# Bootstrap methods in bounding discrete Radon operators

Wojciech Słomian  Faculty of Pure and Applied Mathematics, Wrocław University
of Science and Technology
Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland<EMAIL_ADDRESS>
###### Abstract.

The aim of this paper is to develop bootstrap arguments to establish maximal,
oscillation, variational and jump inequalities for the discrete averaging
Radon operators on $\ell^{p}(\mathbb{Z}^{d})$.

The author was partially supported by the National Science Centre in Poland,
grant Opus 2018/31/B/ST1/00204.

†† The paper is a part of author’s doctoral thesis written under the
supervision of Professor Mariusz Mirek.

##  1\. Introduction

Various kinds of seminorms have been intensively studied since Bourgain’s
seminal papers [2, 3, 4], where oscillation and variation seminorms were used
to prove the pointwise convergence of the ergodic averages along polynomials.
As many authors generalised and extended Bourgain’s work, oscillation,
variational and jump seminorms became widely used tools in harmonic analysis
and ergodic theory. This paper is motivated by the work of Mirek, Stein and
Zorin-Kranich [23] in which they proved, among others, that the so-called
bootstrap approach can be used to prove variational and jump inequalities for
continuous Radon operators. Our purpose is to show that one can deal with
maximal, oscillation, variational and jump inequalities in the context of
discrete Radon operators by using an analogous technique. This approach allows
us to prove seminorms inequalities without referring to vector-valued or
square function estimates as it was in previously known proofs. By using this
technique we give new proofs of well-known variational and jump results from
[21] and [24], as well as a new proof of the oscillation inequality, recently
proven in [19].
Let us consider a polynomial mapping

(1.1)
$\mathcal{P}=(\mathcal{P}_{1},\dots,\mathcal{P}_{d})\colon\mathbb{Z}^{k}\to\mathbb{Z}^{d},$

where each $\mathcal{P}_{j}\colon\mathbb{Z}^{k}\to\mathbb{Z}$ is a polynomial
of $k$ variables with integer coefficients such that $\mathcal{P}_{j}(0)=0$.
Let $\Omega$ be a bounded convex open subset of $\mathbb{R}^{k}$. For any
$t>0$ we set

$\Omega_{t}:=\\{x\in\mathbb{R}^{k}:t^{-1}x\in\Omega\\}.$

Moreover, we assume that $B(0,c_{\Omega})\subseteq\Omega\subseteq B(0,1)$ for
some $c_{\Omega}\in(0,1)$, where $B(x,t)$ is the open Euclidean ball in
$\mathbb{R}^{k}$ with radius $t>0$ centered at a point $x\in\mathbb{R}^{k}$.
Here one can think that $\Omega_{t}$ is the open ball of radius $t$ associated
with some norm on $\mathbb{R}^{k}$.

For finitely supported functions $f\colon\mathbb{Z}^{d}\to\mathbb{C}$ and
$t>0$, we define the discrete averaging Radon operator by setting

(1.2) $\displaystyle
M_{t}^{\mathcal{P}}f(x):=\frac{1}{\lvert\Omega_{t}\cap\mathbb{Z}^{k}\rvert}\sum_{y\in\Omega_{t}\cap\mathbb{Z}^{k}}f(x-\mathcal{P}(y)),\quad
x\in\mathbb{Z}^{d}.$

In the case of seminorms we will follow the notation from [19] and [24]. Let
$\mathbb{I}\subseteq\mathbb{R}$. For a given increasing sequence
$I=(I_{j}:j\in\mathbb{N})\subseteq\mathbb{I}$, the truncated oscillation
seminorm is defined for any function $f\colon\mathbb{I}\to\mathbb{C}$ by
setting

(1.3) $\displaystyle
O^{2}_{I,N}(f):=O_{I,N}^{2}(f(t):t\in\mathbb{I}):=\Big{(}\sum_{j=1}^{N}\sup_{\begin{subarray}{c}I_{j}\leq
t<I_{j+1}\\\ t\in\mathbb{I}\end{subarray}}\lvert
f(t)-f(I_{j})\rvert^{2}\Big{)}^{1/2}\quad\text{for all}\quad
N\in\mathbb{N}\cup\\{\infty\\}.$

A closely related concept to the oscillation seminorm is the $r$-variation
seminorm. Recall that for any $r\geq 0$ the $r$-variation seminorm $V^{r}$ of
a function $f\colon\mathbb{I}\to\mathbb{C}$ is defined by

(1.4) $\displaystyle
V^{r}(f):=V^{r}(f(t):t\in\mathbb{I}):=\sup_{J\in\mathbb{N}}\sup_{\begin{subarray}{c}t_{0}<\cdots<t_{J}\\\
t_{j}\in\mathbb{I}\end{subarray}}\Big{(}\sum_{j=0}^{J-1}\lvert
f(t_{j+1})-f(t_{j})\rvert^{r}\Big{)}^{1/r},$

where the latter supremum is taken over all finite increasing sequences in
$\mathbb{I}$. In the case of $r=\infty$ we have an obvious modification.
Finally, for any $\lambda>0$ and $\mathbb{I}\subseteq\mathbb{R}$, the
_$\lambda$ -jump counting function_ of a function
$f\colon\mathbb{I}\to\mathbb{C}$ is defined by

(1.5) $\displaystyle
N_{\lambda}(f):=N_{\lambda}(f(t):t\in\mathbb{I}):=\sup\\{J\in\mathbb{N}\,|\,\exists_{\begin{subarray}{c}t_{0}<\dotsb<t_{J}\\\
t_{j}\in\mathbb{I}\end{subarray}}:\min_{0<j\leq J}\lvert
f(t_{j})-f(t_{j-1})\rvert\geq\lambda\\}.$

The aim of this article is to give new proofs of seminorm inequalities from
[19], [21] and [24] by using the discrete Littlewood–Paley theory (see [18,
Theorem 3.3]) and the bootstrap arguments similar in spirit to these from
[10]. This approach is much in the spirit of the methods used in the
continuous setting in which we usually use Littlewood–Paley operators, see
[23]. It is also more “elementary” since it does not require any external
knowledge like vector-valued estimates from [20] or the estimates for
appropriate square functions like in [24]. Our main result is to give a
bootstrapping proof of the following theorem.

######  Theorem 1.6.

Let $d,k\geq 1$ and let $\mathcal{P}$ be a polynomial mapping (1.1). Then for
any $p\in(1,\infty)$ there is a constant $C_{p,d,k,{\rm deg}\mathcal{P}}>0$
such that for any $f\in\ell^{p}(\mathbb{Z}^{d})$ we have

(1.7)
$\displaystyle\big{\lVert}\sup_{t>0}|M_{t}^{\mathcal{P}}f|\big{\rVert}_{\ell^{p}(\mathbb{Z}^{d})}$
$\displaystyle\leq C_{p,d,k,{\rm
deg}\mathcal{P}}\|f\|_{\ell^{p}(\mathbb{Z}^{d})},$ (1.8)
$\displaystyle\sup_{\lambda>0}\big{\lVert}\lambda
N_{\lambda}(M_{t}^{\mathcal{P}}f:t\in\mathbb{R}_{+})^{1/2}\big{\rVert}_{\ell^{p}(\mathbb{Z}^{d})}$
$\displaystyle\leq C_{p,d,k,{\rm
deg}\mathcal{P}}\|f\|_{\ell^{p}(\mathbb{Z}^{d})},$ (1.9)
$\displaystyle\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{R}_{+})}\big{\lVert}O_{I,N}^{2}(M_{t}^{\mathcal{P}}f:t\in\mathbb{R}_{+})\big{\rVert}_{\ell^{p}(\mathbb{Z}^{d})}$
$\displaystyle\leq C_{p,d,k,{\rm deg}\mathcal{P}}\lVert
f\rVert_{\ell^{p}(\mathbb{Z}^{d})},$

where $\mathfrak{S}_{N}(\mathbb{R}_{+})$ is the set of all strictly increasing
sequences of length $N+1$ contained in $\mathbb{R}_{+}$ (see Section 2.1).
Moreover, for any $r\in(2,\infty)$ there is a constant $C_{p,d,k,r,{\rm
deg}\mathcal{P}}>0$ such that

(1.10)
$\big{\lVert}V^{r}(M_{t}^{\mathcal{P}}f:t\in\mathbb{R}_{+})\big{\rVert}_{\ell^{p}(\mathbb{Z}^{d})}\leq
C_{p,d,k,r,{\rm deg}\mathcal{P}}\|f\|_{\ell^{p}(\mathbb{Z}^{d})},\quad
f\in\ell^{p}(\mathbb{Z}^{d}).$

In particular, the constants mentioned above are independent of the
coefficients of the polynomial mapping $\mathcal{P}$.

A few comments are in order. The maximal inequality (1.7) was proven in [25]
and it is a generalisation of Bourgain’s result from [4]. The jump inequality
(1.8) was proven in [24]. The oscillation inequality (1.9) was recently proven
in [19] and the first proof of the $r$-variation estimate (1.10) in a full
range $r>2$ was given in [21] (see also [33] for previous results). It is
worth to note that the inequality (1.8) implies $r$-variation estimates for
$r\in(2,\infty)$, and each of these, in turn, implies the maximal inequality
(1.7) – for details see Section 2.6. Only the oscillation inequality (1.9) is
not implied by the former ones. Hence, one could just prove inequalities (1.8)
and (1.9) and conclude that the other inequalities are implied by them.
However, in this paper we would like to show that the Littlewood–Paley theory
and the bootstrap approach can be used in the case of all mentioned seminorms.
Fortunately, it does not greatly affect the length of the article.

###  1.1. Historical background

The oscillation seminorm (1.3) was introduced in Bourgain’s articles [2, 3, 4]
to prove that for any dynamical system $(X,\mathcal{B},\mu,T)$ with a measure
preserving transformation $T$ the averages along the squares

$T_{N}f(x):=\frac{1}{N}\sum_{n=1}^{N}{f(T^{n^{2}}x)}$

converge $\mu$-almost everywhere as $N$ goes to infinity for all $f\in
L^{p}(X,\mu)$ with $p\in(1,\infty)$. The usual approach to prove pointwise
convergence requires $L^{p}$ boundedness for the corresponding maximal
function and pointwise convergence on a dense class of functions in
$L^{p}(X,\mu)$. In the case of the operator $T_{N}$ it is not easy to find an
appropriate dense class for which the pointwise convergence is a priori known.
Fortunately, Bourgain managed to bypass that problem by using the oscillation
seminorm. He proved that for a lacunary sequence $I=(I_{j}:j\in\mathbb{N})$
and for any $J\in\mathbb{N}$ one has

(1.11)
$\big{\lVert}O_{I,J}^{2}(T_{N}f:N\in\mathbb{N})\big{\rVert}_{L^{2}(X)}\lesssim_{I}J^{c}\lVert
f\rVert_{L^{2}(X)}$

for some a constant $c<1/2$. From the above inequality one may deduce that
$T_{N}f$ converges pointwise for $f\in L^{2}(X,\mu)$. Later, many authors
followed Bourgain’s approach and studied oscillation inequalities in various
contexts, see [6, 14, 19].
To prove (1.11) Bourgain used $r$-variation seminorms $V^{r}$, a well known
object from the martingale theory. The $r$-variations for a family of bounded
martingales $(\mathfrak{f}_{n}\colon X\to\mathbb{C}:n\in\mathbb{N})$ were
studied by Lépingle [16] who showed that for all $r\in(2,\infty)$ and
$p\in(1,\infty)$ there is a constant $C_{p,r}>0$ such that the following
inequality

(1.12) $\displaystyle\lVert
V^{r}(\mathfrak{f}_{n}:n\in\mathbb{N})\rVert_{L^{p}(X)}\leq
C_{p,r}\sup_{n\in\mathbb{N}}\lVert\mathfrak{f}_{n}\rVert_{L^{p}(X)}$

holds with sharp ranges of exponents, see also [13] for a counterexample at
$r=2$. We also refer to [4, 22, 27] for generalisations and different proofs
of (1.12). The inequality (1.12) is an extension of Doob’s maximal inequality
for martingales and is a quantitative form of the martingale convergence
theorem. The $r$-variation is closely related to the oscillation seminorm
since by Hölder’s inequality, for $r\geq 2$, we have

(1.13) $O_{I,J}^{2}(T_{N}f:N\in\mathbb{N})\leq
J^{1/2-1/r}V^{r}(T_{N}f:N\in\mathbb{N}).$

Bourgain approximated the operator $T_{N}$ by a suitable martingale and then
used Lépingle’s variational inequality to show that for $r>2$

$\|V^{r}(T_{N}f:N\in\mathbb{N})\|_{L^{p}(X)}\lesssim\|f\|_{L^{p}(X)}.$

Having this, one easily obtains (1.11) using the inequality (1.13).

