Abstract
We prove the extensions of Birkhoff’s and Cotlar’s ergodic theorems to multi-dimensional polynomial subsets of prime numbers \(\mathbb {P}^k\). We deduce them from \(\ell ^p\big (\mathbb {Z}^d\big )\)-boundedness of r-variational seminorms for the corresponding discrete operators of Radon type, where \(p > 1\) and \(r > 2\).
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1 Introduction
Let \((X, \mathcal {B}, \mu )\) be a \(\sigma \)-finite measure space with \(d_0\) invertible commuting and measure preserving transformations \(T_1, \ldots , T_{d_0}: X \rightarrow X\). Let \(\mathcal {P}= \big (\mathcal {P}_1, \ldots , \mathcal {P}_{d_0}\big ):\mathbb {R}^k \rightarrow \mathbb {R}^{d_0}\) denote a polynomial mapping such that each \(\mathcal {P}_j\) is a polynomial on \(\mathbb {R}^k\) having integer coefficients without a constant term. Let B be an open bounded convex subset in \(\mathbb {R}^k\) containing the origin such that for some \(\iota > 0\) and all \(N \in \mathbb {N}\),
where for \(\lambda > 0\), we have set
In this paper we consider the following averages
where \(k = k'+k''\), \(\mathbb {P}\) denotes the set of prime numbers, and
One of the results of this article establishes the following theorem.
Theorem A
Assume that \(p \in (1, \infty )\). For every \(f \in L^p(X, \mu )\) there exists \(f^* \in L^p(X, \mu )\) such that
for \(\mu \)-almost all \(x \in X\).
Sums over prime numbers are irregular, thus it is more convenient to work with weighted averaging operators,
where
Then the pointwise convergence of \((\mathscr {A}_N f : N \in \mathbb {N})\) can be deduced from the properties of \((\mathscr {M}_N f : N \in \mathbb {N})\), see Proposition 2.1 for details.
Next to the averaging operators we also study pointwise convergence of truncated discrete singular operators. To be more precise, let \(K \in \mathcal {C}^1\big (\mathbb {R}^k \setminus \{0\} \big )\) be a Calderón–Zygmund kernel satisfying the differential inequality
for all \(x \in \mathbb {R}^k\) with \(|{x} | \ge 1\), and the cancellation condition
for every \(0 < \lambda ' \le \lambda \). Then the truncated discrete singular operator \(\mathscr {H}^\mathcal {P}_N\) is defined as
The logarithmic weights in \(\mathscr {M}_N^\mathcal {P}\) and \(\mathscr {H}_N^\mathcal {P}\) correspond to the density of prime numbers. In this article we prove the following theorem, which may be thought as an extension of Cotlar’s ergodic theorem, see [4].
Theorem B
Assume that \(p \in (1, \infty )\). For every \(f \in L^p(X, \mu )\) there exists \(f^* \in L^p(X, \mu )\) such that
for \(\mu \)-almost all \(x \in X\).
The classical approach to the pointwise convergence in \(L^p(X, \mu )\) proceeds in two steps. Namely, one needs to show \(L^p(X, \mu )\) boundedness of the corresponding maximal function reducing the problem to showing the convergence on some dense class of \(L^p(X, \mu )\) functions. However, finding such a class may be a difficult task. This is the case of one dimensional averages along \((n^2 : n \in \mathbb {N})\) studied by Bourgain in [2]. To overcome this issue Bourgain introduced the oscillation seminorm defined for a given lacunary sequence \((N_j : j \in \mathbb {N})\) and a sequence of complex numbers \((a_n : n \in \mathbb {N})\) as
Then the pointwise convergence of \((\mathscr {A}_N f : N \in \mathbb {N})\) is reduced to showing that
while J tends to infinity. In place of the oscillation seminorm, we investigate r-variational seminorm. Let us recall that r-variational seminorm of a sequence \((a_n : n\in \mathbb {N})\) is defined by
In fact, r-variational seminorm controls \(\mathcal {O}_J\) as well as the maximal function. Indeed, for any \(r \ge 2\), by Hölder’s inequality we have
Moreover, for any \(n_0 \in \mathbb {N}\),
Nevertheless, the main motivation to study \(L^p(X, \mu )\) boundedness of r-variational seminorm is the following observation: if \(V_r(a_n : n \in \mathbb {N}) < \infty \) for any \(r \ge 1\) then the sequence \((a_n : n \in \mathbb {N})\) converges. Therefore, we can deduce Theorems A and B from the following result.
Theorem C
For every \(p \in (1, \infty )\) there is \(C_p > 0\) such that for all \(r \in (2, \infty )\) and all \(f \in L^p(X, \mu )\),
and
The constant \(C_p\) is independent of the coefficients of the polynomial mapping \(\mathcal {P}\).
The variational estimates for discrete averaging operators have been the subject of many papers, see [8, 10, 12, 13, 15, 16, 26]. In [10], Krause studied the case \(d_0 = k = k' = 1\) and has obtained the inequality (1.4) for \(p \in (1, \infty )\) and \(r > \max \{p, p'\}\). On the other hand, Zorin-Kranich in [26] for the same case obtained (1.4) for all \(r \in (2, \infty )\) but for p in some vicinity of 2. Only recently in [12] the variational estimates have been established in the full range of parameters, that is \(p \in (1, \infty )\) and \(r \in (2, \infty )\), covering the case \(k'' = 0\). In [26], Zorin-Kranich has proved (1.4) also for the averaging operators modeled on prime numbers, that is when \(d_0 = k = k '' = 1\) with a polynomial \(P(n) = n\). It is worth mentioning that the variational estimates for discrete operators are based on a priori estimates for their continuous counterparts developed in [9], see also [12, Appendix].
The variational estimates for discrete singular operators have been studied in [3, 12, 13, 16]. In [16], the authors obtained the inequality (1.5), for the truncated Hilbert transform modeled on prime numbers, which corresponds to \(d_0 = k = k'' = 1\) and a polynomial \(P(n) = n\). In fact, discrete singular operators of Radon type required a new approach. An important milestone has been laid by Ionescu and Wainger in [7]. Ultimately, the complete development of the discrete singular operators of Radon type has been obtained in [12].
Concerning pointwise ergodic theorems over prime numbers, there are some results using oscillation seminorms. In [1], Bourgain has shown pointwise convergence for the averages along prime numbers for functions from \(L^2(X, \mu )\). Then his result was extended to all \(L^p(X, \mu )\), \(p > 1\), by Wierdl in [24], see also [2, Section 9]. Not long afterwards, Nair in [18] has proved Theorem A for \(L^2(X, \mu )\), \(d_0 = k = k'' = 1\), and any integer-valued polynomial. Nair also studied ergodic averages for functions in \(L^p(X, \mu )\) for \(p \ne 2\), however, [19, Lemma 14] contains an error. In fact, the estimates on the multipliers \(W_N\) are insufficient to show that the sum considered at the end of the proof has bounds independent of \(\left|{\alpha -a/b} \right|\). Lastly, the extension of Cotlar’s ergodic theorem to prime numbers has been established in [14], see also [16].
