# Oscillation estimates for truncated singular 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.

In this paper we prove uniform oscillation estimates on $L^{p}$, with
$p\in(1,\infty)$, for truncated singular integrals of the Radon type
associated with the Calderón–Zygmund kernel, both in continuous and discrete
settings. In the discrete case we use the Ionescu–Wainger multiplier theorem
and the Rademacher–Menshov inequality to handle the number-theoretic nature of
the discrete singular integral. The result we obtained in the continuous
setting can be seen as a generalisation of the results of Campbell, Jones,
Reinhold and Wierdl for the continuous singular integrals of the
Calderón–Zygmund type.

###### Key words and phrases:

Radon transform, singular integral,oscillation seminorm

###### 2020 Mathematics Subject Classification:

42B25, 42B20, 42B15

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.

###### Contents

  1. 1 Introduction
    1. 1.1 Historical background
  2. 2 Preliminaries
    1. 2.1 Basic notation
    2. 2.2 Norms and Fourier transform
    3. 2.3 Ionescu–Wainger multiplier theorem
  3. 3 Proof of Theorem 1.7 for the canonical mapping $(x)^{\Gamma}$
    1. 3.1 Estimates for long oscillations
      1. 3.1.1 Minor arcs
      2. 3.1.2 Major arcs and multiplier approximation
      3. 3.1.3 Gaussian multiplier and scale distinction
      4. 3.1.4 Estimates for small scales
      5. 3.1.5 Estimates for large scales
  4. 4 Proof of theorem 1.9 for the canonical mapping $(x)^{\Gamma}$
    1. 4.1 Dyadic martingales on the homogeneous spaces
    2. 4.2 Oscillation inequality for the operator $\mathcal{H}_{t}$

##  1\. Introduction

The purpose of this paper is to give a proof of oscillation inequalities for
truncated singular integrals of the Radon type, both in continuous and
discrete settings.

Let $k\geq 1$ be a natural number and let $\Gamma$ be a fixed finite subset of
$\mathbb{N}_{0}^{k}\setminus\\{0\\}$ equipped with the lexicographic order.
The space of tuples of real numbers and integer numbers labeled by multi-
indices $\gamma=(\gamma_{1},\dots,\gamma_{k})\in\Gamma$ will be denoted by
$\mathbb{R}^{\Gamma}$ and $\mathbb{Z}^{\Gamma}$, respectively. Clearly, one
has $\mathbb{R}^{\Gamma}\cong\mathbb{R}^{|\Gamma|}$ and
$\mathbb{Z}^{\Gamma}\cong\mathbb{Z}^{|\Gamma|}$. The canonical polynomial
mapping associated with the set $\Gamma$ 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 $x^{\gamma}=x_{1}^{\gamma_{1}}x_{2}^{\gamma_{2}}\cdots
x_{k}^{\gamma_{k}}$. The symbol $B(x,t)$ stands for an open Euclidean ball in
$\mathbb{R}^{k}$ with radius $t>0$ centered at a point $x\in\mathbb{R}^{k}$.
Let $\Omega\subset\mathbb{R}^{k}$ be a bounded convex open set such that
$B(0,c_{\Omega})\subseteq\Omega\subseteq B(0,1)$ for some
$c_{\Omega}\in(0,1)$. For $t>0$, the dilates of the set $\Omega$ are defined
as

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

Let $K\colon\mathbb{R}^{k}\setminus\\{0\\}\to\mathbb{C}$ be the
Calderón–Zygmund kernel satisfying the following conditions.

  1. (1)

The size condition. For every $x\in\mathbb{R}^{k}\setminus\\{0\\}$, we have

(1.1) $\lvert K(x)\rvert\leq\lvert x\rvert^{-k}.$

  2. (2)

The cancellation condition. For every $0<r<R<\infty$, we have

(1.2) $\int_{\Omega_{R}\setminus\Omega_{r}}K(y)\mathrm{d}y=0.$

  3. (3)

The Hölder continuity condition. For some $\sigma\in(0,1]$ and every
$x,y\in\mathbb{R}^{k}\setminus\\{0\\}$ with $|y|\leq|x|/2$, we have

(1.3) $\lvert K(x)-K(x+y)\rvert\leq\lvert y\rvert^{\sigma}\lvert
x\rvert^{-k-\sigma}.$

Now, for finitely supported functions
$f\colon\mathbb{Z}^{\Gamma}\to\mathbb{C}$ and $t>0$, we may define the
discrete truncated Radon singular operator by setting

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

where $K$ is the Calderón–Zygmund kernel which satisfies the conditions listed
above. The operator $H_{t}$ is a discrete counterpart of the continuous
singular Radon transform. Recall that for a given smooth compactly supported
function $f\colon\mathbb{R}^{\Gamma}\to\mathbb{C}$ the continuous singular
Radon transform of $f$ is defined as

(1.5) $\mathcal{H}_{t}f(x):={\rm
p.v.}\int_{\Omega_{t}}f(x-(y)^{\Gamma})K(y){\rm d}y,\quad
x\in\mathbb{R}^{\Gamma}.$

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 family of measurable functions
$(\mathfrak{a}_{t}(x):t>0)\subseteq\mathbb{C}$ by setting

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

Our result in the case of the discrete singular operator reads as follows.

######  Theorem 1.7.

Let $k\geq 1$ and let consider
$\Gamma\subset\mathbb{N}_{0}^{k}\setminus\\{0\\}$ such that $|\Gamma|<\infty$.
Then for any $p\in(1,\infty)$ there is a constant $C_{p,k,|\Gamma|}>0$ such
that

(1.8)
$\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{R}_{+})}\big{\lVert}O_{I,N}^{2}(H_{t}f:t\in\mathbb{R}_{+})\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\leq
C_{p,k,|\Gamma|}\lVert f\rVert_{\ell^{p}(\mathbb{Z}^{\Gamma})},\quad
f\in\ell^{p}(\mathbb{Z}^{\Gamma}),$

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).

A similar result holds in the continuous case.

######  Theorem 1.9.

Let $k\geq 1$ and let consider
$\Gamma\subset\mathbb{N}_{0}^{k}\setminus\\{0\\}$ such that $|\Gamma|<\infty$.
Then for any $p\in(1,\infty)$ there is a constant $C_{p,k,|\Gamma|}>0$ such
that

(1.10)
$\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{R}_{+})}\big{\lVert}O_{I,N}^{2}(\mathcal{H}_{t}f:t\in\mathbb{R}_{+})\big{\rVert}_{L^{p}(\mathbb{R}^{\Gamma})}\leq
C_{p,k,|\Gamma|}\lVert f\rVert_{L^{p}(\mathbb{R}^{\Gamma})},\quad f\in
L^{p}(\mathbb{R}^{\Gamma}).$

It is worth to note that Theorem 1.7 does not follow directly from Theorem 1.9
since the discrete nature and number-theoretic complications related to the
mapping $(x)^{\Gamma}$ prevents the use of the standard transference
arguments.

By a standard lifting argument (see [21, Lemma 2.1] and [25, Section 11, pp.
483–485]) Theorems 1.7 and 1.9 imply more general results. Namely, let
$d,k\geq 1$ be fixed natural numbers and consider the mapping

(1.11)
$\mathcal{P}=(\mathcal{P}_{1},\dots,\mathcal{P}_{d})\colon\mathbb{Z}^{k}\to\mathbb{Z}^{d},$

where each $\mathcal{P}_{j}\colon\mathbb{Z}^{k}\to\mathbb{Z}$ is a polynomial
with integer coefficients such that $\mathcal{P}_{j}(0)=0$. For an appropriate
test function $f\colon\mathbb{Z}^{d}\to\mathbb{C}$ (or
$f\colon\mathbb{R}^{d}\to\mathbb{C}$) we define Radon operators associated to
the mapping $\mathcal{P}$ by setting

(1.12) $\displaystyle H_{t}^{\mathcal{P}}f(x)$
$\displaystyle:=\sum_{y\in\Omega_{t}\cap\mathbb{Z}^{k}\setminus\\{0\\}}f(x-\mathcal{P}(y))K(y),\quad
x\in\mathbb{Z}^{d},$ (1.13) $\displaystyle\mathcal{H}_{t}^{\mathcal{P}}f(x)$
$\displaystyle:={\rm p.v.}\int_{\Omega_{t}}f(x-\mathcal{P}(y))K(y){\rm
d}y,\quad x\in\mathbb{R}^{d}.$

The above results may be used to easily deduce the following.

