Abstract
We prove the uniform oscillation and jump inequalities for the polynomial ergodic averages modeled over multi-dimensional subsets of primes. This is a contribution to the Rosenblatt–Wierdl conjecture (Lond Math Soc Lect Notes 205:3–151, 1995, Problem 4.12, p. 80) with averages taken over primes. These inequalities provide endpoints for the r-variational estimates obtained by Trojan (Math Ann 374:1597–1656, 2019).
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1 Introduction
The aim of this paper is to prove uniform oscillation inequalities and \(\lambda \)-jump inequalities in the context of polynomial ergodic averages and truncated singular operators of the Cotlar type modeled on multi-dimensional subset of primes. We extend the known results of Trojan [37] for the r-variation seminorm \(V^r\) with \(r>2\) to endpoint cases expressed in terms of the uniform jump and oscillation inequalities. This provides a fuller quantitative description of the pointwise convergence of the mentioned averages.
1.1 Statement of results
Let \((X,\mathcal {B},\mu )\) be a \(\sigma \)-finite measure space endowed with a family of invertible commuting and measure preserving transformations \(S_1,\ldots , S_d:X\rightarrow X\). Let \(\Omega \) be a bounded convex open subset of \(\mathbb {R}^k\) such that \(B(0,c_{\Omega }) \subseteq \Omega \subseteq B(0,1)\) for some \(c_{\Omega }\in (0, 1)\), where B(0, u) is the open Euclidean ball in \(\mathbb {R}^k\) with radius \(u>0\) centered at \(0\in \mathbb {R}^k\). For any \(t>0\), we set
We consider a polynomial mapping
where each \(\mathcal {P}_j:\mathbb {Z}^k\rightarrow \mathbb {Z}\) is a polynomial of k variables with integer coefficients such that \(\mathcal {P}_j(0)=0\). Let \(k',k''\in \{0,1,\ldots ,k\}\) with \(k=k'+k''\). For \(f\in L^\infty (X,\mu )\), we define the associated ergodic averages by
where \(\pm \mathbb {P}\) denotes the set of positive and negative prime numbers and
is the Chebyshev function. We also consider the Cotlar type ergodic averages given by
where \(K:\mathbb {R}^{k}{\setminus }\{0\} \rightarrow \mathbb {C}\) is a Calderón–Zygmund kernel satisfying the following conditions:
-
1.
The size condition: For every \(x\in \mathbb {R}^k{\setminus }\{0\}\), we have
$$\begin{aligned} |K(x)| \lesssim |x|^{-k}. \end{aligned}$$(1.4) -
2.
The cancellation condition: For every \(0<r<R<\infty \), we have
$$\begin{aligned} \int _{\Omega _{R}{\setminus } \Omega _{r}} K(y) \text {d}y = 0. \end{aligned}$$(1.5) -
3.
The Lipschitz continuity condition: For every \(x, y\in \mathbb {R}^k{\setminus }\{0\}\) with \(2|y|\le |x|\), we have
$$\begin{aligned} |K(x)-K(x+y)| \lesssim |y| |x|^{-(k+1)}. \end{aligned}$$(1.6)
We recall the definitions of the oscillation seminorm and \(\lambda \)-jump counting function. Let \(\mathbb {I}\subseteq \mathbb {R}\). For an increasing sequence \(I=(I_j: j\in \mathbb {N})\subseteq \mathbb {I}\) and \(N\in \mathbb {N}\cup \{\infty \}\), the truncated oscillation seminorm of a function \(f:\mathbb {I}\rightarrow \mathbb {C}\) is defined by
For any \(\lambda >0\) and \(\mathbb {I}\subseteq \mathbb {R}\), the \(\lambda \)-jump counting function of a function \(f:\mathbb {I}\rightarrow \mathbb {C}\) is defined by
We can now state the main result of this paper.
Theorem 1
Let \(d, k\ge 1\) and let \(\mathcal {P}\) be a polynomial mapping as in (1.1). Let \(k',k''\in \{0,1,\ldots ,k\}\) with \(k'+k''=k\) and let \(\mathcal {M}_t^{\mathcal {P},k',k''}\) be either \(\mathcal {A}_t^{\mathcal {P},k',k''}\) or \(\mathcal {H}_t^{\mathcal {P},k',k''}\). Then, for any \(p\in (1,\infty )\), there is a constant \(C_{p,d,k,\deg \mathcal {P}}>0\) such that
for any \(f\in L^p(X,\mu )\). Here, \(\mathfrak {S}_N(\mathbb {R}_+)\) is the set of all strictly increasing sequences in \(\mathbb {R}_+\) of length \(N+1\) (see Sect. 2.2). The constant \(C_{p,d,k,\deg \mathcal {P}}\) is independent of the coefficients of the polynomial mapping \(\mathcal {P}\).
In the proof of the above theorem, we use methods developed in [24, 28, 37] and very recently in [21, 34]. We follow Bourgain’s approach [6] to use the Calderón transference principle [7] which reduce the problem to the integer shift system (see Sect. 2.3) and then exploit the Hardy–Littlewood circle method to analyze the appropriate Fourier multipliers. The main tools used to handle the estimates for the multiplier operators are: an appropriate generalization of Weyl’s inequality (Proposition 6); the Ionescu–Wainger multiplier theorem (see [12, 28] and [36]) combined with the Rademacher–Menshov inequality (see [24]) and standard multiplier approximations (Lemma 8); the Magyar–Stein–Wainger sampling principle [23] and [26].