In view of the Calderón transference principle, while proving Theorem C, we may work with the model dynamical system, namely, \(\mathbb {Z}^{d_0}\) with the counting measure and the shift operators. Let us denote by \(M_N^\mathcal {P}\) and \(H_N^\mathcal {P}\), the corresponding operators, namely,
and
We now give some details about the method of the proof of Theorem C for the model dynamical system. To simplify the exposition we restrict attention to the averaging operators. Let us denote by \(\mathfrak {m}_N\) the discrete Fourier multiplier corresponding to \(M_N^\mathcal {P}\). To deal with r-variational estimates we apply the method recently used [13], see also [26]. Namely, given \(\rho \in (0, 1)\) we consider the set \(\mathcal {D}_\rho = \{N_n : n \in \mathbb {N}\}\), where \(N_n = \big \lfloor 2^{n^\rho }\big \rfloor \). Then in view of (5.6) we can split the r-variation into two parts: long variations and short variations, and study them separately. For each \(p \in (1, \infty )\) we can choose \(\rho \) so that the estimate for \(\ell ^p\)-norm of short variations is straightforward. Next, to control long variations we adopt the partition of unity constructed in [12], that is
for some parameter \(\beta \in \mathbb {N}_0\). Each projector \(\Xi _{n, s}^{\beta }\) is supported by a finite union of disjoint cubes centered at rational points belonging to \(\mathscr {R}_s^\beta \). In this way, we distinguish the part of the multiplier where we can identify the asymptotic from the highly oscillating piece. The oscillating part is controlled by a multi-dimensional version of Weyl–Vinogradov’s inequality with a logarithmic loss together with \(\ell ^p\big (\mathbb {Z}^d\big )\) estimates for multipliers of Ionescu–Wainger type. By the triangle inequality, to control the first part it is enough to show
First, by the circle method of Hardy and Littlewood, we find the asymptotic of the multiplier \(\mathfrak {m}_{N_n}\). Here we encounter the main difference from [12]. Namely, for \(\xi \) sufficiently close to the rational point a / q we have
provided that \(1 \le q \le (\log N_n)^{\beta '}\), where G(a / q) is the Gaussian sum and \(\Phi _{N_n}\) is an integral version of \(\mathfrak {m}_{N_n}\). The limitation on the size of the denominator is a consequence of the fact that for larger q the Siegel–Walfisz theorem has an additional term due to the possible exceptional zero of the exceptional quadratic character. The second issue is the slower decay of the error term in (1.9). In particular, the later has its impact on the size of the cubes in the partition of unity. Both facts made the analysis of the approximating multipliers \(\nu _{N_n}^s\) harder. To overcome this we directly work with \(\mathfrak {m}_N\). Moreover, we get completely unified approach to the variational estimates for the averaging operators and the truncated discrete singular operators.
Going back to the sketch of the proof, in order to show (1.8), we divide the variation into two parts: \(s < n \le 2^{\kappa _s}\) and \(2^{\kappa _s} < n\), where \(\kappa _s \simeq (s+1)^{\rho /10}\). For large scales \(2^{\kappa _s} < n\), we transfer a priori estimates on \(L^p\)-norm for r-variation of the related continuous multipliers. Since the Gaussian sums satisfies \(\left|{G(a/q)} \right| \lesssim q^{-\delta }\) for some \(\delta > 0\), we gain a decay \((s+1)^{-\delta \beta \rho }\) on \(\ell ^2\). Consequently, by interpolation the \(\ell ^p\) norm of r-variation for large scales is bounded by \((s+1)^{-2}\) provided that \(\beta \) is sufficiently large. In the case of small scales \(s < n \le 2^{\kappa _s}\), the estimate on \(\ell ^2\) is obtained with a help of the numerical inequality (2.3). We again show that \(\ell ^2\) norm is bounded by \((s+1)^{-\delta \beta \rho + 1}\). Because of the weaker asymptotic (1.9), to obtain \(\ell ^p\) bounds for r-variations over small scales required a new approach. We further divide the index set into dyadic blocks, then on each block we construct a good approximation to the multiplier giving bounds on \(\ell ^p\) norm independent of the block. At the cost of additional factor of \(\kappa _s^2\), we control \(\ell ^p\) norm of r-variation. Again, by interpolation combined with a choice of \(\beta \) large enough we can make the \(\ell ^p\) norm bounded by \((s+1)^{-2}\).
Let us briefly describe the structure of the article. In Sect. 2.1 we collect basic properties of the variational seminorm. In Sect. 2.2, we show how to deduce Theorem A from r-variational estimates (1.4) and (1.5). Then we present the lifting procedure, which allows us to replace any polynomial mapping \(\mathcal {P}\) by a canonical one \(\mathcal {Q}\). In the next section, we describe multipliers of Ionescu–Wainger type whose \(\ell ^p\) norm estimates are essential to our argument. In Sect. 3, we show a multi-dimensional version of Weyl–Vinogradov’s inequality with a logarithmic loss. Moreover, we prove the estimate on the Gaussian sums of a mixed type. Sections 4.1 and 4.2 are devoted to study the asymptotic behavior of multipliers \(M_N\) and \(H_N\), respectively. Finally, to get completely unified approach to the variational estimates for the averaging operators and truncated singular operators, at the beginning of Sect. 5, we list the properties shared by them which are sufficient to prove Theorem C. In the next two sections we show the estimates on long and short variations.
Notation
Throughout the whole article, we write \(A \lesssim B\) (\(A \gtrsim B\)) if there is an absolute constant \(C>0\) such that \(A\le CB\) (\(A\ge CB\)). Moreover, C stand for a large positive constant whose value may vary from occurrence to occurrence. If \(A \lesssim B\) and \(A\gtrsim B\) hold simultaneously then we write \(A \simeq B\). Lastly, we write \(A \lesssim _{\delta } B\) (\(A \gtrsim _{\delta } B\)) to indicate that the constant C depends on some \(\delta > 0\). Let \(\mathbb {N}_0 = \mathbb {N}\cup \{0\}\). For a vector \(x \in \mathbb {R}^d\), we set \(|{x} |_\infty = \max \{|x_j| : 1 \le j \le d\}\). Given a subset \(A \subseteq \mathbb {Z}\) and \(x \in \mathbb {R}\), we set \(A_x = A \cap [0, x]\).
2 Preliminaries
2.1 Variational norm
Let \(r \in [1, \infty )\). For a sequence \((a_j : j \in A)\), \(A \subseteq \mathbb {Z}\), we define r-variational seminorm by
The function \(r \mapsto V_r(a_j : j \in A)\) is non-decreasing, thus
and by Minkowski’s inequality
Moreover, for any \(j_0 \in A\),
Finally, for any increasing sequence \((u_k : 0 \le k \le K)\), we have
The following lemma is essential in studying variational seminorms.
Lemma 1
[15, Lemma 1] If \(r \ge 2\) then for any sequence \((a_j : 0 \le j \le 2^s)\) of complex numbers
2.2 Pointwise ergodic theorems
In this section we show how to deduce the pointwise ergodic theorem (Theorem A) from a priori r-variational estimates for \(\mathscr {M}_N^\mathcal {P}\).
Proposition 2.1
Let \(p \in (1, \infty )\) and \(r \in (2, \infty )\). Suppose that there is \(C > 0\) such that for all \(f \in L^p(X, \mu )\),
Then there is \(C > 0\) such that for all \(f \in L^p(X, \mu )\),
and the averages \(\big (\mathscr {A}_N f(x) : N \in \mathbb {N}\big )\) converges for \(\mu \)-almost all \(x \in X\).
Proof
Let us fix \(N \in \mathbb {N}\). For each \(m \in \{1, \ldots , N\}\) and \(s \in \{1, \ldots , k''\}\), we set
and \(S^{(0)}_{N, N} f = \vartheta _B(N) \mathscr {M}_N^\mathcal {P}f\). For \(0 \le s < k''\), by the partial summation we obtain
Hence,
where we have used the trivial estimate
which is a consequence of (1.1) and the prime number theorem. Observe that
thus by repeated application of (2.5), we arrive at the conclusion that
because the prime number theorem implies that \(\vartheta _B(N) \simeq N^k\). In particular, by taking \(f = {\mathbb {1}_{{X}}}\) and \(p = \infty \) in (2.6) we get
Hence, for any \(p \in [1, \infty ]\) and \(f \in L^p(X, \mu )\),
Next, if \(p > 1\) then we can write
In view of (2.1), a priori estimate (2.4) entails that
Hence, while proving \(\mu \)-almost everywhere convergence of the averages \(\big (\mathscr {A}_N f : N \in \mathbb {N}\big )\) for \(f \in L^p(X, \mu )\), we may assume that the function f is bounded. By (2.7), for \(p = \infty \), we can write
Therefore, the convergence of \(\big (\mathscr {M}_N^\mathcal {P}f(x) : N \in \mathbb {N}\big )\) implies the convergence of \(\big (\mathscr {A}_N^\mathcal {P}f(x) : N \in \mathbb {N}\big )\) to the same limit. \(\square \)
Thanks to the Calderón’s transference principle we can restrict attention to the model dynamical system, that is, \(\mathbb {Z}^{d_0}\) with the counting measure and the shift operator. Hence, it suffices to study the operators (1.6) and (1.7) on \(\ell ^p\big (\mathbb {Z}^{d_0}\big )\).
