\delta\,p^{\prime}}.$

We shall show that (4.1) still holds by using an auxiliary result, the high-
low decomposition. While the idea for this decomposition has been around for a
long time, our inspiration stems from [Bou99].

######  Lemma 4.4.

Let $p\in(1,2)$. Then there exists $N_{p}>0$ such that for all $N>N_{p}$ and
$1\leq m\leq N^{1/p^{\prime}}$ we can decompose $K_{N}f=H_{m}+L_{m}$ where

$\displaystyle\langle H_{m}\rangle_{E,2}\lesssim m^{-1/p}\,\langle
f\rangle^{1/2}_{E,1}$ $\displaystyle\langle L_{m}\rangle_{E,\infty}\lesssim
m^{1/p^{\prime}}\,\langle f\rangle_{E,1}^{1/p}$

######  Remark 4.5.

We can explicitly define $N_{p}$ as the smallest positive integer that
satisfies

(4.6) $2\leq N^{\frac{(p-1)^{2}}{p}-\delta}$

for the same value of $\delta>0$ that we have chosen for our trivial estimate
in equation (4.2).

Using this lemma we get that

$\frac{1}{N}(K_{N}f,g)\lesssim m^{-1+\frac{1}{p^{\prime}}}\,(\langle
f\rangle_{E,1}\langle g\rangle_{E,1})^{1/2}+m^{1/p^{\prime}}\,\langle
f\rangle_{E,1}^{1/p}\langle g\rangle_{E,1}$

Optimizing over $m$, that is $m\sim\langle
f\rangle_{E,1}^{\frac{1}{2}-\frac{1}{p}}\,\langle
g\rangle_{E,1}^{-\frac{1}{2}}$, and plugging this back to our estimate
completes the proof of (4.1). We point out that this value of $m$ is an
allowed choice within range thanks to (4.3), (4.6), and the way we have chosen
$\delta$. Indeed

$\displaystyle m$ $\displaystyle\leq(2^{-p^{\prime}}N^{-\delta
p^{\prime}})^{-\frac{1}{p}}\leq
N^{\frac{p-1}{p}-\frac{\delta}{p-1}+\frac{\delta}{p-1}}=N^{\frac{1}{p^{\prime}}}.$

Thus all we are missing is the proof of the Lemma.

###### Proof of lemma 4.4.

We shall decompose our operator using the decomposition of Theorem 3.4 for
$P=m$ and $\varepsilon=1/p$. Then set
$H_{m}=\mathcal{F}^{-1}(\widehat{r}_{N,m}\,\widehat{f})$, as in (3.5). The
desired $\ell^{2}$-estimate follows immediately using Theorem 3.4.

It is now obvious that $L_{m}$ has to be defined as the remaining
approximating multipliers
$L_{m}=\mathcal{F}^{-1}\left(\widehat{K}_{N}f-\widehat{H}_{m}\right)$. Note
that from Young’s convolution inequality we get

$\displaystyle|\Gamma_{N}*\check{\chi}_{s}|\lesssim N^{-1}.$

Using Lemma 3.2, we can therefore compute

$\displaystyle|L_{m}(x)|$
$\displaystyle=\left|\sum_{Q=1}^{m}L_{Q,N}*f(x)\right|$
$\displaystyle\lesssim\sum_{Q=1}^{m}\sum_{y=1}^{N}\frac{|c_{Q}(y)|}{Q}|\Gamma_{N}*\check{\chi}_{s}\,(y)|f(x-y)$
$\displaystyle\lesssim\frac{1}{N}\sum_{Q=1}^{m}\sum_{y=1}^{2N}\frac{|c_{Q}(y)|}{Q}f(x-y)$
$\displaystyle\lesssim\left(\frac{1}{2N}\sum_{y=1}^{2N}\left|\sum_{Q=1}^{m}\frac{|c_{Q}(y)|}{Q}\right|^{p^{\prime}}\right)^{1/p^{\prime}}\left(\frac{1}{2N}\sum_{y=1}^{2N}f(x-y)\right)^{1/p}$
by Hölder’s Inequality $\displaystyle\lesssim m^{1/p^{\prime}}\langle
f\rangle_{E,1}^{1/p}$

