Cotlar’s Ergodic Theorem Along the Prime Numbers

Abstract

The aim of this paper is to prove Cotlar’s ergodic theorem modeled on the set of primes.

1 Introduction

Let $(X, \mathcal {B}, \mu , S)$ be a dynamical system on a measure space $X$ endowed with a $\sigma$-algebra $\mathcal {B}$, a $\sigma$-finite measure $\mu$ and an invertible measure preserving transformation $S:X \rightarrow X$. In 1955 Cotlar (see [4]) established the almost everywhere convergence of the ergodic truncated Hilbert transform

$$\begin{aligned} \lim _{N\rightarrow \infty }\sum _{1 \le {\left|{n} \right|} \le N}\frac{f(S^nx)}{n} \end{aligned}$$

for all $f\in L^r(\mu )$ with $1\le r<\infty$. The aim of the present paper is to obtain the corresponding result for the set of prime numbers $\mathbb {P}$. Let $\mathbb {P}_N = \mathbb {P}\cap (1, N]$. We prove

Theorem 1

For a given dynamical system $(X, \mathcal {B}, \mu , S)$ the almost everywhere convergence of the ergodic truncated Hilbert transform along $\mathbb {P}$

$$\begin{aligned} \lim _{N\rightarrow \infty } \sum _{p\in \pm \mathbb {P}_N} \frac{f(S^p x)}{p}\log {\left|{p} \right|} \end{aligned}$$

holds for all $f\in L^r(\mu )$ with $1< r<\infty$.

In view of Calderón’s transference principle, it is more convenient to work with the set of integers rather than an abstract measure space $X$. In these settings we consider discrete singular integrals with Calderón–Zygmund kernels. Given $K \in C^1\big (\mathbb {R}\setminus \{0\}\big )$ satisfying

$$\begin{aligned} {\left|{x} \right|} {\left|{K(x)} \right|} + {\left|{x} \right|}^2 {\left|{K'(x)} \right|} \le 1 \end{aligned}$$

(1)

for ${\left|{x} \right|} \ge 1$, together with a cancellation property

$$\begin{aligned} \sup _{\lambda \ge 1} \bigg |\int \limits _{1 \le {\left|{x} \right|} \le \lambda } K(x) dx \bigg |\le 1 \end{aligned}$$

(2)

a singular transform $T$ along the set of prime numbers is defined for a finitely supported function $f: \mathbb {Z}\rightarrow \mathbb {C}$ as

$$\begin{aligned} T f(n) = \sum _{p \in \pm \mathbb {P}} f(n - p) K(p) \log {\left|{p} \right|}. \end{aligned}$$

Let $T_N$ denote the truncation of $T$, i.e.

$$\begin{aligned} T_N f(n) = \sum _{p \in \pm \mathbb {P}_N} f(n-p) K(p) \log {\left|{p} \right|}. \end{aligned}$$

We show

Theorem 2

The maximal function

$$\begin{aligned} T^* f(n) = \sup _{N \in \mathbb {N}} \big |T_N f(n) \big |\end{aligned}$$

is bounded on $\ell ^r(\mathbb {Z})$ for any $1 < r < \infty$. Moreover, the pointwise limit

$$\begin{aligned} \lim _{N \rightarrow \infty } T_N f(n) \end{aligned}$$

exists and coincides with the Hilbert transform $Tf$ which is also bounded on $\ell ^r(\mathbb {Z})$ for any $1 < r < \infty$.

For $r = 2$, the proof of Theorem 2 is based on the Hardy and Littlewood circle method which allows us to construct appropriate approximating multipliers (see for instance (13)) and control the error terms as in Proposition 3.2. These ideas were pioneered by Bourgain (see [1–3]) in the context of pointwise ergodic theorems along integer valued polynomials. For $r \ne 2$, we shall compare the discrete norm $\Vert \cdot \Vert _{\ell ^r}$ of our approximating multipliers with the continuous norm $\Vert \cdot \Vert _{L^r}$ of certain multipliers which are a priori bounded on $L^r$, we refer to the proof of Proposition 3.3 and Theorem 3. Initially we wanted to follow elegant arguments from [23] which used very specific features of the set of prime numbers. However, we identified an issue in [23] (see Appendix 1) which made the proof incomplete. Instead, we propose an approach (see Lemmas 1 and 2) which rectifies Wierdl’s proof (see Appendix 1 for details) as well as simplifies Bourgain’s arguments.

Bourgain’s works have inspired many authors to investigate discrete analogues of classical operators with arithmetic features (see e.g. [5–7, 12–14, 17–19]). Nevertheless, not many have been proved for the operators and maximal functions modelled on the set of primes (see e.g. [9, 10, 23]). To the authors best knowledge, there are no other results dealing with maximal functions corresponding with truncated discrete singular integrals.

It is worth mentioning that Theorem 2 extends the result of Ionescu and Wainger [6] to the set of prime numbers. However, our approach is different and provides a stronger result since we study maximal functions corresponding with truncations of discrete singular integral rather than the whole singular integral. Furthermore, we are able to define the singular integral as a pointwise limit of its truncations. Theorem 2 encourages us to study maximal functions associated with truncations of the Radon transforms from [6]. For more details we refer the reader to the forthcoming article [8].

1.1 Notation

Throughout the paper, unless otherwise stated, $C > 0$ stands for a large positive constant whose value may vary from occurrence to occurrence. We will say that $A\lesssim B$ ($A\gtrsim B$) if there exists an absolute constant $C>0$ such that $A\le CB$ ($A\ge CB$). If $A\lesssim B$ and $A\gtrsim B$ hold simultaneously then we will shortly write that $A\simeq B$. We will write $A\lesssim _{\delta } B$ ($A\gtrsim _{\delta } B$) to indicate that the constant $C>0$ depends on some $\delta >0$. We always assume zero belongs to the set of natural numbers $\mathbb {N}$.

2 Preliminaries

We start by recalling some basic facts from number theory. A general reference is [11]. Given $q \in \mathbb {N}$ we define $A_q$ to be the set of all $a \in \mathbb {Z}\cap [1, q]$ such that $(a, q) = 1$. By $\mu$ we denote Möbius function, i.e. for $q=p_1^{\alpha _1} \cdot p_2^{\alpha _2}\cdot \ldots \cdot p_n^{\alpha _n}$ where $p_1,\ldots ,p_n\in \mathbb {P}$

$$\begin{aligned} \mu (q) = \left\{ {\begin{array}{ll} (-1)^n &{} \quad \text {if} \; \alpha _1 = \alpha _2 = \cdots = \alpha _n = 1,\\ 0 &{} \quad \text {otherwise,} \end{array} }\right. \end{aligned}$$

and $\mu (1) = 1$. In what follows, a significant role will be played by Ramanujan’s identity

$$\begin{aligned} \mu (q) = \sum _{r \in A_q} e^{2 \pi i r a/ q} \ \ \text{ if } (a, q)=1, \end{aligned}$$

and the Möbius inversion formula

$$\begin{aligned} \sum _{a \in A_q} F(a/q)=\sum _{d \mid q}\mu (q/d)\sum _{a=1}^d F(a/d) \end{aligned}$$

(3)

satisfied by any function $F$. Let $\varphi$ be the Euler’s totient function, i.e. for $q \in \mathbb {N}$ the value $\varphi (q)$ is equal to the number of elements in $A_q$. Then for every $\epsilon > 0$ there is a constant $C_{\epsilon } > 0$ such that

$$\begin{aligned} \varphi (q) \ge C_{\epsilon } q^{1-\epsilon }. \end{aligned}$$

(4)

If we denote by $d(q)$ the number of divisors of $q\in \mathbb {N}$, then for every $\epsilon > 0$ there is a constant $C_{\epsilon } > 0$ such that

$$\begin{aligned} d(q) \le C_{\epsilon } q^{\epsilon }. \end{aligned}$$

(5)

3 Maximal Function on $\mathbb {Z}$

The measure space $\mathbb {Z}$ with the counting measure and the bilateral shift operator will be our model dynamical system which permits us to prove Theorem 1.

From now on, all the maximal functions will be defined on non-negative finitely supported functions $f:\mathbb {Z}\rightarrow \mathbb {R}$ and unless otherwise stated $f$ always has a finite support.

Let us fix $\tau \in (1, 2]$ and define a set $\Lambda = \{\tau ^j: j \in \mathbb {N}\}$. Given a kernel $K \in C^1(\mathbb {R}\setminus \{0\})$ satisfying (1) and (2) we consider a sequence $\left( {K_j}: {j \in \mathbb {N}}\right)$ where

$$\begin{aligned} K_j(x) = \left\{ {\begin{array}{ll} K(x) &{} \quad \text {if} \; {\left|{x} \right|} \in (\tau ^j, \tau ^{j+1}],\\ 0 &{} \quad \text {otherwise.} \end{array} }\right. \end{aligned}$$

Let $\mathcal {F}$ denote the Fourier transform on $\mathbb {R}$ defined for any function $f \in L^1(\mathbb {R})$ as

$$\begin{aligned} \mathcal {F} f(\xi ) = \int _\mathbb {R}f(x) e^{2\pi i \xi x} dx. \end{aligned}$$

If $f \in \ell ^1(\mathbb {Z})$ we set

$$\begin{aligned} \hat{f}(\xi ) = \sum _{n \in \mathbb {Z}} f(n) e^{2\pi i \xi n}, \end{aligned}$$

then for $\Phi _j = \mathcal {F} K_j$, integration by parts shows that

$$\begin{aligned} {\left|{\Phi _j(\xi )} \right|} \lesssim {\left|{\xi } \right|}^{-1} \tau ^{-j}, \end{aligned}$$

(6)

for $\xi \in \mathbb {R}$. We define a sequence $\left( {m_j}: {j \in \mathbb {N}}\right)$ of multipliers

$$\begin{aligned} m_j(\xi ) = \sum _{p \in \pm \mathbb {P}} e^{2\pi i \xi p} K_j(p) \log {\left|{p} \right|}. \end{aligned}$$

3.1 $\ell ^2$-Approximation

To approximate the multiplier $m_j$ we adopt the argument introduced by Bourgain [3] (see also Wierdl [23]) which is based on the Hardy–Littlewood circle method (see e.g. [20]).

For any $\alpha > 0$ and $j \in \mathbb {N}$ major arcs are defined by

$$\begin{aligned} \mathfrak {M}_j^\alpha = \bigcup _{1\le q \le j^\alpha } \bigcup _{a \in A_q} \mathfrak {M}^\alpha _j(a/q) \end{aligned}$$

where

$$\begin{aligned} \mathfrak {M}_j^\alpha (a/q) = \big \{ \xi \in [0, 1]: {\left|{\xi - a/q} \right|} \le \tau ^{-j} j^\alpha \big \}. \end{aligned}$$

Here and subsequently we will treat the interval $[0, 1]$ as the circle group $\Pi =\mathbb {R}/\mathbb {Z}$ identifying $0$ and $1$.

Proposition 3.1

For $\xi \in \mathfrak {M}^\alpha _j(a/q) \cap \mathfrak {M}_j^\alpha$

$$\begin{aligned} \Bigg |m_j(\xi ) - \frac{\mu (q)}{\varphi (q)} \Phi _j(\xi - a/q) \Bigg |\le C_\alpha j^{-\alpha }. \end{aligned}$$

The constant $C_\alpha$ depends only on $\alpha$.