2.3 Lifting lemma
For the polynomial mapping \(\mathcal {P}= \big (\mathcal {P}_1, \ldots , \mathcal {P}_{d_0}\big )\), let us define
It is convenient to work with the set
equipped with the lexicographic order. Then each \(\mathcal {P}_j\) can be expressed as
for some \(c_{j, \gamma } \in \mathbb {Z}\). The cardinality of the set \(\Gamma \) is denoted by d. We identify \(\mathbb {R}^d\) with \(\mathbb {R}^\Gamma \). Let A be a diagonal \(d \times d\) matrix such that for all \(\gamma \in \Gamma \) and \(v \in \mathbb {R}^d\),
For \(t > 0\), we set
Finally, we introduce the canonical polynomial mapping,
by setting \(\mathcal {Q}_\gamma (x) = x^\gamma \). Now, if we define \(L: \mathbb {R}^d \rightarrow \mathbb {R}^{d_0}\) to be the linear transformation such that for \(v \in \mathbb {R}^d\),
then \(L \mathcal {Q}= \mathcal {P}\). The following lemma allows us to reduce the problems to studying the canonical polynomial mappings.
Lemma 2
[11, Lemma 2.1] Let \(R^\mathcal {P}_N\) be any of the operators \(M_N^\mathcal {P}\) or \(H_N^\mathcal {P}\). Suppose that for some \(p \in (1, \infty )\) and \(r \in (2, \infty )\),
then
In the rest of the article by \(M_N\) and \(H_N\) we denote the averaging and the truncated discrete singular operator for the canonical polynomial mapping \(\mathcal {Q}\), that is \(M_N = M_N^\mathcal {Q}\) and \(H_N = H_N^\mathcal {Q}\).
2.4 Ionescu–Wainger type multipliers
Let \(\mathcal {F}\) denote the Fourier transform on \(\mathbb {R}^d\), that is for any \(f \in L^1\big (\mathbb {R}^d\big )\),
If \(f \in \ell ^1\big (\mathbb {Z}^d\big )\), then we set
To simplify the notation, by \(\mathcal {F}^{-1}\) we denote the inverse Fourier transform on \(\mathbb {R}^d\) as well as the inverse Fourier transform on the d-dimensional torus identified with \((0, 1]^d\). We also fix \(\eta : \mathbb {R}^d \rightarrow \mathbb {R}\), a smooth function such that \(0 \le \eta \le 1\), and
We additionally assume that \(\eta \) is a convolution of two non-negative smooth functions with supports contained inside \(\left[ -\tfrac{1}{8d}, \tfrac{1}{8d}\right] ^d\).
Next, let us recall necessary notation to define auxiliary multipliers of Ionescu–Wainger type. For details we refer to [11]. The following construction depends on a parameter \(\beta \in \mathbb {N}\).
For \(n \in \mathbb {N}\), we set \(n_0 = \lfloor n^{1/20}\rfloor \) and \(Q_0 = (n_0!)^D\) where \(D = 20 \beta + 1\). We define
wherein for \(k \in \{1, \ldots , D\}\) we have set
Let
Notice that \(\mathbb {N}_{n^\beta } \subseteq P_n \subseteq \mathbb {N}_{e^{n^{1/10}}}\). For \(q \in \mathbb {N}\), let us define
and
Lastly, we set
Given \((\Theta _j : j \in \mathbb {Z})\) a sequence of multipliers on \(\mathbb {R}^d\) such that for each \(r \in (1, \infty )\) there is \(A_r > 0\) such that for all \(f \in L^2\big (\mathbb {R}^d\big ) \cap L^r\big (\mathbb {R}^d\big )\),
its discrete counterpart is given by the formula
where \(\mathcal {E}_n\) being a diagonal \(d \times d\) matrix with positive entries \((\epsilon _{n, \gamma } : \gamma \in \Gamma )\) such that \(\epsilon _{n, \gamma } \le \exp \big (-n^{1/5}\big )\). Then by [13, Theorem 2.1], for each \(p \in (1, \infty )\) and any finitely supported function \(f: \mathbb {Z}^d \rightarrow \mathbb {C}\),
where \(r = \max \big \{\lceil p/2 \rceil , \lceil p'/2 \rceil \big \}\). The scalar-valued version of (2.10) was proved in [7], see also [11]. The vector-valued extension was recently observed in [13]. Essentially its proof follows the same line as scalar-valued except that in place of Marcinkiewicz–Zygmund inequality one uses Kahane’s vector-valued extension of Khinchine’s inequality, see [13, Theorem 2.1] for details.
3 Trigonometric sums
3.1 Weyl–Vinogradov sum
We say that a subset of integers \(\mathcal {A}\) is polynomially regular, if for all \(\alpha , \alpha _1 > 0\), there are \(\beta _0 > 0\) and a constant \(C > 0\) so that for any integer \(1 \le Q \le (\log N)^{\alpha _1}\), \(\beta > \beta _0\) and any polynomial P of a form
for some coprime integers a and q, such that \(1 \le a \le q\), and
we have
for all \(r \in \{1, \ldots , Q\}\) and \(N \in \mathbb {N}\).
Let us check that \(\mathbb {Z}\) is polynomially regular. We write
where
Set \(M = \lfloor N/Q \rfloor \) and \(a'/q' = Q^d a/q\) with \((a', q') = 1\). Then
and hence, by Weyl estimates with logarithmic loss (see e.g. [25, Remark after Theorem 1.5]),
Therefore, for \(\beta > \beta _0 = (1+\alpha )(2 d^2 - 2d +1) +d \alpha _1\), by (3.2), we conclude that
proving the claim. Another example of polynomially regular sets is the set of prime numbers. This is a consequence of [6, Theorem 10].
Our aim is to understand exponential sums over Cartesian products of polynomially regular sets. Let us fix a function \(\phi : \mathbb {R}^k \rightarrow \mathbb {C}\) satisfying
The main result of this section is the following theorem.
Theorem 1
Let \(\mathcal {A}_1, \ldots \mathcal {A}_k\) be polynomially regular subsets of \(\mathbb {Z}\). For all \(\alpha > 0\) there are \(\beta _0 > 0\) and a constant \(C > 0\) so that for all \(\beta > \beta _0\) and any polynomial P of a form
wherein for some \(0 < \left|{\gamma _0} \right| \le d\),
for some coprime integers a and q such that \(1 \le a \le q\), and
we have
The constant C depends on \(\alpha \), d and a constant in (3.3).
Proof
Let us first assume that \(\phi \equiv 1\). The proof consists of three steps.