As a consequence of Theorem 1, we can state the following quantitative form of the ergodic theorem concerning the averages \(\mathcal {A}_t^{\mathcal {P},k',k''}\) and \(\mathcal {H}_t^{\mathcal {P},k',k''}\).
Corollary 2
Let \((X,\mathcal {B},\mu )\) be a \(\sigma \)-finite measure space. Let \(d, k\ge 1\) and let \(\mathcal {P}\) be a polynomial mapping as in (1.1). Let \(k',k''\in \{0,1,\ldots ,k\}\) with \(k'+k''=k\) and let \(\mathcal {M}_t^{\mathcal {P},k',k''}\) be either \(\mathcal {A}_t^{\mathcal {P},k',k''}\) or \(\mathcal {H}_t^{\mathcal {P},k',k''}\). Let \(p\in (1,\infty )\) and \(f\in L^p(X,\mu )\). Then we have:
-
(i)
(Mean ergodic theorem) the averages \(\mathcal {M}_t^{\mathcal {P},k',k''}f\) converge in \(L^p(X,\mu )\) norm as \(t\rightarrow \infty \);
-
(ii)
(Pointwise ergodic theorem) the averages \(\mathcal {M}_t^{\mathcal {P},k',k''}f\) converge pointwise \(\mu \)-almost everywhere on X as \(t\rightarrow \infty \);
-
(iii)
(Maximal ergodic theorem) the following maximal estimate holds:
$$\begin{aligned} \Bigg \Vert \sup _{t>0}|\mathcal {M}_t^{\mathcal {P},k',k''}f|\Bigg \Vert _{L^p(X, \mu )}\lesssim _{d,k,p, \deg \mathcal P}\Vert f\Vert _{L^p(X, \mu )}; \end{aligned}$$(1.11) -
(iv)
(Oscillation ergodic theorem) the following uniform oscillation inequality holds:
$$\begin{aligned} \sup _{N\in \mathbb {N}}\sup _{I\in \mathfrak S_N(\mathbb {R}_+) }\Bigg \Vert O_{I, N}^2(\mathcal {M}_t^{\mathcal {P},k',k''}f: t>0)\Bigg \Vert _{L^p(X, \mu )}\lesssim _{d,k,p, \deg \mathcal P}\Vert f\Vert _{L^p(X, \mu )}; \end{aligned}$$(1.12) -
(v)
(Variational ergodic theorem) for any \(r\in (2,\infty )\), the following r-variational inequality holds (see Sect. 2.2 for the definition of \(V^r\)):
$$\begin{aligned} \Bigg \Vert V^r(\mathcal {M}_t^{\mathcal {P},k',k''}f: t>0)\Bigg \Vert _{L^p(X, \mu )}\lesssim _{d,k,p, r, \deg \mathcal P}\Vert f\Vert _{L^p(X, \mu )}; \end{aligned}$$(1.13) -
(vi)
(Jump ergodic theorem) the following jump inequality holds:
$$\begin{aligned} \sup _{\lambda>0}\Bigg \Vert \lambda N_{\lambda }(\mathcal {M}_t^{\mathcal {P},k',k''}f:t>0)^{1/2}\Bigg \Vert _{L^p(X,\mu )}\lesssim _{d,k,p, \deg \mathcal P}\Vert f\Vert _{L^p(X, \mu )}. \end{aligned}$$(1.14)
The implicit constants in (1.11), (1.12), (1.13), and (1.14) are independent of the coefficients of the polynomial mapping \(\mathcal {P}\).
A few comments are in order.
-
1.
Corollary 2 is the most general quantitative version of the one parameter ergodic theorem for both averages \(\mathcal {A}_t^{\mathcal {P},k',k''}\) and \(\mathcal {H}_t^{\mathcal {P},k',k''}\) (cf. [22, Theorem 1.20]), which concludes the work of many authors over last decades—see Sect. 1.2 for details.
-
2.
The mean ergodic theorem (i) easily follows from (ii) and (iii) by Lebesgue’s dominated convergence theorem. Each inequality from (iv), (v), and (vi) individually implies pointwise convergence (ii) and the maximal estimate (iii). The jump inequality (vi) implies the variational ergodic theorem (v) in the full range \(r\in (2,\infty )\). Hence, the inequality (1.14) can be seen as an \(r=2\) endpoint for (1.13).
-
3.
Unfortunately, we do not know at this moment if the oscillation inequality (1.12) is any kind of endpoint for the variational inequality (1.13). A recent result from [21] shows that the oscillation estimates cannot be interpreted as an endpoint in a way similar to how the jump inequalities are. See the discussion in [21] and [22].
-
4.
The oscillation inequality (1.12) for the ergodic averages \(\mathcal {A}_t^{\mathcal {P},k',k''}\) can be seen as a contribution to a problem posed by Rosenblatt and Wierdl [31, Problem 4.12, p. 80] in the early 1990’s about uniform oscillation inequalities for the classical Birkhoff ergodic averages given by
$$\begin{aligned} \frac{1}{2N+1}\sum _{n=-N}^Nf(S^nx). \end{aligned}$$(1.15)In 1998, Jones, Kaufman, Rosenblatt, and Wierdl [14] gave an affirmative answer to this problem. The inequality (1.10) provides us with the uniform oscillation inequality for the counterpart of (1.15) along the prime numbers given by
$$\begin{aligned} \frac{1}{2|\mathbb {P}_N|}\sum _{n=-N}^N{f(S^nx)\mathbbm {1}_{\mathbb {P}}(|n|)}, \end{aligned}$$where \(\mathbb {P}_N=\mathbb {P}\cap [1,N]\). Moreover, the inequality (1.10) is much more general than the originally posted problem since it concerns the multi-dimensional averages along arbitrary polynomials with integer coefficients.