Proof

Since for a prime number $p$, $p \mid q$ if and only if $(p\ \mathrm {mod}\ q, q) > 1$, we have

$$\begin{aligned} \left|\sum _{\genfrac{}{}{0.0pt}2{1 \le r \le q}{(r,q) > 1}} \sum _{\genfrac{}{}{0.0pt}2{p \in \mathbb {P}}{q \mid (p -r)}} e^{2\pi i \xi p} K_j(p) \log p \right|\le \tau ^{-j} \sum _{\genfrac{}{}{0.0pt}2{p \in \mathbb {P}}{p \mid q}} \log p \lesssim \tau ^{-j} \log j. \end{aligned}$$

(7)

Let $\theta = \xi - a/q$. If $p \equiv r \pmod q$ then

$$\begin{aligned} \xi p \equiv \theta p + ra/q \pmod 1 \end{aligned}$$

and consequently

$$\begin{aligned} \sum _{r \in A_q} \sum _{\genfrac{}{}{0.0pt}2{p \in \mathbb {P}}{q \mid (p - r)}} e^{2\pi i \xi p} K_j(p) \log p = \sum _{r \in A_q} e^{2 \pi i r a/q} \sum _{\genfrac{}{}{0.0pt}2{p \in \mathbb {P}}{q \mid (p - r)}} e^{2\pi i \theta p} K_j(p) \log p. \end{aligned}$$

(8)

Using the summation by parts (see e.g. [11, p. 304]) for the inner sum on the right hand side in (8) we obtain

$$\begin{aligned} \sum _{\genfrac{}{}{0.0pt}2{n \in N_j}{q \mid (n-r)}} e^{2 \pi i \theta n} K(n) {1\!\!1_{{\mathbb {P}}}}(n) \log n&= \psi (\tau ^{j+1}; q, r)e^{2 \pi i \theta \tau ^{j+1}} K(\tau ^{j+1})\nonumber \\&\quad - \; \psi (\tau ^{j}; q, r)e^{2 \pi i \theta \tau ^{j}} K(\tau ^{j})\nonumber \\&\quad - \int _{\tau ^j}^{\tau ^{j+1}} \psi (t; q, r) \frac{d}{dt} \left( e^{2\pi i \theta t}K(t) \right) dt \end{aligned}$$

(9)

where $N_j = \mathbb {N}\cap (\tau ^j, \tau ^{j+1}]$ and for $x \ge 2$ we have set

$$\begin{aligned} \psi (x; q, r) = \sum _{\genfrac{}{}{0.0pt}2{p \in \mathbb {P}_x}{q \mid (p-r)}} \log p. \end{aligned}$$

Similar reasoning gives

$$\begin{aligned} \sum _{n \in N_j} e^{2\pi i \theta n} K(n)&=\tau ^{j+1}e^{2 \pi i \theta \tau ^{j+1}} K\big (\tau ^{j+1}\big ) - \tau ^{j}e^{2 \pi i \theta \tau ^{j}} K\big (\tau ^{j}\big )\nonumber \\&\qquad - \int _{\tau ^j}^{\tau ^{j+1}} t \frac{d}{dt} \left( e^{2\pi i \theta t}K(t)\right) dt. \end{aligned}$$

(10)

By Siegel–Walfisz theorem (see [16, 22]) we know that for every $\alpha >0$ and $x \ge 2$

$$\begin{aligned} \bigg |\psi (x; q, r) - \frac{x}{\varphi (q)} \bigg |\lesssim x (\log x)^{-3\alpha } \end{aligned}$$

(11)

where the implied constant depends only on $\alpha$. Therefore (9) and (10) combined with the estimates (1) and (11) yield

$$\begin{aligned}&\Bigg |\sum _{\genfrac{}{}{0.0pt}2{p \in \mathbb {P}}{q \mid (p -r)}} e^{2\pi i\theta p} K_j(p) \log p -\frac{1}{\varphi (q)} \sum _{n \in \mathbb {N}} e^{2\pi i \theta n} K_j(n) \Bigg |\\&\quad \lesssim \bigg |\psi (\tau ^{j+1};q,r) - \frac{\tau ^{j+1}}{\varphi (q)} \bigg |\big |K(\tau ^{j+1})\big | +\bigg |\psi (\tau ^{j};q,r) - \frac{\tau ^{j}}{\varphi (q)} \bigg ||K(\tau ^{j})|\\&\qquad + \int _{\tau ^j}^{\tau ^{j+1}} \bigg |\psi (t;q,r) - \frac{t}{\varphi (q)} \bigg |\big (t^{-1} {\left|{\theta } \right|} + t^{-2}\big ) dt\\&\quad \lesssim j^{-3\alpha }+ \int _{\tau ^j}^{\tau ^{j+1}} (\log t)^{-3\alpha } \big ( {\left|{\theta } \right|} + t^{-1}\big ) dt \end{aligned}$$

what is bounded by $j^{-2\alpha }$. Finally, by (8),

$$\begin{aligned}&\Bigg |\sum _{r \in A_q} \sum _{\genfrac{}{}{0.0pt}2{p \in \mathbb {P}}{q \mid (p -r)}} e^{2 \pi i \xi p} K_j(p) \log p -\frac{\mu (q)}{\varphi (q)} \sum _{n \in \mathbb {N}} e^{2\pi i \theta n} K_j(n) \Bigg |\nonumber \\&\quad = \!\Bigg |\!\sum _{r \in A_q} e^{2 \pi i ra/q} \! \Bigg ( \!\sum _{\genfrac{}{}{0.0pt}2{p \in \mathbb {P}}{q \mid (p\! -\!r)}} e^{2 \pi i \!\theta p}K_j(p) \log p \!-\! \frac{1}{\varphi (q)} \!\sum _{n \in \mathbb {N}} e^{2\pi i \theta n} K_j(n) \Bigg ) \Bigg |\lesssim q j^{\!-\!2\alpha } \le j^{\!-\!\alpha }.\nonumber \\ \end{aligned}$$

(12)

Next, we can substitute an integral for the sum since for $n_0 = \lceil \tau ^j \rceil$ and $n_1 = \lfloor \tau ^{j+1} \rfloor$ we have

$$\begin{aligned} \int _{\tau ^j}^{\tau ^{j+1}} e^{2\pi i \theta t} K(t) dt&= \int _{\tau ^j}^{n_0} e^{2\pi i \theta t} K(t) dt + \sum _{n = n_0}^{n_1-1} \int _0^1 e^{2\pi i \theta (n + t)} K(n+t) dt\\&\quad + \int _{n_1}^{\tau ^{j+1}} e^{2\pi i \theta t} K(t) dt. \end{aligned}$$

Since ${\left|{\theta } \right|} \le \tau ^{-j} j^\alpha$ we get

$$\begin{aligned}&\Bigg |\sum _{n = n_0}^{n_1-1} \bigg (e^{2\pi i \theta n} K(n) - \int _0^1 e^{2\pi i \theta (n+t)} K(n+t) dt \bigg ) \Bigg |\\&\quad \le \sum _{n=n_0}^{n_1-1}\int _0^1 {\left|{1-e^{-2\pi i \theta t}} \right|}{\left|{K(n)} \right|} dt + \sum _{n=n_0}^{n_1-1} \int _0^1 {\left|{K(n) - K(n+t)} \right|} dt \lesssim \tau ^{-j} j^\alpha . \end{aligned}$$

Hence, by (7) and (12) we obtain

$$\begin{aligned} \Bigg |\sum _{p \in \mathbb {P}} e^{2\pi i \xi p} K_j(p) \log p - \frac{\mu (q)}{\varphi (q)} \int _0^\infty e^{2\pi i \theta t} K_j(t) dt \Bigg |\lesssim \tau ^{-j} j^\alpha + j^{-\alpha }. \end{aligned}$$

Repeating all the steps with $p$ replaced by $-p$ we finish the proof. $\square$

For $s \in \mathbb {N}$ we set

$$\begin{aligned} \fancyscript{R}_s = \big \{ a/q \in [0, 1]\cap \mathbb {Q}: 2^s \le q < 2^{s+1} \; \text {and}\; (a, q) =1 \big \}. \end{aligned}$$

Since we treat $[0, 1]$ as the circle group identifying $0$ and $1$ we treat $\fancyscript{R}_0=\{1\}$. Let us consider

$$\begin{aligned} \nu _j^s(\xi ) = \sum _{a/q \in \fancyscript{R}_s} \frac{\mu (q)}{\varphi (q)} \Phi _j(\xi - a/q) \eta _s(\xi -a/q) \end{aligned}$$

(13)

where $\eta _s(\xi ) = \eta (A^{s+1} \xi )$ and $\eta : \mathbb {R}\rightarrow \mathbb {R}$ is a smooth function such that $0 \le \eta (x) \le 1$ and

$$\begin{aligned} \eta (x) = \left\{ {\begin{array}{ll} 1 &{} \quad \text {for} \; {\left|{x} \right|} \le 1/4,\\ 0 &{} \quad \text {for} \; {\left|{x} \right|} \ge 1/2. \end{array} }\right. \end{aligned}$$

The value of $A$ is chosen to satisfy (18). Additionally, we may assume (this will be important in Lemma 1) that $\eta$ is a convolution of two smooth functions with compact supports contained in $[-1/2, 1/2]$. Let $\nu _j = \sum _{s \in \mathbb {N}} \nu _j^s$. For any $s\in \mathbb {N}$ the multiplier $\nu _j^s$ is meant to be $1$-periodic.

Proposition 3.2

For every $\alpha > 16$

$$\begin{aligned} \big |m_j(\xi ) - \nu _j(\xi ) \big |\le C_\alpha j^{-\alpha /4}. \end{aligned}$$

The constant $C_\alpha$ depends only on $\alpha$.

Proof

First of all notice that for a fixed $s \in \mathbb {N}$ and $\xi \in [0, 1]$ the sum (13) consists of the single term. Otherwise, there would be $a/q, a'/q' \in \fancyscript{R}_s$ such that $\eta _s(\xi - a/q) \ne 0$ and $\eta _s(\xi -a'/q') \ne 0$. Therefore,

$$\begin{aligned} 2^{-2 s - 2} \le \frac{1}{qq'} \le \Bigg |\frac{a}{q}-\frac{a'}{q'} \Bigg |\le \Bigg |\xi - \frac{a}{q} \Bigg |+ \Bigg |\xi - \frac{a'}{q'} \Bigg |\le A^{-s-1} \end{aligned}$$

which is not possible whenever $A > 4$, as it was assumed in (18).