Step 1. We consider the case when \(k = 1\) and \(\left|{\gamma _0} \right| = d\). Take \(\alpha > 0\) and \(\alpha _1 > 0\), and let \(\beta > \beta _0 = 3 \beta _1 + 3 d \alpha \), where \(\beta _1\) is the value of \(\beta _0\) determined by \(\mathcal {A}_1\) for \(\alpha \) and \(\alpha _1\). Suppose that a and q are coprime integers such that \(1 \le a \le q\), and
with \((\log N)^\beta \le q \le N^d (\log N)^{-\beta }\). By Dirichlet’s principle, there are coprime integers \(a'\) and \(q'\) such that \(1 \le a' \le q' \le N^d (\log N)^{-\frac{1}{3}\beta }\), and
If \(a'/q' \ne a/q\) then
Hence, we obtain
Thus
Observe that the last estimate is also valid if \(q' = q\). Let Q be an integer such that \(1 \le Q \le (\log N)^{\alpha _1}\). Given \(r \in \{1, \ldots , Q\}\), we set
where
We first show that
for all \(\beta > \beta _0\). If \(1 \le N' \le N (\log N)^{-\alpha }\), then there is nothing to be proven. For \(N (\log N)^{-\alpha } \le N' \le N\), we have
thus, by (3.1), we obtain
proving (3.4). We now set \(\theta = \xi _d - a'/q'\) and apply the partial summation to get
Since
by (3.4), we obtain
which finishes the proof of Step 1.
Step 2. We next consider \(k \ge 2\) and \(\gamma _0 \ne (0, \ldots , 0 , \ell , 0, \ldots , 0)\) for any \(\ell \le d\). Without loss of generality we may assume that \(\gamma _0(1) \ge 1\). By the triangle inequality followed by Cauchy–Schwarz inequality we get
Next, we have
which, by another application of Cauchy–Schwarz inequality, is bounded by
Finally,
where
and
Notice that the set \(\Theta \) is a convex subset of a cube \([-N, N]^{2k}\). Moreover, the polynomial \(Q(x, x')\) has degree at least \(\left|{\gamma _0} \right|\) having a coefficient \(\xi _{\gamma _0}\) in front of the monomial \(x^{\gamma _0}\). Therefore, by [11, Theorem 3.1], there are \(\beta _0 > 0\) and \(C > 0\) such that
provided that \(\beta > \beta _0\). Hence, by (3.5), (3.6) and (3.7) we obtain
Step 3. Suppose that \(k \ge 1\) and \(\gamma _0 = (0, \ldots , 0, \ell , 0, \ldots , 0)\) for \(1 \le \ell \le d\). Without loss of generality we may assume that \(\gamma _0 = (\ell , \ldots , 0)\). The proof is by a backward induction over \(\ell \in \{1, \ldots , d\}\). We write
If \(\ell = d\) the conclusion follows by Step 1. Suppose that \(\ell < d\). In view of Step 2 and the inductive hypothesis, the estimate holds for any \(\left|{\gamma _0} \right| = j\), \(\ell < j \le d\). Let \(\beta _1\) be the largest value of \(\beta _0\) among those that were determined in Step 2 and resulting from the inductive hypothesis. By Dirichlet’s principle, for each \(\ell < \left|{\gamma } \right| \le d\), we select coprime integers \(a_\gamma \) and \(q_\gamma \), such that \(1 \le a_\gamma \le q_\gamma \le N^{\left|{\gamma } \right|} (\log N)^{-\beta _1}\), satisfying
If for some \(\gamma \in \Gamma \), \(\ell < \left|{\gamma } \right| \le d\) we have \((\log N)^{\beta _1} \le q_\gamma \), then the conclusion follows by the inductive hypothesis or Step 2. Otherwise, we set \(\theta _\gamma = \xi _\gamma - a_\gamma /q_\gamma \) and \(Q = {\text {lcm}}\{q_\gamma : \ell < \left|{\gamma } \right| \le d \}\). We have
and
where \(\alpha _1 = \beta _1 \cdot \#\{\gamma \in \mathbb {N}_0^k : \ell < \left|{\gamma } \right| \le d \}\). We have
Setting
we can write
Thus
where
and
To estimate the inner sum on the right-hand side of (3.11), we apply the partial summation. Setting
we can write
By (3.9), for \((n_1, \tilde{n}) \in \Omega \) we have
Recall that \(\gamma _0 = (\ell , 0, \ldots , 0)\) and
thus, by Step 1 applied to \(S_{n_1, \tilde{n}}^{(r)}\) we obtain
whenever \(\beta > \beta _2\), where \(\beta _2\) is the value of \(\beta _0\) determined in Step 1 for \(\alpha +\beta _1\) and \(\alpha _1\). Hence,
Consequently, by (3.8), (3.10) and (3.11) we get
provided that \(\beta > \beta _0 = \max \{\beta _1, \beta _2\}\).
Finally, we deal with a general \(\phi \). Given \(\alpha \), let \(\beta _0\) be such that
We divide the cube \([-N, N]^k\) into J closed cubes \((Q_j : 1 \le j \le J)\) with sides parallel to the axes and having side lengths \(\mathcal {O}\big (N (\log N)^{-\alpha -1} \big )\). Thus
By \(Q_j^\mathrm {o}\) we denote the interior of \(Q_j\). We assume that \(Q_j^\mathrm {o}\) are disjoint with the axes. Let \(n_j\) be the vertex of \(Q_j\) at the largest distance to the origin. Then by the mean value theorem and (3.3), we have
thus
On the other hand, in view of (3.12), we get
hence, by (3.13),
which together with (3.14) completes the proof. \(\square \)
We next apply Theorem 1 to get the following variant of Weyl–Vinogradov’s inequality.
Theorem 2
Let \(\xi \in \mathbb {T}^d\). Assume that there is a multi-index \(\gamma _0 \in \Gamma \), such that
for some coprime integers a and q such that \(1 \le a \le q\). Then for all \(\alpha > 0\), there is \(\beta _\alpha > 0\), so that for any \(\beta > \beta _\alpha \), if
then
The constant C depends on \(\alpha \), d and a constant in (3.3).
Proof
We claim that the following holds true.
Claim 1
For all \(\alpha > 0\), there is \(\beta _\alpha > 0\), such that for all \(\beta > \beta _\alpha \), \(N \in \mathbb {N}\), and \(r \in \{0, \ldots , k''\}\), if there is a multi-index \(\gamma _0 \in \Gamma \), such that
for some coprime integers a and q, such that \(1 \le a \le q\), and
then
The proof is by a backward induction over r. For \(r = k''\) the assertion follows by Theorem 1. For \(r \in \{1, \ldots , k''\}\), \(N \in \mathbb {N}\) and \(m \in \{1, \ldots , N\}\), we set
and
where \(\Omega \) is a convex subset of \([-N, N]^k\). For \(0 \le r < k''\), by the partial summation, we can write
Hence, by the inductive hypothesis we get
proving the claim. Now, the theorem follows by Claim 1 for \(r = 0\). \(\square \)
3.2 Gaussian sums
Given \(q \in \mathbb {N}\) and \(a \in \mathbf {A}_q\), the Gaussian sum is
where \(\varphi \) is Euler’s totient function, i.e \(\varphi (q)\) equals to the number of elements in \(A_q\). The following theorem provides a very useful estimate on the Gaussian sums.