-
5.
Parts (i), (ii), (iii), and (v) for the standard averages \(\mathcal {A}_t^{\mathcal {P},k',k''}\) with \(k''\ge 1\) in the presented generality were first obtained by Trojan [37]. In the case with \(k''=0\) (excluding the prime numbers from the summation), the first proof of the variational inequality (1.12) in the full range \(r\in (2,\infty )\) was given by Mirek, Stein, and Trojan [24].
-
6.
In the case of the Cotlar ergodic averages \(\mathcal {H}_t^{\mathcal {P},k',k''}\) with \(k''\ge 1\), the ergodic theorems (i), (ii), (iii) and (v) were proven by Trojan [37] under the gradient condition
$$\begin{aligned} |x|^{k+1}|\nabla K(x)|\lesssim 1, \end{aligned}$$(1.16)but Trojan’s argument can be adapted with small changes to deal with Calderón–Zygmund kernels which satisfy the more general condition (1.6). In this case, the results for \(k''\ge 1\) seem to be completely new. For \(\mathcal {H}_t^{\mathcal {P},k,0}\), the jump inequality was obtained by Mirek, Stein, and Zorin-Kranich [28], and the oscillation ergodic theorem was obtained by the second author [34].
-
7.
The oscillation inequality (1.12) and the jump inequality (1.14) are completely new results for both types of averages when \(k''\ge 1\) and follows by Theorem 1. When \(k''=0\), the corresponding results for the jump inequalities are known due to the work of Mirek, Stein, and Zorin-Kranich [28]. The uniform oscillation inequality was proven by Mirek, Słomian, and Szarek [21] in the case of the averages \(\mathcal {A}_t^{\mathcal {P},k,0}\) and by the second author [34] in the case of \(\mathcal {H}_t^{\mathcal {P},k,0}\).
1.2 Historical background
In 1931, Birkhoff [3] and von Neumann [38] proved that the averages
converge pointwise \(\mu \)-almost everywhere on X and in \(L^p(X,\mu )\) norm respectively for any \(f\in L^p(X,\mu )\), \(p \in [1,\infty )\), as \(N\rightarrow \infty \). In 1955, Cotlar [9] established the pointwise \(\mu \)-almost everywhere convergence on X as \(N\rightarrow \infty \) of the ergodic Hilbert transform given by
for any \(f\in L^p(X,\mu )\). In 1968, Calderón [7] made an important observation (now called the Calderón transference principle) that some results in ergodic theory can be easily deduced from known results in harmonic analysis. Namely, the convergence of the Birkhoff averages \(M_N\) can be deduced from the boundedness of the Hardy–Littlewood maximal function, and the convergence of Cotlar’s averages \(H_N\) follows from the boundedness of the maximal function for the truncated discrete Hilbert transform. As we will see ahead, this observation has had a huge impact in the study of convergence problems in ergodic theory.
At the beginning of the 1980’s, Bellow [2] and independently Furstenberg [11] posed the problem about pointwise convergence of the averages along squares given by
Despite its similarity to Birkhoff’s theorem, the problem of pointwise convergence of the \(T_N\) averages has a totally different nature from that of its linear counterpart. In particular, the standard approach is insufficient in this case.
We briefly sketch the classical approach of handling the problem of pointwise convergence. It consists of two steps:
-
(a)
Establish \(L^p\)-boundedness for the corresponding maximal function.
-
(b)
Find a dense class of functions in \(L^p(X,\mu )\) for which the pointwise convergence holds.
In the case of Birkhoff’s averages \(M_N\), the Calderón transference principle allows one to deduce the estimate
for \(p\in (1,\infty ]\) from the estimate for the discrete Hardy–Littlewood maximal function (and we have a weak-type estimate for \(p=1\)). In turn, estimates for the discrete Hardy–Littlewood maximal function follow easily from those for the continuous one. This establishes the first step (a). For the second step, one can use the idea of Riesz decomposition [30] to analyze the space \({\mathbb {I}}_S\oplus {\mathbb {T}}_S\subseteq L^2(X,\mu )\), where
We see that \(M_Nf=f\) for \(f\in {\mathbb {I}}_S\) and, for \(g=h\circ S - h\in {\mathbb {T}}_S\), we have
by telescoping. Consequently, we see that \(M_Ng\rightarrow 0\) as \(N\rightarrow \infty \). This establishes \(\mu \)-almost everywhere pointwise convergence of \(M_N\) on \({\mathbb {I}}_S\oplus {\mathbb {T}}_S\), which is dense in \(L^2(X,\mu )\). Since \(L^2(X,\mu )\) is dense in \(L^p(X,\mu )\) for every \(p\in [1,\infty )\), this establishes (b).
In the case of the quadratic averages \(T_N\), the matter is more complicated. For the first step, by the Calderón transference principle, it is enough to establish \(\ell ^p\) bounds for the maximal function given by
The \(\ell ^p\) estimate for the above maximal function does not follow directly from the continuous counterpart and requires completely new methods. However, a more serious problem arises in connection with the second step. Namely, the idea of von Neumann fails in this case because the averages \(T_N g\) do not possess the telescoping property for \(g\in \mathbb {T}_S\).