In [4] Bourgain also considered the $\lambda$-jump counting function which
turns out to be even more fundamental object than the $r$-variation $V^{r}$.
It is not hard to obtain that for every $r\geq 1$ one has

(1.14) $\displaystyle\sup_{\lambda>0}\|\lambda
N_{\lambda}(T_{N}f:N\in\mathbb{N})^{1/r}\|_{L^{p}(X)}\leq\|V^{r}(T_{N}f:N\in\mathbb{N})\|_{L^{p}(X)}.$

The remarkable feature of the $\lambda$-jumps, observed by Bourgain [4], is
that, in some sense, the inequality (1.14) can be reversed. Namely, a priori
uniform $\lambda$-jump estimates

(1.15) $\displaystyle\sup_{\lambda>0}\|\lambda
N_{\lambda}(T_{N}f:N\in\mathbb{N})^{1/2}\|_{L^{p}(X)}\leq
C_{p}\|f\|_{L^{p}(X)},$

for some $p\in[1,\infty)$ imply weak $r$-variational estimates

(1.16) $\displaystyle\|V^{r}(T_{N}f:N\in\mathbb{N})\|_{L^{p,\infty}(X)}\leq
C_{p,r}\|f\|_{L^{p}(X)},$

for the same value of $p$ and for all $r\in(2,\infty]$. For more details we
refer to [15] and [22, 24], and the references given there.

Bourgain’s papers [2, 3, 4] were the starting point of comprehensive
investigations in ergodic theory and harmonic analysis, which resulted in many
papers, see for instance [1, 13, 15, 21, 22, 23, 24, 25].
The discrete Radon averages $M_{t}^{\mathcal{P}}$ arise upon applying
Calderón’s transference principle [5] to the ergodic averages. Let
$(X,\mathcal{B},\mu)$ be a $\sigma$-finite measure space with a family of
invertible, commuting and measure preserving transformations
$T_{1},T_{2},\ldots,T_{d}$. For a given polynomial mapping of the form (1.1)
we define the ergodic averages

$A_{t}^{\mathcal{P}}f(x):=\frac{1}{|\Omega_{t}\cap\mathbb{Z}^{k}|}\sum_{y\in\Omega_{t}\cap\mathbb{Z}^{k}}f(T_{1}^{\mathcal{P}_{1}(y)}T_{2}^{\mathcal{P}_{1}(y)}\cdots
T_{d}^{\mathcal{P}_{d}(y)}x).$

If one wants to prove, for example, the maximal inequality

$\big{\lVert}\sup_{t>0}|A_{t}^{\mathcal{P}}f|\big{\rVert}_{L^{p}(X)}\lesssim\|f\|_{L^{p}(X)},$

then by Calderón’s transference principle it is enough to prove the
corresponding maximal inequality for appropriate Radon averages
$M_{t}^{\mathcal{P}}$ on $\mathbb{Z}^{d}$. Nonetheless, the above observation
is not the only reason to consider Radon averages. Namely, the operators
$M_{t}^{\mathcal{P}}$ can be seen as discrete counterparts of the continuous
Radon operators

$\mathcal{M}_{t}^{\mathcal{P}}f(x):=\frac{1}{|\Omega_{t}|}\int_{\Omega_{t}}f(x-\mathcal{P}(y)){\rm
d}y$

being, in turn, a natural generalisation of the Hardy–Littlewood operators.
This direction originates in some problems related to curvatures, see [9, 11,
30]. It is well known (see for instance [29, Chapter 11]) that the maximal
operator $\sup_{t>0}|\mathcal{M}_{t}^{\mathcal{P}}|$ is bounded on
$L^{p}(\mathbb{R}^{d})$ with $p>1$. In 2008, Jones, Seeger and Wright [15]
managed to prove the jump inequality for the operators
$\mathcal{M}_{t}^{\mathcal{P}}$ over Euclidean balls, see also the appendix in
[21]. For operators associated with arbitrary convex bodies the jump
inequality was proven by Mirek, Stein and Zorin-Kranich [23] by using
bootstrap arguments. Finally, the oscillation inequality was recently proven
in [19] by the author in collaboration with Mirek and Szarek.
The idea of the bootstrap approach originates in the work of Nagel, Stein and
Wainger [26] where they studied the problem of differentiation in lacunary
directions. Their argument relied heavily on some geometrical considerations.
Later, Duoandikoetxea and Rubio de Francia [10] used the ideas of bootstrap
from [26] (see Lemma 3.43 below) to prove $L^{p}$ bounds for maximal Radon
transform. At the same time, Christ formulated the bootstrap argument from
[26] in a fairly abstract way, which was used and published by Carbery [7, 8].
Finally, Mirek, Stein and Zorin-Kranich [23] managed to use the bootstrap
argument to establish jump inequalities in a very abstract setting.

In order to prove Theorem 1.6 we exploit a procedure introduced in [21] in the
context of $r$-variations. The key ingredient is the vector-valued
Ionescu–Wainger multiplier theorem from [24] and the discrete Littlewood–Paley
theory which follows from it (see [18]). Finally, we use the bootstrap lemma
from [23] which allows us to bound the appropriate square function by the
power of the appropriate seminorm.

##  2\. Preliminaries

###  2.1. Basic notation

We denote $\mathbb{N}:=\\{1,2,\ldots\\}$, $\mathbb{N}_{0}:=\\{0,1,2,\ldots\\}$
and $\mathbb{R}_{+}:=(0,\infty)$. For $d\in\mathbb{N}$ the sets
$\mathbb{Z}^{d}$, $\mathbb{R}^{d}$, $\mathbb{C}^{d}$ and
$\mathbb{T}^{d}\equiv[-1/2,1/2)^{d}$ have the usual meaning. For every
$N\in\mathbb{N}$ we define

$\mathbb{N}_{N}:=\\{1,\ldots,N\\}.$

For any $x\in\mathbb{R}$ the floor function is defined by

$\lfloor x\rfloor:=\max\\{n\in\mathbb{Z}:n\leq x\\}.$

We write $A\lesssim B$ to indicate that $A\leq CB$ with a constant $C>0$. The
constant $C$ may change from line to line. We write $\lesssim_{\delta}$ if the
implicit constant depends on $\delta$. For two functions $f\colon
X\to\mathbb{C}$ and $g\colon X\to[0,\infty)$, we write $f=\mathcal{O}(g)$ if
there is a constant $C>0$ such that $|f(x)|\leq Cg(x)$ for all $x\in X$.

For $N\in\mathbb{N}\cup\\{\infty\\}$ we denote by
$\mathfrak{S}_{N}(\mathbb{I})$ the family of all strictly increasing sequences
of length $N+1$ contained in $\mathbb{I}$.

###  2.2. Norms and inner product

In the paper we use the standard inner product on $\mathbb{R}^{d}$ denoted by
$x\cdot\xi$. Moreover, for any $x\in\mathbb{R}^{d}$ we use the the
$\ell^{2}$-norm and the maximum norm defined by

$\displaystyle\lvert x\rvert:=\lvert x\rvert_{2}:=\sqrt{x\cdot x},\qquad\text{
and }\qquad|x|_{\infty}:=\max_{1\leq k\leq d}|x_{k}|.$

For any multi-index
$\gamma=(\gamma_{1},\dots,\gamma_{k})\in\mathbb{N}^{k}_{0}$, by abuse of
notation we will write $|\gamma|:=\gamma_{1}+\cdots+\gamma_{k}$. This should
never cause any confusions since the multi-indices are always denoted by Greek
letter $\gamma$.

###  2.3. Fourier transform

Let $\mathbb{G}=\mathbb{R}^{d}$ or $\mathbb{G}=\mathbb{Z}^{d}$ and let
$\mathbb{G}^{*}$ denote the dual group of $\mathbb{G}$. For every
$z\in\mathbb{C}$ we set $\bm{e}(z):=e^{2\pi{\bm{i}}z}$, where
${\bm{i}}^{2}=-1$. Let $\mathcal{F}_{\mathbb{G}}$ denote the Fourier transform
on $\mathbb{G}$ defined for any $f\in L^{1}(\mathbb{G})$ as

$\displaystyle\mathcal{F}_{\mathbb{G}}f(\xi):=\int_{\mathbb{G}}f(x)\bm{e}(x\cdot\xi){\rm
d}\mu(x),\quad\xi\in\mathbb{G}^{*},$

where $\mu$ is the Haar measure on $\mathbb{G}$. For any bounded function
$\mathfrak{m}\colon\mathbb{G}^{*}\to\mathbb{C}$ we define the Fourier
multiplier operator by

(2.1) $\displaystyle
T_{\mathbb{G}}[\mathfrak{m}]f(x):=\int_{\mathbb{G}^{*}}\bm{e}(-\xi\cdot
x)\mathfrak{m}(\xi)\mathcal{F}_{\mathbb{G}}f(\xi){\rm d}\xi,\quad\text{ for
}\quad x\in\mathbb{G}.$

Here, we assume that $f\colon\mathbb{G}\to\mathbb{C}$ is a compactly supported
function on $\mathbb{G}$ (and smooth if $\mathbb{G}=\mathbb{R}^{d}$) or any
other function for which (2.1) makes sense.

###  2.4. Ionescu–Wainger multiplier theorem

The important tool in dealing with discrete singular integral will be the
Ionescu–Wainger multiplier theorem. The following result is the
multidimensional vector-valued Ionescu–Wainger multiplier theorem from [24,
Section 2].

######  Theorem 2.2.

For every $\varrho>0$, there exists a family $(P_{\leq N})_{N\in\mathbb{N}}$
of subsets of $\mathbb{N}$ such that:

  1. (i)

One has $\mathbb{N}_{N}\subseteq P_{\leq
N}\subseteq\mathbb{N}_{\max\\{N,e^{N^{\varrho}}\\}}$.

  2. (ii)

If $N_{1}\leq N_{2}$, then $P_{\leq N_{1}}\subseteq P_{\leq N_{2}}$.

  3. (iii)

If $q\in P_{\leq N}$, then all factors of $q$ also lie in $P_{\leq N}$.

  4. (iv)

One has $\operatorname{lcm}{(P_{N})}\leq 3^{N}$.

Furthermore, for every $p\in(1,\infty)$, there exists
$0<C_{p,\varrho,d}<\infty$ such that, for every $N\in\mathbb{N}$, the
following holds.

Let $0<\varepsilon_{N}\leq e^{-N^{2\varrho}}$, and let
$\Theta\colon\mathbb{R}^{d}\to L(H_{0},H_{1})$ be a measurable function
supported on $\varepsilon_{N}\mathbf{Q}$, where $\mathbf{Q}:=[-1/2,1/2)^{d}$
is a unit cube, with values in the space $L(H_{0},H_{1})$ of bounded linear
operators between separable Hilbert spaces $H_{0}$ and $H_{1}$. Let
$0\leq\mathbf{A}_{p}\leq\infty$ denote the smallest constant such that, for
every function $f\in L^{2}(\mathbb{R}^{d};H_{0})\cap
L^{p}(\mathbb{R}^{d};H_{0})$, we have

(2.3)
$\displaystyle\big{\lVert}T_{\mathbb{R}^{d}}[\Theta]f\big{\rVert}_{L^{p}(\mathbb{R}^{d};H_{1})}\leq\mathbf{A}_{p}\lVert
f\rVert_{L^{p}(\mathbb{R}^{d};H_{0})}.$

Then the multiplier

(2.4) $\Delta_{N}(\xi):=\sum_{b\in\Sigma_{\leq N}}\Theta(\xi-b),$

where $\Sigma_{\leq N}$ is $1$-periodic subsets of $\mathbb{T}^{d}$ defined by

(2.5) $\displaystyle\Sigma_{\leq
N}:=\Big{\\{}\frac{a}{q}\in\mathbb{Q}^{d}\cap\mathbb{T}^{d}:q\in P_{\leq
N}\text{ and }{\rm gcd}(a,q)=1\Big{\\}},$

satisfies for every $f\in L^{p}(\mathbb{Z}^{d};H_{0})$ the following
inequality

(2.6)
$\displaystyle\big{\lVert}T_{\mathbb{Z}^{d}}[\Delta_{N}]f\big{\rVert}_{\ell^{p}(\mathbb{Z}^{d};H_{1})}\leq
C_{p,\varrho,d}(\log N)\mathbf{A}_{p}\lVert
f\rVert_{\ell^{p}(\mathbb{Z}^{d};H_{0})}.$

An important feature of Theorem 2.2 is that one can directly transfer square
function estimates from the continuous to the discrete setting. By property
(i) we see that

(2.7) $\displaystyle|\Sigma_{\leq N}|\lesssim e^{(d+1)N^{\varrho}}.$

A detailed proof of Theorem 2.2 (in the spirit of [18]) can be found in [24,
Section 2]. It turns out that factor $\log N$ may be removed from (2.6), see
[32].