Theorem 3
There are \(C > 0\) and \(\delta > 0\) such that for all \(q \in \mathbb {N}\) and \(a \in \mathbf {A}_q\),
Proof
Let us recall that for \(a, q \in \mathbb {N}\), (see e.g. [17, Theorem 4.1])
wherein \(\mu (q)\) is Möbius function defined for \(q = p_1^{j_1} \cdots p_m^{j_m}\), \(p_j\) are distinct prime numbers, as
For each \(\epsilon > 0\) there is \(C_\epsilon > 0\), such that (see e.g. [17, Theorem 2.9])
We start the proof of the theorem by considering \(d = 1\). Then
Suppose that \(k' \ge 1\). If \(G(a/q) \ne 0\) then \(q \mid a_\gamma \) for all \(\gamma = (\gamma ', 0) \in \Gamma \). Since \(a \in \mathbf {A}_q\), we must have \(k'' \ge 1\). For \(\gamma = (0, \gamma '') \in \Gamma \), we set \(b_\gamma /q_\gamma = a_\gamma /q\), where \((b_\gamma , q_\gamma ) = 1\). By (3.15), \(G(a/q) \ne 0\) entails that each \(q_\gamma \) is square-free. Since for any p, a prime factor of q, there is \(\gamma = (0, \gamma '') \in \Gamma \) such that \(p \not \mid q/q_\gamma \), we conclude that q is square-free. Because \(q = {\text {lcm}}\big (q_\gamma : \gamma = (0, \gamma '') \in \Gamma \big )\),
which together with (3.16) gives
Next, let us consider the case \(d \ge 2\). For a given polynomial P on \(\mathbb {R}^k\) with integral coefficients we define
Let
where \(a \in \mathbf {A}_q\). Our aim is to show that there are \(C > 0\) and \(\delta > 0\) such that for all \(q \in \mathbb {N}\) and \(a \in \mathbf {A}_q\),
First, observe that for \(q = q_1 q_2\), \((q_1, q_2) = 1\), we have
Therefore, if \(q = p_1^{j_1} \cdots p_m^{j_m}\) for some distinct prime numbers \(p_j\), then
where
Since \(\omega (q)\), the number of distinct prime factors of q, satisfies (see e.g. [17, Theorem 2.10])
we have
Hence, it is enough to proof (3.17) for \(q = p^j\) with p being a prime number and \(j \ge 1\). Since for any arithmetic function F, we have
if \(j \ge 2\) we write
where for \(\sigma \in \{0, 1\}^{k''}\), we have set
Fix \(\sigma \in \{0, 1\}^{k''}\). For each \(\gamma \in \Gamma \), we define
where \((b_\gamma , q_\gamma ) = 1\). Let
Observe that
entails that \(q = Q\). To obtain a contradiction, let us suppose that \(q < Q\). Let \(\gamma _0 \in \Gamma \), \(\left|{\gamma _0} \right| = 1\) be such that \(q_{\gamma _0} = Q\). Thus \(q \mid p^{j-\sigma _1}\). For any \(r \in \mathbb {N}_q^k\) we can write
where
Thus (3.18) implies that \(q_{\gamma _0} \mid b_{\gamma _0} q\), which is impossible. Hence, \(q = Q\).
Now, let \(\gamma _0 \in \Gamma \), \(\left|{\gamma _0} \right| \ge 2\), be such that \(q_{\gamma _0} = Q\). Then
and thus
Suppose that \(Q < p^j\). Since \(a \in \mathbf {A}_q\), we must have \(\sigma \ne 0\). Then for \(j \le D = \max \{\left|{\gamma ''} \right| : \gamma \in \Gamma \}\), by a trivial estimate we have
provided \(0< \delta _1 < (k D)^{-1}\). Since \(Q \ge p^{j-D}\), for \(j \ge D+1\) we have
whenever \(0< \epsilon < (d (D+1))^{-1}\). Hence, by (3.19),
Obviously, the last estimate is also valid for \(Q = p^j\). Since \(\Omega ^\sigma \subseteq \mathbb {N}_{p^j}^k\), by [21, Proposition 3], there are \(C > 0\) and \(\delta _2 > 0\) such that
which finishes the proof of (3.17) for \(q=p^j\), and the theorem follows. \(\square \)
4 Multipliers
In this section we develop some estimates on discrete Fourier multipliers corresponding to operators \(M_N\) and \(H_N\).
4.1 Averaging operators
For a function \(f: \mathbb {Z}^d \rightarrow \mathbb {C}\) with a finite support we have
where \(\mathfrak {m}_N\) is the discrete Fourier multiplier
wherein \(\vartheta _B\) is the Chebyshev function
By (1.1) and the prime number theorem,
Next, let us define
where \(\left|{B} \right|\) denotes Euclidean measure of B. By a multi-dimensional version of van der Corput’s lemma (see [22, Proposition 2.1]) we have
where A is the matrix defined in (2.8). Moreover,
Therefore, for \(N < N' \le 2N\), we have
We start with the following proposition.
Proposition 4.1
For each \(\beta ' > 0\) there are \(C, c > 0\) such that for all \(N \in \mathbb {N}\), and \(\xi \in \mathbb {T}^d\) satisfying
where \(1 \le q \le (\log N)^{\beta '}\), \(a \in \mathbf {A}_q\), and \(1 \le L \le \exp \big (c \sqrt{\log N}\big )\), we have
The constant c is absolute.
Proof
Observe that for a prime number p, \(p \mid q\) if and only if \((p \bmod q, q) > 1\). Hence, for each \(s \in \{1, \ldots , k''\}\), we have
Let \(\theta = \xi - a/q\). Then by (4.4),
Since for \((u, p) \in \mathbb {N}^{k'} \times \mathbb {P}^{k''}\) such that \(u \equiv r' \bmod q\), and \(p \equiv r'' \bmod q\),
we have
Let us fix \(u \in \mathbb {N}^{k'}\), \(\tilde{p} \in \mathbb {P}^{k''-1}\) and \(r_1'' \in A_q\). Then
for some \(0 \le V_0 \le V_1 \le N\). Let \(\tilde{V}_0 = \max \big \{N^{1/2}, V_0\big \}\) and \(\tilde{V}_1 = \max \big \{N^{1/2}, V_1\big \}\). We have
By the partial summation we obtain
where for \(x \ge 2\), we have set
Analogously, we can write
Furthermore, in view of the Siegel–Walfisz theorem ([20, 23], see also [17, Corollary 11.21]), there are \(C, c > 0\) such that for all \(x \ge 2\), \((r, q) = 1\) and \(1 \le q \le (\log x)^{2 \beta '}\),
Hence, by (4.7), (4.8) and (4.5), we obtain
Thus,
In view of (4.1), similar arguments applied to the sums over \(p_2, \ldots , p_{k''}\) lead to
By [11, Proposition 3.1], the number of lattice points in \(B_N\) at the distance \(<q\) from the boundary of \(B_N\) is \(\mathcal {O}(q N^{k-1})\). Moreover, for each \((x, y) \in [0, 1]^k\), and \((q u + q x, v + y) \in B_N\), we have
Finally, another application of the mean value theorem allows us to replace the sums by the corresponding integrals. Indeed, we have
which is again bounded by \(q N^{k-1} L\). Therefore,
In particular, taking \(\xi = 0\), \(a = 0\) and \(L = 1\), we obtain
This completes the proof. \(\square \)
Lemma 3
For each \(\alpha > 0\) there is \(C > 0\) such that for all \(N \in \mathbb {N}\), and \(\xi \in \mathbb {T}^d\) satisfying
where \(1 \le q \le L\), \(a \in \mathbf {A}_q\), and \(1 \le L \le \exp \big (c \sqrt{\log N}\big ) (\log N)^{-\alpha }\), we have
Proof
Given \(\alpha > 0\), let \(\beta ' \ge d \beta _\alpha \), where \(\beta _\alpha \) is the value determined in Theorem 2.
Suppose that (4.11) holds for some \((\log N)^{\beta '} < q \le L\) and \(a \in \mathbf {A}_q\). For each \(\gamma \in \Gamma \), by Dirichlet’s principle there are coprime integers \(a'_\gamma \) and \(q'_\gamma \) such that \(1 \le a_\gamma ' \le q_\gamma ' \le N^{\left|{\gamma } \right|} L^{-1} (\log N)^{-\beta '/d}\), and satisfying
Assume that for some \(\gamma \in \Gamma \), \((\log N)^{\beta '/d} \le q_\gamma ' \le N^{\left|{\gamma } \right|} L^{-1} (\log N)^{-\beta '/d}\). Then, by Theorem 2, we have
If for all \(\gamma \in \Gamma \), \(1 \le q_\gamma ' \le (\log N)^{\beta '/d}\), then we set \(q'' = {\text {lcm}}\big (q_\gamma ' : \gamma \in \Gamma \big )\) and \(a''_\gamma = a_\gamma ' q''/q_\gamma '\) getting \(1 \le q'' \le (\log N)^{\beta '}\) and \(a'' \in \mathbf {A}_{q''}\) with
Since \(a'/q' \ne a/q\),
which is possible only for finite number of N’s.