Major arcs estimates: $\xi \in \mathfrak {M}_j^\alpha (a/q) \cap \mathfrak {M}_j^\alpha$. Let $s_0$ be such that

$$\begin{aligned} 2^{s_0} \le q < 2^{s_0+1}. \end{aligned}$$

(14)

We choose $s_1$ satisfying

$$\begin{aligned} 2^{s_1+1} \le \tau ^j j^{-2\alpha } < 2^{s_1+2}. \end{aligned}$$

If $s < s_1$ then for any $a'/q' \in \fancyscript{R}_s$, $a'/q' \ne a/q$ we have

$$\begin{aligned} \Bigg |\xi - \frac{a'}{q'} \Bigg |\ge \frac{1}{qq'} - \Bigg |\xi - \frac{a}{q} \Bigg |\ge 2^{-s-1} j^{-\alpha } - \tau ^{-j} j^\alpha \ge \tau ^{-j} j^\alpha . \end{aligned}$$

Therefore, using (6)

$$\begin{aligned} {\left|{\Phi _j\big (\xi - a'/q'\big )} \right|} \lesssim \big ({\left|{\xi - a'/q'} \right|} \tau ^j\big )^{-1} \lesssim j^{-\alpha }. \end{aligned}$$

Combining the last estimate with (4), we obtain that for any $0 < \delta _1 < 1$

$$\begin{aligned} I_1=\Bigg |\sum _{s = 0}^{s_1-1} \sum _{\genfrac{}{}{0.0pt}2{a'/q' \in \fancyscript{R}_s}{a'/q' \ne a/q}} \frac{\mu (q')}{\varphi (q')} \Phi _j\big (\xi - a'/q'\big ) \eta _s\big (\xi - a'/q'\big ) \Bigg |\lesssim j^{-\alpha } \sum _{s=0}^{s_1-1} 2^{-\delta _1 s}. \end{aligned}$$

Moreover, if $\eta _{s_0}(\xi -a/q) < 1$ then ${\left|{\xi - a/q} \right|} \ge 4^{-1} A^{-s_0-1}$. By (14) we have $2^{s_0} \le j^\alpha$. Hence, (5) together with (6) implies

$$\begin{aligned} I_2=\Bigg |\frac{\mu (q)}{\varphi (q)} \Phi _j(\xi - a/q) \big (1 - \eta _{s_0}(\xi -a/q)\big ) \Bigg |\lesssim A^{s_0+1} \tau ^{-j} \lesssim j^{-\alpha }. \end{aligned}$$

In the last estimate it is important that the implied constant does not depend on $s_0$. Since $\Phi _j$ is bounded uniformly with respect to $j \in \mathbb {N}$, by (4) and the definition of $s_1$ we have

$$\begin{aligned} I_3\!=\!\Bigg |\sum _{s \!=\! s_1}^\infty \sum _{\genfrac{}{}{0.0pt}2{a'/q' \in \fancyscript{R}_s}{a'/q' \ne a/q}} \!\frac{\mu (q')}{\varphi (q')} \Phi _j (\xi - a'/q') \eta _s(\xi \!-\! a'/q') \Bigg |\lesssim \sum _{s\! =\! s_1}^\infty 2^{\!-\!\delta _2 s} \lesssim \big (\tau ^{-j}j^{2\alpha }\big )^{\delta _2}\lesssim j^{\!-\!\alpha } \end{aligned}$$

for appropriately chosen $\delta _2>0$. Finally, in view of Proposition 3.1 and definitions of $s_0$ and $s_1$ we conclude

$$\begin{aligned} \big |m_j (\xi ) - \nu _j(\xi ) \big |\le C_\alpha j^{-\alpha }+ I_1+I_2+I_3\lesssim j^{-\alpha }. \end{aligned}$$

Minor arcs estimates $\xi \not \in \mathfrak {M}_j^\alpha$. Firstly, by the summation by parts, we get

$$\begin{aligned} \Big |\sum _{p \in \mathbb {P}} e^{2\pi i \xi p} K_j (p) \log p \Big |&\le {\left|{F_{\tau ^{j+1}}(\xi )} \right|}\Big |K\big (\tau ^{j+1}\big )\Big | +{\left|{F_{\tau ^{j}}(\xi )} \right|}\Big |K\big (\tau ^{j}\big )\Big |\nonumber \\&\quad + \int _{\tau ^j}^{\tau ^{j+1}} {\left|{F_t(\xi )} \right|} |K'(t) |dt \end{aligned}$$

(15)

where

$$\begin{aligned} F_x(\xi ) = \sum _{p \in \mathbb {P}_x} e^{2 \pi i \xi p} \log p. \end{aligned}$$

Using Dirichlet’s principle there are $(a, q) = 1$, $j^\alpha \le q \le \tau ^j j^{-\alpha }$ such that

$$\begin{aligned} {\left|{\xi - a/q} \right|} \le q^{-1} \tau ^{-j} j^\alpha \le q^{-2}. \end{aligned}$$

Thus, by Vinogradov’s theorem (see [21, Theorem 1, Chapter IX] or [11, Theorem 8.5]) we get

$$\begin{aligned} {\left|{F_t(\xi )} \right|} \lesssim j^4 \Big (\tau ^j q^{-1/2} + \tau ^{4 j/5} + \tau ^{j/2} q^{1/2}\Big ) \lesssim \tau ^j j^{4 - \alpha /2} \end{aligned}$$

for $t \in [\tau ^j, \tau ^{j+1}]$. Combining ${\left|{K'(t)} \right|} \lesssim \tau ^{-2j}$ with the last bound and (15) we conclude

$$\begin{aligned} {\left|{m_j(\xi )} \right|} \lesssim j^{4 - \alpha /2}\lesssim j^{-\alpha /4} \end{aligned}$$

since $\alpha > 16$. In order to estimate the $\nu _j$ let us define $s_2$ by setting

$$\begin{aligned} 2^{s_2} \le j^{\alpha /2} < 2^{s_2+1}. \end{aligned}$$

If $a/q \in \fancyscript{R}_s$ for $s < s_2$ then $q < j^{\alpha }$ and

$$\begin{aligned} \Bigg |\xi - \frac{a}{q} \Bigg |\ge 2^{-s - 1} \tau ^{-j} j^{\alpha } \gtrsim \tau ^{-j} j^{\alpha /2}. \end{aligned}$$

Again, by (6) we obtain

$$\begin{aligned} {\left|{\Phi _j(\xi - a/q)} \right|} \lesssim \big ({\left|{\xi - a/q} \right|} \tau ^j\big )^{-1} \lesssim j^{-\alpha /2}. \end{aligned}$$

Therefore, the first part of the sum may be majorized by

$$\begin{aligned} \Bigg |\sum _{s = 0}^{s_2-1} \nu _j^s(\xi ) \Bigg \vert \lesssim j^{-\alpha /2} \sum _{s = 0}^{\infty } 2^{-\delta _1 s}, \end{aligned}$$

as for $I_1$. For the second part we proceed as for $I_3$ to get

$$\begin{aligned} \Bigg |\sum _{s = s_2}^\infty \nu _j^s(\xi )\Bigg |\lesssim \sum _{s=s_2}^\infty 2^{-\delta s} \lesssim j^{-\delta _2 \alpha /2}\lesssim j^{-\alpha /4}. \end{aligned}$$

A suitable choice of $\delta _1, \delta _2>0$ in both estimates above was possible thanks to (4). $\square$

3.2 $\ell ^r$-Theory

We start the section by proving two lemmas which will play a crucial role.

Lemma 1

There is a constant $C > 0$ such that for all $s \in \mathbb {N}$ and $u\in \mathbb {R}$

$$\begin{aligned}&\bigg ||\int _{-\frac{1}{2}}^{\frac{1}{2}} e^{-2\pi i \xi j} \eta _s(\xi ) d\xi \bigg ||_{\ell ^1(j)} \le C,\end{aligned}$$

(16)

$$\begin{aligned}&\bigg ||\int _{-\frac{1}{2}}^{\frac{1}{2}} e^{-2\pi i \xi j} \big (1-e^{2\pi i \xi u}\big )\eta _s(\xi )d\xi \bigg ||_{\ell ^1(j)} \le C |u| A^{-s-1}. \end{aligned}$$

(17)

Proof

We only show (17) for $u\in \mathbb {R}$, since the proof of (16) is almost identical. Recall, $\eta = \phi * \psi$ for $\psi , \phi$ smooth functions with supports inside $[-1/2, 1/2]$. Hence, $\eta _s = A^{s+1} \phi _s * \psi _s$ and

$$\begin{aligned} A^{-s-1} \int _{-\frac{1}{2}}^{\frac{1}{2}} e^{-2\pi i \xi j}\big (1 - e^{2\pi i \xi u}\big ) \eta _s(\xi ) d\xi&=\mathcal {F}^{-1} \phi _s (j) \mathcal {F}^{-1} \psi _s(j)\\&\quad - \mathcal {F}^{-1} \phi _s(j-u) \mathcal {F}^{-1} \psi _s(j-u). \end{aligned}$$

By Cauchy–Schwarz’s inequality and Plancherel’s theorem

$$\begin{aligned}&\sum _{j \in \mathbb {Z}} {\left|{\mathcal {F}^{-1} \phi _s(j)} \right|} {\left|{\mathcal {F}^{-1} \psi _s(j)-\mathcal {F}^{-1} \psi _s(j-u)} \right|}\\&\quad \le {\left\| \mathcal {F}^{-1} \phi _s \right\| }_{\ell ^2} \bigg ||\int _\mathbb {R}e^{-2\pi i \xi j} \big (1-e^{2\pi i \xi u}\big ) \psi _s(\xi ) d\xi \bigg ||_{\ell ^2(j)}\\&\quad = {\left\| \phi _s \right\| }_{L^2} {\left\| \big (1-e^{2\pi i \xi u}\big ) \psi _s(\xi ) \right\| }_{L^2(d\xi )}. \end{aligned}$$

Moreover, since

$$\begin{aligned} \int _\mathbb {R}{\left|{1 - e^{-2\pi i \xi u}} \right|}^2 {\left|{\psi _s(\xi )} \right|}^2 d\xi \le u^2 \int _\mathbb {R}{\left|{\xi } \right|}^2 {\left|{\psi _s(\xi )} \right|}^2 d\xi \lesssim u^2 A^{-3(s+1)} {\left\| \psi \right\| }_{L^2}^2 \end{aligned}$$

we obtain

$$\begin{aligned} \sum _{j \in \mathbb {Z}} {\left|{\mathcal {F}^{-1} \phi _s(j)} \right|} {\left|{\mathcal {F}^{-1} \psi _s(j)- \mathcal {F}^{-1} \psi _s(j-u)} \right|} \lesssim |u| A^{-2(s+1)} {\left\| \phi \right\| }_{L^2} {\left\| \psi \right\| }_{L^2} \end{aligned}$$

which finishes the proof of (17). $\square$

Lemma 2

Let $r \ge 1$. For all $q \in [2^s, 2^{s+1})$, $s \ge r$ and $l \in \{1, 2, \ldots , q\}$

$$\begin{aligned} \Bigg ||\mathcal {F}^{-1} \big (\eta _s \hat{f}\big )(q j + l) \Bigg ||_{\ell ^{r}(j)} \simeq q^{-1/r} \Bigg ||\mathcal {F}^{-1} \big (\eta _s \hat{f}\big ) \Bigg ||_{\ell ^{r}}. \end{aligned}$$

Proof

We define a sequence $\big (J_1, J_2, \ldots , J_q\big )$ by

$$\begin{aligned} J_l = \Bigg ||\mathcal {F}^{-1} \big (\eta _s \hat{f} \big )(qj+l) \Bigg ||_{\ell ^r(j)}. \end{aligned}$$