###  2.5. Lifting procedure

By a standard lifting argument it suffices to prove Theorem 1.6 only for
canonical polynomial mappings. Let $\mathcal{P}$ be a polynomial mapping
(1.1). We define

${\rm deg}\,\mathcal{P}:=\max\\{{\rm deg}\,\mathcal{P}_{j}:1\leq j\leq d\\}.$

Let us consider the set of multi-indexes

$\Gamma:=\big{\\{}\gamma\in\mathbb{N}_{0}^{k}\setminus\\{0\\}:0<|\gamma|\leq{\rm
deg}\,\mathcal{P}\big{\\}}$

equipped with the lexicographic order. By $\mathbb{R}^{\Gamma}$ we mean the
space of tuples of real numbers labeled by multi-indices
$\gamma=(\gamma_{1},\dots,\gamma_{k})$. It can be easily noted that
$\mathbb{R}^{\Gamma}\cong\mathbb{R}^{|\Gamma|}$ and
$\mathbb{Z}^{\Gamma}\cong\mathbb{Z}^{|\Gamma|}$. Recall that the canonical
polynomial mapping is defined by

$\mathbb{R}^{k}\ni
x=(x_{1},\dots,x_{k})\mapsto(x)^{\Gamma}:=(x^{\gamma}\colon\gamma\in\Gamma)\in\mathbb{R}^{\Gamma},$

where $x^{\gamma}=x_{1}^{\gamma_{1}}x_{2}^{\gamma_{2}}\cdots
x_{k}^{\gamma_{k}}$. In the case of the averaging Radon operator associated
with with the canonical mapping we will write

$M_{t}f(x):=\frac{1}{|\Omega_{t}\cap\mathbb{Z}^{k}|}\sum_{y\in\Omega_{t}\cap\mathbb{Z}^{k}}f(x-(y)^{\Gamma}).$

By invoking the lifting procedure for the Radon averages described in [20,
Lemma 2.2] (see also [29, Section 11]) it is enough to prove Theorem 1.6 only
for the canonical mappings.

###  2.6. Properties of the considered seminorms

The proof of Theorem 1.6 will be conducted simultaneously for all considered
(quasi-)seminorms, pointing out the places where the distinction should be
made. Due to technical reasons we will consider the following seminorm

(2.8) $\sup_{t>0}|M_{t}f-M_{0}f|$

instead of the usual maximal function. We could prove Theorem 1.6 for the
usual maximal function but it would require a few minor changes which would
affect the length of the article. The seminorm (2.8) is more consistent with
the rest of the seminorms and allows us to write results in a more elegant
way.

Now let $\mathbb{I}\subseteq\mathbb{R}$. For a given family of measurable
functions
$(\mathfrak{a}_{t}:t\in\mathbb{I})\subset\ell^{p}(\mathbb{Z}^{\Gamma})$ we
consider the following seminorms: the supremum seminorm

(2.9)
$\big{\lVert}\sup_{t\in\mathbb{I}}|\mathfrak{a}_{t}-\mathfrak{a}_{\inf\mathbb{I}}|\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

the oscillation seminorm

(2.10)
$\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{I})}\big{\lVert}O_{I,N}^{2}(\mathfrak{a}_{t}:t\in\mathbb{I})\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

the jump quasi-seminorm

(2.11) $\sup_{\lambda}\big{\lVert}\lambda
N_{\lambda}(\mathfrak{a}_{t}:t\in\mathbb{I})^{1/2}\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

and, for $r\in[2,\infty)$, the $r$-variation seminorm

(2.12)
$\big{\lVert}V^{r}(\mathfrak{a}_{t}:t\in\mathbb{I})\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

For clarity, for all of the above seminorms we will use one common symbol

$\mathcal{S}_{p}(\mathfrak{a}_{t}:t\in\mathbb{I}).$

Below we state some basic proprieties of the seminorms (2.9)–(2.12). For any
$r\in[2,\infty]$ we have the following pointwise estimate for the maximal
function

(2.13) $\sup_{t\in\mathbb{I}}|\mathfrak{a}_{t}|\leq
V^{r}(\mathfrak{a}_{t})+|\mathfrak{a}_{t_{0}}|.$

A similar result holds in the case of the oscillation seminorm

(2.14)
$\lVert\sup_{t\in\mathbb{I}\setminus\sup\mathbb{I}}|\mathfrak{a}_{t}|\rVert_{\ell^{p}(\mathbb{Z}^{\Gamma})}\leq\sup_{t\in\mathbb{I}}\lVert\mathfrak{a}_{t}\rVert_{\ell^{p}(\mathbb{Z}^{\Gamma})}+\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{I})}\big{\lVert}O_{I,N}^{2}(\mathfrak{a}_{t}:t\in\mathbb{I})\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

The jump quasi-seminorm (2.11), in turn, provides the weak type estimate for
the $r$-variation seminorm (see [22, Lemma 2.5]), that is, for any $r>2$ one
has

(2.15)
$\big{\|}V^{r}\big{(}\mathfrak{a}_{t}:t\in\mathbb{I}\big{)}\big{\|}_{\ell^{p,\infty}(\mathbb{Z}^{\Gamma})}\lesssim_{p,r}\sup_{\lambda>0}\big{\|}\lambda
N_{\lambda}(\mathfrak{a}_{t}:t\in\mathbb{I})^{1/2}\big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

Let us note that for any countable family
$(a_{n}:n\in\mathbb{J})\subset\ell^{p}(\mathbb{Z}^{\Gamma})$ the following
inequality holds

(2.16) $\mathcal{S}_{p}(a_{n}:n\in\mathbb{J})\leq
2\big{\lVert}\big{(}\sum_{n\in\mathbb{J}}|\mathfrak{a}_{n}|^{2}\big{)}^{1/2}\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

######  Proposition 2.17.

Let $p\in(1,\infty)$ and $\mathbb{I}\subseteq\mathbb{R}$. Then the seminorm
$\mathcal{S}_{p}$ is subadditive up to some positive constant, that is,

$\mathcal{S}_{p}(\mathfrak{a}_{t}+\mathfrak{b}_{t}:t\in\mathbb{I})\lesssim\mathcal{S}_{p}(\mathfrak{a}_{t}:t\in\mathbb{I})+\mathcal{S}_{p}(\mathfrak{b}_{t}:t\in\mathbb{I}),$

where the implied constant is independent of the set $\mathbb{I}$, and the
families $(\mathfrak{a}_{t}:t\in\mathbb{I})$ and
$(\mathfrak{b}_{t}:t\in\mathbb{I})$. Moreover, for $-\infty\leq
u<w<v\leq\infty$, we have

(2.18)
$\displaystyle\mathcal{S}_{p}(\mathfrak{a}_{t}:t\in[u,v])\lesssim\mathcal{S}_{p}(\mathfrak{a}_{t}\in[u,w+1])+\mathcal{S}_{p}(\mathfrak{a}_{t}:t\in[w,v]),$
(2.19)
$\displaystyle\mathcal{S}_{p}(\mathfrak{a}_{t}\mathds{1}_{(u,\infty)}(t):t\in[0,w])\lesssim\mathcal{S}_{p}(\mathfrak{a}_{t}:t\in[u,w])+\|\mathfrak{a}_{u}\|_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

where implied constants depend only on the seminorm $\mathcal{S}_{p}$.

###### Proof.

The only difficult part of the proof may be subadditivity of the jump quasi-
seminorm. However, by [22, Corollary 2.2] we know that it admits an equivalent
subadditive seminorm which yields the desired result. ∎

The next result is a well-known decomposition into the long “jumps” and the
short variations from [15] (see also [21, Lemma 8.1]).

######  Proposition 2.20 ([15, Lemma 1.3]).

Let $p\in[1,\infty)$ and $\mathbb{I}\subseteq\mathbb{R}_{+}$. Then we have the
following decomposition into the long “jumps” and short variations

(2.21)
$\mathcal{S}_{p}(\mathfrak{a}_{t}:t\in\mathbb{I})\lesssim\mathcal{S}_{p}(\mathfrak{a}_{2^{n}}:n\in\mathbb{Z})+\Big{\lVert}\Big{(}\sum_{n\in\mathbb{Z}}V^{2}(\mathfrak{a}_{t}:t\in[2^{n},2^{n+1})\cap\mathbb{I})^{2}\Big{)}^{1/2}\Big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

The next numerical inequality is crucial in our investigations.

######  Remark 2.22 (Rademacher–Menshov inequality).

Since all of the seminorms (2.9)–(2.12) are dominated by the $2$-variation
seminorm we deduce that the Rademacher–Menshov inequality [23, Lemma 2.5, p.
534] holds for $\mathcal{S}_{p}$. Let $b$ and $s$ be fixed positive integers.
Then for any complex-valued sequence $(a_{j}:b\leq j\leq 2^{s})$ one has

(2.23) $\mathcal{S}_{p}(a_{n}:b\leq n\leq
2^{s})\leq\sqrt{2}\Big{\lVert}\sum_{i=1}^{s}\Big{(}\sum_{j}{|a_{u_{j+1}^{i}}-a_{u_{j}^{i}}|^{2}}\Big{)}^{1/2}\Big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

where $[u_{j}^{i},u_{j+1}^{i})$ are dyadic intervals of the form
$[j2^{i},(j+1)2^{i})$ for some $0\leq i\leq s$, $0\leq j\leq 2^{s-i}-1$,
contained in $[b,2^{s}]$ (in particular, the number of intervals occurring in
the inner sum is finite).

##  3\. Proof of Theorem 1.6

Assume that $p\in(1,\infty)$ and let $f\in\ell^{p}(\mathbb{Z}^{\Gamma})$ be a
compactly supported function. Let
$\mathbb{U}:=\bigcup_{n\in\mathbb{Z}}2^{n}\mathbb{N}$. Let us note that it is
enough to establish the following inequality

(3.1)
$\mathcal{S}_{p}(M_{t}f:t\in\mathbb{U})\lesssim_{\mathcal{S}_{p}}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

where the implied constant may depend on the seminorm $\mathcal{S}_{p}$ and
$p\in(1,\infty)$ but is independent of $f$. Let us choose $p_{0}\in(1,2)$,
close to $1$ such that $p\in(p_{0},p_{0}^{\prime})$. Note that $M_{t}f\equiv
f$ for $t\in(0,1)$. By Proposition 2.20 we can split (3.1) into long “jumps”
and short variations

(3.2)
$\mathcal{S}_{p}(M_{t}f:t\in\mathbb{U})\lesssim\mathcal{S}_{p}(M_{2^{n}}f:n\in\mathbb{N}_{0})+\Big{\lVert}\Big{(}\sum_{l=0}^{\infty}V^{2}\big{(}M_{n}f:n\in[2^{l},2^{l+1}]\cap\mathbb{U}\big{)}^{2}\Big{)}^{1/2}\Big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

We will estimate separately each part of the right hand side of (3.2).

###  3.1. Estimates for the long jumps

The aim of this subsection is to give a proof of the estimate for the long
jumps,

(3.3)
$\mathcal{S}_{p}(M_{2^{n}}f:n\in\mathbb{N}_{0})\lesssim_{\mathcal{S}_{p}}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

where the implicit constant may only depend on the seminorm $\mathcal{S}_{p}$,
but is independent of $f$. For this purpose we will exploit the following
bootstrap argument. For $N\in\mathbb{N}$ let us consider the following cut-off
seminorms

(3.4) $\mathcal{S}_{p}(M_{2^{n}}f:n\in[0,N]\cap\mathbb{N}_{0}).$

By $R_{p}(N)$ we denote the smallest constant $C>0$ for which the following
estimate holds

$\mathcal{S}_{p}(M_{2^{n}}f:n\in[0,N]\cap\mathbb{N}_{0})\leq
C\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})},\quad
f\in\ell^{p}(\mathbb{Z}^{\Gamma}).$

By the estimate (2.16) we know that $R_{p}(N)\lesssim N<\infty$. However, we
will show that there exists a constant $C_{p}>0$ such that $R_{p}(N)\leq
C_{p}$ for any $N\in\mathbb{N}$. If such a constant exists, then by taking
limit as $N\to\infty$ and by using the monotone convergence theorem one easily
obtains (3.3). Without loss of generality we can assume that $R_{p}(N)>1$ and
$N\in\mathbb{N}$ is large.