Finally, in the case when \(1 \le q \le (\log N)^{\beta '}\), by Proposition 4.1, we obtain
which concludes the proof. \(\square \)
Lemma 4
For all \(p \in [1, \infty )\), \(N_1, N_2 \in \mathbb {N}\), \(N_1 < N_2\), and any \(f \in \ell ^p\big (\mathbb {Z}^d\big )\),
Proof
Let us denote by \(m_n\) the convolution kernel corresponding to \(M_n\). Consider \((x, y) \in \mathbb {N}^{k'} \times \mathbb {P}^{k''}\). If \((x, y) \in B_{N_1}\) then
If \((x, y) \in B_{N_2} {\setminus }B_{N_1}\) then by setting
we have
Therefore,
and hence, by Young’s inequality,
which finishes the proof since \(\vartheta _B(N_1) \simeq N_1^k\). \(\square \)
4.2 Truncated discrete singular operators
In this section we investigate the asymptotic of Fourier multipliers corresponding to the truncated discrete singular operators \(H_N\) with a kernel K satisfying (1.2) and (1.3). Let \(\mathfrak {h}_N\) be the Fourier multiplier corresponding to \(H_N\), that is for a finitely supported function \(f: \mathbb {Z}^d \rightarrow \mathbb {C}\),
where
We also define
In view of a multi-dimensional version of van der Corput’s lemma (see [22, Proposition 2.1]), for \(N < N' \le 2N\),
Moreover, by (1.3),
Hence,
We start with a proposition analogous to Proposition 4.1.
Proposition 4.2
For each \(\beta ' > 0\) there \(C, c > 0\) such that for all \(N < N' \le 2N\), and \(\xi \in \mathbb {T}^d\) satisfying
where \(1 \le q \le (\log N)^{\beta '}\), \(a \in \mathbf {A}_q\), and \(1 \le L \le \exp \big (c \sqrt{\log N}\big )\), we have
Proof
For a prime number p, \(p \mid q\) if and only if \(\big (p \bmod q, q \big ) > 1\). Therefore, by (1.1), (1.2), and the prime number theorem, for any \(s \in \{1, \ldots , k''\}\),
To simplify the notations, for \((x, y) \in \mathbb {R}^k{\setminus } \{0\}\), we set
where \(\theta = \xi - a/q\). For any \((u, p) \in \mathbb {N}^{k'} \times \mathbb {P}^{k''}\) such that \(u \equiv r' \bmod q\), and \(p \equiv r'' \bmod q\), we have
thus
Fix \(u \in \mathbb {N}^{k'}\), \(\tilde{p} \in \mathbb {P}^{k''-1}\) and \(r_1'' \in A_q\). Then
for some \(1 \le V_0 \le V_1 \le N' \le 2 N\). Let \(\tilde{V}_0 = \max \big \{N^{1/2}, V_0\big \}\) and \(\tilde{V}_1 = \max \big \{N^{1/2}, V_1\big \}\). We have
By the partial summation
Analogously, we have
Hence, by (4.9) and (1.2), we obtain
Therefore,
By similar reasonings applied to the sums over \(p_2, \ldots , p_{k''}\), one can show that
Since for each \((x, y) \in [0, 1]^k\) and \((qu+q x, v+y) \in B_{N'} {\setminus } B_N\), we have
and
thus by the mean value theorem, we obtain
Moreover, in view of [11, Proposition 3.1], the number of lattice points in \(B_N\) of distance \(<q\) from the boundary of \(B_N\) is \(\mathcal {O}(q N^{k-1})\). Therefore,
Lastly, we can replace the sums by the corresponding integrals because
which is bounded by \(q N^{-1} L\). \(\square \)
Analogously to Lemma 3, we can prove the following statement.
Lemma 5
For each \(\alpha > 0\) there is \(C > 0\) such that for all \(N \le N' \le 2N\), and \(\xi \in \mathbb {T}^d\) satisfying
where \(1 \le q \le L\), \(a \in \mathbf {A}_q\), and \(1 \le L \le \exp \big (c \sqrt{\log N} \big )(\log N)^{-\alpha }\),
Lemma 6
For all \(p \in [1, \infty )\), \(N_1, N_2 \in \mathbb {N}\), \(N_1 < N_2\), and any \(f \in \ell ^p\big (\mathbb {Z}^d\big )\),
Proof
Let \(h_n\) denote the convolution kernel corresponding to \(H_n\). Observe that for \((x, y) \in \mathbb {Z}^{k'} \times (\pm \mathbb {P})^{k''}\), if \((x, y) \in B_{N_2} {\setminus } B_{N_1}\) then
otherwise the sum equals zero. Thus, by (1.2), we obtain
hence, by Young’s inequality,
which completes the proof. \(\square \)
5 Variational estimates
In this section we present the estimates for \(\ell ^p\big (\mathbb {Z}^d\big )\) norm of the r-variational seminorm for the averaging operators \((M_N : N \in \mathbb {N})\) and the truncated discrete singular operators \((H_N : N \in \mathbb {N})\). In order to give a unified approach, we set \((Y_N : N \in \mathbb {N})\) to be any of them. By \((\mathfrak {y}_N : N \in \mathbb {N})\) we denote the corresponding discrete Fourier multipliers and by \((\Upsilon _N : N \in \mathbb {N})\) its continuous counterparts. We start by listing properties that are sufficient to obtain r-variational estimates. Let \(\rho \in (0, 1)\) and set \(N_n = \big \lfloor 2^{n^\rho } \big \rfloor \).
Property 1
In view of [12] (see also [9]) for each \(p \in (1, \infty )\) there is \(C_p > 0\) such that for all \(r \in (2, \infty )\) and any function \(f \in L^p\big (\mathbb {R}^d \big ) \cap L^2\big (\mathbb {R}^d \big )\),
and
Property 2
By (4.3) and (4.12), for each \(n \in \mathbb {N}\),
where A is the matrix defined in (2.8).
Property 3
By Lemmas 4 and 6 we deduce that for each \(p \in (1, \infty )\) and any \(f \in \ell ^p\big (\mathbb {Z}^d\big )\),
because by (4.10),
In particular,
Property 4
By Theorem 2 and partial summation for each \(\alpha > 0\), there is \(\beta _\alpha > 0\) so that for any \(\beta > \beta _\alpha \), and \(n \in \mathbb {N}\), if there is \(\gamma _0 \in \Gamma \), such that
for some coprime numbers a and q such that \(1 \le a \le q\), and \((\log N_n)^\beta \le q \le N_n^{\left|{\gamma _0} \right|} (\log N_n)^{-\beta }\), then
Property 5
By Propositions 4.1 and 4.2, for each \(\beta ' > 0\) there is \(C > 0\) such that for all \(n \in \mathbb {N}\), and \(\xi \in \mathbb {T}^d\), satisfying
where \(1 \le q \le (\log N_n)^{\beta '}\), \(a \in \mathbf {A}_q\), and \(1 \le L \le \exp \big (c\sqrt{\log {N_n}}\big )\), we have
Property 6
By Lemmas 3 and 5, for each \(\alpha > 0\), all \(n \in \mathbb {N}\), and \(\xi \in \mathbb {T}^d\), satisfying
where \(1 \le q \le L\), \(a \in \mathbf {A}_q\), and \(1 \le L \le \exp \big (c\sqrt{\log {N_n}}\big ) (\log N_n)^{-\alpha }\), we have
Before we embark on proving variational estimates, we show the following auxiliary result.