Then $J_1^r + J_2^r + \cdots + J_q^r = I^r$ where $I = \big ||\mathcal {F}^{-1} \big (\eta _s \hat{f} \big ) \big ||_{\ell ^r(j)}$. Since $\eta _s = \eta _s \eta _{s-1}$, by Minkowski’s inequality we obtain

$$\begin{aligned}&\Bigg ||\mathcal {F}^{-1} \big (\eta _s \hat{f} \big )(qj+l) -\mathcal {F}^{-1}\big (\eta _s \hat{f} \big )(qj+l') \Bigg ||_{\ell ^r(j)}\\&\quad = \Bigg ||\int _{-\frac{1}{2}}^{\frac{1}{2}} e^{-2\pi i \xi (q j+l)} \big (1 - e^{2\pi i\xi (l-l')}\big ) \eta _s(\xi ) \hat{f}(\xi ) d\xi \Bigg ||_{\ell ^r(j)}\\&\quad \le \bigg ||\int _{-\frac{1}{2}}^{\frac{1}{2}} e^{-2\pi i \xi j} \big (1 - e^{2\pi i\xi (l-l')}\big ) \eta _{s-1}(\xi ) d\xi \bigg ||_{\ell ^1(j)} I, \end{aligned}$$

what, by (17), is bounded by $C q A^{-s} I$. We notice, the constant $C > 0$ depends only on $\eta$. Hence, for all $l,l' \in \{1,2,\ldots , q\}$

$$\begin{aligned} J_l \le J_{l'} + C q A^{-s} I. \end{aligned}$$

Since $q < 2^{s+1}$ taking

$$\begin{aligned} A > 32 \max \{1, C\} \end{aligned}$$

(18)

we obtain $C q A^{-s} \le 2^{-4s+1}$ thus

$$\begin{aligned} J_l^r \le 2^{r-1}J_{l'}^r + 2^{r-1}\big (C q A^{-s}\big )^r I^r \le 2^{r-1}J_{l'}^r + 2^{2r-4s-1} I^r. \end{aligned}$$

(19)

Therefore,

$$\begin{aligned} I^r = J_1^r + J_2^r + \cdots + J_q^r \le 2^{r-1}q J_l^r + q2^{2r-4s-1} I^r\le 2^{r-1}q J_l^r + 2^{3r-3s-1} I^r \end{aligned}$$

and using $s \ge r$, we get $I^r \le 2^{r} q J_l^r$. For the converse inequality, we use again (19) to conclude

$$\begin{aligned} q J_l^r \le 2^{r-1}\big (J_1^r + J_2^r + \cdots + J_q^r\big ) + q 2^{2r-4s-1} I^r \le 2^r I^r. \end{aligned}$$

$\square$

Proposition 3.3

For $r > 1$ and $s\in \mathbb {N}$

$$\begin{aligned} \Bigg ||\sup _{k \in \mathbb {N}} \big |\mathcal {F}^{-1} \big (\Psi _k \eta _s \hat{f} \big ) \big |\Bigg ||_{\ell ^r} \le C_r \big \Vert \mathcal {F}^{-1} \big (\eta _s \hat{f} \big )\big \Vert _{\ell ^r} \end{aligned}$$

where $\Psi _k = \sum _{j = 0}^k \Phi _j$.

Proof

Since $\eta _s=\eta _{s-1}\eta _s$ thus by Hölder’s inequality we have

$$\begin{aligned}&\sup _{k \in \mathbb {N}} \big |\mathcal {F}^{-1} \big (\Psi _k \eta _s \hat{f} \big ) (m) \big |^r \le \bigg (\int _{\mathbb {R}} \sup _{k \in \mathbb {N}} \big |\mathcal {F}^{-1} \big (\Psi _k \eta _s\hat{f} \big )(t)\big |\big |\mathcal {F}^{-1}\eta _{s-1}(m - t)\big |dt\bigg )^r\\&\quad \le \int _{\mathbb {R}} \sup _{k \in \mathbb {N}} \big |\mathcal {F}^{-1} \big (\Psi _k \eta _s\hat{f} \big )(t) \big |^r \big |\mathcal {F}^{-1} \eta _{s-1}(m \!-\! t) \big |dt\ {\left\| \mathcal {F}^{-1} \eta _{s-1} \right\| }_{L^{1}}^{r-1}. \end{aligned}$$

Now we note that ${\left\| \mathcal {F}^{-1} \eta _{s-1} \right\| }_{L^{1}}\lesssim 1$ and

$$\begin{aligned} \sum _{m \in \mathbb {Z}} \big |\mathcal {F}^{-1}\eta _{s-1} (m-t) \big |&\lesssim A^{-s}\sum _{m \in \mathbb {Z}} \frac{1}{1 + (A^{-s}(m - t))^2}\\&\lesssim A^{-s}\bigg (1+\int _{\mathbb {R}} \frac{dx}{1 + (A^{-s}x)^2 }\bigg )\lesssim A^{-s}(1+A^s)\lesssim 1 \end{aligned}$$

and the implied constants are independent of $A$. Thus we obtain

$$\begin{aligned} \Bigg ||\sup _{k \in \mathbb {N}} \big |\mathcal {F}^{-1} \big (\Psi _k \eta _s\hat{f} \big )\big |\Bigg ||_{\ell ^r} \lesssim \Bigg ||\sup _{k \in \mathbb {N}} \big |\mathcal {F}^{-1} \big (\Psi _k \eta _s\hat{f} \big )\big |\Bigg ||_{L^r}\lesssim \big \Vert \mathcal {F}^{-1} \big (\eta _s\hat{f}\big )\big \Vert _{L^r}, \end{aligned}$$

(20)

where the last inequality is a consequence of [15]. The proof will be completed if we show

$$\begin{aligned} \big \Vert \mathcal {F}^{-1} \big (\eta _s\hat{f}\big ) \big \Vert _{L^r}\lesssim \big \Vert \mathcal {F}^{-1} \big (\eta _s\hat{f}\big )\big \Vert _{\ell ^r}. \end{aligned}$$

For this purpose we use (17) from Lemma 1. Indeed,

$$\begin{aligned}&\big \Vert \mathcal {F}^{-1} \big (\eta _s\hat{f}\big )\big \Vert _{L^r}^r \\&\quad = \sum _{j\in \mathbb {Z}}\int _{0}^{1} \big |\mathcal {F}^{-1} \big (\eta _s\hat{f}\big )(x+j)\big |^rdx\\&\quad \le 2^{r-1}\big \Vert \mathcal {F}^{-1} \big (\eta _s\hat{f}\big )\big \Vert _{\ell ^r}^r + 2^{r-1}\sum _{j\in \mathbb {Z}}\int _{0}^{1} \big |\mathcal {F}^{-1} \big (\eta _s\hat{f}\big )(x+j)-\mathcal {F}^{-1} \big (\eta _s\hat{f}\big )(j)\big |^rdx\\&\quad = 2^{r-1}\big \Vert \mathcal {F}^{-1} \big (\eta _s\hat{f}\big )\big \Vert _{\ell ^r}^r +2^{r-1}\int _{0}^{1} \bigg \Vert \int _{-\frac{1}{2}}^{\frac{1}{2}} e^{-2\pi i \xi j} \big (1-e^{-2\pi i \xi x}\big )\eta _s(\xi )\hat{f}(\xi )d\xi \bigg \Vert _{\ell ^r(j)}^rdx\\&\quad \le 2^{r-1}\big \Vert \mathcal {F}^{-1} \big (\eta _s\hat{f}\big )\big \Vert _{\ell ^r}^r +2^{r-1}\int _{0}^{1} \bigg \Vert \int _{-\frac{1}{2}}^{\frac{1}{2}} e^{-2\pi i \xi j} \big (1-e^{-2\pi i \xi x}\big )\eta _{s-1}(\xi )d\xi \bigg \Vert _{\ell ^1(j)}^r \big \Vert \mathcal {F}^{-1} \big (\eta _s\hat{f}\big )\big \Vert _{\ell ^r}^rdx\\&\quad \lesssim \big \Vert \mathcal {F}^{-1} \big (\eta _s\hat{f}\big )\big \Vert _{\ell ^r}^r. \end{aligned}$$

This finishes the proof of the proposition. $\square$

Theorem 3

For each $r > 1$ there are $\delta _r > 0$ and $C_r > 0$ such that

$$\begin{aligned} \Bigg ||\sup _{k \in \mathbb {N}} \Big |\sum _{j = 0}^k \mathcal {F}^{-1} (\nu ^s_j \hat{f}) \Big |\Bigg ||_{\ell ^r} \le C_r 2^{-\delta _r s} {\left\| f \right\| }_{\ell ^r} \end{aligned}$$

for all $f \in \ell ^r(\mathbb {Z})$.

There is an interesting question about the endpoint estimate for $r=1$ in Theorem 3. Unfortunately, our method does not settle this issue. However, we hope to return to this problem at some point.

Proof

Let us fix $r > 1$. For $s < r$, by Proposition 3.3 we have

$$\begin{aligned} \Bigg ||\sup _{k \in \mathbb {N}} \Big |\sum _{j = 0}^k \mathcal {F}^{-1} (\nu ^s_j \hat{f}) \Big |\Bigg ||_{\ell ^r} \le \sum _{a/q \in \fancyscript{R}_s} \frac{1}{\varphi (q)} \Bigg ||\sup _{k \in \mathbb {N}} \Big |\mathcal {F}^{-1} \big (\Psi _k \hat{f} (\cdot + a/q) \big ) \Big |\Bigg ||_{\ell ^r} \le C_r {\left\| f \right\| }_{\ell ^r}. \end{aligned}$$

Next, we consider $s \ge r$. Let $q \in [2^s, 2^{s+1})$ be fixed. We are going to show that for every $\epsilon >0$ we have

$$\begin{aligned} \bigg ||\sup _{k \in \mathbb {N}} \Big |\sum _{a \in A_q} \mathcal {F}^{-1} \big (\Psi _k(\cdot -a/q) \eta _s(\cdot -a/q) \hat{f}\big ) \Big |\bigg ||_{\ell ^r}\le C_{\epsilon } q^{\epsilon }\Vert f\Vert _{\ell ^r}. \end{aligned}$$

(21)

By Möbius inversion formula (3) we see that

$$\begin{aligned} \sum _{a \in A_q} \mathcal {F}^{-1} \big (\Psi _k(\cdot -a/q) \eta _s(\cdot -a/q) \hat{f}\big )(x) =\sum _{b \mid q} \mu (q/b) \sum _{a=1}^b e^{-2\pi i ax/b} \mathcal {F}^{-1} \big (\Psi _k \eta _s \hat{f}(\cdot +a/b)\big )(x).\nonumber \\ \end{aligned}$$

(22)

Moreover, for $x \equiv l \pmod {q}$ we may write

$$\begin{aligned} \sum _{a=1}^b e^{-2\pi i a x/b} \mathcal {F}^{-1}\big (\Psi _k \eta _s \hat{f} (\cdot + a/b)\big )(x) =\mathcal {F}^{-1}\big (\Psi _k \eta _s F_b(\cdot \ ; l)\big ) (x) \end{aligned}$$