At the end of the 1980’s, Bourgain established the pointwise convergence of the averages \(T_N\) in a series of groundbreaking articles [4,5,6]. By using the Hardy–Littlewood circle method from analytic number theory, he established \(\ell ^p\)-bounds for the maximal function (1.18), which establishes step (a). He then bypassed the problem of finding the requisite dense class of functions by using the oscillation seminorm (1.7). Bourgain [6] proved that, for any \(\lambda >1\) and any sequence of integers \(I=(I_j:{j\in \mathbb {N}})\) with \(I_{j+1}>2I_j\) for all \(j\in \mathbb {N}\), we have
for any \(f\in L^2(X,\mu )\) with \(\lim _{N\rightarrow \infty } N^{-1/2}C_{I, \lambda }(N)=0\). This non-uniform inequality (1.19) suffices to establish the pointwise convergence of the averaging operators \(T_Nf\) for any \(f\in L^2(X, \mu )\). In the same series of papers, by similar methods, Bourgain established the pointwise convergence of the averages along primes
for \(f\in L^p(X,\mu )\) with \(p>\frac{1}{2}(1+\sqrt{3})\). In the same year, Wierdl [40] extended Bourgain’s result to \(p\in (1,\infty )\).
In order to establish the inequality (1.19), Bourgain used the Hardy–Littewood circle method and r-variation seminorms \(V^r\). The r-variations were introduced by Lépingle [17] in the context of families of bounded martingales. In 1976, he proved that, for all \(r\in (2, \infty )\), \(p\in (1, \infty )\), and any family of bounded martingales \((\mathfrak f_n:X\rightarrow \mathbb {C}:n\in \mathbb {N})\), we have
with the implicit constant depending only on p and r. The above inequality is sharp in the sense that it fails for \(r=2\), see [13] for a counterexample.
Bourgain observed that the \(V^r\) seminorm can be used to obtain (1.19). This is because, by Hölder’s inequality, we have
for \(r\ge 2\). In order to prove the r-variational inequality for the averages \(T_N\), Bourgain used the \(\lambda \)-jump counting function. It can easily be seen that
for every \(r\ge 2\). The above inequality can be reversed in some sense [6]. Namely, for any \(p\in (1,\infty )\) and any \(r\in (2,\infty )\), we have
For more details about oscillation, variation, and jump seminorms, we refer to [16, 21, 22].
The above arguments demonstrate that the problem of proving pointwise convergence can be reduced to proving an appropriate r-variational estimate or jump inequality. However, an intriguing question was the issue of uniformity in the inequality (1.19). Shortly after the groundbreaking work of Bourgain, Lacey [31, Theorem 4.23, p. 95] improved inequality (1.19) showing that, for every \(\lambda >1\), there is a constant \(C_{\lambda }>0\) such that
where \(\mathbb {L}_{\tau }:=\{\tau ^n:n\in \mathbb {N}\}\). This result motivated the question about uniform estimates independent of \(\lambda >1\) in (1.20). In the case of Birkhoff’s averages, this question was explicitly formulated in [31, Problem 4.12, p. 80].
In 1998, Jones, Kaufman, Rosenblatt, and Wierdl [14] established the uniform oscillation inequality on \(L^p(X,\mu )\) for the standard Birkhoff averages \(M_N\). Two years later, Campbell, Jones, Reinhold, and Wierdl [8] established the uniform oscillation inequality for the ergodic Hilbert transform. In 2003, Jones, Rosenblatt, and Wierdl [15] proved uniform oscillation inequalities on \(L^p(X,\mu )\) with \(p\in (1,2]\) for the Birkhoff averages over cubes. However, the case of polynomial averages, even one-dimensional, was open until recent works [21, 34], and the case of averages along primes was open until this paper.
In 2015, Mirek and Trojan [19], using the ideas of Bourgain and Wierdl, established \(\mu \)-almost everywhere pointwise convergence of the Cotlar averages along the primes,
They proved that the corresponding maximal function is bounded on \(L^p(X,\mu )\) with \(p>1\) and showed that the analogue of Bourgain’s non-uniform oscillation inequality (1.19) holds for those averages.
In the same year, Zorin-Kranich [41] established the pointwise convergence of the averages related to the polynomial mapping given by
Namely, he proved that, for any \(r>2\) and \( \left| \frac{1}{p}-\frac{1}{2}\right| <\frac{1}{2(d+1)}\), we have the following r-variational estimate
As a consequence, the averages \(\mathcal {A}_N^{\tilde{\mathcal {P}},1,0}f\) converge \(\mu \)-almost everywhere for any \(f\in L^p(X,\mu )\).
In 2016, Mirek and Trojan [20] established the pointwise convergence for the averages (1.2) taken over cubes with \(k'=k\), that is
There, Mirek and Trojan noted for the first time that the Rademacher–Menshov inequality (2.5) may be used to establish r-variational estimates. For \(p\in (1,\infty )\) and \(r>\max \{p, p/(p-1)\}\), they proved that
Unfortunately, the methods introduced by Bourgain had limitations. These work perfectly fine in the case of the \(L^2\) estimates, but, in the case of an \(L^p\) estimates with \(p\ne 2\), there arise difficulties which are hard to overcome concerning the fractions around which major arcs are defined. However, Ionescu and Wainger [12], in their groundbreaking 2005 work about discrete singular Radon operators, introduced a set of fractions for which the circle method can be applied towards \(L^p\) estimates with \(p\ne 2\).