######  Theorem 1.14.

Let $d,k\geq 1$ and let $\mathcal{P}$ be a polynomial mapping (1.11). For any
$p\in(1,\infty)$ there is a constant $C_{p,d,k,{\rm deg}\mathcal{P}}>0$ such
that

(1.15)
$\displaystyle\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{R}_{+})}\big{\lVert}O_{I,N}^{2}(H_{t}^{\mathcal{P}}f:t\in\mathbb{R}_{+})\big{\rVert}_{\ell^{p}(\mathbb{Z}^{d})}$
$\displaystyle\leq C_{p,d,k,{\rm deg}\mathcal{P}}\lVert
f\rVert_{\ell^{p}(\mathbb{Z}^{d})},\quad f\in\ell^{p}(\mathbb{Z}^{d}),$ (1.16)
$\displaystyle\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{R}_{+})}\big{\lVert}O_{I,N}^{2}(\mathcal{H}_{t}^{\mathcal{P}}f:t\in\mathbb{R}_{+})\big{\rVert}_{L^{p}(\mathbb{R}^{d})}$
$\displaystyle\leq C_{p,d,k,{\rm deg}\mathcal{P}}\lVert
f\rVert_{L^{p}(\mathbb{R}^{d})},\quad f\in L^{p}(\mathbb{R}^{d}).$

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

By the properties of the oscillation seminorm inequalities (1.15) and (1.16)
implies the maximal estimates

$\displaystyle\big{\lVert}\sup_{t>0}|H_{t}^{\mathcal{P}}f|\big{\rVert}_{\ell^{p}(\mathbb{Z}^{d})}\lesssim\|f\|_{\ell^{p}(\mathbb{Z}^{d})}\quad\text{and}\quad\big{\lVert}\sup_{t>0}|\mathcal{H}_{t}^{\mathcal{P}}f|\big{\rVert}_{L^{p}(\mathbb{R}^{d})}\lesssim\|f\|_{L^{p}(\mathbb{R}^{d})}.$

Furthermore, from the oscillation inequality one may deduce the pointwise
convergence. Namely, the inequality (1.16) implies that

$\lim_{t\to\infty}\mathcal{H}_{t}^{\mathcal{P}}f(x)\qquad\text{and}\qquad\lim_{t\to
0}\mathcal{H}_{t}^{\mathcal{P}}f(x)$

exists for almost all $x\in\mathbb{R}^{d}$, every $f\in L^{p}(\mathbb{R}^{d})$
and every $p\in(1,\infty)$.
It turns out that the inequality (1.15) has an ergodic theoretical
interpretation. Let $(X,\mathcal{B},\mu)$ be a $\sigma$-finite measure space
endowed with a family of invertible commuting and measure preserving
transformations $S_{1},\ldots,S_{d}$. Let us define the following singular
operator

(1.17)
$\mathcal{T}_{t}^{\mathcal{P}}f(x)=\sum_{y\in\Omega_{t}\cap\mathbb{Z}^{k}\setminus\\{0\\}}f(S_{1}^{\mathcal{P}_{1}(y)}\circ
S_{2}^{\mathcal{P}_{2}(y)}\circ\cdots\circ S_{d}^{\mathcal{P}_{d}(y)}x)K(y).$

Then by appealing to the Calderón transference principle [4] we see that by
using (1.15) one can deduce the following result.

######  Theorem 1.18.

Let $d,k\geq 1$ and let $\mathcal{T}_{t}^{\mathcal{P}}$ be defined as in
(1.17). Then for any $p\in(1,\infty)$ there is a constant $C_{p,d,k,{\rm
deg}\mathcal{P}}>0$ such that

(1.19)
$\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{R}_{+})}\big{\lVert}O_{I,N}^{2}(\mathcal{T}_{t}^{\mathcal{P}}f:t\in\mathbb{R}_{+})\big{\rVert}_{L^{p}(X)}\leq
C_{p,d,k,{\rm deg}\mathcal{P}}\lVert f\rVert_{L^{p}(X)},\quad f\in
L^{p}(X,\mu).$

The implied constant is independent of the coefficients of the polynomial
mapping $\mathcal{P}$.

Again, the inequality (1.19) implies the maximal estimate

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

and pointwise convergence, that is

$\lim_{t\to\infty}\mathcal{T}_{t}^{\mathcal{P}}(x)$

exists $\mu$-almost everywhere on $X$ for every $f\in L^{p}(X,\mu)$ and
$p\in(1,\infty)$.

###  1.1. Historical background

The systematic study of singular integrals began with the Hilbert transform,
which is given by

$\mathcal{H}f(x):=\frac{1}{\pi}{\rm p.v.}\int_{\mathbb{R}}\frac{f(x-y)}{y}{\rm
d}y,\quad x\in\mathbb{R}.$

The first questions concerning the operator $\mathcal{H}$ were about it
boundedness on $L^{p}(\mathbb{R})$ with $p\in(1,\infty)$. First answer to that
question was given by M. Riesz in mid 1920’s. His approach relied heavily on
some properties of analytic functions. In early 1950’s Calderón and Zygmund in
their groundbreaking paper [5] developed a real-variable method which allowed
to study singular integrals in higher dimensions and resulted in introducing
Calderón–Zygmund operators of the form

$Tf(x):={\rm p.v.}\int_{\mathbb{R}^{d}}f(x-y)K(y){\rm d}y,\quad
x\in\mathbb{R}^{d},$

where $K\colon\mathbb{R}^{d}\setminus\\{0\\}\to\mathbb{R}$ is called a
Calderón–Zygmund kernel.

An important tool in studying the singular integrals is their truncated form
and its convergence. Namely, for $t>0$ we define the truncated Hilbert
transform

(1.20) $\mathcal{H}_{t}f(x):={\rm
p.v.}\frac{1}{\pi}\int_{|y|<t}\frac{f(x-y)}{y}{\rm d}y,\quad x\in\mathbb{R}$

as well as the truncated Calderón–Zygmund operator

$T_{t}^{\rm ball}f(x):={\rm p.v.}\int_{|y|<t}f(x-y)K(y){\rm d}y,\quad
x\in\mathbb{R}^{d}.$

As is well known, one of the most important steps in proving convergence of a
family of operators is the boundedness on $L^{p}$ of maximal functions
associated with it. In our case, this means showing the following inequalities

$\big{\lVert}\sup_{t>0}|\mathcal{H}_{t}f|\big{\rVert}_{L^{p}(\mathbb{R})}\lesssim_{p}\|f\|_{L^{p}(\mathbb{R})}\quad\text{and}\quad\big{\lVert}\sup_{t>0}|T_{t}^{\rm
ball}f|\big{\rVert}_{L^{p}(\mathbb{R}^{d})}\lesssim_{p,d}\|f\|_{L^{p}(\mathbb{R}^{d})}$

hold for every $f\in L^{p}$. It is known that the above inequalities hold for
$p\in(1,\infty)$.

It turns out that the question about convergence and the boundedness of the
maximal function can be reduced to study only the oscillation inequality. The
oscillation seminorm (1.6) arises in the work of Bourgain [1, 2, 3] on the
convergence almost everywhere of the ergodic averages along polynomials. The
connection established by Bourgain between the convergence and the oscillation
seminorm resulted in investigation of the oscillation inequalities (and other
seminorm inequalities) for various operators, see [13, 14, 15, 16, 20] and the
references given there. In particular, motivated by Bourgain’s work, Campbell,
Jones, Reinhold and Wierdl [6, Theorem 1.1, p. 59] investigated oscillation
inequalities for the truncated Hilbert transform. They proved that for any
$p\in(1,\infty)$ there is a constant $C_{p}>0$ such that

(1.21)
$\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{R}_{+})}\big{\lVert}O_{I,N}^{2}(\mathcal{H}_{t}f:t\in\mathbb{R}_{+})\big{\rVert}_{L^{p}(\mathbb{R})}\leq
C_{p}\lVert f\rVert_{L^{p}(\mathbb{R})},\quad f\in L^{p}(\mathbb{R}).$

In [7, Theorem A, p. 2116] the authors managed to extend the above result to
the case of multidimensional singular integrals of the Calderón–Zygmund type.
Namely, for every $p\in(1,\infty)$, there is a constant $C_{p,d}>0$ such that

(1.22)
$\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{R}_{+})}\big{\lVert}O_{I,N}^{2}(T_{t}^{\rm
ball}f:t\in\mathbb{R}_{+})\big{\rVert}_{L^{p}(\mathbb{R}^{d})}\leq
C_{p,d}\lVert f\rVert_{L^{p}(\mathbb{R}^{d})},\quad f\in
L^{p}(\mathbb{R}^{d}).$

In the proof of the inequality (1.22) the authors used the symmetry of
standard Euclidean ball and the fact that the operator $T_{t}^{\rm ball}$ is
related to the group of dilations given by

$\delta_{t}(x):=(tx_{1},tx_{2},\ldots,tx_{d}),\quad x\in\mathbb{R}^{d}$

to reduce matters to the inequality (1.21). In order to prove (1.21) they
approximated $\mathcal{H}_{t}$ by the standard dyadic martingale and used the
fact that the oscillation inequality for martingales is known.