We start with writing operator $M_{2^{n}}$ in terms of the discrete Fourier
transform. One has

$M_{2^{n}}f(x)=T_{\mathbb{Z}^{\Gamma}}[m_{2^{n}}]f(x),\quad
x\in\mathbb{Z}^{\Gamma},$

where

$m_{N}(\xi):=\frac{1}{|\Omega_{N}\cap\mathbb{Z}^{k}|}\sum_{y\in\Omega_{N}\cap\mathbb{Z}^{k}}{e((y)^{\Gamma}\cdot\xi)},\quad
N\in\mathbb{N}_{0},\xi\in\mathbb{T}^{\Gamma}.$

In order to prove (3.3) we require several appropriately chosen parameters.
Let $\alpha>0$ be such that

(3.5)
$\alpha>100\left(\frac{1}{p_{0}}-\frac{1}{2}\right)\left(\frac{1}{p_{0}}-\frac{1}{\min\\{p,p^{\prime}\\}}\right)^{-1}.$

Fix $\chi\in(0,1/10)$ and let $u\in\mathbb{N}$ be a large natural number which
will be specified later. Now we are able to define the so-called
Ionsescu–Wainger projections. For this purpose, we introduce a diagonal matrix
$A$ of size $|\Gamma|\times|\Gamma|$ such that
$(Av)_{\gamma}:=\lvert\gamma\rvert v_{\gamma}$ for any $\gamma\in\Gamma$ and
$v\in\mathbb{R}^{\Gamma}$, and for any $t>0$ we also define corresponding
dilations by setting
$t^{A}x=\big{(}t^{|\gamma|}x_{\gamma}:\gamma\in\Gamma\big{)}$ for every
$x\in\mathbb{R}^{\Gamma}$. Let $\eta\colon\mathbb{R}^{\Gamma}\to[0,1]$ be a
smooth function such that

(3.6) $\eta(x)=\begin{cases}1,&|x|\leq 1/(16|\Gamma|),\\\ 0,&|x|\geq
1/(8|\Gamma|).\end{cases}$

Let us set $\varrho:=(10u)^{-1}$ and recall the family of rational fractions
$\Sigma_{\leq n^{u}}$ related to the parameter $\varrho$ described in Section
2.4. For each $n\in\mathbb{N}$ we define the following function

(3.7) $\Xi_{n}(\xi):=\sum_{a/q\in\Sigma_{\leq
n^{u}}}\eta^{2}\big{(}2^{n(A-\chi I)}(\xi-a/q)\big{)},$

where $I$ is the $|\Gamma|\times|\Gamma|$ identity matrix. By Theorem 2.2 we
have that

(3.8)
$\big{\|}T_{\mathbb{Z}^{\Gamma}}[\Xi_{n}]f)\big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim_{u,\,p}\log(n+1)\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

since for large $n$ one has $\varepsilon_{n}=2^{-n(|\gamma|-\chi)}\leq
e^{-n^{1/5}}=e^{-(n^{u})^{2\varrho}}$. We use projections defined in (3.7) to
partition the multiplier $m_{2^{n}}$ and estimate (3.4) by

(3.9)
$\displaystyle\mathcal{S}_{p}\big{(}T_{\mathbb{Z}^{\Gamma}}[m_{2^{n}}\Xi_{n}]f):n\in[0,N]\cap\mathbb{N}_{0}\big{)}+\mathcal{S}_{p}\big{(}T_{\mathbb{Z}^{\Gamma}}[(1-\Xi_{n})m_{2^{n}}]f):n\in[0,N]\cap\mathbb{N}_{0}\big{)}.$

Remark that the first and second term in the above inequality correspond to
the major and minor arcs in the Hardy–Littlewood circle method.

####  3.1.1. Estimates for the minor arcs

Now, our aim is to prove that

(3.10)
$\mathcal{S}_{p}\big{(}T_{\mathbb{Z}^{\Gamma}}[(1-\Xi_{n})m_{2^{n}}]f):n\in[0,N]\cap\mathbb{N}_{0}\big{)}\lesssim\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

We will obtain the above estimate by exploiting the standard approach–by using
Weyl’s inequality. Note that each seminorm (2.9)–(2.12) is bounded by the
$2$-variation seminorm. Moreover, since the $r$-variation seminorms are non-
increasing in $r$ we may estimate $V^{2}$ by the $V^{1}$ and hence

(3.11) $\displaystyle{\rm LHS}\eqref{eq:minor-
arcs}\lesssim\sum_{n=0}^{\infty}\big{\lVert}T_{\mathbb{Z}^{\Gamma}}[(1-\Xi_{n})m_{2^{n}}]f\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

Therefore, it is enough to show

(3.12)
$\big{\lVert}T_{\mathbb{Z}^{\Gamma}}[(1-\Xi_{n})m_{2^{n}}]f\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim(n+1)^{-2}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

In order to prove the estimate above we appeal to Weyl’s inequality and
proceed as in [24, Lemma 3.29, p. 34].

####  3.1.2. Major arcs and scale distinction

Now we return to the major arcs. Let $\tilde{\eta}(x)=\eta(x/2)$. We define
new multipliers by setting

$\Xi_{n}^{s}(\xi):=\sum_{a/q\in\Sigma_{s^{u}}}\eta^{2}\big{(}2^{n(A-\chi
I)}(\xi-a/q)\big{)}\tilde{\eta}^{2}\big{(}2^{s(A-\chi I)}(\xi-a/q)\big{)},$

where $\Sigma_{s^{u}}=\Sigma_{\leq(s+1)^{u}}\setminus\Sigma_{\leq s^{u}}$ for
$s\in\mathbb{N}$ and $\Sigma_{0^{u}}=\Sigma_{\leq 1}$. It is easy to see that
one has $\Xi_{n}(\xi)=\sum_{s=0}^{n-1}\Xi_{n}^{s}(\xi)$. Next, by (2.19) we
obtain the following estimate

$\displaystyle\mathcal{S}_{p}\big{(}T_{\mathbb{Z}^{\Gamma}}[m_{2^{n}}\Xi_{n}]f):n\in[0,N]\cap\mathbb{N}_{0}\big{)}\lesssim\sum_{s=0}^{N}\mathcal{S}_{p}\big{(}T_{\mathbb{Z}^{\Gamma}}[m_{2^{n}}\Xi_{n}^{s}]f):n\in[s,N]\cap\mathbb{N}_{0}\big{)}+\big{\lVert}T_{\mathbb{Z}^{\Gamma}}[m_{2^{s}}\Xi_{s}^{s}]f\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

Now, for $s\in\mathbb{N}_{0}$ we set $\kappa_{s}=20d\lceil(s+1)^{1/10}\rceil$
and by (2.18) we see that the expression under the sum is bounded by

$\displaystyle\mathcal{S}_{p}\big{(}T_{\mathbb{Z}^{\Gamma}}[m_{2^{n}}\Xi_{n}^{s}]f):n\in[s,N]\cap\mathbb{N}_{0},n\leq
2^{\kappa_{s}+1}\big{)}$
$\displaystyle+\mathcal{S}_{p}\big{(}T_{\mathbb{Z}^{\Gamma}}[m_{2^{n}}\Xi_{n}^{s}]f):n\in\mathbb{N}_{0},n>2^{\kappa_{s}}\big{)}$
$\displaystyle+\big{\lVert}T_{\mathbb{Z}^{\Gamma}}[m_{2^{s}}\Xi_{s}^{s}]f\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

The first term corresponds to small scales and the second one to large scales.
For $p\in(1,\infty)$ we will show the following bounds:

(3.13)
$\displaystyle\big{\|}T_{\mathbb{Z}^{\Gamma}}[m_{2^{s}}\Xi_{s}^{s}]f\big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}$
$\displaystyle\lesssim(s+1)^{-3}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})},$ (3.14)
$\displaystyle\mathcal{S}_{p}\big{(}T_{\mathbb{Z}^{\Gamma}}[m_{2^{n}}\Xi_{n}^{s}]f:n\in\mathbb{N}_{0},n>2^{\kappa_{s}}\big{)}$
$\displaystyle\lesssim(s+1)^{-3}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

Moreover, in the case of $p\in(1,2]$, we prove that the inequality

(3.15)
$\mathcal{S}_{p}\big{(}T_{\mathbb{Z}^{\Gamma}}[m_{2^{n}}\Xi_{n}^{s}]f):n\in[s,N]\cap\mathbb{N}_{0},n\leq
2^{\kappa_{s}+1}\big{)}\lesssim
R_{p}(N)^{\beta(p)}(s+1)^{-3}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

holds with some $\beta(p)\in[0,1)$. If we show the above inequalities, then
for $p\in(1,2]$ the inequality (3.10) gives us

$R_{p}(N)\lesssim
1+\sum_{s=0}^{\infty}{(s+1)^{-3}(R_{p}(N)^{\beta(p)}+1)}\lesssim
R_{p}(N)^{\beta(p)},$

since $R_{p}(N)\geq 1$ and $\beta(p)\in[0,1)$. This gives
$R_{p}(N)\lesssim_{p}1$ and thus the proof is complete in the case of
$p\in(1,2]$. When $p\in(2,\infty)$ a minor change is required for the estimate
(3.15). Namely, in this case we show that

(3.16)
$\mathcal{S}_{p}\big{(}T_{\mathbb{Z}^{\Gamma}}[m_{2^{n}}\Xi_{n}^{s}]f):n\in[s,N]\cap\mathbb{N},n\leq
2^{\kappa_{s}+1}\big{)}\lesssim
R_{p^{\prime}}(N)^{\beta^{\prime}(p)}(s+1)^{-3}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

where $1/p+1/p^{\prime}=1$ and $\beta^{\prime}(p)\in(0,1)$. Since
$p^{\prime}\in(1,2)$, by the first part one has that
$R_{p^{\prime}}(N)\lesssim_{p^{\prime}}1$ and consequently

$R_{p}(N)\lesssim
1+\sum_{s=0}^{\infty}{(s+1)^{-3}(R_{p^{\prime}}(N)^{\beta^{\prime}(p)}+1)}\lesssim_{p}1$

which finishes the proof in the case of $p\in(2,\infty)$.

####  3.1.3. Estimate for (3.13)

We prove the estimate (3.13) by approximating the multiplier
$m_{2^{s}}\Xi_{s}^{s}$ by a suitably chosen integral. At first, note that by
Theorem 2.2, for any $p\in(1,\infty)$, we obtain the estimate

(3.17)
$\big{\lVert}T_{\mathbb{Z}^{\Gamma}}[m_{2^{s}}\Xi_{s}^{s}]f\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim\log(s+1)\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

For further reference, for $N\in\mathbb{N}_{0}$ we denote

$\displaystyle\Phi_{N}(\xi)$
$\displaystyle:=\frac{1}{|\Omega_{N}|}\int_{\Omega_{N}}{e(\xi\cdot(t)^{\Gamma}){\rm
d}t}\quad{\rm and}\quad
G(a/q):=\frac{1}{q^{k}}\sum_{r\in\mathbb{N}_{q}^{k}}{e((a/q)\cdot(r)^{\Gamma})}.$

We have the following estimates for the function $\Phi_{N}$:

(3.18)
$|\Phi_{N}(\xi)|\lesssim|N^{A}\xi|_{\infty}^{-1/|\Gamma|}\quad\text{and}\quad|\Phi_{N}(\xi)-1|\lesssim|N^{A}\xi|_{\infty},$

where the first estimate follows from the multidimensional van der Corput
lemma [29, Proposition 5, p. 342] and the second one is a simple consequence
of the mean value theorem. Note that by the multidimensional version of Weyl’s
inequality [31, Proposition 3] one has

(3.19) $|G(a,q)|\lesssim_{k}q^{-\delta}$

for some $\delta>0$.

By appealing to [24, Proposition 4.18] and proceeding as in [24, Lemma 3.38,
p. 36] one can prove that

(3.20)
$(m_{2^{s}}\Xi_{s}^{s})(\xi)=\sum_{a/q\in\Sigma_{s^{u}}}G(a/q)\Phi_{2^{s}}(\xi-a/q)\eta^{2}\big{(}2^{s(A-\chi
I)}(\xi-a/q)\big{)}+\mathcal{O}(2^{-s/2}).$

By Plancherel’s theorem, the above estimate together with the estimate for the
Gauss sum (3.19) yield

(3.21)
$\displaystyle\big{\lVert}T_{\mathbb{Z}^{\Gamma}}[m_{2^{s}}\Xi_{s}^{s}]f\big{\rVert}_{\ell^{2}(\mathbb{Z}^{\Gamma})}\lesssim(s+1)^{-\alpha}\|f\|_{\ell^{2}(\mathbb{Z}^{\Gamma})}$

provided that $u\in\mathbb{N}$ satisfies $u>\alpha\delta^{-1}$. If we
interpolate (3.17) for $p=p_{0}$ with (3.21) we get that (3.13) holds.

####  3.1.4. Estimates for the large scales

Now we focus on proving the estimate for the large scales,

$\mathcal{S}_{p}\big{(}T_{\mathbb{Z}^{\Gamma}}[m_{2^{n}}\Xi_{n}^{s}]f:n\in\mathbb{N}_{0},n>2^{\kappa_{s}}\big{)}\lesssim(s+1)^{-3}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

Again, by [24, Proposition 4.18] (compare with (3.20)) we have

$\displaystyle
m_{2^{n}}\Xi_{n}^{s}(\xi)=h_{s}^{n}(\xi)+\mathcal{O}(2^{-n/2}),$

where

$h_{s}^{n}(\xi):=\sum_{a/q\in\Sigma_{s^{u}}}{G(a,q)\Phi_{2^{n}}(\xi-a/q)\eta^{2}(2^{n(A-\chi
I)}(\xi-a/q))\tilde{\eta}^{2}(2^{s(A-\chi I)}(\xi-a/q))}$

and we see that is suffices to prove that

(3.22)
$\mathcal{S}_{p}\big{(}T_{\mathbb{Z}^{\Gamma}}[h_{n}^{s}]f):n\in\mathbb{N}_{0},n>2^{\kappa_{s}}\big{)}\lesssim(s+1)^{-3}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

Proceeding as in [24, Section 3.6, pp. 40–41] and invoking transference
principle [17, Corollary 2.1, p. 196] we conclude that the estimate (3.22)
follows by continuous seminorm inequalities. In the case of the jump seminorm
we use [22, Theorem 1.3].