(23)

where for $b \mid q$ we have set

$$\begin{aligned} F_b(\xi ; l) = \sum _{a=1}^b \hat{f}(\xi + a/b) e^{-2\pi i l a/b}. \end{aligned}$$

Therefore, by formulas (22) and (23) we have

$$\begin{aligned}&\Bigg ||\sup _{k \in \mathbb {N}} \Big |\sum _{a \in A_q} \mathcal {F}^{-1} \big (\Psi _k(\cdot -a/q) \eta _s(\cdot -a/q) \hat{f}\big ) \Big |\Bigg ||_{\ell ^r} \\&\qquad \le \sum _{b \mid q} \bigg ( \sum _{l=1}^q \Big ||\sup _{k \in \mathbb {N}} \big |\mathcal {F}^{-1} \big (\Psi _k \eta _s F_b(\cdot \ ; l) \big ) (q j + l) \big |\Big ||_{\ell ^r(j)}^r \bigg )^{1/r}. \end{aligned}$$

Thus in view of (5) it will suffice to prove that

$$\begin{aligned} \Bigg ( \sum _{l=1}^q \Big ||\sup _{k \in \mathbb {N}} \big |\mathcal {F}^{-1} \big (\Psi _k \eta _s F_b(\cdot \ ; l) \big ) (q j + l) \big |\Big ||_{\ell ^r(j)}^r \Bigg )^{1/r} \le C_r {\left\| f \right\| }_{\ell ^r} \end{aligned}$$

(24)

where the constant does not depend on $b$. For the proof let us fix $f \in \ell ^r(\mathbb {Z})$ and consider a sequence $(J_1, J_2, \ldots , J_q)$ defined by

$$\begin{aligned} J_l = \Big ||\sup _{k \in \mathbb {N}} \big |\mathcal {F}^{-1} \big (\Psi _k \eta _s \hat{f} \big )(q j + l) \big |\Big ||_{\ell ^r(j)}. \end{aligned}$$

By Proposition 3.3, we have

$$\begin{aligned} J_1^r + J_2^r + \cdots + J_q^r =I^r= \Big ||\sup _{k \in \mathbb {N}} \big |\mathcal {F}^{-1} \big (\Psi _k \eta _s \hat{f} \big ) \big |\Big ||_{\ell ^r}^r \lesssim \big ||\mathcal {F}^{-1} \big (\eta _s \hat{f}\big ) \big ||_{\ell ^r}^r. \end{aligned}$$

Also for any $l, l' \in \{1,2,\ldots ,q\}$

$$\begin{aligned} \bigg ||\sup _{k \in \mathbb {N}} \bigg |\int _{-\frac{1}{2}}^{\frac{1}{2}} e^{-2\pi i \xi (q j + l)} \big (1 - e^{2\pi i \xi (l-l')}\big ) \Psi _k(\xi )\eta _s(\xi ) \hat{f}(\xi ) d\xi \bigg |\bigg ||_{\ell ^r(j)}\\ \lesssim \bigg ||\int _{-\frac{1}{2}}^{\frac{1}{2}} e^{-2\pi i \xi j} \big (1 - e^{2\pi i \xi (l-l')}\big ) \eta _s(\xi ) \hat{f}(\xi ) d\xi \bigg ||_{\ell ^r(j)}. \end{aligned}$$

Since $\eta _s = \eta _s \eta _{s-1}$, by Minkowski’s inequality and Lemma 1 we obtain that the last expression can be dominated by

$$\begin{aligned} \bigg ||\int _{\!-\!\frac{1}{2}}^{\frac{1}{2}} e^{-2\pi i \xi j} \big (1 \!-\! e^{2\pi i \xi (l-l')}\big ) \eta _{s-1}(\xi ) d\!\xi \bigg ||_{\ell ^1(j)} \!\big ||\mathcal {F}^{-1} \!\big (\eta _s \hat{f}\big ) \big ||_{\ell ^r} \le C q A^{-s} \big ||\mathcal {F}^{-1} \big (\eta _s \hat{f}\big ) \big ||_{\ell ^r}. \end{aligned}$$

Therefore, by (18)

$$\begin{aligned} J_l \le J_{l'} + q^{-1} \big ||\mathcal {F}^{-1} \big (\eta _s \hat{f}\big ) \big ||_{\ell ^r}. \end{aligned}$$

Summing up over all $l' \in \{1, 2, \ldots , q\}$ we obtain

$$\begin{aligned} q J_l^r \le 2^{r-1} I^r + C 2^{r-1} q^{1-r} \big ||\mathcal {F}^{-1} \big (\eta _s \hat{f}\big ) \big ||_{\ell ^r}^r \lesssim \big ||\mathcal {F}^{-1} \big (\eta _s \hat{f}\big ) \big ||_{\ell ^r}^r. \end{aligned}$$

Finally, by Lemma 2 we conclude

$$\begin{aligned} \Big ||\sup _{k \in \mathbb {N}} \big |\mathcal {F}^{-1} \big (\Psi _k \eta _s \hat{f} \big )(q j + l) \big |\Big ||_{\ell ^r(j)} \lesssim \big ||\mathcal {F}^{-1} \big (\eta _s \hat{f}\big )(qj+l) \big ||_{\ell ^r(j)}. \end{aligned}$$

(25)

Next, we resume the analysis of (24). Using (25) we get

$$\begin{aligned}&\bigg ( \sum _{l=1}^q \Big ||\sup _{k \in \mathbb {N}} \big |\mathcal {F}^{-1} \big (\Psi _k \eta _s F_b(\cdot \ ; l) \big ) (q j + l) \big |\Big ||_{\ell ^r(j)}^r \bigg )^{1/r}\\&\quad \lesssim \bigg ( \sum _{l=1}^q \Big ||\mathcal {F}^{-1} \big (\eta _s F_b(\cdot \ ; l) \big ) (q j + l) \Big ||_{\ell ^r(j)}^r \bigg )^{1/r}. \end{aligned}$$

We observe that by the change of variables

$$\begin{aligned} \mathcal {F}^{-1}\big (\eta _s F_b(\cdot \ ; l)\big )(qj+l) =\sum _{a=1}^b \mathcal {F}^{-1}\big (\eta _s(\cdot - a/b) \hat{f}\big )(qj+l). \end{aligned}$$

Thus by Minkowski’s inequality

$$\begin{aligned} \Bigg ( \sum _{l=1}^q \Big ||\mathcal {F}^{-1} \big (\eta _s F_b(\cdot \ ; l) \big ) (q j + l) \Big ||_{\ell ^r(j)}^r \Bigg )^{1/r} \le \Bigg ||\mathcal {F}^{-1} \Bigg (\sum _{a=1}^b \eta _s(\cdot - a/b) \Bigg ) \Bigg ||_{\ell ^1} {\left\| f \right\| }_{\ell ^r}. \end{aligned}$$

Since for $j \in \mathbb {Z}$

$$\begin{aligned} \sum _{a=1}^b e^{-2\pi i j a/b} = \left\{ {\begin{array}{ll} b &{} \quad \text {if} \; b \mid j,\\ 0 &{} \quad \text {otherwise} \end{array} }\right. \end{aligned}$$

we conclude

$$\begin{aligned} \Bigg ||\mathcal {F}^{-1} \Bigg (\sum _{a=1}^b \eta _s(\cdot - a/b) \Bigg ) \Bigg ||_{\ell ^1} = \Bigg ||\mathcal {F}^{-1} \eta _s(j) \sum _{a=1}^b e^{-2\pi i j a/b} \Bigg ||_{\ell ^1(j)} =b {\left\| \mathcal {F}^{-1} \eta _s(bj) \right\| }_{\ell ^1(j)}. \end{aligned}$$

Now Lemmas 1 and 2 imply

$$\begin{aligned} b {\left\| \mathcal {F}^{-1} \eta _s(bj) \right\| }_{\ell ^1(j)} \lesssim {\left\| \mathcal {F}^{-1} \eta _s \right\| }_{\ell ^1} \lesssim 1. \end{aligned}$$

This completes the proof of (24). Finally, by (4) and (21) we obtain that

$$\begin{aligned} \Bigg ||\sup _{k \in \mathbb {N}} \Big |\sum _{j=0}^k \mathcal {F}^{-1} (\nu _j^s \hat{f}) \Big |\Bigg ||_{\ell ^r} \lesssim 2^{\epsilon s} {\left\| f \right\| }_{\ell ^r} \end{aligned}$$

(26)

for any $\epsilon > 0$ and $s \in \mathbb {N}$. If $r = 2$ we may refine the estimate (26) (see also [1]). Let

$$\begin{aligned} G_q(\xi ) = \sum _{a \in A_q} \eta _{s-1}(\xi - a/q) \hat{f}(\xi ). \end{aligned}$$

and note that

$$\begin{aligned} \sum _{a \in A_q} \mathcal {F}^{-1} \big (\Psi _k(\cdot -a/q) \eta _s(\cdot -a/q) \hat{f} \big ) =\sum _{a \in A_q} \mathcal {F}^{-1} \big (\Psi _k(\cdot - a/q) \eta _s(\cdot -a/q) G_q\big ) \end{aligned}$$

since $\eta _s = \eta _s \eta _{s-1}$, and the supports of $\eta _s(\cdot -a/q)$’s are disjoint when $a/q$ varies. By (21) we have

$$\begin{aligned} \Bigg ||\sup _{k \in \mathbb {N}} \Big |\sum _{a \in A_q} \mathcal {F}^{-1} (\Psi _k(\cdot -a/q) \eta _s(\cdot -a/q) G_q) \Big |\Bigg ||_{\ell ^2} \lesssim q^{\epsilon } {\left\| \mathcal {F}^{-1} G_q \right\| }_{\ell ^2} \end{aligned}$$

whereas by (4), we have

$$\begin{aligned} \!\Bigg ||\sup _{k \in \mathbb {N}} \Big |\!\sum _{j=0}^k \mathcal {F}^{-1}\big (\nu ^s_j \hat{f}\big ) \Big |\Bigg ||_{\ell ^2} \le \sum _{q \!=\! 2^s}^{2^{s+1}-1}q^{-1\!+\!\epsilon }\Bigg ||\sup _{k \in \mathbb {N}} \Big |\sum _{a \in A_q} \mathcal {F}^{-1} (\Psi _k(\cdot \!-\!a/q) \eta _s(\cdot \!-\!a/q) \hat{f}) \Big |\Bigg ||_{\ell ^2}. \end{aligned}$$

These two bounds yield

$$\begin{aligned} \Bigg ||\sup _{k \in \mathbb {N}} \Big |\sum _{j=0}^k \mathcal {F}^{-1}\big (\nu ^s_j \hat{f}\big ) \Big |\Bigg ||_{\ell ^2} \lesssim \sum _{q = 2^s}^{2^{s+1}-1} q^{-1+2\epsilon } {\left\| \mathcal {F}^{-1} G_q \right\| }_{\ell ^2}\\ \lesssim 2^{-s/2 + 2\epsilon s} \Bigg ( \sum _{a/q \in \fancyscript{R}_s} \Big ||\mathcal {F}^{-1} \big (\eta _{s-1}(\cdot -a/q) \hat{f}\big ) \Big ||_{\ell ^2}^2 \Bigg )^{1/2}, \end{aligned}$$

where the last estimate follows from Cauchy–Schwarz inequality and the definition of $G_q$. Finally, by Plancherel’s theorem we may write

$$\begin{aligned} \sum _{a/q \in \fancyscript{R}_s} \Bigg ||\mathcal {F}^{-1} \big (\eta _{s-1}(\cdot -a/q) \hat{f}\big ) \Bigg ||_{\ell ^2}^2 = \sum _{a/q \in \fancyscript{R}_s} \int _{\mathbb {R}} {\left|{\eta _{s-1}(\xi - a/q)} \right|}^2 \big |\hat{f}(\xi ) \big |^2 d\xi \end{aligned}$$

which is majorized by ${\left\| f \right\| }_{\ell ^2}^2$. Thus for appropriately chosen $\epsilon >0$ we obtain

$$\begin{aligned} \Bigg ||\sup _{k \in \mathbb {N}} \Big |\sum _{j=0}^k \mathcal {F}^{-1}\big (\nu _j^s \hat{f}\big ) \Big |\Bigg ||_{\ell ^2} \le 2^{-s/4} {\left\| f \right\| }_{\ell ^2}. \end{aligned}$$

(27)

Next, for $r \ne 2$ we can use Marcinkiewicz interpolation theorem and interpolate between (26) and (27) to conclude the proof. $\square$

3.3 Maximal Function

We have gathered necessary tools to illustrate the proof of Theorem 2. First, we show the boundedness on $\ell ^r(\mathbb {Z})$ of the maximal function $T^*$.