The last inequality comes from Lemma 2.2 with $M=2N$, $J=m$, $k=p^{\prime}$,
and $\varepsilon=1/p^{\prime}$. Note that this estimate requires $N>N_{p}$. ∎

##  5\. Sparse Bounds

Let us now turn our attention to the $(p,p)$-sparse bound for $p\in(1,2)$, and
specifically the proof of Lemma 5.2. A recursive argument can then complete
the proof of the sparse bound in Theorem 1.3. This recursive argument can be
found in [KLM20], in the proof of Theorem 1.2, using Lemma 2.1. A similar
treatment of the proof can be found in [HLY19]. As we have an open condition,
we can again focus our attention on indicator functions. One can easily
generalize to all functions using Lemma 4.1 from [KLM20].

Suppose $E$ is an interval of length $2^{n_{0}}$. Let $f=\mathbbm{1}_{F}$ be
supported on $E$ and $g=\mathbbm{1}_{G}$ be supported on $G\subseteq E$. Now,
consider a choice of stopping time $\tau:E\to\\{2^{n}\,:\,1\leq n\leq
n_{0}\\}.$ Our goal is to prove an estimate for a particular type of stopping
times, the definition of which can also be found in [HKLY20].

######  Definition 5.1.

A stopping time $\tau$ is admissible if and only if for any interval
$I\subseteq E$ such that $\langle f\rangle_{3I,1}>100\langle f\rangle_{E,1}$,
there holds $\inf\\{\tau(x)\,:\,x\in I\\}>|I|$.

######  Lemma 5.2.

For all admissible stopping times, and for all $1<p<2$ we have that

(5.3) $\left(K_{\tau}f,g\right)\lesssim\left(\langle f\rangle_{E,1}\;\langle
g\rangle_{E,1}\right)^{1/p}\,|E|$

The proof of Lemma 5.2 follows the idea of [HKLY20]. We will again use the
auxiliary high-low construction, slightly modified this time. Once we have the
individual estimates, the final result can be obtained using the same argument
as in the previous section, and is therefore omitted. For integers $M=2^{m}$,
we decompose $K_{\tau}f\leq H+L$ where

(5.4) $\displaystyle\langle H\rangle_{E,2}$ $\displaystyle\lesssim
M^{-\frac{1}{p}+\epsilon}\langle f\rangle^{1/2}_{E,1}$ (5.5)
$\displaystyle\langle L\rangle_{E,\infty}$ $\displaystyle\lesssim
M^{\frac{1}{p^{\prime}}}\langle f\rangle^{1/p}_{E,1}$

for some $\epsilon>0$. Note that while this is a slightly different estimate
that the one in the previous section, however it does not change our
computations at all, since $\epsilon$ can be chosen to be appropriately small
depending on $M$ so that all our calculations still hold.

In this case, the trivial bound is given by

$\langle
K_{\tau}f\rangle_{E,\infty}\lesssim\sup_{x}\tau^{\delta}(x)\,\cdot\,\langle
f\rangle_{3E,1}.$

for some $\delta>0$. We pick $\delta$ so that $2(p^{\prime}+1)\delta\ll 1/2$.
It is then obvious that we need only study the operator on a set
$D=\\{x\,:\,\tau^{\delta}(x)\geq C_{p}\cdot M^{1/p^{\prime}}\\}$, where
$C_{p}$ is a constant that depends only on $p$. Outside our set, the result
holds as a consequence of the trivial bound.