The operator $\mathcal{H}_{t}$ defined in (1.5) is a truncated form of the
full continuous singular Radon transform given by

$\mathcal{H}f(x)={\rm p.v.}\int_{\mathbb{R}^{k}}f(x-(y)^{\Gamma})K(y){\rm
d}y.$

It is well known that the operator $\mathcal{H}$ is bounded on
$L^{p}(\mathbb{R}^{d})$ with $p\in(1,\infty)$. The singular Radon transforms
can be seen as a natural extension of the Calderón–Zygmund theory of singular
integrals. It originates in some problems related to curvatures and parabolic
differential equations, see [9, 11, 26, 27]. Therefore, the inequality (1.16)
can be seen as a generalisation of the result of Campbell, Jones, Reinhold and
Wierdl to the context of singular integrals of the Calderón–Zygmund type
associated with the polynomial mappings and convex sets.

The operator $\mathcal{H}$, as well as it truncated form (1.5), is related to
the following group of dilations

(1.23)
$\delta_{t}(x):=\big{(}t^{|\gamma|}x_{\gamma}:\gamma\in\Gamma\big{)},\quad
x\in\mathbb{R}^{\Gamma}.$

As one can see those dilations acts different on the each coordinate. This
fact prevents from reducing matters to one dimensional case. Fortunately,
Jones, Seeger and Wright [17] in their study of jumps inequalities and
$r$-variations managed to overcome this obstacle. Instead of reducing to the
one-dimensional case and then approximating by martingales they went straight
to approximation by martingales, but this time associated to Christ’s dyadic
cubes (which are related to the group of dilations (1.23)). We follow that
argument in our proof of Theorem 1.9.

The operator $H_{t}^{\mathcal{P}}$ is a truncated form of the full discrete
singular Radon transform given by

(1.24)
$H^{\mathcal{P}}f(x):=\sum_{y\in\mathbb{Z}^{k}\setminus\\{0\\}}f(x-\mathcal{P}(y))K(y).$

The operator $H^{\mathcal{P}}$ is a discrete counterpart of the continuous
singular Radon transform $\mathcal{H}^{\mathcal{P}}$. It was a challenging
problem to establish boundedness of $H^{\mathcal{P}}$ on
$\ell^{p}(\mathbb{Z}^{d})$ with $p\in(1,\infty)$. The first partial answer was
given by Stein and Wainger in [27] where they managed to prove that
$H^{\mathcal{P}}$ is bounded on $\ell^{p}(\mathbb{Z}^{d})$ for $p$ in a
certain neighbourhood of $2$. The full range of $p\in(1,\infty)$ were obtained
by Ionescu and Wainger [12], see also [19] for a different approach.

One needs to be careful while dealing with discrete analogues of the
continuous operators. Some results can be easily transfered form the
continuous to the discrete setting. For example, the $\ell^{p}$-bound for the
discrete version of the usual Calderón–Zygmund operator can be easily deduced
from the bound of its continuous counterpart on $L^{p}$. Unfortunately, this
kind of approach is impossible in the case of Radon operators, since (roughly)
the polynomial $(x)^{\Gamma}$ may have an unbounded gaps. Also the behaviour
of the discrete objects may be totally different then their continuous
cousins. In order to illustrate these differences let us focus our attention
on the one dimensional case and let consider $\Omega=(-1,1)$,
$(x)^{\Gamma}=x^{3}$ and $K(y)=y^{-1}$. Then the Fourier multiplier associated
with the operator $H_{t}$ is of the form

$m_{t}(\xi)=\sum_{y\in(-t,t)\cap\mathbb{Z}\setminus\\{0\\}}e(\xi
y^{3})y^{-1},$

while in the case of $\mathcal{H}_{t}$ the multiplier is given by

$\psi_{t}(\xi)={\rm p.v.}\int_{-t}^{t}e(\xi y^{3})y^{-1}{\rm d}y.$

It can be easily shown that

$\psi_{t}(\xi)=i\frac{2}{3}{\rm sgn}(\xi){\rm Si}(2\pi\xi t^{3}),$

where ${\rm Si}(z)$ is the Sine Integral. Meanwhile in the discrete case we
have $m_{t}(\xi)=0$ when $\xi\in\mathbb{Z}$. As we can notice the sum and the
integral demonstrate completely different behaviours and consequently we need
to use different approach to study discrete analogues of Radon operators.

The oscillation inequality (1.15) for the discrete Radon transform seems to be
new, even in the case $k=d=1$. In the proof of Theorem 1.7 we exploit the
technique based on subexponential sequences and the Rademacher–Menshov
inequality which was introduced in [24], see also [22]. Moreover, we make use
of the vector-valued Ionescu–Wainger multiplier theorem and the square
function estimates from [23].

At the end, it is worth to mention that similar seminorm inequalities, like
variational estimates or the jump inequality, for both $H_{t}^{\mathcal{P}}$
and $\mathcal{H}^{\mathcal{P}}_{t}$, are already known, see [17, 22, 23, 24].
However, those known estimates do not imply our results.

##  2\. Preliminaries

###  2.1. Basic notation

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

$\mathbb{N}_{N}:=\\{1,\ldots,N\\}\text{ and }\mathbb{N}_{\infty}:=\mathbb{N}$

For $\tau\in(0,1)$ and $u\in\mathbb{N}$ we define sets

$\displaystyle\mathbb{D}_{\tau}:=\\{n^{\tau}:n\in\mathbb{N}\\}\quad\text{ and
}\quad 2^{u\mathbb{N}}:=\\{2^{un}\colon n\in\mathbb{N}\\}.$

We write $A\lesssim B$ to indicate that $A\leq CB$ with a constant $C>0$. The
constant $C$ may vary from line to line. We write $\lesssim_{\delta}$ if the
implicit constant depends on $\delta$.

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 Fourier transform

The standard inner product and the corresponding Euclidean norm on
$\mathbb{R}^{d}$ are denoted by $x\cdot\xi$ and $\lvert x\rvert:=\lvert
x\rvert_{2}$, respectively. Furthermore, we need the maximum norm on
$\mathbb{R}^{d}$ defined as

$|x|_{\infty}:=\max_{1\leq k\leq d}|x_{k}|.$

For any $\gamma=(\gamma_{1},\dots,\gamma_{k})\in\mathbb{N}^{k}$ we abuse the
notation by writing $|\gamma|:=\gamma_{1}+\cdots+\gamma_{k}$. However, this
should never cause any confusions since the multi-indices will be always
denoted by Greek letter $\gamma$.

Let $\mathbb{G}=\mathbb{R}^{d}$ or $\mathbb{G}=\mathbb{Z}^{d}$. By the symbol
$\mathbb{G}^{*}$ we denote the dual group of $\mathbb{G}$. The Fourier
transform $\mathcal{F}_{\mathbb{G}}$ of any $f\in L^{1}(\mathbb{G})$ is
defined by

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

where $\mu$ is the Haar measure on $\mathbb{G}$ and
$\bm{e}(\xi):=e^{2\pi{\bm{i}}\xi}$ with ${\bm{i}}^{2}=-1$. The Fourier
multiplier operator associated with the bounded function
$\mathfrak{m}\colon\mathbb{G}^{*}\to\mathbb{C}$ is given by

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

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.

###  2.3. Ionescu–Wainger multiplier theorem

In order to handle the discrete singular integral we make use of the
Ionescu–Wainger multiplier theorem.

######  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 $P_{\leq N}$.

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

The reader can found the proof of Theorem 2.2 in [24, Section 2]. We also
refer to the paper of Tao [29], where he was able to remove the factor $\log
N$ from (2.6).