Proposition 5.1
For each \(p \in (1, \infty )\) there is \(C > 0\), such that for all increasing sequences of integers \((n_j : j \in \mathbb {N})\) and any function \(f \in L^p\big (\mathbb {R}^d\big ) \cap L^2\big (\mathbb {R}^d\big )\),
Proof
For each \(j \in \mathbb {N}\), such that
we write
For every \(j_1, j_2 \in \mathbb {N}\), \(j_1 < j_2\) such that
we estimate
Hence, for some increasing sequence of integers \((m_j : j \in \mathbb {N})\), we have
The conclusion now follows by [5] and Property 1. \(\square \)
The aim of this section is to prove the following theorem.
Theorem 4
For each \(p \in (1, \infty )\) and \(r \in (2, \infty )\) there is \(C > 0\) such that for any finitely supported function \(f: \mathbb {Z}^d \rightarrow \mathbb {C}\),
We split a variational seminorm into two parts long variations\(V_r^L\), and short variations\(V_r^S\), where
and
respectively. Then
We first estimate \(\ell ^p\)-norm of long variations.
5.1 Long variations
Let \(\beta \in \mathbb {N}\) which value will be determined later. Take \(\rho \in (0, 1)\) and \(0< \chi < \tfrac{1}{10}\min \{1, c\}\) where c is the constant from Lemma 3. For each \(n \in \mathbb {N}\), we define the multiplier
where the sets \(\mathscr {U}_{\lfloor n^\rho \rfloor }^\beta \) are given by (2.9) and
We write
We now separately estimate each term on the right-hand side of (5.7). We notice that in view of (5.3) and (2.10), we have
In fact, for \(p = 2\), we can gain some decay in n. Given \(\alpha > 0\), we select \(\beta _\alpha \) to be determined by Property 4. Let \(\beta > d \beta _\alpha \). Take any \(\xi \in \mathbb {T}^d\). By Dirichlet’s principle, for each \(\gamma \in \Gamma \), there are coprime integers \(a_\gamma \) and \(q_\gamma \), such that \(1 \le a_\gamma \le q_\gamma \le N_n^{\left|{\gamma } \right|} (\log N_n)^{-\beta /d}\), and
Suppose that \(1 \le q_\gamma \le (\log N_n)^{\beta /d}\), for all \(\gamma \in \Gamma \). We set \(q' = {\text {lcm}}\big (q_\gamma : \gamma \in \Gamma \big )\) and \(a'_\gamma = a_\gamma q'/q_\gamma \). Observe that for all \(\gamma \in \Gamma \), we have
provided that
which excludes only a finite number of n’s depending on \(\beta \) and \(\rho \). In particular, \(\eta _{n}(\xi - a'/q') = 1\). Since \(1 \le q' \le (\log N_n)^\beta \), \(a' \in \mathbf {A}_{q'}\), we conclude that \(\Xi _{n+1}^{\beta }(\xi ) = 1\). Hence, the condition \(\Xi _{n+1}^{\beta }(\xi ) < 1\) implies that \((\log N_n)^{\beta /d} \le q_\gamma \le N_n^{\left|{\gamma } \right|} (\log N_n)^{-\beta /d}\) for some \(\gamma \in \Gamma \). Now, by Property 4, we obtain
which entails that
Interpolation between (5.8) and (5.9), shows that for each \(p \in (1, \infty )\) and \(\alpha > 0\) there is \(\beta _{p, \alpha } > 0\) such that for all \(\beta > \beta _{p, \alpha }\) and \(n \in \mathbb {N}\), we have
Taking \(\beta > \beta _{p, 2 \rho ^{-1}}\), we get
We now turn to bounding the first term on the right-hand side of (5.7). For each \(n \in \mathbb {N}\) and \(s \in \{0, \ldots , n-1\}\) let us define the multiplier
where \(\mathscr {R}_s^\beta =\mathscr {U}_{\lfloor (s+1)^\rho \rfloor }^\beta {\setminus } \mathscr {U}_{\lfloor s^\rho \rfloor }^\beta \). By the triangle inequality we can write
Thus, the aim is to show that for each \(\beta \in \mathbb {N}\), \(p \in (1, \infty )\), \(s \in \mathbb {N}_0\), and \(r \in (2, \infty )\),
We split the variational seminorm into two parts: \(s < n \le 2^{\kappa _s}\) and \(2^{\kappa _s} < n\), where
We begin with \(p = 2\) and \(s < n \le 2^{\kappa _s}\).
Theorem 5
For each \(\beta \in \mathbb {N}\) there is \(C > 0\) such that for all \(s \in \mathbb {N}_0\), \(r \in (2, \infty )\) and any finitely supported function \(f: \mathbb {Z}^d \rightarrow \mathbb {C}\), we have
where \(\delta \) is determined in Theorem 3.
Proof
First, let us see that for each \(m > s\), supports of functions \(\eta _{m}(\cdot - a/q)\) are disjoint while a / q varies over \(\mathscr {R}_s^\beta \). Indeed, otherwise there would be \(a/q, a'/q' \in \mathscr {R}_s^\beta \), \(a'/q' \ne a/q\) and \(\xi \in \mathbb {T}^d\), such that \(\eta _{m}(\xi - a/q) > 0\) and \(\eta _{m}(\xi - a'/q') > 0\). Hence,
which is impossible.
Next, we consider the following multiplier
Let us see that \(\Lambda _{n, s}^\beta \) is sufficiently close to \((\mathfrak {y}_{N_n} - \mathfrak {y}_{N_{n-1}}) \Xi _{j, s}^{\beta }\). For each \(a/q \in \mathscr {R}^\beta _s\), we have \(q \le \exp \big (\tfrac{c}{2} \sqrt{\log N_n} \big )\), thus by (5.5), on the support of \(\eta _n(\cdot - a/q)\) we can write
Therefore,
and hence,
Therefore, our task is reduced to showing boundedness of the first term on the right-hand side of (5.12). Observe that for \(n > s\), \(\eta _n = \eta _n \eta _s\), thus we can write
where
and
Now, in view of Lemma 1,
where \(I^i_j = \big \{j 2^i, j2^i+1, \ldots , (j+1)2^i-1\big \}\). Let us consider a fixed \(i \in \{0, \ldots , \kappa _s\}\). To bound the norm of the square function on the right-hand side of (5.13), we first study its continuous counterpart, that is
If \(\eta _m(\xi ) < 1\) then
thus by Property 2,
Therefore,
Now, by Proposition 5.1, we have
thus, in view of (2.10), we conclude that
Therefore, by (5.13), we arrive at the
Finally, by Plancherel’s theorem
and hence, by Theorem 3,
which together with (5.14) and (5.12) concludes the proof. \(\square \)
Theorem 6
For each \(\beta \in \mathbb {N}\) and \(p \in (1, \infty )\) there is \(C > 0\), such that for all \(s \in \mathbb {N}_0\), \(r \in (2, \infty )\), and any finitely supported function \(f: \mathbb {Z}^d \rightarrow \mathbb {C}\), we have
Proof
For the proof, let us consider the following multiplier
Fix \(s< n_1 < n_2 \le \min \big \{2^{\kappa _s}, 2 n_1\big \}\). Let \(J_{n_1} = N_{n_1} 2^{-3 \chi \sqrt{\log N_{n_1}}}\). We claim the following holds true.
Claim 2
For each \(\beta \in \mathbb {N}\) and \(p \in (1, \infty )\) there is \(C > 0\), such that for all \(n_1 \le n \le n_2 \le 2n_1\),
The constant C is independent of \(n_1\) and \(n_2\).