####  3.1.5. Small scales and the discrete Littlewood–Paley theory: estimates
for (3.15) and (3.16)

We begin with writing inequalities (3.15) and (3.16) in a more convenient
form, namely

(3.23)
$\mathcal{S}_{p}\big{(}T_{\mathbb{Z}^{\Gamma}}[m_{2^{n}}\Xi_{n}^{s}]f):n\in[s,N]\cap\mathbb{N}_{0},n\leq
2^{\kappa_{s}+1}\big{)}\lesssim
B_{p}(N)(s+1)^{-3}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

where for $p\in(1,2]$ the constant $B_{p}(N)=R_{p}(N)^{\beta(p)}$ and for
$p\in(2,\infty)$ we have $B_{p}(N)=R_{p^{\prime}}(N)^{\beta^{\prime}(p)}$ with
$\beta(p),\beta^{\prime}(p)\in[0,1)$. By Remark 2.22 we can apply the
Rademacher–Menshov inequality (2.23) and estimate

(3.24) ${\rm
LHS}\eqref{eq:sq1}\lesssim\sum_{i=0}^{\kappa_{s}+1}\Big{\lVert}\Big{(}\sum_{j=0}^{2^{\kappa_{s}+1-i}-1}\big{|}\sum_{n\in
I_{j}^{i}}T_{\mathbb{Z}^{\Gamma}}\big{[}(m_{2^{n+1}}\Xi_{n+1}^{s}-m_{2^{n}}\Xi_{n}^{s})\big{]}f\big{|}^{2}\Big{)}^{1/2}\Big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

where
$I_{j}^{i}:=[j2^{i},(j+1)2^{i})\cap[s,\min\\{N,2^{\kappa_{s}+1}\\})\cap\mathbb{N}_{0}$
since the inner sum telescopes. Now, by the triangle inequality one has

(3.25) $\displaystyle{\rm RHS}\eqref{eq:sq2}$
$\displaystyle\leq\sum_{i=0}^{\kappa_{s}+1}\Big{\lVert}\Big{(}\sum_{j}\big{|}\sum_{n\in
I_{j}^{i}}T_{\mathbb{Z}^{\Gamma}}\big{[}(m_{2^{n+1}}-m_{2^{n}})\Xi_{n}^{s}\big{]}f\big{|}^{2}\Big{)}^{1/2}\Big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}$
(3.26)
$\displaystyle\,\,+\sum_{i=0}^{\kappa_{s}+1}\Big{\lVert}\Big{(}\sum_{j}\big{|}\sum_{n\in
I_{j}^{i}}T_{\mathbb{Z}^{\Gamma}}\big{[}m_{2^{n+1}}(\Xi_{n+1}^{s}-\Xi_{n}^{s})\big{]}f\big{|}^{2}\Big{)}^{1/2}\Big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

Here and later on we will omit the limits of summation in $j$ for the sake of
clarity. Now we invoke Khintchine’s inequality to (3.25) and (3.26) and as a
consequence we see that the estimate (3.23) will follow if we show that
inequalities

(3.27) $\displaystyle\Big{\lVert}\sum_{j}\sum_{n\in
I_{j}^{i}}T_{\mathbb{Z}^{\Gamma}}\big{[}\varepsilon_{j}(m_{2^{n+1}}-m_{2^{n}})\Xi_{n}^{s}\big{]}f\Big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}$
$\displaystyle\lesssim
B_{p}(N)(s+1)^{-5}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})},$ (3.28)
$\displaystyle\Big{\lVert}\sum_{j}\sum_{n\in
I_{j}^{i}}T_{\mathbb{Z}^{\Gamma}}\big{[}\varepsilon_{j}m_{2^{n+1}}(\Xi_{n+1}^{s}-\Xi_{n}^{s})]f\Big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}$
$\displaystyle\lesssim(s+1)^{-5}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

hold for every $i\leq\kappa_{s}+1$ and any sequence $(\varepsilon_{j}\colon
j\leq 2^{\kappa_{s}+1-i}-1)\subseteq\\{-1,1\\}$. Finally, for estimates (3.27)
and (3.28) it is enough to show that for any interval
$I\subseteq[s,\min\\{N,2^{\kappa_{s}+1}\\})\cap\mathbb{N}$ and for any
sequence $(\varepsilon_{n}\colon n\in I)\subseteq\\{-1,1\\}$ one has

(3.29) $\displaystyle\Big{\lVert}\sum_{n\in
I}T_{\mathbb{Z}^{\Gamma}}\big{[}\varepsilon_{n}(m_{2^{n+1}}-m_{2^{n}})\Xi_{n}^{s}]f\Big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}$
$\displaystyle\lesssim
B_{p}(N)(s+1)^{-5}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})},$ (3.30)
$\displaystyle\Big{\lVert}\sum_{n\in
I}T_{\mathbb{Z}^{\Gamma}}\big{[}\varepsilon_{n}m_{2^{n+1}}(\Xi_{n+1}^{s}-\Xi_{n}^{s})\big{]}f\Big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}$
$\displaystyle\lesssim(s+1)^{-5}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

To prove (3.30), by the triangle inequality, it is enough to establish

(3.31)
$\big{\|}T_{\mathbb{Z}^{\Gamma}}[m_{2^{n+1}}(\Xi_{n+1}^{s}-\Xi_{n}^{s})]f\big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim(s+1)^{-5}(n+1)^{-3}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

For any $p\in(1,\infty)$ by Theorem 2.2 we have

(3.32)
$\big{\|}T_{\mathbb{Z}^{\Gamma}}[m_{2^{n+1}}(\Xi_{n+1}^{s}-\Xi_{n}^{s})]f\big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim\log(s+1)\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

In the case of $p=2$ we use [24, Proposition 4.18] (compare with (3.20), the
only difference is the error term which is a consequence of the inequality
$2^{-n/2}\lesssim 2^{-(n+s)/4}$ since $n\geq s$) to get

(3.33)
$m_{2^{n+1}}(\xi)=G(a/q)\Phi_{2^{n+1}}(\xi-a/q)+\mathcal{O}(2^{-(n+s)/4}),$

where $a/q$ is the rational approximation of $\xi$ such that for every
$\gamma\in\Gamma$ holds $|\xi_{\gamma}-a_{\gamma}/q|\lesssim
2^{-n(|\gamma|-\chi)}$. Next, we note that the function
$\Xi_{n+1}^{s}-\Xi_{n}^{s}$ is nonzero only for $\xi$ such that
$|2^{(n+1)(A-\chi I)}(\xi-a/q)|\gtrsim 1$ and $|2^{n(A-\chi
I)}(\xi-a/q)|\lesssim 1$, where $a/q\in\Sigma_{s^{u}}$. Hence, by Plancherel’s
theorem the estimate (3.19) and by the van der Corput estimate in (3.18) one
gets

$\big{\|}T_{\mathbb{Z}^{\Gamma}}[m_{2^{n+1}}(\Xi_{n+1}^{s}-\Xi_{n}^{s})]f\big{\|}_{\ell^{2}(\mathbb{Z}^{\Gamma})}\lesssim(s+1)^{-\alpha}2^{-n\chi/|\Gamma|}\|f\|_{\ell^{2}(\mathbb{Z}^{\Gamma})}.$

Interpolating the above inequality with (3.32) for $p=p_{0}$ yields (3.31).
Now we focus our attention on the proof of the estimate (3.29). For this
purpose we introduce new multipliers of the form

$\Xi_{n}^{s,j}(\xi):=\sum_{a/q\in\Sigma_{s^{u}}}\eta^{2}\big{(}2^{nA+jI}(\xi-a/q)\big{)}\tilde{\eta}^{2}\big{(}2^{s(A-\chi
I)}(\xi-a/q)\big{)},\quad j\in\mathbb{Z}.$

We have the following decomposition

$\Xi_{n}^{s}(\xi)=\sum_{-\lfloor\chi n\rfloor\leq
j<n}\big{(}\Xi_{n}^{s,j}(\xi)-\Xi_{n}^{s,j+1}(\xi)\big{)}+\big{(}\Xi_{n}^{s,-\chi
n}(\xi)-\Xi_{n}^{s,-\lfloor\chi n\rfloor}(\xi)\big{)}+\Xi_{n}^{s,n}(\xi),$

since the sum above telescopes. By using the new multipliers one may write

$\displaystyle{\rm LHS}\eqref{kin7}$ $\displaystyle\leq\big{\|}\sum_{n\in
I}\sum_{-\lfloor\chi n\rfloor\leq
j<n}T_{\mathbb{Z}^{\Gamma}}\big{[}\varepsilon_{n}(m_{2^{n+1}}-m_{2^{n}})(\Xi_{n}^{s,j}-\Xi_{n}^{s,j+1})\big{]}f\big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}$
$\displaystyle\,\,+\big{\|}\sum_{n\in
I}T_{\mathbb{Z}^{\Gamma}}\big{[}\varepsilon_{n}(m_{2^{n+1}}-m_{2^{n}})\big{(}(\Xi_{n}^{s,-\chi
n}-\Xi_{n}^{s,-\lfloor\chi
n\rfloor})+\Xi_{n}^{s,n}\big{)}\big{]}f\big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

Consequently, to obtain (3.29) it is enough to show two inequalities:

(3.34) $\big{\|}\sum_{n\in I}\sum_{-\lfloor\chi n\rfloor\leq
j<n}T_{\mathbb{Z}^{\Gamma}}\big{[}\varepsilon_{n}(m_{2^{n+1}}-m_{2^{n}})(\Xi_{n}^{s,j}-\Xi_{n}^{s,j+1})\big{]}f\big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim
B_{p}(N)(s+1)^{-5}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}$

and

(3.35) $\big{\|}\sum_{n\in
I}T_{\mathbb{Z}^{\Gamma}}\big{[}\varepsilon_{n}(m_{2^{n+1}}-m_{2^{n}})\big{(}(\Xi_{n}^{s,-\chi
n}-\Xi_{n}^{s,-\lfloor\chi
n\rfloor})+\Xi_{n}^{s,n}\big{)}\big{]}f\big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim(s+1)^{-5}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

In order to show (3.35) we apply the triangle inequality and we see it will
follow from

$\big{\|}T_{\mathbb{Z}^{\Gamma}}\big{[}(m_{2^{n+1}}-m_{2^{n}})\big{(}(\Xi_{n}^{s,-\chi
n}-\Xi_{n}^{s,-\lfloor\chi
n\rfloor})+\Xi_{n}^{s,n}\big{)}\big{]}f\big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim(s+1)^{-5}(n+1)^{-3}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

By using Theorem 2.2, estimates (3.18) and (3.19) one can prove the above
inequality in the same way as (3.31) – we omit the details.
Now we may return to (3.34). Let $I_{j}:=\big{\\{}n\in\mathbb{N}_{0}\colon
n\in I,n\geq\max\\{-j/\chi,\,j-1\\}\big{\\}}$. By changing the order of
summation, we see that to prove (3.34) it is enough to show that

(3.36) $\Big{\|}\sum_{n\in
I_{j}}T_{\mathbb{Z}^{\Gamma}}\big{[}\varepsilon_{n}(m_{2^{n+1}}-m_{2^{n}})(\Xi_{n}^{s,j}-\Xi_{n}^{s,j+1})\big{]}f\Big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim(s+1)^{-5}B_{p}(N)2^{-|j|\beta}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}$

holds for some $\beta=\beta_{p}>0$. Remark that one has

$\left(\Xi_{n}^{s,j}-\Xi_{n}^{s,j+1}\right)(\xi)=\Delta_{n,\,s}^{j,\,1}(\xi)\Delta_{n,\,s}^{j,\,2}(\xi),$

where

$\displaystyle\Delta_{n,\,s}^{j,\,1}(\xi)$
$\displaystyle:=\sum_{a/q\in\Sigma_{s^{u}}}\Big{[}\eta\left(2^{nA+(j-1)I}(\xi-a/q)\right)-\eta\left(2^{nA+(j+2)I}(\xi-a/q)\right)\Big{]}\tilde{\eta}(2^{s(A-\chi
I)}(\xi-a/q)),$ $\displaystyle\Delta_{n,\,s}^{j,\,2}(\xi)$
$\displaystyle:=\sum_{a/q\in\Sigma_{s^{u}}}\Big{[}\eta^{2}\big{(}2^{nA+jI}(\xi-a/q)\big{)}-\eta^{2}\big{(}2^{nA+(j+1)I}(\xi-a/q)\big{)}\Big{]}\tilde{\eta}(2^{s(A-\chi
I)}(\xi-a/q)).$

Now we will derive from the discrete Littlewood–Paley theory which originates
in [18, Theorem 3.3]. Let $j,n\in\mathbb{Z}$ and let
$\Phi_{j,\,n}(\xi)=\Phi(2^{nA+jI}\xi)$, where $\Phi$ is a Schwartz function
such that $\Phi(0)=0$. Observe that one has