##  3\. Proof of Theorem 1.7 for the canonical mapping $(x)^{\Gamma}$

Assume that $p\in(1,\infty)$ and let $f\in\ell^{p}(\mathbb{Z}^{\Gamma})$ be a
function with a compact support. By using the monotone convergence theorem and
standard density arguments to prove (1.8) it is enough to establish

(3.1) $\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{I})}\lVert
O_{I,N}^{2}(H_{t}f:t\in\mathbb{I})\rVert_{\ell^{p}(\mathbb{Z}^{\Gamma})}\leq
C_{p,k,|\Gamma|}\lVert f\rVert_{\ell^{p}(\mathbb{Z}^{\Gamma})}$

for every finite subset $\mathbb{I}\subset\mathbb{R}_{+}$ with a constant
$C_{p,k,|\Gamma|}>0$ that is independent of the set $\mathbb{I}$. Let us
choose $p_{0}\in(1,2)$, close to $1$ such that $p\in(p_{0},p_{0}^{\prime})$.
Then we take $\tau\in(0,1)$ such that

(3.2) $\tau<\frac{1}{2}\min\\{p_{0}-1,1\\}.$

We split (3.1) (cf. [17, Lemma 1.3]) into long oscillations and short
variations

(3.3)
$\displaystyle\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{I})}\lVert
O_{I,N}^{2}(H_{t}f:t\in\mathbb{I})\rVert_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim$
$\displaystyle\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{D}_{\tau})}\big{\lVert}O_{I,N}^{2}(H_{2^{n^{\tau}}}f:n\in\mathbb{N}_{0})\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}$
$\displaystyle+\Big{\lVert}\Big{(}\sum_{n=0}^{\infty}V^{2}\big{(}H_{t}f:t\in[2^{n^{\tau}},2^{(n+1)^{\tau}})\cap\mathbb{I}\big{)}^{2}\Big{)}^{1/2}\Big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

The short variations were estimated in [24, Section 3.1]. Hence we are reduced
to prove only the estimate for long oscillations.

###  3.1. Estimates for long oscillations

Our aim is to establish the following inequality

(3.4)
$\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{D}_{\tau})}\big{\lVert}O_{I,N}^{2}(H_{2^{n^{\tau}}}f:n\in\mathbb{N}_{0})\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

Let us observe that for any $x\in\mathbb{Z}^{\Gamma}$ one has
$H_{2^{n^{\tau}}}f(x)=T_{\mathbb{Z}^{\Gamma}}[m_{{n^{\tau}}}]f(x),$ where

$m_{n^{\tau}}(\xi)=\sum_{y\in\Omega_{2^{n^{\tau}}}\cap\mathbb{Z}^{k}\setminus\\{0\\}}e(\xi\cdot(y)^{\Gamma})K(y),\quad\xi\in\mathbb{T}^{\Gamma}.$

Now, let $\chi\in(0,1/10)$. The proof of (3.4) will require several
appropriately chosen parameters. Choose $\alpha>0$ such that

$\alpha>10\left(\frac{1}{p_{0}}-\frac{1}{2}\right)\left(\frac{1}{p_{0}}-\frac{1}{\min\\{p,p^{\prime}\\}}\right)^{-1}.$

Let $u\in\mathbb{N}$ be a large natural number which will be specified later.
We set

(3.5) $\varrho:=\min\left\\{\frac{1}{10u},\frac{\delta}{8\alpha}\right\\},$

where $\delta>0$ is from the estimate for the Gauss sum (3.14). Now, let us
consider $S_{0}=\max\\{t>0:t\in 2^{u\mathbb{N}}\cap[1,n^{\tau u}]\\}$. We
recall the family of rational fractions $\Sigma_{\leq S_{0}}$ related to
parameter $\varrho$, defined in Section 2.3. For simplicity we will write

$\Sigma_{\leq n^{\tau u}}:=\Sigma_{\leq S_{0}}.$

Next, for dyadic integers $S\in 2^{u\mathbb{N}}$ we define

$\Sigma_{S}=\begin{cases}\Sigma_{\leq S},&{\rm if}\quad S=2^{u},\\\
\Sigma_{\leq S}\setminus\Sigma_{\leq S/2^{u}},&{\rm if}\quad
S>2^{u}.\end{cases}$

It is easy to see that

(3.6) $\Sigma_{\leq n^{\tau u}}=\bigcup_{\begin{subarray}{c}S\leq n^{\tau
u},\\\ S\in 2^{u\mathbb{N}}\end{subarray}}\Sigma_{S}.$

Now, we define the Ionescu–Wainger projection multipliers. For this purpose,
we introduce a diagonal matrix $A$ of size $|\Gamma|\times|\Gamma|$ given by
the condition $(Av)_{\gamma}:=\lvert\gamma\rvert v_{\gamma}$ for any
$\gamma\in\Gamma$ and $v\in\mathbb{R}^{\Gamma}$. Moreover, 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

$\eta(x)=\begin{cases}1,&|x|\leq 1/(32|\Gamma|),\\\ 0,&|x|\geq
1/(16|\Gamma|).\end{cases}$

For any $n\in\mathbb{N}$ we set

(3.7) $\Pi_{\leq n^{\tau}}(\xi):=\sum_{a/q\in\Sigma_{\leq n^{\tau
u}}}\eta\big{(}2^{n^{\tau}(A-\chi
I)}(\xi-a/q)\big{)},\quad\xi\in\mathbb{T}^{\Gamma}.$

It is easy to check that Theorem 2.2 holds for the multiplier $\Pi_{\leq
n^{\tau}}$ since one has $2^{-n^{\tau}(|\gamma|-\chi)}\leq
e^{-n^{\tau/10}}\leq e^{-S_{0}^{\varrho}}$ for sufficiently large
$n\in\mathbb{N}$.
Using the functions defined in (3.7) we can partition the multiplier
$m_{{n^{\tau}}}$ and estimate the left hand side of (3.4) by

(3.8)
$\displaystyle\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{D}_{\tau})}\Big{\lVert}O_{I,N}^{2}\Big{(}T_{\mathbb{Z}^{\Gamma}}\big{[}\sum_{j=1}^{n}(m_{{j^{\tau}}}-m_{(j-1)^{\tau}})\Pi_{\leq
j^{\tau}}\big{]}f:n\in\mathbb{N}_{0}\Big{)}\Big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}$
(3.9)
$\displaystyle+\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{D}_{\tau})}\Big{\lVert}O_{I,N}^{2}\Big{(}T_{\mathbb{Z}^{\Gamma}}\big{[}\sum_{j=1}^{n}(m_{{j^{\tau}}}-m_{(j-1)^{\tau}})(1-\Pi_{\leq
j^{\tau}})\big{]}f:n\in\mathbb{N}_{0}\Big{)}\Big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

Here we use convection that for $n=0$ the sum is equal to $0$. The expressions
in (3.8) and (3.9) correspond to major and minor arcs from the
Hardy–Littlewood circle method, respectively.

####  3.1.1. Minor arcs

First, we note that the oscillation seminorm (3.9) is controlled by
1-variation $V^{1}$ and by the following estimate

$\displaystyle\Big{\lVert}V^{1}\big{(}T_{\mathbb{Z}^{\Gamma}}\big{[}\sum_{j=1}^{n}(m_{{j^{\tau}}}-m_{(j-1)^{\tau}})$
$\displaystyle(1-\Pi_{\leq
j^{\tau}})\big{]}f:n\in\mathbb{N}_{0}\big{)}\Big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}$
$\displaystyle\lesssim\sum_{n=0}^{\infty}\lVert
T_{\mathbb{Z}^{\Gamma}}[(m_{{(n+1)^{\tau}}}-m_{n^{\tau}})(1-\Pi_{\leq(n+1)^{\tau}})]f\rVert_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

Consequently, it is enough to show

(3.10)
$\|T_{\mathbb{Z}^{\Gamma}}[(m_{{(n+1)^{\tau}}}-m_{n^{\tau}})(1-\Pi_{\leq(n+1)^{\tau}})]f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim(n+1)^{-2}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

To prove the above estimate we use Weyl’s inequality [24, Theorem A.1, p. 49]
and proceed as in [24, Lemma 3.29, p. 34] (provided that $u\in\mathbb{N}$ is
large enough).

####  3.1.2. Major arcs and multiplier approximation

Now we need to prove that the estimate

(3.11)
$\displaystyle\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{D}_{\tau})}\Big{\lVert}O_{I,N}^{2}\Big{(}T_{\mathbb{Z}^{\Gamma}}\big{[}\sum_{j=1}^{n}(m_{{j^{\tau}}}-m_{(j-1)^{\tau}})\Pi_{\leq
j^{\tau}}\big{]}f:n\in\mathbb{N}_{0}\Big{)}\Big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim\lVert
f\rVert_{\ell^{p}(\mathbb{Z}^{\Gamma})}$

holds. At first we show that the multiplier

$\sum_{a/q\in\Sigma_{\leq j^{\tau
u}}}\big{(}m_{j^{\tau}}-m_{(j-1)^{\tau}}\big{)}(\xi)\eta(2^{j^{\tau}(A-\chi
I)}(\xi-a/q)).$

is, up to an acceptable error term, equal to

(3.12) ${\bm{m}}_{j}(\xi)=\sum_{a/q\in\Sigma_{\leq j^{\tau
u}}}G(a/q)\big{(}\Psi_{2^{j^{\tau}}}-\Psi_{2^{(j-1)^{\tau}}}\big{)}(\xi-a/q)\eta(2^{j^{\tau}(A-\chi
I)}(\xi-a/q))$

where $\Psi_{t}$ is a continuous version of the multiplier $m_{t}$ given by

$\displaystyle\Psi_{t}(\xi)$ $\displaystyle:={\rm
p.v.}\int_{\Omega_{t}}{e(\xi\cdot(y)^{\Gamma})K(y){\rm d}y}$

and $G(a/q)$ is the Gauss sum defined by

$G(a/q):=\frac{1}{q^{k}}\sum_{r\in\mathbb{N}_{q}^{k}}{e((a/q)\cdot(r)^{\Gamma})}.$