In 2015, Mirek [18] built a discrete counterpart of the Littlewood–Paley theory using the Ionescu–Wainger multiplier theorem and used it to reprove the main result from [12]. In 2017, Mirek, Stein, and Trojan [24, 25] further exploited these ideas together with the Rademacher–Menshov inequality from [20] to obtain an \(L^p\) estimate for the r-variation seminorm for both \(\mathcal {A}_t^{\mathcal {P},k,0}\) and \(\mathcal {H}_t^{\mathcal {P},k,0}\) associated with convex sets in the full range of parameters. Namely, they showed that
for \(p\in (1,\infty )\) and \(r\in (2,\infty )\), where \(\mathcal {M}_t^{\mathcal {P},k,0}\) is either \(\mathcal {A}_t^{\mathcal {P},k,0}\) or \(\mathcal {H}_t^{\mathcal {P},k,0}\). There, the operators \(\mathcal {H}_t^{\mathcal {P},k,0}\) are related to Calderón–Zygmund kernels satisfying the gradient condition (1.16).
In 2019, Trojan [37] proved an \(L^p\) estimate for the r-variation seminorm for both \(\mathcal {A}_t^{\mathcal {P},k',k''}\) and \(\mathcal {H}_t^{\mathcal {P},k',k''}\) with \(k',k''\in \{0,1,\ldots ,k\}\) such that \(k'+k''=k\). Namely, he showed that
for \(p\in (1,\infty )\) and \(r\in (2,\infty )\), where \(\mathcal {M}_t^{\mathcal {P},k',k''}\) is either \(\mathcal {A}_t^{\mathcal {P},k',k''}\) or \(\mathcal {H}_t^{\mathcal {P},k',k''}\). A straightforward consequence of the inequality (1.22) is the \(\mu \)-almost everywhere convergence of the averages \(\mathcal {M}_t^{\mathcal {P},k',k''}f\). Again, the operators \(\mathcal {H}_t^{\mathcal {P},k',k''}\) there are related to Calderón–Zygmund kernels satisfying the gradient condition (1.16).
In 2020, Mirek, Stein, and Zorin-Kranich [28] further refined the methods developed in [24, 25] and proved a uniform \(L^p\) estimate for the \(\lambda \)-jump counting function. They proved that
for any \(p\in (1,\infty )\) and any \(f\in L^p(X,\mu )\), where \(\mathcal {M}_t^{\mathcal {P},k,0}f\) is either \(\mathcal {A}_t^{\mathcal {P},k,0}f\) or \(\mathcal {H}_t^{\mathcal {P},k,0}f\). There, the operators \(\mathcal {H}_t^{\mathcal {P},k,0}\) are associated with Calderón–Zygmund kernels satisfying the Hölder continuity condition generalizing (1.6): For some \(\sigma \in (0,1]\) and for every \(x, y\in \mathbb {R}^k{\setminus }\{0\}\) with \(2|y|\le |x|\), we have
It is worth noting that the inequality (1.23) implies the r-variation inequality (1.21).
In 2021, the second author in collaboration with Mirek and Szarek [21] established the oscillation inequality
and, recently, the second author [34] proved the counterpart of (1.25) in the case of the operators \(\mathcal {H}_t^{\mathcal {P},k,0}\) related to Calderón–Zygmund kernels satisfying (1.24).
2 Notation and necessary tools
2.1 Basic notation
We denote \(\mathbb {N}:=\{1, 2, \ldots \}\), \(\mathbb {N}_0:=\{0,1,2,\ldots \}\), and \(\mathbb {R}_+:=(0, \infty )\). For \(d\in \mathbb {N}\), the sets \(\mathbb {Z}^d\), \(\mathbb {R}^d\), \(\mathbb {C}^d\), and \(\mathbb {T}^d = (\mathbb {R}/\mathbb {Z})^d \equiv [-1/2, 1/2)^d\) have the standard meanings. For each \(N\in \mathbb {N}\), we set
For any \(x\in \mathbb {R}\), we set
For \(u\in \mathbb {N}\), we define the set
For two non-negative numbers A and B, we write \(A \lesssim B\) to indicate that \(A\le CB\) for some \(C>0\) that may change from line to line, and we may write \(\lesssim _{\delta }\) if the implicit constant depends on \(\delta \).
We denote the standard inner product on \(\mathbb {R}^d\) by \( x\cdot \xi \). Moreover, for any \(x\in \mathbb {R}^d\), we denote the \(\ell ^2\)-norm and the maximum norm respectively by
For a multi-index \(\gamma =(\gamma _1,\dots ,\gamma _k)\in \mathbb {N}^k_0\), we abuse the notation to write \(|\gamma |:=\gamma _1+\cdots +\gamma _k\). No confusion should arise since all multi-indices will be denoted by \(\gamma \).