$|\Phi_{n,j}(\xi)|\lesssim\min\\{|2^{nA+jI}\xi|,|2^{nA+jI}\xi|^{-1}\\}.$

Moreover, for any $p\in(1,\infty)$ there is a constant $C_{p}>0$ such that

$\big{\lVert}\sup_{n\in\mathbb{Z}}|T_{\mathbb{R}^{\Gamma}}[|\Phi_{n,j}|]f|\big{\rVert}_{L^{p}(\mathbb{R}^{\Gamma})}\leq
C_{p}\|f\|_{L^{p}(\mathbb{R}^{\Gamma})}.$

Hence, by [10, Theorem B] for any $-\infty\leq M_{1}\leq M_{2}\leq\infty$ we
have

$\big{\lVert}\big{(}\sum_{M_{1}\leq n\leq
M_{2}}|T_{\mathbb{R}^{\Gamma}}[\Phi_{j,n}]f|^{2}\big{)}^{1/2}\big{\rVert}_{L^{p}(\mathbb{R}^{\Gamma})}\lesssim_{p}\|f\|_{L^{p}(\mathbb{R}^{\Gamma})},$

where the implied constant is independent of $j,M_{1}$ and $M_{2}$. Therefore,
by Theorem 2.2 the multiplier

(3.37)
$\Omega_{s}^{j,\,n}(\xi):=\sum_{a/q\in\Sigma_{s^{u}}}{\Phi_{j,\,n}(\xi-a/q)\tilde{\eta}(2^{s(A-\chi
I)}(\xi-a/q))}$

satisfies

(3.38) $\Big{\|}\big{(}\sum_{M_{1}\leq n\leq
M_{2}}|T_{\mathbb{Z}^{\Gamma}}[\Omega_{s}^{j,\,n}]f|^{2}\big{)}^{1/2}\Big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim\log(s+1)\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

Moreover, if $\Phi$ is a real valued function, then the dual version of the
inequality (3.38) also holds, namely

(3.39) $\big{\|}\sum_{M_{1}\leq n\leq
M_{2}}|T_{\mathbb{Z}^{\Gamma}}[\Omega_{s}^{j,\,n}]f_{n})\big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim\log(s+1)\Big{\|}\big{(}\sum_{M_{1}\leq
n\leq
M_{2}}{|f_{n}|^{2}}\big{)}^{1/2}\Big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

where $(f_{n}:M_{1}\leq n\leq M_{2})$ is a sequence of functions such that

$\big{\lVert}\big{(}\sum_{M_{1}\leq n\leq
M_{2}}|f_{n}|^{2}\big{)}^{1/2}\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}<\infty.$

It is easy to see that multipliers $\Delta_{n,\,s}^{j,\,1}$ and
$\Delta_{n,\,s}^{j,\,2}$ can be written as (3.37). Hence, by applying the
inequality (3.39) to the multiplier $\Delta_{n,\,s}^{j,\,1}$ we get

$\displaystyle{\rm LHS}\eqref{ex6.1}\lesssim\log(s+1)\Big{\|}\big{(}\sum_{n\in
I_{j}}|T_{\mathbb{Z}^{\Gamma}}[(m_{2^{n+1}}-m_{2^{n}})\Delta_{n,\,s}^{j,\,2}]f|^{2}\big{)}^{1/2}\Big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

Consequently, the estimate (3.36) will follow if we prove that

(3.40) $\Big{\|}\big{(}\sum_{n\in
I_{j}}|T_{\mathbb{Z}^{\Gamma}}[(m_{2^{n+1}}-m_{2^{n}})\Delta_{n,\,s}^{j,\,2}]f|^{2}\big{)}^{1/2}\Big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim(s+1)^{-10}B_{p}(N)2^{-|j|\beta}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

for any $p\in(1,\infty)$.

####  3.1.6. Bootstrap estimates for the square function in (3.40)

We start with proving some estimates in the case of $p=2$. For simplicity, we
denote

$\psi_{n,\,j}^{a/q,\,s}(\xi):=\big{[}\eta^{2}\big{(}2^{nA+jI}(\xi-a/q)\big{)}-\eta^{2}\big{(}2^{nA+(j+1)I}(\xi-a/q)\big{)}\big{]}\tilde{\eta}\big{(}2^{s(A-\chi
I)}(\xi-a/q)\big{)}.$

Observe that $\psi_{n,\,j}^{a/q,\,s}$ is nonzero only if
$2^{-(j+2)}\lesssim|2^{nA}(\xi-a/q)|_{\infty}\lesssim 2^{-j}$. Hence
$|\xi_{\gamma}-a_{\gamma}/q|\leq 2^{-n(|\gamma|-\chi)}$ for $\gamma\in\Gamma$
since $n\geq-j/\chi$. By [24, Proposition 4.18] we have

(3.41) $m_{2^{n}}(\xi)=G(a,q)\Phi_{2^{n}}(\xi-a/q)+\mathcal{O}(2^{-n/2}),$

where $a/q$ satisfy $|\xi_{\gamma}-a_{\gamma}/q|\lesssim
2^{-n(|\gamma|-\chi)}$ for every $\gamma\in\Gamma$. To simplify notation we
denote
$w_{n}(\xi):=\min\\{|2^{nA}\xi|_{\infty},|2^{nA}\xi|_{\infty}^{-1/|\Gamma|}\\}$.
By using estimates (3.41), (3.18) and (3.19) one may conclude that

$\displaystyle|(m_{2^{n+1}}-m_{2^{n}})(\xi)|$
$\displaystyle\lesssim(s+1)^{-u\delta}w_{n}(\xi-a/q)+(s+1)^{-u\delta}\mathcal{O}(2^{-n/4}),$
$\displaystyle|(m_{2^{n+1}}-m_{2^{n}})(\xi)|\psi_{n,\,j}^{a/q,\,s}(\xi)$
$\displaystyle\lesssim(s+1)^{-u\delta}2^{-|j|/d}+(s+1)^{-u\delta}\mathcal{O}(2^{-n/4}).$

Hence, by using the above estimates we obtain

$\displaystyle\sum_{n\in
I_{j}}\sum_{a/q\in\Sigma_{s^{u}}}|(m_{2^{n+1}}-m_{2^{n}})(\xi)\psi_{n,\,j}^{a/q,\,s}(\xi)|^{2}\lesssim(s+1)^{-2u\delta}2^{-|j|\beta},$

for some $\beta>0$. Hence, by Plancherel’s theorem

(3.42) $\displaystyle\Big{\|}\big{(}\sum_{n\in
I_{j}}|T_{\mathbb{Z}^{\Gamma}}[(m_{2^{n+1}}-m_{2^{n}})\Delta_{n,\,s}^{j,\,2}]f|^{2}\big{)}^{1/2}\Big{\|}_{\ell^{2}(\mathbb{Z}^{\Gamma})}$
$\displaystyle\lesssim(s+1)^{-\delta
u}2^{-|j|\beta/2}\|f\|_{\ell^{2}(\mathbb{Z}^{\Gamma})}.$

Let us note that the above estimate together with the estimate for the large
scales (3.14) proves that $R_{2}(N)<\infty$ for any $N\in\mathbb{N}$ so we
have proven estimate (3.3) in the case of $p=2$. In order to handle other
values of $p$ we will relate to the bootstrap argument from [23]. This
argument originates in [26] and [10].

######  Lemma 3.43.

[23, Lemma 2.8] Suppose that $(X,\mathcal{B},m)$ is a $\sigma$-finite measure
space and $(B_{k})_{k\in\mathbb{J}}$ is a sequence of linear operators on
$L^{1}(X)+L^{\infty}(X)$ indexed by a countable set $\mathbb{J}$. The
corresponding maximal operator is defined by

$B_{\ast,\,\mathbb{J}}f\colon=\sup_{k\in\mathbb{J}}\sup_{|g|\leq|f|}|B_{k}g|,$

where the supremum is taken in the lattice sense. Let
$q_{0},q_{1}\in[1,\infty]$ and $0\leq\vartheta\leq 1$ with
$\frac{1}{2}=\frac{1-\vartheta}{q_{0}}$ and $q_{0}\leq q_{1}$. Let
$q_{\vartheta}\in[q_{0},q_{1}]$ be given by
$\frac{1}{q_{\vartheta}}=\frac{1-\vartheta}{q_{0}}+\frac{\vartheta}{q_{1}}=\frac{1}{2}+\frac{1-q_{0}/2}{q_{1}}$.
Then

$\Big{\|}\big{(}\sum_{k\in\mathbb{J}}|B_{k}g_{k}|^{2}\big{)}^{1/2}\Big{\|}_{L^{q_{\vartheta}}(X)}\leq(\sup_{k\in\mathbb{J}}\|B_{k}\|_{L^{q_{0}}\to
L^{q_{0}}})^{1-\vartheta}\|B_{\ast,\,\mathbb{J}}\|^{\vartheta}_{L^{q_{1}}\to
L^{q_{1}}}\Big{\|}\big{(}\sum_{k\in\mathbb{J}}|g_{k}|^{2}\big{)}^{1/2}\Big{\|}_{L^{q_{\vartheta}}(X)}.$

Here we have to make some distinction between the considered seminorms
(2.9)–(2.12).
_Case of the supremum, oscillation and $r$-variational seminorm._ At first let
$p\in(1,2]$. We will apply Lemma 3.43 with the set
$\mathbb{J}=I_{j}\subseteq[0,N)$, parameters $q_{0}=1$, $q_{1}=p$,
$\vartheta=1/2$, operators $B_{n}=M_{2^{n+1}}-M_{2^{n}}$ and functions
$g_{n}=T_{\mathbb{Z}^{\Gamma}}[\Delta_{n,s}^{j,2}]f$. Now, since the norm of
the operator $M_{2^{n}}$ is uniformly bounded we see that for every
$q\in(1,\infty)$ one has

$\displaystyle\sup_{n<N}\|B_{n}\|_{\ell^{q}\to\ell^{q}}$
$\displaystyle\lesssim 1.$

When we consider the supremum seminorm,

$\mathcal{S}_{p}(M_{2^{n}}f:n\in[0,N]\cap\mathbb{N}_{0})=\big{\lVert}\sup_{0\leq
n\leq N}|(M_{2^{n}}-M_{1})f|\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

then it is easy to check that for every $q\in(1,\infty)$

$\displaystyle\|B_{\ast,n<N}\|_{\ell^{q}\to\ell^{q}}\lesssim R_{q}(N)$

since the norm of the operator $M_{t}$ is uniformly bounded. Similarly, in the
case of the oscillation seminorm,

$\mathcal{S}_{p}(M_{2^{n}}f:n\in[0,N]\cap\mathbb{N}_{0})=\sup_{K\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{K}([0,N]\cap\mathbb{N}_{0})}\big{\lVert}O_{I,K}^{2}(M_{2^{n}}f:n\in[0,N]\cap\mathbb{N}_{0})\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

by the inequality (2.14) we have that the estimate

$\displaystyle\|B_{\ast,n<N}\|_{\ell^{q}\to\ell^{q}}\lesssim R_{q}(N)$

yields for any $q\in(1,\infty)$. Finally, in the case of $r$-variational
seminorm,

$\mathcal{S}_{p}(M_{2^{n}}f:n\in[0,N]\cap\mathbb{N}_{0})=\big{\lVert}V^{r}(M_{2^{n}}f:n\in[0,N]\cap\mathbb{N}_{0})\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

by inequality (2.13) we see that for any $r\in(2,\infty)$ the estimate

$\displaystyle\|B_{\ast,n<N}\|_{\ell^{q}\to\ell^{q}}\lesssim R_{q}(N)$

also holds for $q\in(1,\infty)$. Hence, in the case of the supremum,
oscillation and $r$-variational seminorm by Lemma 3.43 we may write

(3.44)
$\displaystyle\Big{\|}\big{(}\sum_{n\in\mathbb{J}}|B_{n}T_{\mathbb{Z}^{\Gamma}}[\Delta_{n,\,s}^{j,\,2}]f|^{2}\big{)}^{1/2}\Big{\|}_{\ell^{q_{1/2}}(\mathbb{Z}^{\Gamma})}$
$\displaystyle\lesssim
R_{p}(N)^{1/2}\Big{\|}\big{(}\sum_{n\in\mathbb{J}}|T_{\mathbb{Z}^{\Gamma}}[\Delta_{n,\,s}^{j,\,2}]f|^{2}\big{)}^{1/2}\Big{\|}_{\ell^{q_{1/2}}(\mathbb{Z}^{\Gamma})}$
$\displaystyle\lesssim
R_{p}(N)^{1/2}\log(s+1)\|f\|_{\ell^{q_{1/2}}(\mathbb{Z}^{\Gamma})},$

where in the last inequality we have used (3.38). Since $q_{1/2}<p\leq 2$,
there exists $t\in[0,1)$ such that
$\frac{1}{p}=\frac{t}{q_{1/2}}+\frac{1-t}{2}$. If we use definition of
$q_{1/2}$ from Lemma 3.43 we see that $t=2-p$. Hence, by interpolating (3.42)
with (3.44) one has

$\displaystyle\Big{\|}\big{(}\sum_{n\in\mathbb{J}}|B_{n}T_{\mathbb{Z}^{\Gamma}}[\Delta_{n,\,s}^{j,\,2}]f|^{2}\big{)}^{1/2}\Big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim(s+1)^{-u\delta(1-t)}2^{-|j|\frac{\beta}{2}(1-t)}R_{p}(N)^{(2-p)/2}(s+1)\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

Since $u\in\mathbb{N}$ can be large, we get that (3.40) is satisfied with
$B_{p}(N)=R_{p}(N)^{(2-p)/2}$. Hence, we see that for $p\in(1,2]$ the
inequality (3.15) holds with $\beta(p)=\frac{2-p}{2}\in[0,1)$.