Let us state some properties of the function $\Psi_{t}$ and the Gauss sum
$G(a/q)$ which will be used later on. For a fixed $c\in(0,1)$ and any real
number $t>0$ we have the following estimates for the function $\Psi_{t}$:

(3.13)
$|\Psi_{t}(\xi)-\Psi_{ct}(\xi)|\lesssim|t^{A}\xi|_{\infty}\quad\text{and}\quad|\Psi_{t}(\xi)-\Psi_{ct}(\xi)|\lesssim|t^{A}\xi|_{\infty}^{-\sigma/|\Gamma|},$

where the first estimate follows from the cancellation condition (1.2) and the
second one is a consequence of van der Corput’s lemma (see [23, p. 21] for
details). It is well known (see [24, Lemma 4.14, p. 44]) that for some
$\delta>0$ one has

(3.14) $|G(a/q)|\lesssim_{k}q^{-\delta}.$

Now, we claim that one has

(3.15)
$\displaystyle\Big{\lVert}T_{\mathbb{Z}^{\Gamma}}\big{[}\sum_{j=1}^{n}(m_{j^{\tau}}-m_{(j-1)^{\tau}})\Pi_{\leq
j^{\tau}}-{\bm{m}}_{j}\big{]}f\Big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim
2^{-j^{\tau}\sigma/4}\lVert f\rVert_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

To see that (3.15) holds we follow the approach taken in [24, Lemma 3.38, p.
36] by appealing to conditions (1.1) and (1.3) and Theorem 2.2. As a result,
to prove (3.11) it suffices to establish that

(3.16)
$\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{D}_{\tau})}\big{\lVert}O_{I,N}^{2}\big{(}T_{\mathbb{Z}^{\Gamma}}\big{[}\sum_{j=1}^{n}{\bm{m}}_{j}\big{]}f:n\in\mathbb{N}_{0}\big{)}\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim\lVert
f\rVert_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

In the next step, we use (3.6) to write that

(3.17) ${\bm{m}}_{j}(\xi)=\sum_{\begin{subarray}{c}S\leq j^{\tau u},\\\ S\in
2^{u\mathbb{N}}\end{subarray}}{\bm{m}}_{S}^{j}(\xi),$

where ${\bm{m}}_{S}^{j}$ is defined as

(3.18)
${\bm{m}}_{S}^{j}(\xi):=\sum_{a/q\in\Sigma_{S}}G(a/q)\big{(}\Psi_{2^{j^{\tau}}}-\Psi_{2^{(j-1)^{\tau}}}\big{)}(\xi-a/q)\eta\big{(}2^{j^{\tau}(A-\chi
I)}(\xi-a/q)\big{)}.$

Consequently, we see that to prove (3.16) it is enough to show that

(3.19)
$\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{D}_{\tau}^{S})}\big{\lVert}O_{I,N}^{2}(T_{\mathbb{Z}^{\Gamma}}\big{[}\sum_{\begin{subarray}{c}1\leq
j\leq n\\\ S^{1/u}\leq
j^{\tau}\end{subarray}}{\bm{m}}_{S}^{j}\big{]}f:n^{\tau}\geq
S^{1/u})\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim
S^{-3\varrho}\lVert f\rVert_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

where $\mathbb{D}_{\tau}^{S}=\\{n\in\mathbb{N}:n^{\tau}\geq S^{1/u}\\}$ since
$S^{-3\varrho}$ is summable in $S\in 2^{u\mathbb{N}}$.

####  3.1.3. Gaussian multiplier and scale distinction

In order to apply Theorem 2.2 we need to get rid of the Gaussian part $G(a/q)$
in the multiplier (3.18). For this purpose let us define
$\tilde{\eta}(x):=\eta(x/2)$ and two new multipliers

$\displaystyle v_{S}^{j}(\xi)$
$\displaystyle:=\sum_{a/q\in\Sigma_{S}}(\Psi_{2^{j^{\tau}}}-\Psi_{2^{(j-1)^{\tau}}})(\xi-a/q)\eta\big{(}2^{j^{\tau}(A-\chi
I)}(\xi-a/q)\big{)},$ $\displaystyle\mu_{S}(\xi)$
$\displaystyle:=\sum_{a/q\in\Sigma_{S}}G(a/q)\tilde{\eta}\big{(}2^{S^{1/u}(A-\chi
I)}(\xi-a/q)\big{)}.$

We note that one has ${\bm{m}}_{S}^{j}=v_{S}^{j}\mu_{S}$ and consequently to
prove (3.19) it suffices to show that for every $p\in(1,\infty)$ one has

(3.20)
$\displaystyle\big{\lVert}T_{\mathbb{Z}^{\Gamma}}[\mu_{S}]f\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}$
$\displaystyle\lesssim S^{-6\varrho}\lVert
f\rVert_{\ell^{p}(\mathbb{Z}^{\Gamma})},$ (3.21)
$\displaystyle\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{D}_{\tau}^{S})}\big{\lVert}O_{I,N}^{2}\big{(}T_{\mathbb{Z}^{\Gamma}}\big{[}\sum_{\begin{subarray}{c}1\leq
j\leq n\\\ S^{1/u}\leq j^{\tau}\end{subarray}}v_{S}^{j}\big{]}f:n^{\tau}\geq
S^{1/u}\big{)}\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}$
$\displaystyle\lesssim S^{3\varrho}\lVert
f\rVert_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

The Gauss sum estimates (3.20) were established in [24, Lemma 3.47 and
estimate (3.49) pp. 38–39]. Hence we may focus on (3.21). Let
$\kappa_{S}=\lceil S^{2\varrho}\rceil$. We split the left hand side of (3.21)
at point $2^{\kappa_{S}}$ and write

$\displaystyle{\rm LHS}(\ref{eq:23})\lesssim$
$\displaystyle\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{D}^{\tau}_{\leq
S})}\big{\lVert}O_{I,N}^{2}\big{(}T_{\mathbb{Z}^{\Gamma}}\big{[}\sum_{\begin{subarray}{c}1\leq
j\leq n\\\ S^{1/(\tau u)}\leq
j\end{subarray}}v_{S}^{j}\big{]}f:n\in[S^{1/(\tau
u)},2^{\kappa_{S}+1}]\big{)}\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}$
$\displaystyle+\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{D}_{\geq
S})}\big{\lVert}O_{I,N}^{2}\big{(}T_{\mathbb{Z}^{\Gamma}}\big{[}\sum_{\begin{subarray}{c}1\leq
j\leq n\\\ 2^{\kappa_{S}}\leq j\end{subarray}}v_{S}^{j}\big{]}f:n\geq
2^{\kappa_{S}})\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

where $\mathbb{D}^{\tau}_{\leq S}:=\\{n\in\mathbb{N}:n\in[S^{1/(\tau
u)},2^{\kappa_{S}+1}]\\}$ and $\mathbb{D}_{\geq S}:=\\{n\in\mathbb{N}:n\geq
2^{\kappa_{S}}\\}$. We handle with small scales by using the
Rademacher–Menschov inequality and Theorem 2.2. In the case of large scales we
make use of the transference principle due to Magyar–Stein–Wainger from [18]
to obtain estimates from the continuous case.