Let us first observe that, by (5.3), we can write
thus, by (2.10),
We can improve the estimate for \(p = 2\). Namely, we are going to show that for each \(\alpha > 0\), and \(n_1 \le n \le n_2 \le 2n_1\),
Given \(\alpha > 0\), let c be the minimal value among those determined in Lemmas 3 and 5. Then for each \(a/q \in \mathscr {R}_s^\beta \),
If \(\eta _{s}(\xi - a/q) - \eta _{n}(\xi - a/q) \ne 0\), then
thus, by Property 2,
Moreover, if \(\eta _{n}(\xi - a/q) > 0\) then
hence, by (4.2), we obtain
Therefore,
Since the functions \(\eta _{s}(\cdot - a/q)\) have disjoint supports provided that \(a \in \mathbf {A}_q\) and \(1 \le q \le e^{(s+1)^{\rho /10}}\), by (5.18) and Plancherel’s theorem we conclude (5.17). Now, by interpolation between (5.17) and (5.16) we arrive at (5.15).
With a help of Claim 2, we obtain
Hence,
with an implied constant independent of \(n_1\). We next claim that the following holds true.
Claim 3
For each \(\beta \in \mathbb {N}\) and \(p \in (1, \infty )\) there is \(C > 0\), such that for all \(s \in \mathbb {N}_0\), we have
Let us see that (5.20) suffices to finish the proof of the theorem. Indeed, (5.19) together with (5.20) imply that
Therefore, by (2.2) and Minkowski’s inequality
It remains to prove Claim 3. By Lemma 1, we can write
where \(I_j^i = \big \{j 2^i, j 2^i+1, \ldots , (j+1) 2^i-1\big \}\). Let us fix \(i \in \{0, 1, \ldots \kappa _s\}\). In view of Proposition 5.1,
where the implied constant is independent of i. Hence, by (2.10), we obtain
which together with (5.21) implies (5.20). \(\square \)
We now turn to studying the part of the variational seminorm where \(2^{\kappa _s} < n\). For \(s \in \mathbb {N}_0\) we set
Theorem 7
For each \(\beta \in \mathbb {N}\) there is \(C > 0\), such that for all \(r \in (2, \infty )\), \(s \in \mathbb {N}_0\), and any finitely supported function \(f: \mathbb {Z}^d \rightarrow \mathbb {C}\), we have
where \(\delta \) is determined in Theorem 3.
Proof
Let us define
where
Our first goal is to show that the multipliers \(\Omega _{n, s}^\beta \) approximate \((\mathfrak {y}_{N_n} - \mathfrak {y}_{N_{n-1}}) \Xi _{n, s}^{\beta }\) well.
Claim 4
For each \(\beta \in \mathbb {N}\) there is \(C > 0\), such that for all \(s \in \mathbb {N}_0\), and \(n > 2^{\kappa _s}\),
Since \(n > 2^{\kappa _s}\), for each \(a/q \in \mathscr {R}_s^\beta \) we have \(q \le \log N_n\). Therefore, by (5.4), we obtain
Next, if \(\varrho _s(\xi - a/q) - \eta _{n}(\xi -a/q) \ne 0\), then
and thus, by (5.1), we have
Hence,
Since the functions \(\eta _{s}(\cdot - a/q)\) have disjoint supports while a / q varies over \(\mathscr {R}_s^\beta \), by Plancherel’s theorem we obtain (5.22).
Now, by applying Claim 4,
thus
Our next task is to show that there is \(C > 0\) such that
For the proof, let us define
and
By Plancherel’s theorem, for any \(u \in \mathbb {N}^d_{Q_s}\) and \(a/q \in \mathscr {R}_s^\beta \), we have
because in view of (5.1), for each \(\xi \in \mathbb {T}^d\),
Therefore,
Since the set \(\mathscr {R}_s^\beta \) has at most \(e^{(d+1)(s+1)^{\rho /10}}\) elements, and
we obtain
Hence,
Let us observe that the functions \(x \mapsto I(x, y)\) and \(x \mapsto J(x, y)\) are \(Q_s\mathbb {Z}^d\)-periodic. Therefore, by repeated change of variables, we get
By [11, Proposition 4.1] (see also [15, Proposition 3.2]), Property 1 entails that for each \(u \in \mathbb {N}_{Q_s}^d\), we have
Observe that
Since by Theorem 3 and disjointness of supports of \(\varrho _s(\cdot - a/q)\) while a / q varies over \(\mathscr {R}_s^\beta \), we get
we obtain
which together with (5.24) implies (5.23) and the proof of theorem is completed. \(\square \)
Theorem 8
For each \(\beta \in \mathbb {N}\) and \(p \in (1, \infty )\) there is \(C > 0\), such that for all \(s \in \mathbb {N}_0\), \(r \in (2, \infty )\), and any finitely supported function \(f: \mathbb {Z}^d \rightarrow \mathbb {C}\),
Proof
First, we are going to refine Claim 4.
Claim 5
For each \(\beta \in \mathbb {N}\) and \(p \in (1, \infty )\) there is \(c_p > 0\) such that for all \(s \in \mathbb {N}_0\), and \(n > 2^{\kappa _s}\),
We notice the following trivial bound
thus, by (5.3) and (2.10), we also have
Now, interpolation between (5.26) and (5.22) leads to (5.25).
Next, using Claim 5, we obtain
Hence,
and the proof is reduced to showing the following claim.
Claim 6
For each \(\beta \in \mathbb {N}\) and \(p \in (1, \infty )\) there is \(C > 0\) such that for all \(r \in (2, \infty )\), and \(s \in \mathbb {N}_0\),
For any \(a/q \in \mathscr {R}_s^\beta \), \(x \in \mathbb {Z}^k\) and \(m \in \mathbb {N}_{Q_s}^k\), we have
Therefore,
where
By [11, Proposition 4.2] (see also [15, Proposition 3.2]), we can write
Therefore, the problem is reduced to showing
For the proof, let \(N = \lfloor e^{(s+1)^{\rho /10}} \rfloor +1\) and \(J = 2^N\). We write
In view of (2.10), we have
Next, we have the following trivial bound
We want to improve the above estimate for \(p =2\). We have
Since each fraction a / q belonging to \(\mathscr {R}_s^\beta \) has its denominator \(q \le e^{(s+1)^{\rho /10}} \le \log J\), by Proposition 4.1,
If \(\varrho _s(\xi - a/q) > 0\) then
thus, by (4.2), we get
Hence, (5.31) takes the following form
Therefore, by (5.30), we get
Now, interpolating (5.29) with (5.32), we obtain
which together with (5.28) implies (5.27), and the proof of the theorem is completed. \(\square \)
Theorem 9
For each \(p \in (1, \infty )\) and \(\rho \in (0, 1)\), there is \(C > 0\) such that for all \(r \in (2, \infty )\) and any finitely supported function \(f: \mathbb {Z}^d \rightarrow \mathbb {C}\),
where \(N_n = \big \lfloor 2^{n^\rho } \big \rfloor \).
Proof
In view of (5.7), (5.10) and (5.11), we have
provided \(\beta > \beta _{p, 2 \rho ^{-1}}\). Next, we split the index set
By interpolation between Theorems 5 and 6, and between Theorems 7 and 8, for \(\beta \) sufficiently larger we get
and
and the theorem follows. \(\square \)
5.2 Short variations
Theorem 10
For each \(p \in (1, \infty )\) there are \(\rho \in (0, 1)\) and \(C > 0\) such that for all \(r \in (2, \infty )\) and any finitely supported function \(f : \mathbb {Z}^d \rightarrow \mathbb {C}\), we have
Proof
Let \(u = \min \{2, p\}\). By monotonicity and Minkowski’s inequality, we get
which together with (5.2) gives
which is bounded whenever \(0< \rho < \frac{u-1}{u}\). \(\square \)
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Communicated by Loukas Grafakos.
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Trojan, B. Variational estimates for discrete operators modeled on multi-dimensional polynomial subsets of primes. Math. Ann. 374, 1597–1656 (2019). https://doi.org/10.1007/s00208-018-1777-6
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DOI: https://doi.org/10.1007/s00208-018-1777-6