Theorem 4

The maximal function $T^*$ is bounded on $\ell ^r(\mathbb {Z})$ for each $1 < r < \infty$.

Proof

Let us observe that for a non-negative function $f$

$$\begin{aligned} T^* f(n) \lesssim \sup _{k \in \mathbb {N}} \Bigg |\sum _{j = 0}^k \mathcal {F}^{-1}\big (m_j \hat{f}\big )(n) \Bigg |+ \mathcal {M} f(n) \end{aligned}$$

where $\mathcal {M} f = \sup _{N \in \mathbb {N}} {\left|{A_N f} \right|}$ is a maximal function corresponding with Bourgain–Wierdl’s averages

$$\begin{aligned} A_N f(n) = N^{-1} \sum _{p \in \pm \mathbb {P}_N} f(n-p) \log {\left|{p} \right|}. \end{aligned}$$

Indeed, suppose $\tau ^k \le N < \tau ^{k+1}$ for $k \in \mathbb {N}$. Then

$$\begin{aligned} T_N f(n) = \sum _{j=0}^k \sum _{p \in \pm \mathbb {P}} f(n-p) K_j(p) \log {\left|{p} \right|} -\sum _{p \in \pm R_N} f(n-p) K(p) \log {\left|{p} \right|}. \end{aligned}$$

where $R_N = \mathbb {P}\cap (N, \tau ^{k+1})$. Therefore, by (1), we see

$$\begin{aligned} \Bigg |\sum _{p \in R_N} f(n-p) K(p) \log {\left|{p} \right|} \Bigg |\lesssim \tau ^{-k} \sum _{p \in \pm \mathbb {P}_{\tau ^{k+1}}} f(n-p) \log |p| \lesssim A_{\tau ^{k+1}} f(n). \end{aligned}$$

Since the maximal function $\mathcal {M}$ is bounded on $\ell ^r(\mathbb {Z})$ for any $r > 1$ (see [3] or Appendix 1) thus we have reduced the boundedness of $T^*$ to proving

$$\begin{aligned} \Bigg ||\sup _{k \in \mathbb {N}} \Big |\sum _{j=0}^k \mathcal {F}^{-1} \big (m_j \hat{f} \big ) \Big |\Bigg ||_{\ell ^r} \lesssim {\left\| f \right\| }_{\ell ^r}. \end{aligned}$$

Let us consider $f \in \ell ^r(\mathbb {Z})$ for $r>1$. By Theorem 3 we know that for $j\in \mathbb {N}$

$$\begin{aligned} \Bigg ||\mathcal {F}^{-1} \big (\nu _j \hat{f} \big ) \big ||_{\ell ^r}&\le \sum _{s \in \mathbb {N}} \Bigg ||\mathcal {F}^{-1} \big (\nu _j^s \hat{f}\big ) \big ||_{\ell ^r} \le \sum _{s \in \mathbb {N}}\Bigg \Vert \sup _{k\in \mathbb {N}}\Big |\sum _{j=0}^k \mathcal {F}^{-1} \big (\nu _j^s \hat{f}\big )-\sum _{j=0}^{k-1} \mathcal {F}^{-1} \big (\nu _j^s \hat{f}\big )\Big | \Bigg \Vert _{\ell ^r}\\&\lesssim \sum _{s \in \mathbb {N}}\Bigg \Vert \sup _{k\in \mathbb {N}}\Big |\sum _{j=0}^k \mathcal {F}^{-1} \big (\nu _j^s \hat{f}\big )\Big | \Bigg \Vert _{\ell ^r} \lesssim \sum _{s \in \mathbb {N}} 2^{-\delta s} {\left\| f \right\| }_{\ell ^r} \lesssim {\left\| f \right\| }_{\ell ^r}. \end{aligned}$$

If $f$ is non-negative then

$$\begin{aligned} \Big |\sum _{p \in \pm \mathbb {P}} f(x-p) K_j(p) \log {\left|{p} \right|} \Big |\le \tau ^{-j} \sum _{p \in \pm \mathbb {P}_{\tau ^{j+1}}} f(x-p) \log {\left|{p} \right|} \end{aligned}$$

thus by Prime Number Theorem,

$$\begin{aligned} \big ||\mathcal {F}^{-1} \big (m_j \hat{f}\big ) \big ||_{\ell ^r} \le \tau ^{-j} \Bigg (\sum _{p \in \mathbb {P}_{\tau ^{j+1}}} \log p \Bigg ) {\left\| f \right\| }_{\ell ^r} \lesssim {\left\| f \right\| }_{\ell ^r}. \end{aligned}$$

Hence,

$$\begin{aligned} \Big ||\mathcal {F}^{-1} \big ((m_j - \nu _j)\hat{f} \big ) \Big ||_{\ell ^r} \lesssim {\left\| f \right\| }_{\ell ^r}. \end{aligned}$$

(28)

For $r = 2$ we use Proposition 3.2 to get

$$\begin{aligned} \Big ||\mathcal {F}^{-1}\big ((m_j - \nu _j) \hat{f} \big ) \Big ||_{\ell ^2} \le {\left\| m_j - \nu _j \right\| }_{L^\infty } {\left\| f \right\| }_{\ell ^2} \lesssim j^{-\alpha } {\left\| f \right\| }_{\ell ^2} \end{aligned}$$

(29)

for any $\alpha >0$ big enough. If $r \ne 2$ we apply Marcinkiewicz interpolation theorem to interpolate between (28) and (29) and obtain

$$\begin{aligned} \Big ||\mathcal {F}^{-1} \big ((m_j - \nu _j)\hat{f} \big ) \Big ||_{\ell ^r} \lesssim j^{-2} {\left\| f \right\| }_{\ell ^r}. \end{aligned}$$

(30)

Since

$$\begin{aligned} \Bigg ||\sup _{k \in \mathbb {N}} \Big |\sum _{j=0}^k \mathcal {F}^{-1} \big ((m_j - \nu _j)\hat{f} \big ) \Big |\Bigg ||_{\ell ^r} \le \sum _{j \in \mathbb {N}} \Big ||\mathcal {F}^{-1} \big ((m_j - \nu _j) \hat{f} \big ) \Big ||_{\ell ^r} \end{aligned}$$

by (30) and Theorem 3 we finish the proof. $\square$

Next, we demonstrate the pointwise convergence of $\left( {T_N}: {N \in \mathbb {N}}\right)$.

Proposition 3.4

If $f \in \ell ^r(\mathbb {Z})$, $1 < r < \infty$ then for every $n \in \mathbb {Z}$

$$\begin{aligned} \lim _{N \rightarrow \infty } T_N f(n) = T f (n) \end{aligned}$$

(31)

and $T$ is bounded on $\ell ^r(\mathbb {Z})$.

Proof

If $N \in \mathbb {N}$ we define an operator $T^N$ by setting

$$\begin{aligned} T^N f(n) = \sum _{\genfrac{}{}{0.0pt}2{p\in \pm \mathbb {P}}{|p|> N}} f(n-p) K(p) \log |p| \end{aligned}$$

for any $f \in \ell ^r(\mathbb {Z})$. By Hölder’s inequality we see that for every $n \in \mathbb {Z}$

$$\begin{aligned} {\left|{T^Nf(n)} \right|} \le 2 \Big ( \sum _{\genfrac{}{}{0.0pt}2{p \in \mathbb {P}}{p > N}} \big ( p^{-1} \log p \big )^{r'} \Big )^{1/{r'}} {\left\| f \right\| }_{\ell ^r} \end{aligned}$$

where $r'$ stands for the conjugate exponent to $r$, i.e. $1/r+1/r'=1$. The last inequality shows that, on the one hand, $T$ is well defined for any $f\in \ell ^r(\mathbb {Z})$, on the other proves (31). Next, Fatou’s lemma with boundedness of $T^*$ yield

$$\begin{aligned} {\left\| Tf \right\| }_{\ell ^r} = {\left\| \liminf _{N\rightarrow \infty }T_N f \right\| }_{\ell ^r} \le \liminf _{N\rightarrow \infty } \Big ||T_Nf \Big ||_{\ell ^r} \le \Big ||T^*f \Big ||_{\ell ^r} \lesssim _r {\left\| f \right\| }_{\ell ^r} \end{aligned}$$

which completes the proof. $\square$

3.4 Oscillatory Norm for $H_N$

Let $\left( {N_j}: {j \in \mathbb {N}}\right)$ be a strictly increasing sequence of $\Lambda$ elements. We set $N_j = \tau ^{k_j}$ and $\Lambda _j = \Lambda \cap (N_j, N_{j+1}]$. In this Section we consider the kernel $K(x) = x^{-1}$. Since each $K_j$ for $j \in \mathbb {N}$ has mean zero we have

$$\begin{aligned} {\left|{\Phi _j(\xi )} \right|} \le \int _{\mathbb {R}} {\left|{1 - e^{2\pi i \xi x}} \right|} {\left|{K_j(x)} \right|} dx \lesssim {\left|{\xi } \right|} \tau ^j. \end{aligned}$$

(32)

Let $H_N$ denote the truncated Hilbert transform

$$\begin{aligned} H_N f(n) = \sum _{p \in \pm \mathbb {P}_N} \frac{f(n - p)}{p} \log {\left|{p} \right|}. \end{aligned}$$

The following argument is based on [1, Section 7].