####  3.1.4. Estimates for small scales

We closely follow approach taken in [24] to prove the following estimate

(3.22)
$\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{D}^{\tau}_{\leq
S})}\big{\lVert}O_{I,N}^{2}\big{(}T_{\mathbb{Z}^{\Gamma}}\big{[}\sum_{\begin{subarray}{c}1\leq
j\leq n\\\ S^{1/(\tau u)}\leq
j\end{subarray}}v_{S}^{j}\big{]}f:n\in[S^{1/(\tau
u)},2^{\kappa_{S}+1}]\big{)}\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim\kappa_{S}\log(S)\lVert
f\rVert_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

By applying the Rademacher–Menshov inequality [23, Lemma 2.5, p. 534] to the
left hand side of (3.22) we see that

${\rm
LHS}(\ref{eq:14})\lesssim\sum_{i=0}^{\kappa_{S}+1}\bigg{\lVert}\Big{(}\sum_{j}\big{|}\sum_{k\in
I_{j}^{i}}T_{\mathbb{Z}^{\Gamma}}[v_{S}^{k}]f\big{|}^{2}\Big{)}^{1/2}\bigg{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

where $I_{j}^{i}:=[j2^{i},(j+1)2^{i})\cap[S^{1/(\tau u)},2^{\kappa_{S}+1}]$.
Hence it suffices to prove that for every $i\leq\kappa_{S}+1$ we have

(3.23) $\displaystyle\bigg{\lVert}\Big{(}\sum_{j}\big{|}\sum_{k\in
I_{j}^{i}}T_{\mathbb{Z}^{\Gamma}}[v_{S}^{k}]f\big{|}^{2}\Big{)}^{1/2}\bigg{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim\log(S)\lVert
f\rVert_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

By Theorem 2.2, the estimate (3.23) is a consequence of its continuous
counterpart

$\displaystyle\bigg{\lVert}\Big{(}\sum_{j}\big{|}\sum_{k\in
I_{j}^{i}}T_{\mathbb{R}^{\Gamma}}\big{[}(\Psi_{2^{k^{\tau}}}-\Psi_{2^{(k-1)^{\tau}}})\eta\big{(}2^{k^{\tau}(A-\chi
I)}\cdot\big{)}\big{]}f\big{|}^{2}\Big{)}^{1/2}\bigg{\rVert}_{L^{p}(\mathbb{R}^{\Gamma})}\lesssim\lVert
f\rVert_{L^{p}(\mathbb{R}^{\Gamma})}.$

The above square function estimate follows by appealing to (3.13) and standard
arguments from the Littlewood–Paley theory. We refer to [23] for more details,
see also discussion below [24, Theorem 4.3, p. 42].

####  3.1.5. Estimates for large scales

The last thing to show is the estimate for the large scales,

(3.24) $\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{D}_{\geq
S})}\big{\lVert}O_{I,N}^{2}\big{(}T_{\mathbb{Z}^{\Gamma}}\big{[}\sum_{\begin{subarray}{c}1\leq
j\leq n\\\ 2^{\kappa_{S}}\leq j\end{subarray}}v_{S}^{j}\big{]}f:n\geq
2^{\kappa_{S}})\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim\log(S)\lVert
f\rVert_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

We would like to exploit the almost telescoping nature of the multipliers
appearing in (3.24). We do this by introducing new approximating multipliers.
Let

$\tilde{v}_{S}^{j}(\xi):=\sum_{a/q\in\Sigma_{S}}(\Psi_{2^{j^{\tau}}}-\Psi_{2^{(j-1)^{\tau}}})(\xi-a/q)\eta(2^{2^{\tau\kappa_{S}}(A-\chi)}(\xi-a/q)).$

Since $j\geq 2^{\kappa_{S}}$, the expression

$\eta(2^{j^{\tau}(A-\chi I)}(\xi-a/q))-\eta(2^{2^{\tau\kappa_{S}}(A-\chi
I)}(\xi-a/q))$

is nonzero only when $|\xi_{\gamma}-a_{\gamma}/q|\gtrsim
2^{-j^{\tau}(|\gamma|-\chi)}$ for some $\gamma\in\Gamma$. Hence, by the van
der Corput estimate in (3.13) we get

$\big{\lVert}T_{\mathbb{Z}^{\Gamma}}[v_{S}^{j}-\tilde{v}_{S}^{j}]f\big{\rVert}_{\ell^{2}(\mathbb{Z}^{\Gamma})}\lesssim
2^{-j^{\tau}\chi\sigma/|\Gamma|}\|f\|_{\ell^{2}(\mathbb{Z}^{\Gamma})},$

whereas for any $p\neq 2$, by property (i), we have

$\big{\lVert}T_{\mathbb{Z}^{\Gamma}}[v_{S}^{j}-\tilde{v}_{S}^{j}]f\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim\big{|}\Sigma_{\leq
j^{\tau u}}\big{|}\lVert f\rVert_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim
e^{(|\Gamma|+1)j^{\tau/10}}\lVert f\rVert_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

By interpolating the above inequalities we get

(3.25)
$\big{\lVert}T_{\mathbb{Z}^{\Gamma}}[v_{S}^{j}-\tilde{v}_{S}^{j}]f)\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim
2^{-j^{\tau}\varepsilon}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}$

with some $\varepsilon>0$. Consequently, the inequality (3.24) will follow if
we show

(3.26)
$\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{D}^{\tau}_{\geq
S})}\big{\lVert}O_{I,N}^{2}\big{(}T_{\mathbb{Z}^{\Gamma}}[\Delta_{S}^{n}]f):n\geq
2^{\kappa_{S}})\big{\rVert}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim\log(S)\lVert
f\rVert_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

where

$\Delta_{S}^{n}(\xi)=\sum_{a/q\in\Sigma_{S}}\Psi_{2^{n^{\tau}}}(\xi-a/q)\eta(2^{2^{\tau\kappa_{S}}(A-\chi
I)}(\xi-a/q)).$

By using the standard transference argument (see [24, Section 3.6, pp. 40–41])
the estimate (3.26) is reduced to showing for every $p\in(1,\infty)$ and $f\in
L^{p}(\mathbb{R}^{\Gamma})$ the following uniform oscillation inequality

(3.27)
$\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{R}_{+})}\big{\lVert}O_{I,N}^{2}(T_{\mathbb{R}^{\Gamma}}[\Psi_{t}]f:t\in\mathbb{R}_{+})\big{\rVert}_{L^{p}(\mathbb{R}^{\Gamma})}\lesssim\lVert
f\rVert_{L^{p}(\mathbb{R}^{\Gamma})}$

which follows by Theorem 1.9. This completes the proof of Theorem 1.7.

##  4\. Proof of theorem 1.9 for the canonical mapping $(x)^{\Gamma}$

The proof of the inequality

(4.1)
$\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{R}_{+})}\big{\lVert}O_{I,N}^{2}(\mathcal{H}_{t}f:t\in\mathbb{R}_{+})\big{\rVert}_{L^{p}(\mathbb{R}^{\Gamma})}\lesssim\|f\|_{L^{p}(\mathbb{R}^{\Gamma})}$

follows the same lines as the proofs of [17, Theorem 1.1, Theorem 1.2] and
[22, Theorem A.2]. We will split (4.1) into long oscillations and short
variations. The estimate for long oscillations is obtained by approximation
with a suitable dyadic martingale, whereas short variations are estimated by
using the Littlewood–Paley theory. In the case of long oscillations the key
ingredient is the oscillation inequality for Christ’s dyadic martingales which
follows from the result of Jones, Kaufman, Rosenblatt and Wierdl [13, Theorem
6.4, p. 930] (see also [20, Proposition 2.8])

###  4.1. Dyadic martingales on the homogeneous spaces

In the context of homogeneous spaces and dyadic martingales we will follow the
notation introduced in [17]. Let $A$ be a $d\times d$ matrix whose eigenvalues
have positive real parts. For any $t>0$ we consider the dilation given by

(4.2) $t^{A}:=\exp(A\log t).$

We say that a quasi-norm $\rho\colon\mathbb{R}^{d}\rightarrow[0,\infty)$ is
homogeneous with respect to the group of dilations $(t^{A}:t>0)$ if
$\rho(t^{A}x)=t\rho(x)$ for any $x\in\mathbb{R}^{d}$ and $t>0$. Recall, that
for a given group of of dilations $(t^{A}:t>0)$ by [26, Proposition 1.7,
Definition 1.8] there exists a quasi-norm $\rho$ which is homogeneous with
respect to that group. We note that $\mathbb{R}^{d}$ equipped with a
homogeneous quasi-norm $\rho$ and Lebesgue measure is a space of homogeneous
type with the quasi-metric induced by $\rho$. As it was shown by Christ [8,
Theorem 11] for any given space of homogeneous type there exists a system of
$D$-dyadic cubes $\\{Q_{\alpha}^{k}:k\in\mathbb{Z},\alpha\in I_{k}\\}$, with
$D>1$. For a locally integrable function $f$ the martingale sequence
associated with the system of dyadic cubes $\\{Q_{\alpha}^{k}\\}$ is of the
form

(4.3)
$\mathbb{E}_{k}f(x):=\mathbb{E}[f|\mathcal{F}_{k}](x):=\frac{1}{|Q_{\alpha}^{k}|}\int_{Q_{\alpha}^{k}}f(y){\>\rm
d}y$

provided $Q_{\alpha}^{k}$ is the unique dyadic cube containing
$x\in\mathbb{R}^{d}$.

######  Theorem 4.4 ([13, Theorem 6.4, p. 930]).

For every $p\in(1,\infty)$ there exists a constant $C_{p}>0$ such that

$\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{Z})}\big{\lVert}O^{2}_{I,N}(\mathbb{E}_{n}f:n\in\mathbb{Z})\big{\rVert}_{L^{p}(\mathbb{R}^{d})}\leq
C_{p}\lVert f\rVert_{L^{p}(\mathbb{R}^{d})}.$

The next result which follows from [17] concern the approximation by
martingales associated with Christ’s dyadic cubes.