We need to know what happens to $K_{\lfloor n^{1/\delta}\rfloor}f$ for
$n>C_{p}M^{1/p^{\prime}}$. For convenience, let $\ell=n^{\delta
p^{\prime}/\varepsilon_{1}}$, where $\varepsilon_{1}=2(p^{\prime}+1)\delta\ll
1$ by the way we chose $\delta$. Therefore, $\ell\leq\sqrt{n}$, and by Theorem
3.4 we get

$\displaystyle\widehat{K}_{\lfloor n^{1/\delta}\rfloor}=\sum_{1\leq
Q\leq\lfloor\ell^{1/\delta}\rfloor}\widehat{L}_{Q,\lfloor
n^{1/\delta}\rfloor}+\widehat{r}_{\lfloor
n^{1/\delta}\rfloor,\lfloor\ell^{1/\delta}\rfloor}$

with $N=\lfloor n^{1/\delta}\rfloor$, $P=\lfloor\ell^{1/\delta}\rfloor$, and
$\varepsilon=\varepsilon_{1}$. This means that $\|\widehat{r}_{\lfloor
n^{1/\delta}\rfloor,\lfloor\ell^{1/\delta}\rfloor}\|_{\infty}\lesssim\ell^{-\varepsilon_{1}/\delta}\lesssim
n^{-p^{\prime}}$, by the way we chose $\ell$.

The high-low decomposition for this argument is a bit more involved.
Particularly, when we decompose the operator into the approximating
multipliers and the remainder, the remainder goes to the high pass term, as it
did in the previous section. However, in this case we need to break the
approximating multipliers further, into two pieces, one of which will become
the low-pass term, and the other will be added in the high-pass term. That is,
our decomposition for the operator now becomes $K_{\tau}\leq H_{1}+H_{2}+L$,
where now $H=H_{1}+H_{2}$.

So we let $H_{1}=|r_{\lfloor
n^{1/\delta}\rfloor,\lfloor\ell^{1/\delta}\rfloor}*f|$. In that case

$\displaystyle\langle H_{1}\rangle_{E,2}^{2}$ $\displaystyle\leq\sum_{n\geq
C_{p}M^{1/p^{\prime}}}\langle r_{\lfloor
n^{1/\delta}\rfloor,\lfloor\ell^{1/\delta}\rfloor}*f\rangle_{E,2}^{2}\;\;\lesssim\sum_{n\geq
C_{p}M^{1/p^{\prime}}}n^{-2p^{\prime}}\langle f\rangle_{E,2}^{2}\;\;\lesssim
M^{-2+\frac{2}{p^{\prime}}}\langle f\rangle_{E,2}^{2}$

using a square function argument. The second contribution to the high-pass
term is the following

$\displaystyle H_{2}$
$\displaystyle=\sup_{n}\left|\sum_{2^{m}<Q\leq\lfloor\ell^{1/\delta}\rfloor}L_{Q,\lfloor
n^{1/\delta}\rfloor}*f\right|\;\;\leq\sum_{k=m}^{\log_{2}\lfloor\ell^{1/\delta}\rfloor}\sup_{n}\left|\sum_{2^{k}<Q\leq
2^{k+1}}L_{Q,\lfloor n^{1/\delta}\rfloor}*f\right|$

To estimate the $\ell^{2}$ norm of $H_{2}$ we use Bourgain’s Multi-frequency
Lemma [Bou89], a fundamental result in discrete Harmonic Analysis. This
particular form can be found in [Kra18, Prop. 5.11], and provides an
$\ell^{2}\to\ell^{2}$ bound for each of the individual terms in the sum.
Specifically,

$\displaystyle\left\|\sup_{n}\left|\sum_{2^{k}\leq Q<2^{k+1}}L_{Q,\lfloor
n^{1/\delta}\rfloor}*f\right|\,\right\|_{\ell^{2}}$ $\displaystyle\lesssim
k\,2^{-\frac{k}{p}}\|f\|_{\ell^{2}}$

Summing over $k$ immediately yields the desired result.