Proposition 3.5

There is $C > 0$ such that for every $J \in \mathbb {N}$ and $s\in \mathbb {N}$ we have

$$\begin{aligned} \sum _{j=0}^J \Bigg ||\sup _{\tau ^k \in \Lambda _j} \Big |\mathcal {F}^{-1}\big ((\Psi _k -\Psi _{k_j}) \eta _s \hat{f} \big ) \Big |\Bigg ||_{\ell ^2}^2 \le C \big \Vert \mathcal {F}^{-1}\big (\eta _s \hat{f}\big )\big \Vert _{\ell ^2}^2. \end{aligned}$$

Proof

Let $B_j = \{x \in (-1/2, 1/2): {\left|{x} \right|} \le N_j^{-1}\}$. By Plancherel’s theorem we have

$$\begin{aligned}&\sum _{j=0}^J \Bigg ||\sup _{\tau ^k \in \Lambda _j} \Big |\mathcal {F}^{-1} \big ((\Psi _k - \Psi _{k_j}) {1\!\!1_{{B_{j+1}}}} \eta _s \hat{f}\big ) \Big |\Bigg ||_{\ell ^2}^2 \\&\quad \le \sum _{j=0}^J \sum _{k = k_j}^{k_{j+1}} \Bigg ||\mathcal {F}^{-1}\big ((\Psi _k - \Psi _{k_j}) {1\!\!1_{{B_{j+1}}}} \eta _s \hat{f}\big ) \Bigg ||_{\ell ^2}^2\\&\quad \le \Bigg ||\sum _{j=0}^J {1\!\!1_{{B_{j+1}}}} \sum _{k=k_j}^{k_{j+1}} {\left|{\Psi _k - \Psi _{k_j}} \right|}^2 \Bigg ||_{L^\infty } \big \Vert \mathcal {F}^{-1}\big (\eta _s \hat{f}\big )\big \Vert _{\ell ^2}^2. \end{aligned}$$

By (32) we have

$$\begin{aligned} {\left|{\Psi _k(\xi ) - \Psi _{k_j}(\xi )} \right|} = \Bigg |\sum _{l = k_j+1}^k \Phi _l(\xi ) \Bigg |\lesssim {\left|{\xi } \right|} \tau ^k. \end{aligned}$$

Hence,

$$\begin{aligned}&\sum _{j=0}^J {1\!\!1_{{B_{j+1}}}}(\xi ) \sum _{k = k_j}^{k_{j+1}} {\left|{\Psi _k(\xi ) - \Psi _{k_j}(\xi )} \right|}^2 \\&\quad \lesssim {\left|{\xi } \right|}^2 \sum _{j=0}^J {1\!\!1_{{B_{j+1}}}}(\xi ) \sum _{k = k_j}^{k_{j+1}} \tau ^{2k} \lesssim {\left|{\xi } \right|}^2 \sum _{j: N_{j+1} \le {\left|{\xi } \right|}^{-1}} N_{j+1}^2 \lesssim 1. \end{aligned}$$

Therefore, we obtain

$$\begin{aligned} \sum _{j=0}^J \Bigg ||\sup _{\tau ^k \in \Lambda _j} \big |\mathcal {F}^{-1} \big ((\Psi _k - \Psi _{k_j}) {1\!\!1_{{B_{j+1}}}} \eta _s \hat{f}\big ) \big |\Bigg ||_{\ell ^2}^2 \lesssim \Big \Vert \mathcal {F}^{-1}\big (\eta _s \hat{f}\big )\Big \Vert _{\ell ^2}^2. \end{aligned}$$

Similar for $B_j^c$, replacing $\Psi _{k_j}$ by $\Psi _{k_{j+1}}$ under the supremum, we can estimate

$$\begin{aligned}&\sum _{j=0}^J \Bigg ||\sup _{\tau ^k \in \Lambda _j} \big |\mathcal {F}^{-1} \big ((\Psi _k - \Psi _{k_j}) {1\!\!1_{{B_j^c}}} \eta _s \hat{f}\big ) \big |\Bigg ||_{\ell ^2}^2 \\&\quad \le \sum _{j=0}^J \sum _{k = k_j}^{k_{j+1}} \Bigg ||\mathcal {F}^{-1}\big ((\Psi _{k_{j+1}} - \Psi _k) {1\!\!1_{{B_j^c}}} \eta _s \hat{f}\big ) \Bigg ||_{\ell ^2}^2\\&\quad \le \Bigg ||\sum _{j=0}^J {1\!\!1_{{B_j^c}}} \sum _{k=k_j}^{k_{j+1}} {\left|{\Psi _{k_{j+1}} - \Psi _k} \right|}^2 \Bigg ||_{L^\infty } \Big \Vert \mathcal {F}^{-1}\big (\eta _s \hat{f}\big )\Big \Vert _{\ell ^2}^2. \end{aligned}$$

Now, using (6) we get

$$\begin{aligned} {\left|{\Psi _{k_{j+1}}(\xi ) - \Psi _k(\xi )} \right|} \lesssim {\left|{\xi } \right|}^{-1} \tau ^{-k} \end{aligned}$$

thus

$$\begin{aligned} \sum _{j=0}^J {1\!\!1_{{B_j^c}}}(\xi ) \sum _{k = k_j}^{k_{j+1}} {\left|{\Psi _{k_{j+1}}(\xi ) - \Psi _k(\xi )} \right|}^2\lesssim & {} {\left|{\xi } \right|}^{-2} \sum _{j=0}^J {1\!\!1_{{B_j^c}}}(\xi ) \sum _{k = k_j}^{k_{j+1}} \tau ^{-2k}\\\lesssim & {} {\left|{\xi } \right|}^{-2} \sum _{j: N_j \ge {\left|{\xi } \right|}^{-1}} N_j^{-2} \lesssim 1. \end{aligned}$$

Therefore, we conclude

$$\begin{aligned} \sum _{j=0}^J \Bigg ||\sup _{\tau ^k \in \Lambda _j} \Big |\mathcal {F}^{-1} \big ((\Psi _k - \Psi _{k_j}) {1\!\!1_{{B_j^c}}} \eta _s \hat{f}\big ) \Big |\Bigg ||_{\ell ^2}^2 \lesssim \big \Vert \mathcal {F}^{-1}\big (\eta _s \hat{f}\big )\big \Vert _{\ell ^2}^2. \end{aligned}$$

Finally, by Proposition 3.3

$$\begin{aligned} \sum _{j=0}^J \Bigg ||\sup _{\tau ^k \in \Lambda _j} \big |\mathcal {F}^{-1}\big ((\Psi _k - \Psi _{k_j}) {1\!\!1_{{B_j}}} {1\!\!1_{{B_{j+1}^c}}} \eta _s \hat{f} \big ) \big |\Bigg ||_{\ell ^2}^2 \lesssim \sum _{j=0}^J {\left\| \mathcal {F}^{-1} \big ({1\!\!1_{{B_j}}} {1\!\!1_{{B_{j+1}^c}}} \eta _s\hat{f}\big ) \right\| }_{\ell ^2}^2 \end{aligned}$$

which is bounded by $\big \Vert \mathcal {F}^{-1}\big (\eta _s \hat{f}\big )\big \Vert _{\ell ^2}^2$. $\square$

Theorem 5

For every $J \in \mathbb {N}$ there is $C_J$ such that

$$\begin{aligned} \sum _{j=0}^J \Bigg ||\sup _{\tau ^k \in \Lambda _j} \big |H_{\tau ^k } f - H_{N_j} f \big |\Bigg ||_{\ell ^2}^2 \le C_J {\left\| f \right\| }_{\ell ^2}^2 \end{aligned}$$

and $\lim _{J \rightarrow \infty } C_J/J = 0$.

Proof

By Proposition 3.2, we have

$$\begin{aligned} \sum _{j=0}^J \Bigg ||\sup _{\tau ^k \in \Lambda _j} \Big |\sum _{l = k_j+1}^k \mathcal {F}^{-1} \big ((m_l - \nu _l)\hat{f}\big ) \Big |\Bigg ||_{\ell ^2}^2 \lesssim \Bigg ( \sum _{j=0}^J \sum _{l = k_j+1}^{k_{j+1}} l^{-2} \Bigg ) {\left\| f \right\| }_{\ell ^2}^2\lesssim {\left\| f \right\| }_{\ell ^2}^2. \end{aligned}$$

Consequently, it is enough to demonstrate

$$\begin{aligned} \sum _{j = 0}^J \Big ||\sup _{\tau ^k \in \Lambda _j} \Big |\sum _{l = k_j+1}^k \mathcal {F}^{-1}\big (\nu _l \hat{f}\big ) \Big |\Big ||_{\ell ^2}^2 \le C_J {\left\| f \right\| }_{\ell ^2}^2 \end{aligned}$$

where $\lim _{J \rightarrow \infty } C_J/J = 0$.

Let $s_0 \in \mathbb {N}$ be defined as $2^{s_0} \le J^{1/3} < 2^{s_0+1}$. By Theorem 3 we have

$$\begin{aligned} \Big ||\sup _{\tau ^k \in \Lambda _j} \Big |\sum _{s = s_0}^{\infty } \sum _{l=k_j+1}^k \mathcal {F}^{-1}\big (\nu _l^s \hat{f} \big ) \Big |\Big ||_{\ell ^2} \lesssim \sum _{s = s_0}^\infty \Big ||\sup _{k \in \mathbb {N}} \Big |\sum _{l=0}^k \mathcal {F}^{-1}\big (\nu _l^s \hat{f} \big ) \Big |\Big ||_{\ell ^2} \lesssim J^{-\delta /3} {\left\| f \right\| }_{\ell ^2}. \end{aligned}$$

We set

$$\begin{aligned} D_J = \sum _{s=0}^{s_0-1} \sum _{a/q \in \fancyscript{R}_s} \frac{1}{\varphi (q)^2}. \end{aligned}$$

By the change of variables, Cauchy–Schwarz inequality and by Proposition 3.5 we get

$$\begin{aligned}&\sum _{j=0}^J \Bigg ||\sup _{\tau ^k \in \Lambda _j} \Big |\sum _{s = 0}^{s_0-1} \sum _{l=k_j+1}^k \mathcal {F}^{-1}\big (\nu _l^s \hat{f} \big ) \Big |\Bigg ||_{\ell ^2}^2\\&\quad \le \sum _{j = 0}^J\Bigg (\sum _{s = 0}^{s_0-1}\sum _{a/q \in \fancyscript{R}_s}\frac{1}{\varphi (q)} \Bigg ||\sup _{\tau ^k \in \Lambda _j} \Big |\sum _{l=k_j+1}^k \mathcal {F}^{-1}\big (\Phi _l \eta _s \hat{f}(\cdot + a/q)\big ) \Big |\Bigg ||_{\ell ^2}\Bigg )^2\\&\quad \le D_J \sum _{s = 0}^{s_0-1} \sum _{a/q \in \fancyscript{R}_s} \sum _{j = 0}^J \Bigg ||\sup _{\tau ^k \in \Lambda _j} \Big |\mathcal {F}^{-1}\big ((\Psi _k-\Psi _{k_j}) \eta _s \hat{f}(\cdot + a/q)\big ) \Big |\Bigg ||_{\ell ^2}^2\\&\quad \lesssim D_J \sum _{s = 0}^{s_0-1} \sum _{a/q \in \fancyscript{R}_s} \Big \Vert \mathcal {F}^{-1}\big (\eta _s(\cdot - a/q) \hat{f}\big ) \Big \Vert _{\ell ^2}^2\lesssim D_J s_0 \Vert f\Vert _{\ell ^2}^2. \end{aligned}$$

By the definition of $\fancyscript{R}_s$ we see that $D_J\lesssim 2^{s_0}\le J^{1/3}$ thus we achieve

$$\begin{aligned} \sum _{j = 0}^J \Bigg ||\sup _{\tau ^k \in \Lambda _j} \Big |\sum _{l = k_j+1}^k \mathcal {F}^{-1}\big (\nu _l \hat{f}\big ) \Big |\Bigg ||_{\ell ^2}^2 \lesssim J \Big (J^{-\delta /3} + J^{-1/3} \log J \Big ) {\left\| f \right\| }_{\ell ^2}^2 \end{aligned}$$

which finishes the proof. $\square$

4 Dynamical Systems

Let $(X, \mathcal {B}, \mu , S)$ be a dynamical system on a measure space $X$. Let $S: X \rightarrow X$ be an invertible measure preserving transformation. For $N > 0$ we set

$$\begin{aligned} \mathcal {H}_N f (x) = \sum _{p \in \pm \mathbb {P}_N} \frac{f(S^{-p} x)}{p}\log {\left|{p} \right|}. \end{aligned}$$

We are going to show Theorem 1. We start from oscillatory norm.