2.2 Seminorms
Let \(\mathbb {I}\subseteq \mathbb {R}\) and \(\lambda > 0\). 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 \(\mathbb {I}\). We already defined the oscillation seminorm (1.7) and the \(\lambda \)-jump counting function (1.8) in the introduction. For any \(r\in [1,\infty )\), the r-variation seminorm \(V^r\) of a function \(f:\mathbb {I}\rightarrow \mathbb {C}\) is defined by
The r-variational seminorm controls the oscillation seminorm and the \(\lambda \)-jump counting function. Indeed, by Hölder’s inequality, we have
for any \(N\in \mathbb {N}\), \(I\in \mathfrak {S}_N(\mathbb {I})\), and \(r \ge 2\). Moreover, for any \(\lambda >0\), we have
We adopt notation to simultaneously handle the oscillation seminorm and the \(\lambda \)-jump counting function for the sake of brevity and to emphasize the required properties. Let E be either of \(\mathbb {R}^d\) or \(\mathbb {Z}^d\) with the usual measures and let \((f_t:t\in \mathbb {I}) \subset L^p(E)\). We write
to represent either of the following quantities:
Proposition 3
Let \(p\in (1,\infty )\) and \(\mathbb {I}\subseteq \mathbb {R}\). The seminorm \(\mathcal {S}_E^p\) is subadditive up to a positive constant, that is,
where the implied constant is independent of \(\mathbb {I}\) and the families \((f_t:t\in \mathbb {I})\) and \((g_t:t\in \mathbb {I})\).
The critical point is that the jump quasi-seminorm admits an equivalent subadditive seminorm, see [28, Corollary 2.11].
Remark 2.4
(Rademacher–Menshov inequality) By inequalities (2.2) and (2.3), we deduce that the Rademacher–Menshov inequality [27, Lemma 2.5, p. 534] holds for \(\mathcal {S}_E^p\). Namely, for any \(k,m\in \mathbb {N}\) with \(k<2^m\) and any sequence of functions \((f_n:n\in \mathbb {N})\subset L^p(E)\), we have
where each \([u_j^i,u_{j+1}^i)\) is a dyadic interval contained in \([k, 2^m]\) of the form \([j2^i,(j+1)2^{i})\) for some \(0\le i\le m\) and \(0\le j\le 2^{m-i}-1\).
For more information about the \(\lambda \)-jump counting function and the oscillation and r-variation seminorms, we refer to [6, 16, 22, 26, 33].
2.3 Reductions: Calderón transference and lifting
By the Calderón transference principle [7], we may restrict attention to the model dynamical system of \(\mathbb {Z}^d\) equipped with the counting measure and the shift operators \(S_j:\mathbb {Z}^d\rightarrow \mathbb {Z}^d\) given by \(S_j(x_1,\ldots ,x_d):=(x_1,\ldots ,x_j-1,\ldots ,x_d)\). We denote the corresponding averaging operators by
and
Moreover, by a standard lifting argument, it suffices to prove Theorem 1 for a canonical case of the polynomial mapping \(\mathcal {P}\). Let \(\mathcal {P}\) be a polynomial mapping as in (1.1). We define
and consider the set of multi-indices
equipped with the lexicographic order. We define the canonical polynomial mapping by
where \(x^\gamma =x_1^{\gamma _1}x_2^{\gamma _2}\cdots x_k^{\gamma _k}\). By invoking the lifting procedure described in [25, Lemma 2.2] (see also [35, Section 11]), the following implies Theorem 1.
Theorem 4
Let \(k\in \mathbb {N}\), let \(\Gamma \subset \mathbb {N}^{k} {\setminus } \{0\}\) be a nonempty finite set, and let \(k',k''\in \{0,1,\ldots ,k\}\) with \(k'+k''=k\). Let \(M_t^{k',k''}\) be either \(A_t^{\mathcal {Q},k',k''}\) or \(H_t^{\mathcal {Q},k',k''}\). For any \(p\in (1,\infty )\), there is a constant \(C_{p,k,|\Gamma |}>0\) such that
2.4 Fourier transform and Ionescu–Wainger multiplier theorem
Let \(\mathbb {G}=\mathbb {R}^d\) or \(\mathbb {G}=\mathbb {Z}^d\) and let \(\mathbb {G}^*\) denote the dual group of \(\mathbb {G}\). For every \(z\in \mathbb {C}\), we set \(\varvec{e}(z):=e^{2\pi {\varvec{i}} z}\), where \({\varvec{i}}^2=-1\). Let \(\mathcal {F}_{\mathbb {G}}\) denote the Fourier transform on \(\mathbb {G}\) defined for any \(f \in L^1(\mathbb {G})\) by
where \(\mu \) is the usual Haar measure on \(\mathbb {G}\). For any bounded function \(\mathfrak m:\mathbb {G}^*\rightarrow \mathbb {C}\), we define the corresponding Fourier multiplier operator by
Here, we assume that \(f:\mathbb {G}\rightarrow \mathbb {C}\) is a compactly supported function on \(\mathbb {G}\) (and smooth if \(\mathbb {G}=\mathbb {R}^d\)) or any other function for which (2.8) makes sense.
An indispensable tool in the proof of Theorem 4 is the vector-valued Ionescu–Wainger multiplier theorem from [28, Section 2] with an improvement by Tao [36].
Theorem 5
For every \(\varrho >0\), there exists a family \((P_{\le N})_{N\in \mathbb {N}}\) of subsets of \(\mathbb {N}\) such that:
-
(i)
\(\mathbb {N}_N\subseteq P_{\le N}\subseteq \mathbb {N}_{\max \{N, e^{N^{\varrho }}\}}\).
-
(ii)
If \(N_1\le N_2\), then \(P_{\le N_1}\subseteq P_{\le N_2}\).