Obviously the remaining terms constitute the low-term, which according to
(3.3) is given in terms of Ramanujan sums and satisfies

(5.6) $\displaystyle
L=\left|\sum_{Q=1}^{M}L_{\tau,Q}*f\right|\lesssim\sum_{Q=1}^{M}\left(\frac{|c_{Q}(\cdot)|}{Q}\,|\Gamma_{\tau}*\check{\chi}_{s}(\cdot)|\right)*f$

with $s\geq 1$, $2^{s}\leq Q<2^{s+1}$ as in the definition of (3.1).

Now let us take a closer look at $\Gamma_{\tau}*\check{\chi}_{s}$. Using (2.8)
we can see that for $|x|\leq 4\tau$

$|\Gamma_{\tau}*\check{\chi}_{s}(x)|\lesssim\frac{1}{\tau}.$

Next, suppose $2^{k}\tau<|x|\leq 2^{k+1}\tau$, for $k\geq 2$, and recall that
$\chi_{s}$ is a dilation of a Schwartz function, where $s\geq 1$ and
$2^{s}\leq Q<2^{s+1}$. Using (2.8) and the fact that $M^{3}\leq
C_{p}M^{\frac{1}{\delta p^{\prime}}}\lesssim\tau$, we obtain

$\displaystyle|\Gamma_{\tau}*\check{\chi}_{s}(x)|$
$\displaystyle\;\lesssim\;\sum_{y\leq\tau}|\Gamma_{\tau}(y)|\,|\check{\chi}_{s}(x-y)|\;\lesssim\;\sum_{y\leq\tau}\frac{1}{\tau}\,\frac{1}{2^{3s}}\left|\check{\chi}\left(\frac{x-y}{2^{3s}}\right)\right|$
$\displaystyle\;\lesssim\frac{1}{2^{3s}\tau}\,\frac{2^{6s}}{2^{2k}\tau}\;\lesssim\;\frac{M^{3}}{2^{2k}\tau^{2}}\;\lesssim\;\frac{1}{2^{2k}\tau}$

Using these estimates in (5.6) we get

$\displaystyle|L(x)|$ $\displaystyle\leq\frac{1}{\tau}\sum_{|y|\leq
4\tau}\sum_{Q=1}^{M}\frac{|c_{Q}(y)|}{Q}*f(x-y)+\sum_{k=2}^{\infty}\frac{1}{4^{k}\tau}\;\sum_{|y|=2^{k}}^{2^{k+1}}\sum_{Q=1}^{M}\frac{|c_{Q}(y)|}{Q}*f(x-y)$
$\displaystyle\lesssim\left[\frac{1}{\tau}\sum_{y=1}^{\tau}\left|\sum_{Q=1}^{M}\frac{c_{Q}(y)}{Q}\right|^{p^{\prime}}\right]^{\frac{1}{p^{\prime}}}\left[\frac{1}{\tau}\sum_{y=1}^{\tau}f(x-y)\right]^{\frac{1}{p}}+$
$\displaystyle\hskip
28.45274pt+\sum_{k=2}^{\infty}\frac{1}{2^{k}}\left[\sum_{y=2^{k}}^{2^{k+1}\tau}\frac{1}{2^{k+1}\tau}\left|\sum_{Q=1}^{M}\frac{|c_{Q}(y)|}{Q}\right|^{p^{\prime}}\right]^{1/p^{\prime}}\left[\frac{1}{2^{k+1}\tau}\sum_{y=2^{k}}^{2^{k+1}\tau}f(x-y)\right]^{\frac{1}{p}}$
$\displaystyle\lesssim M^{1/p^{\prime}}\langle f\rangle^{1/p}_{E,1}$

Here, we have used Hölder’s $p-p^{\prime}$ inequality and Lemma 2.2.
Admissibility of $\tau$ was essential to obtain the estimate in terms of the
average of $f$. Our assumption that $\tau^{\delta}(x)>C_{p}J^{1/p^{\prime}}$
is also unavoidable, if we wish to use Lemma 2.2. Thus the constant $C_{p}$
has to be chosen accordingly, and our proof is complete.

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