######  Lemma 4.5 (cf. [17, Theorem 1.1]).

Let $\phi$ be a Schwartz function such that $\int\phi=1$ and let
$\phi_{D^{k}}(x)=D^{-{\rm tr}(A)k}\phi(D^{-kA}x)$ where $D>1$ is associated
with the system of $D$-dyadic cubes $\\{Q_{\alpha}^{k}\\}$. Then the operator

$\mathcal{S}f(x)=\big{(}\sum_{k\in\mathbb{Z}}|\phi_{D^{k}}\ast
f(x)-\mathbb{E}_{k}f(x)|^{2}\big{)}^{1/2}$

is bounded on $L^{p}(\mathbb{R}^{d})$ for $p\in(1,\infty)$. Moreover, for
$p=1$ the operator $\mathcal{S}$ is of weak type (1,1).

###### Proof.

The proof is a repetition of arguments presented in proofs of [17, Lemma 3.2
and Theorem 1.1 pp. 19–20]. ∎

The next result is a counterpart of [17, Theorem 1.1] in the context of the
oscillation seminorm. Let $\sigma$ be a compactly supported finite Borel
measure on $\mathbb{R}^{d}$. Let us consider dilates of $\sigma$ defined by

(4.6) $\langle\sigma_{t},f\rangle=\int_{\mathbb{R}^{d}}f(t^{A}x){\rm
d}\sigma(x)$

where $t^{A}$ is as in (4.2). We additionally assume that the Fourier
transform satisfies the following size condition

(4.7) $|\hat{\sigma}(\xi)|\lesssim|\xi|^{-a}\qquad\text{for some}\qquad a>0.$

######  Theorem 4.8.

Assume that $\sigma$ is a compactly supported finite Borel measure on
$\mathbb{R}^{d}$ satisfying (4.7) for some $a>0$. Let $\sigma_{t}$ be as in
(4.6). Then for $p\in(1,\infty)$ one has

(4.9) $\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{Z})}\lVert
O_{I,N}^{2}(f\ast\sigma_{D^{k}}:k\in\mathbb{Z})\rVert_{L^{p}(\mathbb{R}^{d})}\lesssim_{p}\|f\|_{L^{p}(\mathbb{R}^{d})},\qquad
f\in L^{p}(\mathbb{R}^{d}),$

where $D>1$ is associated with the system of $D$-dyadic cubes
$\\{Q_{\alpha}^{k}\\}$.

###### Proof.

The proof is a simple repetition of the arguments given in the proof of [17,
Theorem 1.1] but we include it for the sake of completeness. We can assume
that $\int{\rm d}\sigma\neq 0$ since we see that the left hand side of (4.9)
is controlled by the square function
$\big{(}\sum_{k\in\mathbb{Z}}|f\ast\sigma_{D^{k}}(x)|^{2})^{1/2}$, and if
$\hat{\sigma}(0)=0$ then the known estimates from [10, Theorem A, Theorem B]
can be used to control it on $L^{p}(\mathbb{R}^{d})$. Without loss of
generality assume that $\int{\rm d}\sigma=1$. Let $\phi\in C_{\rm
c}^{\infty}(\mathbb{R}^{d})$ be such that $\int\phi=1$. Then one may write the
following decomposition $\sigma=\phi\ast\sigma+(\delta_{0}-\phi)\ast\sigma,$
where $\delta_{0}$ is the Dirac measure at 0. Therefore we may write
$f\ast\sigma_{D^{k}}(x)=\mathcal{L}_{k}f(x)+\mathcal{H}_{k}f(x),$ where

$\mathcal{L}_{k}f(x)=f\ast(\phi\ast\sigma)_{D^{k}}(x)\quad\text{and}\quad\mathcal{H}_{k}f(x)=f\ast\big{(}(\delta_{0}-\phi)\ast\sigma\big{)}_{D^{k}}(x).$

Hence, by the triangle inequality it is enough to prove

(4.10)
$\displaystyle\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{Z})}\lVert
O_{I,N}^{2}(\mathcal{L}_{k}f:k\in\mathbb{Z})\rVert_{L^{p}(\mathbb{R}^{d})}\lesssim_{p}\|f\|_{L^{p}(\mathbb{R}^{d})}$

and

(4.11) $\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{Z})}\lVert
O_{I,N}^{2}(\mathcal{H}_{k}f:k\in\mathbb{Z})\rVert_{L^{p}(\mathbb{R}^{d})}\lesssim_{p}\|f\|_{L^{p}(\mathbb{R}^{d})}.$

At first we handle the estimate (4.11). Since the oscillation seminorm is
controlled by the square function it is enough to prove

(4.12)
$\big{\lVert}\big{(}\sum_{k\in\mathbb{Z}}|\mathcal{H}_{k}f|^{2}\big{)}\big{\rVert}_{L^{p}(\mathbb{R}^{d})}\lesssim_{p}\|f\|_{L^{p}(\mathbb{R}^{d})}.$

We note that $(\delta_{0}-\phi)\ast\sigma$ is a compactly supported measure
with the vanishing mean value which satisfies condition (4.7). Hence by the
known results from [10, Theorem A, Theorem B] we see that (4.12) holds. The
estimates for the low frequency part $\mathcal{L}_{k}f$ will follow from the
martingale estimates. Let $(\mathbb{E}_{k})_{k\in\mathbb{Z}}$ be the dyadic
martingale sequence associated with the dyadic cubes related to the dilation
$t^{A}$. Then we may write

$\displaystyle\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}(\mathbb{Z})}\lVert
O_{I,N}^{2}(\mathcal{L}_{k}f:k\in\mathbb{Z})\rVert_{L^{p}(\mathbb{R}^{d})}\lesssim$
$\displaystyle\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}(\mathbb{Z})}\lVert
O_{I,N}^{2}(\mathcal{D}_{k}f:k\in\mathbb{Z})\rVert_{L^{p}(\mathbb{R}^{d})}$
$\displaystyle+\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}(\mathbb{Z})}\lVert
O_{I,N}^{2}(\mathbb{E}_{k}f:k\in\mathbb{Z})\rVert_{L^{p}(\mathbb{R}^{d})},$

where

$\mathcal{D}_{k}f(x):=f\ast(\phi\ast\sigma)_{D^{k}}(x)-\mathbb{E}_{k}f(x).$

Since by Theorem 4.4 we know that the oscillation inequality holds for the
martingale $\mathbb{E}_{k}f$, we only need to handle the part with
$\mathcal{D}_{k}f$. As before, we are reduced to show that for any
$p\in(1,\infty)$ one has

$\Big{\lVert}\big{(}\sum_{k\in\mathbb{Z}}|\mathcal{D}_{k}f|^{2}\big{)}^{1/2}\Big{\rVert}_{L^{p}(\mathbb{R}^{d})}\lesssim_{p}\|f\|_{L^{p}(\mathbb{R}^{d})},$

which follows by Lemma 4.5 since $\phi\ast\sigma$ is a Schwartz function such
that $\int\phi\ast\sigma=1$. ∎

###  4.2. Oscillation inequality for the operator $\mathcal{H}_{t}$

Now we are ready to give a proof of (4.1). Due to the differential nature of
the oscillation seminorm it is enough to prove (4.1) for the “complement”
Radon transform given by

$\tilde{\mathcal{H}}_{t}f(x):=\int_{\Omega_{t}^{c}}f(x-(y)^{\Gamma})K(y){\rm
d}y.$

As usual in this context we divide our seminorm into long oscillations and
short variations

$\displaystyle\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{R}_{+})}\big{\lVert}O_{I,N}^{2}(\tilde{\mathcal{H}}_{t}f:t\in\mathbb{R}_{+})\big{\rVert}_{L^{p}(\mathbb{R}^{\Gamma})}\lesssim$
$\displaystyle\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{Z})}\big{\lVert}O_{I,N}^{2}(\tilde{\mathcal{H}}_{D^{k}}f:k\in\mathbb{Z})\big{\rVert}_{L^{p}(\mathbb{R}^{\Gamma})}$
$\displaystyle+\Big{\lVert}\Big{(}\sum_{k\in\mathbb{Z}}V^{2}\big{(}\tilde{\mathcal{H}}_{t}f:t\in[D^{k},D^{k+1})\big{)}^{2}\Big{)}^{1/2}\Big{\rVert}_{L^{p}(\mathbb{R}^{\Gamma})}.$