-
(iii)
If \(q \in P_{\le N}\), then all factors of q also lie in \(P_{\le N}\).
-
(iv)
\(\textrm{lcm}(P_N) \le 3^N\).
Furthermore, for every \(p \in (1,\infty )\), there exists \(0<C_{p, \varrho , |\Gamma |}<\infty \) such that, for every \(N\in \mathbb {N}\), the following holds:
Let \(0<\varepsilon _N \le e^{-N^{2\varrho }}\) and let \(\textbf{Q}:=[-1/2, 1/2)^\Gamma \) be a unit cube. Let \(\mathfrak {m}:\mathbb {R}^{\Gamma } \rightarrow L(H_0,H_1)\) be a measurable function supported on \(\varepsilon _{N}\textbf{Q}\) taking values in \(L(H_{0},H_{1})\), the space of bounded linear operators between separable Hilbert spaces \(H_{0}\) and \(H_{1}\). Let \(0 \le \textbf{A}_{p} \le \infty \) denote the smallest constant such that
for every function \(f\in L^2(\mathbb {R}^\Gamma ;H_0)\cap L^{p}(\mathbb {R}^\Gamma ;H_0)\). Then, the multiplier
where \(\Sigma _{\le N}\) is defined by
satisfies
for every \(f\in \ell ^p(\mathbb {Z}^\Gamma ;H_0)\), (cf. [36, Theorem 1.4] which removes the factor of \(\log N\) in the inequality (2.9)).
3 Preliminaries
3.1 General results
In this section, we present some general results concerning the behavior of exponential sums. The following proposition is an enhancement of the variant of Weyl’s inequality due to Trojan [37, Theorem 2] that allows us to estimate exponential sums related to a possibly non-differentiable function \(\phi \), (cf. [28, Theorem A.1]).
Proposition 6
(Weyl’s inequality) Let \(\alpha >0\), \(k\in \mathbb {N}\), and let \(\Gamma \subset \mathbb {N}^{k} {{\setminus }} \{0\}\) be a nonempty finite set. Let \(\Omega '\subseteq \Omega \subseteq B(0,N)\subset \mathbb {R}^k\) be convex sets and let \(\phi :\Omega \cap \mathbb {Z}^{k}\rightarrow \mathbb {C}\). There is \(\beta _\alpha >0\) such that, for any \(\beta > \beta _\alpha \), if there is a multi-index \(\gamma _0\in \Gamma \) with
for some coprime integers a and q with \(1\le a\le q\) and \((\log N)^\beta \le q\le N^{|\gamma _0|}(\log N)^{-\beta }\), then
The implicit constant is independent of the function \(\phi \), the variable \(\xi \), the sets \(\Omega ,\Omega '\), and the numbers a, q, and N.
Proof
We define \(\tilde{\phi }(n,p,A):=\phi (n,p)\mathbbm {1}_{(\Omega {{\setminus }}\Omega ')\cap A}(n,p)\). We partition the cube \([-N,N]^k\) into \(J\lesssim \log (N)^{k\alpha }\) cubes \(Q_j\) with disjoint interiors and side lengths \(C N\log (N)^{-\alpha }\) for some constant \(C>0\). Let \((m_j,p_j)\) be a fixed element of \(Q_j\cap \Omega {\setminus }\Omega '\). Since \(\mathbbm {1}_{\Omega }(x)\mathbbm {1}_{\Omega {\setminus }\Omega '}(x)=\mathbbm {1}_{\Omega {\setminus }\Omega '}(x)\) for any \(x\in \mathbb {R}^k\), we have
where all sums are taken over \((n,p)\in \mathbb {Z}^{k'}\times (\pm \mathbb {P})^{k''}\). Let \((m_j,p_j)\) be a fixed element of \(Q_j\cap \Omega {\setminus }\Omega '\). We estimate the right hand side of (3.1) by
By Trojan’s variant of Weyl’s inequality [37, Theorem 2], the first term is bounded by
for any \(\alpha '>0\). Since \(\mathbbm {1}_{\Omega \cap Q_j}(n,p)=\mathbbm {1}_{\Omega '\cap Q_j}(n,p) +\mathbbm {1}_{(\Omega {\setminus }\Omega ')\cap Q_j}(n,p)\), the second term is bounded by
Choosing an appropriate \(\alpha '>0\) in (3.2) and (3.3) yields the claim. \(\square \)
The next result is a generalization of [37, Proposition 4.1] and [37, Proposition 4.2] in the spirit of [28, Proposition 4.18]. For \(q\in \mathbb {N}\) and \(a\in \mathbb {N}_q^\Gamma \) with \(\textrm{gcd}(a,q)=1\), the Gaussian sum related to the polynomial mapping \(\mathcal {Q}\) is given by
where \(A_q:=\{a\in \mathbb {N}_q:\textrm{gdc}(a,q)=1\}\) and \(\varphi \) is Euler’s totient function. There is \(\delta > 0\) such that
according to [37, Theorem 3].
Lemma 7
Let \(N\in \mathbb {N}\) and let \(\Omega \subseteq B(0,N)\subset \mathbb {R}^k\) be a convex set or a Boolean combination of finitely many convex sets. Let \(\mathcal {K}:\mathbb {R}^k\rightarrow \mathbb {C}\) be a continuous function supported in \(\Omega \). Then, for each \(\beta >0\), there is a constant \(c = c_{\beta }>0\) such that, for any \(q \in \mathbb {N}\) with \(1 \le q \le (\log N)^{\beta }\), \(a \in A_q\), and \(\xi = a/q + \theta \in \mathbb {R}^\Gamma \), we have
The implied constant is independent of \(N,a,q,\xi \) and the kernel \(\mathcal {K}\).