Now let us assume that $p\in(2,\infty)$. Then one has $p^{\prime}\in(1,2)$ and
therefore by applying Lemma 3.43 with $q_{0}=1$, $q_{1}=p^{\prime}$ and
$\vartheta=1/2$, and by inequality (3.38) we obtain

(3.45)
$\displaystyle\Big{\|}\big{(}\sum_{n\in\mathbb{J}}|B_{n}T_{\mathbb{Z}^{\Gamma}}[\Delta_{n,\,s}^{j,\,2}]f|^{2}\big{)}^{1/2}\Big{\|}_{\ell^{q_{1/2}}(\mathbb{Z}^{\Gamma})}\lesssim
R_{p^{\prime}}(N)^{1/2}\log(s+1)\|f\|_{\ell^{q_{1/2}}(\mathbb{Z}^{\Gamma})}$

where $q_{1/2}=2p/(2p-1)$. Now, since $B_{n}=M_{2^{n+1}}-M_{2^{n}}$ is a
convolution operator we see that by duality the inequality (3.45) holds for
$q_{1/2}^{\prime}=2p$. Since $2<p<q_{1/2}^{\prime}$ there exists
$\tau\in[0,1)$ such that
$\frac{1}{p}=\frac{\tau}{q_{1/2}^{\prime}}+\frac{1-\tau}{2}$ and

$\tau=\frac{2-p}{1-p}.$

Hence, by interpolating (3.42) with (3.45) for $q_{1/2}^{\prime}$ we may write

$\displaystyle\Big{\|}\big{(}\sum_{n\in\mathbb{J}}|B_{n}T_{\mathbb{Z}^{\Gamma}}[\Delta_{n,\,s}^{j,\,2}]f|^{2}\big{)}^{1/2}\Big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim(s+1)^{-u\delta(1-\tau)}2^{-|j|\frac{\beta}{2}(1-\tau)}R_{p^{\prime}}(N)^{\frac{2-p}{2(1-p)}}(s+1)\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

Since $u\in\mathbb{N}$ can be large, we get that (3.40) is satisfied with
$B_{p}(N)=R_{p^{\prime}}(N)^{\frac{2-p}{2(1-p)}}$. Hence, we see that for
$p\in(2,\infty)$ the inequality (3.16) holds with
$\beta^{\prime}(p)=\frac{2-p}{2(1-p)}\in[0,1)$.

_Case of the jump quasi-seminorm_ In the context of the jump quasi-seminorm we
need to proceed in a slightly different way since in this case we do not have
a pointwise estimate of the form (2.13) or even an $\ell^{p}$-estimate of the
form (2.14). Fortunately, for $r>2$ one has “weak $\ell^{p}$”-estimate (2.15)
for the $r$-variation which we will use at this moment. As mentioned before we
have already proved that $R_{2}(N)\lesssim 1$. Hence, we may assume that
$p\in(1,2)$. Let us consider $\lambda\in(0,1)$ such that

(3.46) $\lambda>\max\big{\\{}0,\frac{4-3p}{(p-2)^{2}}\big{\\}}.$

We are going to apply Lemma 3.43 with parameters $q_{0}=1$, $q_{1}=\lambda
p+(1-\lambda)2$, $\vartheta=1/2$, operators $B_{n}=M_{2^{n+1}}-M_{2^{n}}$ and
functions $g_{n}=T_{\mathbb{Z}^{\Gamma}}[\Delta_{n,s}^{j,2}]f$. If $\lambda$
satisfy condition (3.46), then one has $q_{1/2}<p<q_{1}<2$. Furthermore, for
any $q\in(1,\infty)$

$\displaystyle\sup_{n<N}\|B_{n}\|_{\ell^{q}\to\ell^{q}}$
$\displaystyle\lesssim
1\quad\text{and}\quad\|B_{\ast,n<N}\|_{\ell^{q}\to\ell^{q}}\lesssim\|V^{3}(M_{2^{n}}:n\in[0,N]\cap\mathbb{N}_{0})\|_{\ell^{q}\to\ell^{q}},$

where the last inequality follows by (2.13). By using the inequality (2.15) we
get weak type estimates

$\displaystyle\big{\lVert}V^{3}(M_{2^{n}}f:n\in[0,N]\cap\mathbb{N}_{0})\big{\rVert}_{\ell^{2,\,\infty}(\mathbb{Z}^{\Gamma})}$
$\displaystyle\lesssim\|f\|_{\ell^{2}(\mathbb{Z}^{\Gamma})},$
$\displaystyle\big{\lVert}V^{3}(M_{2^{n}}f:n\in[0,N]\cap\mathbb{N}_{0})\big{\rVert}_{\ell^{p,\,\infty}(\mathbb{Z}^{\Gamma})}$
$\displaystyle\lesssim R_{p}(N)\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

Since $p<q_{1}<2$, one may use Marcinkiewicz’s interpolation theorem to get

$\|B_{\ast,n<N}\|_{\ell^{q_{1}}\to\ell^{q_{1}}}\lesssim_{p}R_{p}(N)^{\frac{p(2-q_{1})}{q_{1}(2-p)}}.$

Therefore, by Lemma 3.43 and inequality (3.38) one has

(3.47)
$\displaystyle\Big{\|}\big{(}\sum_{n\in\mathbb{J}}|B_{n}T_{\mathbb{Z}^{\Gamma}}[\Delta_{n,\,s}^{j,\,2}]f|^{2}\big{)}^{1/2}\Big{\|}_{\ell^{q_{1/2}}(\mathbb{Z}^{\Gamma})}\lesssim
R_{p}(N)^{\frac{p(2-q_{1})}{2q_{1}(2-p)}}\log(s+1)\|f\|_{\ell^{q_{1/2}}(\mathbb{Z}^{\Gamma})},$

Since $q_{1/2}<p<2$, there exists $t\in(0,1)$ such that
$\frac{1}{p}=\frac{t}{q_{1/2}}+\frac{1-t}{2}$. Hence, by the definition of
$q_{1/2}$ one has

$t\frac{p(2-q_{1})}{2q_{1}(2-p)}=\frac{2-q_{1}}{2}.$

Interpolating (3.42) with (3.47) leads to

$\displaystyle\Big{\|}\big{(}\sum_{n\in\mathbb{J}}|B_{n}T_{\mathbb{Z}^{\Gamma}}[\Delta_{n,\,s}^{j,\,2}]f|^{2}\big{)}^{1/2}\Big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim(s+1)^{-u\delta(1-t)}2^{-|j|\beta(1-t)/2}R_{p}(N)^{\frac{2-q_{1}}{2}}\log(s+1)\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

Since $u\in\mathbb{N}$ can be large, we get that (3.40) in the case of the
jump quasi-seminorm is satisfied with $B_{p}(N)=R_{p}(N)^{(2-q_{1})/2}$.
Hence, we see that for $p\in(1,2]$ the inequality (3.15) holds with
$\beta(p)=\frac{2-q_{1}}{2}\in[0,1)$. In the case of $p\in(2,\infty)$ we again
use the duality to obtain the desired result.

###  3.2. Estimates for short variations

In this section we focus on bounding the short variations, namely we want to
establish the following estimate

(3.48)
$\Big{\lVert}\Big{(}\sum_{l=0}^{\infty}V^{2}\big{(}M_{n}f:n\in[2^{l},2^{l+1}]\cap\mathbb{U}\big{)}^{2}\Big{)}^{1/2}\Big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim_{\mathcal{S}_{p}}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})},\quad
f\in\ell^{p}(\mathbb{Z}^{\Gamma}).$

For this purpose, for $L\in\mathbb{N}$, let us consider the following cut-off
short variations

$\Big{(}\sum_{l=0}^{L}V^{2}\big{(}M_{n}f:n\in[2^{l},2^{l+1}]\cap\mathbb{U}\big{)}^{2}\Big{)}^{1/2}.$

Let $R_{p}(L)$ denote the smallest constant for which the following estimate
holds

(3.49)
$\Big{\|}\Big{(}\sum_{l=0}^{L}V^{2}\big{(}M_{n}f:n\in[2^{l},2^{l+1}]\cap\mathbb{U}\big{)}^{2}\Big{)}^{1/2}\Big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\leq
R_{p}(L)\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})},\quad
f\in\ell^{p}(\mathbb{Z}^{\Gamma}).$

By estimate (2.16) we know that $R_{p}(L)\lesssim_{L,\,p}1$. Using again the
bootstrap argument we will show that $R_{p}(L)\lesssim_{p}1$. The proof will
proceed in a similar way as in the case of the long jumps hence we will omit
some details. Without loss of generality we can assume that $R_{p}(L)>1$ and
$L\in\mathbb{N}$ is large. Let $\chi\in(0,1/10)$ and let $u\in\mathbb{N}$ be a
fixed large number. For each $n\in\mathbb{N}$ we define the following function

(3.50) $\Xi_{n}(\xi):=\sum_{a/q\in\Sigma_{\leq n^{u}}}\eta\big{(}2^{n(A-\chi
I)}(\xi-a/q)\big{)},$

where $\eta$ is a bump function of the form (3.6), $I$ is the
$|\Gamma|\times|\Gamma|$ identity matrix, $A$ is a matrix defined in Section
3.1 and $\Sigma_{\leq n^{u}}$ is the set of the Ionescu–Wainger rational
fractions related to the parameter $\varrho=(10u)^{-1}$. Recall that we may
write $M_{n}f=T_{\mathbb{Z}^{\Gamma}}[m_{n}]f$. Next, we use functions (3.50)
to estimate the left hand side of (3.49) by

(3.51)
$\displaystyle\Big{\|}\Big{(}\sum_{l=0}^{L}V^{2}\big{(}T_{\mathbb{Z}^{\Gamma}}[\Xi_{l}$
$\displaystyle
m_{n}]f:n\in[2^{l},2^{l+1}]\cap\mathbb{U}\big{)}^{2}\Big{)}^{1/2}\Big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}$
(3.52)
$\displaystyle+\Big{\|}\Big{(}\sum_{l=0}^{L}V^{2}\big{(}T_{\mathbb{Z}^{\Gamma}}[(1-\Xi_{l})m_{n}]f:n\in[2^{l},2^{l+1}]\cap\mathbb{U}\big{)}^{2}\Big{)}^{1/2}\Big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

Similar to the case of the long jumps, the first expression corresponds to the
major arcs and the second one to the minor arcs in the Hardy–Littlewood circle
method.

####  3.2.1. Minor arcs

Again, we start with the estimate for the minor arcs since it is relatively
easy and follows the same rule as in the case of the long jumps. By the
triangle inequality it is enough to show

$\big{\|}V^{2}\big{(}T_{\mathbb{Z}^{\Gamma}}[(1-\Xi_{l})m_{n}]f:n\in[2^{l},2^{l+1}]\cap\mathbb{U}\big{)}\big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim(l+1)^{-2}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

We note that for each $l\in\mathbb{N}_{0}$ only a finite set of numbers from
$[2^{l},2^{l+1}]\cap\mathbb{U}$ give a contribution to the above variational
seminorm. Hence it is enough to prove that for some $a(l)\in\mathbb{N}$ we
have

(3.53)
$\big{\|}V^{2}\big{(}T_{\mathbb{Z}^{\Gamma}}[(1-\Xi_{l})m_{n/2^{a(l)}}]f:n\in[2^{l+a(l)},2^{l+1+a(l)}]\cap\mathbb{N}_{0}\big{)}\big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim(l+1)^{-2}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

We can prove the estimate above in the same way as (3.12), for details see
[21, Section 5.1, pp. 689–691]

####  3.2.2. Major arcs

Now, our aim is to estimate (3.51). In this case we will follow the approach
presented in the proof of the estimate for the long jumps. The case of the
short jumps is in some way easier since there is no need to consider small and
large scales. In order to estimate (3.51) we introduce new multipliers

$\Xi_{l}^{j}(\xi)=\sum_{a/q\in\Sigma_{\leq
l^{u}}}\eta\big{(}2^{lA+jI}(\xi-a/q)\big{)},\quad j\in\mathbb{Z}.$

Then one may write (compare with Section 3.1.5)

$\Xi_{l}(\xi)=\sum_{-\lfloor\chi l\rfloor\leq
j<l}\big{(}\Xi_{l}^{j}(\xi)-\Xi_{l}^{j+1}(\xi)\big{)}+\big{(}\Xi_{l}^{-\chi
l}(\xi)-\Xi_{l}^{-\lfloor\chi l\rfloor}(\xi)\big{)}+\Xi_{l}^{l}(\xi).$