The estimate for the short variations follows the same lines as in [22,
Section 9.3], since the Littlewood–Paley theory is still available in the
context of $D$-dyadic numbers – we omit the details. We focus on proving
estimates for the long oscillations. The presented approach is well known (see
[10, 17, 23]) hence we will not go into precise details. At the beginning, we
see that we can express $\tilde{\mathcal{H}}_{D^{k}}$ as a telescoping sum

$\tilde{\mathcal{H}}_{D^{k}}f(x):=\sum_{j\geq k}\mu_{D^{j}}\ast
f(x),\qquad\text{with}\qquad\mu_{D^{k}}\ast
f(x):=\int_{\Omega_{D^{k+1}}\setminus\Omega_{D^{k}}}f(x-(y)^{\Gamma})K(y){\rm
d}y.$

Now, let $\varphi$ be a smooth compactly supported function such that
$\hat{\varphi}(0)=1$. Then we employ the following decomposition (cf. [10,
Theorem E])

(4.13)
$\displaystyle\tilde{\mathcal{H}}_{D^{k}}f=\varphi_{D^{k}}\ast\mathcal{H}f-\big{(}\varphi\ast\sum_{j<0}\mu_{D^{j}}\big{)}_{D^{k}}\ast
f+\big{(}\sum_{j\geq 0}(\delta_{0}-\varphi)\ast\mu_{D^{j}}\big{)}_{D^{k}}\ast
f,$

where

$\displaystyle\mathcal{H}f(x):={\rm
p.v.}\int_{\mathbb{R}^{k}}f(x-(y)^{\Gamma})K(y){\rm d}y.$

The oscillation inequality for the term $\varphi_{D^{k}}\ast\mathcal{H}f$
follows by Theorem 4.8 and by the fact that $\mathcal{H}$ is a bounded
operator on $L^{p}$. For the second term in (4.13) it is enough to show

$\Big{\lVert}\Big{(}\sum_{k\in\mathbb{Z}}\big{|}\big{(}\varphi\ast\sum_{j<0}\mu_{D^{j}}\big{)}_{D^{k}}\ast
f|^{2}\Big{)}^{1/2}\Big{\rVert}_{L^{p}(\mathbb{R}^{\Gamma})}\lesssim\|f\|_{L^{p}(\mathbb{R}^{\Gamma})}.$

To prove the above estimate we note that $\varphi\ast\sum_{j<0}\mu_{D^{j}}$ is
a Schwartz function with mean value zero and use the results from [10, Theorem
A, Theorem B]. To estimate the third term occurring in (4.13) it is enough to
show that for some positive constant $c_{p}$ one has

(4.14)
$\Big{\lVert}\Big{(}\sum_{k\in\mathbb{Z}}\big{|}f\ast\big{(}(\delta_{0}-\varphi)\ast\mu_{D^{j}}\big{)}_{D^{k}}\big{|}^{2}\Big{)}^{1/2}\Big{\rVert}_{L^{p}(\mathbb{R}^{\Gamma})}\lesssim
D^{-c_{p}j}\|f\|_{L^{p}(\mathbb{R}^{\Gamma})}.$

The above estimate can be proved by using the Littlewood–Paley theory, for
details see [17, p. 21]. This ends the proof of Theorem 1.9 for the canonical
mapping $(y)^{\Gamma}$.

## References

  * [1] J. Bourgain. On the maximal ergodic theorem for certain subsets of the integers.  Israel J. Math. 61 (1988), pp. 39–72.
  * [2] J. Bourgain. On the pointwise ergodic theorem on $L^{p}$ for arithmetic sets.  Israel J. Math. 61 (1988), pp. 73–84.
  * [3] J. Bourgain. Pointwise ergodic theorems for arithmetic sets. With an appendix by the author, H. Furstenberg, Y. Katznelson, and D.S. Ornstein.  Inst. Hautes Etudes Sci. Publ. Math. 69 (1989), pp. 5–45.
  * [4] A. Calderón. Ergodic theory and translation invariant operators.  Proc. Natl. Acad. Sci. USA 59 (1968), 349–353.
  * [5] A. Calderón, A. Zygmund. On the existence of certain singular integrals.  Acta Math. 88 (1952),85–139
  * [6] J.T. Campbell, R.L. Jones, K. Reinhold, M. Wierdl. Oscillation and variation for the Hilbert transform.  Duke Math. J. 105 (2000), no. 1, pp. 59–83.
  * [7] J.T. Campbell, R.L. Jones, K. Reinhold, M. Wierdl. Oscillation and Variation for Singular Integrals in Higher Dimension.  Trans. Amer. Math. Soc. 355 (2003), no. 5, pp. 2115–2137
  * [8] M. Christ. A $T(b)$theorem with remarks on analytic capacity and the Cauchy integral.  Colloq. Math. 60/61 (1990), no. 2, pp. 601–628.
  * [9] M. Christ, A. Nagel, E.M. Stein, S. Wainger. Singular and maximal Radon transforms: analysis and geometry.  Ann. of Math. 150 (1999), pp. 489–577.
  * [10] J. Duoandikoetxea, J.L. Rubio de Francia. Maximal and singular integral operators via Fourier transform estimates.  Invent. Math. 84 (1986), pp. 541–-561.
  * [11] A.D. Ionescu, A. Magyar, E.M. Stein, S. Wainger. Discrete Radon transforms and applications to ergodic theory.  Acta Math. 198 (2007), pp. 231–298.
  * [12] A.D. Ionescu, S. Wainger. $L^{p}$ boundedness of discrete singular Radon transforms.  J. Amer. Math. Soc. 19 (2005), no. 2, 357–383.
  * [13] R.L. Jones, R. Kaufman, J.M. Rosenblatt, M. Wierdl. Oscillation in ergodic theory.  Ergodic Theory Dynam. Systems 18 (1998), no. 4, 889–935.
  * [14] R.L. Jones, K. Reinhold. Oscillation and variation inequalities for convolution powers. Ergodic Theory Dynam. Systems 21 (2001), no. 6, pp. 1809–1829.
  * [15] R.L. Jones, J.M. Rosenblatt, M. Wierdl. Oscillation inequalities for rectangles.  Proc. Amer. Math. Soc. 129 (2001), no. 5, 1349–1358.
  * [16] R.L. Jones, J.M. Rosenblatt, M. Wierdl. Oscillation in ergodic theory: Higher dimensional results. Isr. J. Math. 135 (2003), pp.1–27
  * [17] R.L. Jones, A. Seeger, J. Wright. Strong variational and jump inequalities in harmonic analysis.  Trans. Amer. Math. Soc. 360 (2008), no. 12, 6711–6742.
  * [18] A. Magyar, E.M. Stein, S. Wainger. Discrete analogues in harmonic analysis: spherical averages.  Ann. Math. 155 (2002), pp. 189–208.
  * [19] M. Mirek. $\ell^{p}\big{(}\mathbb{Z}^{d}\big{)}$-estimates for discrete Radon transform: square function estimates.  Anal. PDE 11 (2018), no. 3, pp. 583–608.
  * [20] M. Mirek, W. Słomian, T.Z. Szarek. Some remarks on oscillation inequalities.  (preprint) arXiv:2110.01149
  * [21] M. Mirek, E.M. Stein, B. Trojan. $\ell^{p}\big{(}\mathbb{Z}^{d}\big{)}$-estimates for discrete operators of Radon types I: Maximal functions and vector-valued estimates.  J. Funct. Anal. 277 (2019), pp. 2471–2892
  * [22] M. Mirek, E.M. Stein, B. Trojan. $\ell^{p}(\mathbb{Z}^{d})$-estimates for discrete operators of Radon type: Variational estimates. Invent. Math. 209 (2017), no. 3, pp. 665–748.
  * [23] M. Mirek, E.M. Stein, P. Zorin-Kranich. A bootstrapping approach to jump inequalities and their applications.  Anal. PDE 13 (2020), No. 2, pp. 527–558
  * [24] M. Mirek, E.M. Stein, P. Zorin-Kranich. Jump inequalities for translation-invariant operators of Radon type on $\mathbb{Z}^{d}$.  Adv. Math. 365 (2020) 107065
  * [25] E. M. Stein. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals.  Princeton University Press, (1993).
  * [26] E.M. Stein, S. Wainger. Problems in harmonic analysis related to curvature.  Bull. Amer. Math. Soc. 84 (1978), pp. 1239–1295.
  * [27] E.M. Stein, S. Wainger. Discrete analogues of singular Radon transforms.  Bull. Amer. Math. Soc. 23 (1990), pp. 537–544.
  * [28] E.M. Stein, S. Wainger. Discrete analogues in harmonic analysis I: $\ell^{2}$ estimates for singular Radon transforms.  Amer. J. Math. 121 (1999), pp. 1291–1336.
  * [29] T. Tao. The Ionescu–Wainger multiplier theorem and the adeles.  Mathematika, doi:10.1112/mtk.12094