Proposition 4.1

For each $J \in \mathbb {N}$ there is $C_J$ such that

$$\begin{aligned} \sum _{j = 0}^J \Bigg ||\sup _{N \in \Lambda _j} \big |\mathcal {H}_N f - \mathcal {H}_{N_j} f \big |\Bigg ||_{L^2(\mu )}^2 \le C_J {\left\| f \right\| }_{L^2(\mu )}^2 \end{aligned}$$

and $\lim _{J \rightarrow \infty } C_J/J = 0$.

Proof

Let $R \ge N_{J}$. For a fixed $x \in X$ we define a function on $\mathbb {Z}$ by

$$\begin{aligned} \phi (n) = \left\{ {\begin{array}{ll} f(S^n x) &{} \quad {\left|{n} \right|} \le R, \\ 0 &{} \quad \text {otherwise.} \end{array} }\right. \end{aligned}$$

Then for ${\left|{n} \right|} \le R - N$

$$\begin{aligned} \mathcal {H}_N f(S^n x) = \sum _{p \in \pm \mathbb {P}_N} \frac{f(S^{n-p}x)}{p} \log {\left|{p} \right|} = \sum _{p \in \pm \mathbb {P}_N} \frac{\phi (n-p)}{p} \log {\left|{p} \right|} = H_N \phi (n). \end{aligned}$$

Hence,

$$\begin{aligned} \sum _{{\left|{n} \right|} = 0}^{R - N_J} \sup _{N \in \Lambda _j} \big |\mathcal {H}_N f(S^n x) - \mathcal {H}_{N_j} f(S^n x) \big |^2 \le \big ||\sup _{N \in \Lambda _j} \big |H_N \phi - H_{N_j} \phi \big |\big ||_{\ell ^2}^2. \end{aligned}$$

Therefore, by Theorem 5 we can estimate

$$\begin{aligned} \sum _{{\left|{n} \right|} = 0}^{R - N_J} \sum _{j = 0}^J \sup _{N \in \Lambda _j} \big |\mathcal {H}_N f(S^n x) - \mathcal {H}_{N_j} f(S^n x) \big |^2 \le C_J {\left\| \phi \right\| }_{\ell ^2}^2 = C_J \sum _{{\left|{n} \right|}=0}^R {\left|{f(S^n x)} \right|}^2. \end{aligned}$$

Since $S$ is a measure preserving transformation integration with respect to $x \in X$ implies

$$\begin{aligned} (R - N_J) \sum _{j = 0}^J \Bigg ||\sup _{N \in \Lambda _j} \big |\mathcal {H}_N f - \mathcal {H}_{N_j} f \big |\Bigg ||_{L^2(\mu )}^2 \le C_J R {\left\| f \right\| }_{L^2(\mu )}^2. \end{aligned}$$

Finally, if we divide both sides by $R$ and take $R \rightarrow \infty$ we conclude the proof. $\square$

Corollary 1

The maximal function

$$\begin{aligned} \mathcal {H}^* f(x) = \sup _{N \in \mathbb {N}} \big |\mathcal {H}_N f (x)\big |\end{aligned}$$

is bounded on $L^r(\mu )$ for each $1 < r < \infty$.

Next, we show the pointwise convergence of $\left( {\mathcal {H}_N}: {N \in \mathbb {N}}\right)$.

Theorem 6

Let $f \in L^r(\mu )$, $1 < r < \infty$. For $\mu$-almost every $x \in X$

$$\begin{aligned} \lim _{N \rightarrow \infty } \mathcal {H}_N f(x) = \mathcal {H} f(x) \end{aligned}$$

and $\mathcal {H}$ is bounded on $L^r(\mu )$.

Proof

Let $f \in L^2(\mu )$, since the maximal function $\mathcal {H}^*$ is bounded on $L^2(\mu )$ we may assume $f$ is bounded by $1$. Suppose $\left( {\mathcal {H}_N f}: {N \in \mathbb {N}}\right)$ does not converge $\mu$-almost everywhere. Then there is $\epsilon > 0$ such that

$$\begin{aligned} \mu \Big \{x\in X : \limsup _{M, N\rightarrow \infty } \big |\mathcal {H}_N f(x) - \mathcal {H}_{M} f(x) \big |> 4\epsilon \Big \} > 4\epsilon . \end{aligned}$$

Now one can find a strictly increasing sequence of integers $\left( {k_j}: {j\in \mathbb {N}}\right)$ such that for each $j \in \mathbb {N}$

$$\begin{aligned} \mu \Big \{x\in X : \sup _{N_j \le N \le N_{j+1}} \big |\mathcal {H}_N f(x) - \mathcal {H}_{N_j} f(x) \big |> \epsilon \Big \} > \epsilon \end{aligned}$$

where $N_j = \tau ^{k_j}$ and $\tau =1+\epsilon /4$. If $\tau ^k \le N < \tau ^{k+1}$ then setting $P_k = \mathbb {P}\cap (\tau ^k, \tau ^{k+1}]$ we get

$$\begin{aligned} \Big |\mathcal {H}_N f(x) - \mathcal {H}_{\tau ^k} f(x) \Big |\le \tau ^{-k} \sum _{p \in P_k} \log p. \end{aligned}$$

By Siegel–Walfisz theorem we get

$$\begin{aligned} \sum _{p \in \mathbb {P}_N} \log p =N + \mathcal {O}\left( N (\log N)^{-1}\right) \end{aligned}$$

thus there is $C > 0$ such that

$$\begin{aligned} \Bigg |\tau ^{-k} \sum _{p \in P_k} \log p -\tau + 1 \Bigg |\le C k^{-1} (\log \tau )^{-1}. \end{aligned}$$

Hence, whenever $k \ge 4 C \epsilon ^{-1} (\log \tau )^{-1}$ we have

$$\begin{aligned} \big |\mathcal {H}_N f(x) - \mathcal {H}_{\tau ^k} f (x) \big |\le \epsilon /2. \end{aligned}$$

In particular, we conclude

$$\begin{aligned} \mu \Big \{x\in X: \sup _{\tau ^k \in \Lambda _j} \big |\mathcal {H}_{\tau ^k } f(x) - \mathcal {H}_{N_j} f(x)\big |> \epsilon /2 \Big \} > \epsilon \end{aligned}$$

for each $k_j \ge 4C \epsilon ^{-1} (\log \tau )^{-1}$ which contradicts to Proposition 4.1. Indeed,

$$\begin{aligned} \epsilon ^3 \lesssim \frac{1}{J-J_0}\sum _{j = 0}^J \Bigg ||\sup _{\tau ^k \in \Lambda _j} \big |\mathcal {H}_{\tau ^k } f - \mathcal {H}_{N_j} f \big |\Bigg ||_{L^2(\mu )}^2 \le \frac{C_J}{J-J_0} {\left\| f \right\| }_{L^2(\mu )}^2 \end{aligned}$$

where $J_0=\min \{j\in \mathbb {N}: k_j\ge 4C \epsilon ^{-1} (\log \tau )^{-1}\}$. Now, the standard density argument implies pointwise convergence for each $f\in L^r(\mu )$ where $r>1$, and the proof of the theorem is completed. $\square$

Appendix: Boundedness of \(\mathcal {M}\)

In the Appendix we discuss why the maximal function

$$\begin{aligned} \mathcal {M} f(n) = \sup _{N \in \mathbb {N}} \Bigg |N^{-1} \sum _{p \in \pm \mathbb {P}_N} f(n - p) \log {\left|{p} \right|} \Bigg |\end{aligned}$$

is bounded on $\ell ^r(\mathbb {Z})$. This fact was published by Wierdl in [23], however, on p. 331 in the last equality for ** the factor $q$ has the power $1$ in place of $p$. Therefore, it is not sufficient to show an estimate (24) from [23] to conclude the proof. In fact, one has to prove the estimate corresponding to (25) from the present paper.

For the completeness we provide the sketch of the proof based on the method used in Sect. 3. First, we may restrict supremum to dyadic $N$. We modify the definition of the multiplier $m_j$ by setting

$$\begin{aligned} m_j(\xi ) = 2^{-j} \sum _{p \in \pm \mathbb {P}_N} e^{2\pi i \xi p} \log {\left|{p} \right|}. \end{aligned}$$

Hence, it suffices to show that for $r > 1$

$$\begin{aligned} \Bigg ||\sup _{k \in \mathbb {N}} \big |\mathcal {F}^{-1} \big (m_k \hat{f}\big ) \big |\Bigg ||_{\ell ^r} \lesssim {\left\| f \right\| }_{\ell ^r}. \end{aligned}$$

Keeping the definition of the major arcs and setting

$$\begin{aligned} \Psi _j(\xi ) = 2^{-j} \int \limits _{1 \le {\left|{x} \right|} \le 2^j} e^{2\pi i \xi x} dx \end{aligned}$$

Proposition 3.1 holds true. For proof we use the well-known result that for $\xi \in \mathfrak {M}_j^{\alpha }(a/q) \cap \mathfrak {M}_j^{\alpha }$ (see e.g. [11, Lemma 8.3])

$$\begin{aligned} \Bigg |m_{2^j}(\xi ) - 2^{-j} \frac{\mu (q)}{\varphi (q)} \sum _{1 \le {\left|{n} \right|} \le 2^j} e^{2\pi i \theta n} \Bigg |\lesssim j^{-\alpha } \end{aligned}$$

and then, as in the proof of Proposition 3.1, we replace the sum by $\Psi _j$. Also the demonstration of Proposition 3.2 has to be modified. There, the estimate for $\xi \not \in \mathfrak {M}_j^{\alpha }$ is a direct application of Vinogradov’s theorem. In the proof of Proposition 3.3 in the place of (20) we use $L^r$-boundedness of Hardy–Littlewood maximal function. Finally, in the proof of Theorem 4 we replace the sum $\sum _{j=0}^km_j$ with a single term $m_k$.