Next, we use the new multipliers and estimate (3.51) by

(3.54)
$\displaystyle\Big{\|}\Big{(}\sum_{l=0}^{L}V^{2}\big{(}T_{\mathbb{Z}^{\Gamma}}\big{[}m_{n}\big{(}\sum_{-\lfloor\chi
l\rfloor\leq
j<l}\Xi_{l}^{j}-\Xi_{l}^{j+1}\big{)}\big{]}f:n\in[2^{l},2^{l+1}]\cap\mathbb{U}\big{)}^{2}\Big{)}^{1/2}\Big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}$
(3.55)
$\displaystyle+\Big{\|}\Big{(}\sum_{l=0}^{L}V^{2}\big{(}T_{\mathbb{Z}^{\Gamma}}\big{[}(m_{n}-m_{2^{l}})(\Xi_{l}^{-\chi
l}-\Xi_{l}^{-\lfloor\chi
l\rfloor}+\Xi_{l}^{l})\big{]}f:n\in[2^{l},2^{l+1}]\cap\mathbb{U}\big{)}^{2}\Big{)}^{1/2}\Big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

One can prove that $\eqref{ad4}\lesssim\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}$
by proceeding in a similar way as in [21, pp. 692–694].
Now let us go back to (3.54). For $p\in(1,2]$ it is enough to show

(3.56)
$\displaystyle\Big{\|}\Big{(}\sum_{l=0}^{L}V^{2}\big{(}T_{\mathbb{Z}^{\Gamma}}\big{[}m_{n}\big{(}\sum_{-\lfloor\chi
l\rfloor\leq
j<l}\Xi_{l}^{j}-\Xi_{l}^{j+1}\big{)}\big{]}f:n\in[2^{l},2^{l+1}]\cap\mathbb{U}\big{)}^{2}$
$\displaystyle\Big{)}^{1/2}\Big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim
R_{p}(L)^{\frac{2-p}{2}}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

For $p\in(2,\infty)$ we will show that

(3.57) ${\rm LHS}\eqref{pinfs3}\lesssim
R_{p^{\prime}}(L)^{(p-2)/2(p-1)}\|f\|_{L^{p}(\mathbb{Z}^{\Gamma})}$

where $1/p+1/p^{\prime}=1$.

####  3.2.3. Estimates for (3.56) and discrete Littlewood–Paley theory

Now, we take a look at the left hand side of (3.56) in the case of
$p\in(1,\infty)$. Let $\tilde{\eta}(x):=\eta(x/2)$ and define a new multiplier

$\Delta_{l,\,s}^{j}(\xi)=\sum_{a/q\in\Sigma_{s^{u}}}\big{[}\eta\big{(}2^{lA+jI}(\xi-a/q)\big{)}-\eta\big{(}2^{lA+(j+1)I}(\xi-a/q)\big{)}\big{]}\tilde{\eta}\big{(}2^{s(A-\chi
I)}(\xi-a/q)\big{)},$

where $\Sigma_{s^{u}}:=\Sigma_{\leq(s+1)^{u}}\setminus\Sigma_{\leq s^{u}}$ for
$s\in\mathbb{N}$ and $\Sigma_{0^{u}}:=\Sigma_{\leq 1}$. We see that

(3.58)
$\Xi_{l}^{j}(\xi)-\Xi_{l}^{j+1}(\xi)=\sum_{s=0}^{l-1}\Delta_{l,\,s}^{j}(\xi).$

Consequently, if we use (3.58) and change the order of summation we see that
the estimate (3.56) will follow if we prove that

(3.59) $\displaystyle\Big{\|}\Big{(}\sum_{\begin{subarray}{c}0\leq l\leq L,\\\
l\geq\max\\{-j/\chi,\,j-1,s-1,i\\}\end{subarray}}V^{2}\big{(}T_{\mathbb{Z}^{\Gamma}}\big{[}m_{n}\Delta_{l,\,s}^{j}\big{]}f:n\in[2^{l},$
$\displaystyle
2^{l+1}]\cap\mathbb{U}\big{)}^{2}\Big{)}^{1/2}\Big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}$
$\displaystyle\lesssim(s+1)^{-2}B_{p}(L)2^{-|j|\beta}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

for some $\beta=\beta_{p}>0$ where for $p\in(1,2]$ the constant $B_{p}(L)$ is
equal to $R_{p}(L)^{(2-p)/2}$ and for $p\in(2,\infty)$ we have
$B_{p}(L)=R_{p^{\prime}}(L)^{(p-2)/2(p-1)}$. Now, if we apply the
Rademacher–Menshov inequality for the short jumps [23, Lemma 2.5] we see it is
enough to establish

(3.60) $\displaystyle\Big{\|}\Big{(}\sum_{\begin{subarray}{c}0\leq l\leq L,\\\
l\geq\max\\{-j/\chi,\,j-1,s-1,i\\}\end{subarray}}\sum_{m=0}^{2^{i}-1}|T_{\mathbb{Z}^{\Gamma}}\big{[}($
$\displaystyle
m_{2^{l}+2^{l-i}(m+1)}-m_{2^{l}+2^{l-i}m})\Delta_{l,\,s}^{j}\big{]}f|^{2}\Big{)}^{1/2}\Big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}$
$\displaystyle\lesssim(s+1)^{-2}(i+1)^{-2}B_{p}(L)2^{-|j|\beta}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

####  3.2.4. Estimates for square function in (3.60)

For simplicity we denote $B_{l,m}=M_{2^{l}+2^{l-i}(m+1)}-M_{2^{l}+2^{l-i}m}$.
At first we will prove (3.60) in the case of $p=2$. If we approximate discrete
multipliers by their continuous counterparts by using [24, Proposition 4.18]
we can use estimates (3.18) and (3.19), and [24, Proposition 4.16] to prove
that on the support of $\Delta_{l,\,s}^{j}$ we have

(3.61)
$|m_{2^{l}+2^{l-i}(m+1)}-m_{2^{l}+2^{l-i}m}|\lesssim\min\big{\\{}2^{-i},(s+1)^{-u\delta}(2^{-|j|/|\Gamma|}+2^{-l/4})\big{\\}}.$

Consequently, one has

$|m_{2^{l}+2^{l-i}(m+1)}-m_{2^{l}+2^{l-i}m}|^{2}\lesssim
2^{-3i/2}(s+1)^{-\delta u/2}(2^{-|j|/(2|\Gamma|)}+2^{-l/8}).$

Hence, by Parseval’s inequality we get

(3.62)
$\displaystyle\Big{\|}\big{(}\sum_{(l,m)\in\mathbb{J}}|B_{l,m}T_{\mathbb{Z}^{\Gamma}}[\Delta_{l,\,s}^{j}]f|^{2}\big{)}^{1/2}\Big{\|}_{\ell^{2}(\mathbb{Z}^{\Gamma})}\lesssim
2^{-i/4}(s+1)^{-u\alpha/4}2^{-|j|\beta}\|f\|_{\ell^{2}(\mathbb{Z}^{\Gamma})},$

with $\mathbb{J}:=\\{(l,m)\in\mathbb{Z}^{2}:l\in[l_{0},L],v\in[0,2^{i}-1]\\}$
where $l_{0}=\max\\{-j/\chi,\,j-1,\,s-1,i\\}$.
In the case $p\in(1,2]$ we will use discrete Littlewood–Paley theory (see
Section 3.1.5) and the bootstrap Lemma 3.43. We will apply it with parameters
$q_{0}=1$, $q_{1}=p$, $\vartheta=1/2$, to a countable set $\mathbb{J}$, the
operator $B_{l,m}$ and the functions
$g_{l,m}=T_{\mathbb{Z}^{\Gamma}}[\Delta_{l,\,s}^{j}]f$. It is easy to check
that for every $q\in(1,\infty)$ we have

$\displaystyle\sup_{(l,m)\in\mathbb{J}}\|B_{l,m}\|_{L^{1}\to L^{1}}$
$\displaystyle\lesssim
2^{-i}\quad\text{and}\quad\|B_{\ast,\mathbb{J}}\|_{L^{q}\to L^{q}}\lesssim
R_{q}(L).$

The first inequality follows from [24, Proposition 4.16] and the last
inequality follows from the fact that for any $m=0,\ldots,2^{i}-1$ and any
$l\in\mathbb{N}$ one has the following pointwise estimate

$|M_{2^{l}+2^{l-i}(m+1)}f-M_{2^{l}+2^{l-i}m}f|\lesssim
V^{2}\big{(}M_{n}f:n\in[2^{l},2^{l+1}]\cap\mathbb{U}\big{)}.$

Therefore, by applying Lemma 3.43 and inequality (3.38) we obtain

(3.63)
$\displaystyle\Big{\|}\big{(}\sum_{(l,m)\in\mathbb{J}}|B_{l,m}T_{\mathbb{Z}^{\Gamma}}[\Delta_{l,\,s}^{j}]f|^{2}\big{)}^{1/2}\Big{\|}_{\ell^{q_{1/2}}(\mathbb{Z}^{\Gamma})}\lesssim
R_{p}(L)^{1/2}\log(s+1)\|f\|_{\ell^{q_{1/2}}(\mathbb{Z}^{\Gamma})}.$

Since $q_{1/2}<p\leq 2$, we can interpolate (3.62) with (3.63) to get that
(3.60) holds in the case of $p\in(1,2]$. When $p\in(2,\infty)$ the desired
result follows by duality since $B_{l,m}$ is a convolution operator.

###  3.3. Comments and remarks

Let us point out several remarks which can be made after reading our proof:

  1. (1)

The proof of Theorem 1.6 was reduced to proving the estimates for the long
jumps and short variations related to the dyadic integers
$\mathbb{D}:=\\{2^{n}:n\in\mathbb{N}\\}$. Here one could also apply a slightly
different approach. Namely, for a fixed $\tau\in(0,1)$ we consider the
subexponential dyadic integers
$\mathbb{D}_{\tau}:=\\{2^{n^{\tau}}:n\in\mathbb{N}_{0}\\}$ and use Proposition
2.20 to get

$\mathcal{S}_{p}(M_{t}f:t>0)\lesssim_{\mathcal{S}_{p}}\mathcal{S}_{p}(M_{n}f:n\in\mathbb{D}_{\tau})+\Big{\lVert}\Big{(}\sum_{n=0}^{\infty}V^{2}\big{(}M_{t}f:t\in[2^{n^{\tau}},2^{(n+1)^{\tau}}]\big{)}^{2}\Big{)}^{1/2}\Big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

Then the short variations can be estimated by using arguments from [33] (see
also [19, 24]), which reduces matters to showing that

$\mathcal{S}_{p}(M_{n}f:n\in\mathbb{D}_{\tau})\lesssim_{\mathcal{S}_{p}}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

The inequality above can be proved by using techniques presented in the proof
of estimates for the long jumps in Section 3.1. There could occur some
technical nuances in defining the Ionescu–Wainger projections (3.7), for
details see [19, 24]. We decided to take the presented approach since we
wanted to show that the bootstrap approach is possible in the context of the
short variations.

  2. (2)

It is not hard to notice that one could use our approach to prove the seminorm
inequalities

$\mathcal{S}_{p}(M_{2^{n}}f:n\in\mathbb{N})\lesssim_{\mathcal{S}_{p}}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

where the seminorm $\mathcal{S}_{p}$ is not of the form (2.9)–(2.12). It is
enough for $\mathcal{S}_{p}$ to be subadditive, satisfy the square function
estimate (2.16), have some properties of splitting and to satisfy the
Rademacher–Menshov inequality (2.23). Moreover, we need to know that the
appropriate continuous counterpart holds. Besides that, it does not even have
to control the maximal function like (2.13), (2.14) or even (2.15). In that
case, in Lemma 3.43 we can just control the norm of the maximal function
$\|B_{\ast,\mathbb{J}}\|$ by the norm of the usual maximal function
$\sup_{t>0}|M_{t}|$ since we know that the latter one is $\ell^{p}$-bounded.

  3. (3)

Our approach can be modified and applied to the truncated singular Radon
operator,

$H_{t}f(x):=\sum_{y\in\Omega_{t}\cap\mathbb{Z}^{k}\setminus\\{0\\}}f(x-(y)^{\Gamma})K(y),$

where $K\colon\mathbb{R}^{k}\setminus\\{0\\}\to\mathbb{C}$ is a
Calderón–Zygmund kernel satisfying appropriate conditions. We can prove the
inequality

(3.64)
$\mathcal{S}_{p}(H_{t}f:t>0)\lesssim_{\mathcal{S}_{p}}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

where the seminorm $\mathcal{S}_{p}$ is one of (2.10)–(2.12) (we excluded the
maximal seminorm). It is known that the inequality (3.64) holds for all
seminorms (2.9)–(2.12), see [20, 21, 24, 28]. The main difference between
$M_{t}$ and $H_{t}$ is that the operator $H_{t}$ is not positive, hence to use
Lemma 3.43 (i.e. to estimate $\|B_{\ast,\mathbb{J}}\|$) we need to know that

$\|\sup_{t>0}|H_{t}f|\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim_{p}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

This is very consistent with the continuous case (see [23, Theorem 2.28]).

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