Proof
The case when \(k=k'\) was proven in [28, Proposition 4.18], so we assume that \(k>k'\). 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
To simplify the notation, for \((x,y) \in \mathbb {R}^k {\setminus } \{0\}\), we set \(F(x,y):= \varvec{e}(\theta \cdot \mathcal {Q}(x, y))\mathcal {K}(x, y).\) For \((n, p) \in \mathbb {N}^{k'} \times \mathbb {P}^{k''}\) with \(n \equiv r' \bmod q\) and \(p \equiv r'' \bmod q\), we have
Therefore, we have \(\varvec{e}(\xi \cdot \mathcal {Q}(n, p)) = \varvec{e}((a/q) \cdot \mathcal {Q}(r', r''))\varvec{e}(\theta \cdot \mathcal {Q}(n, p)),\) so then
where the error term is the cost for making the summation for \(r''\) over \(A_q^{k''}\) instead of \(N_q^{k''}\). Fix \(u \in \mathbb {N}^{k'}\), \(\tilde{p} \in \mathbb {P}^{k''-1}\), and \(r_1'' \in A_q\). Then \(\left\{ v \in \mathbb {N}: (u, v, \tilde{p}) \in \Omega \right\} = \left( V_0+1, \ldots , V_1\right) \) for some \(0 \le V_0 \le V_1 \le N\). By partial summation, we obtain
where, for \(x \ge 1\), we have set
Similarly, we have
Furthermore, in view of the Siegel–Walfisz theorem ( [32, 39], see also [29, Corollary 11.21]), there are \(C, c' > 0\) such that for all \(x \ge 1\), \((r, q) = 1\) and \(1 \le q \le (\log x)^{\beta '}\),
Hence, by (3.8), (3.9), and (3.10), we obtain
Similar arguments applied to the sums over \(p_2, \ldots , p_{k''}\) give
Let \(\Omega _+:= \Omega \cap [0,\infty )^{k'} \times [1,\infty )^{k''}\). We can estimate the sum by an integral by writing
We use three estimates to control this:
where the last inequality is a consequence of [28, Proposition 4.16], which gives that the number of lattice points in \(\Omega \) at a distance \(<q\) from the boundary of \(\Omega \) is \(\mathcal {O}(q N^{k-1})\). We therefore get a bound for (3.12) of the form
Applying this in (3.11) and combining the error terms appropriately gives
Applying this in (3.7) by summing in \(r'\) and \(r''\) together with (3.6) gives
for any \(c < c'\). In simplifying to get the error term above, note that \(q^{k'} \phi (q)^{k''} \le q^{k} \le (\log N)^{\beta k}\) and
Finally, we note that we can increase the range of integration at (3.13) to the larger \(\Omega \cap [0,\infty )^{k}\) by noting that
is bounded by \(N^{k'}\Vert \mathcal {K}\Vert _{L^\infty (\Omega )}\le N^{k-1}\Vert \mathcal {K}\Vert _{L^\infty (\Omega )}\).
We can repeat the entire proof replacing \(\mathbb {N}_0\) with \(-\mathbb {N}_0\) and/or \(\mathbb {P}\) with \(-\mathbb {P}\) in all the \(2^k\) many possible combinations thereof in \(\mathbb {N}_0^{k'} \times \mathbb {P}^{k''}\). Then, collecting all of the error terms yields the claim. \(\square \)
3.2 Multipliers for the averaging operators
For a function \(f:\mathbb {Z}^\Gamma \rightarrow \mathbb {C}\) with finite support, we have
for the discrete Fourier multipliers
and
Their continuous counterparts are given by
respectively. To present a unified approach, we write \(M_t^{k',k''}\), \(\mathfrak {y}_t\), and \(\Theta _t\) to represent either \(A_t^{\mathcal {Q},k',k''}\), \(\mathfrak {m}_t\), and \(\Phi _t\) or \(H_t^{\mathcal {Q},k',k''}\), \(\mathfrak {n}_t\), and \(\Psi _t\) respectively. We now present the key properties of our multiplier operators that will be used in the proof of Theorem 4. Let \(N_n:=\lfloor 2^{n^\tau } \rfloor \) for \(n\in \mathbb {N}\) and some \(\tau \in (0,1]\) adjusted later.
-
Property 1.
For each \(\alpha > 0\), there is \(\beta _{\alpha } > 0\) such that, for any \(\beta > \beta _{\alpha }\) and \(n \in \mathbb {N}\), if there is a multi-index \(\gamma _0 \in \Gamma \) with
$$\begin{aligned} \bigg |\xi _{\gamma _0} - \frac{a}{q} \bigg | \le \frac{1}{q^2} \end{aligned}$$for some coprime integers a and q with \(1 \le a \le q\) and \((\log N_n)^\beta \le q \le N_n^{|\gamma _0|} (\log N_n)^{-\beta }\), then
$$\begin{aligned} |(\mathfrak {y}_{N_n} - \mathfrak {y}_{N_{n-1}})(\xi )| \lesssim C(\log N_n)^{-\alpha }. \end{aligned}$$This follows from Proposition 6 with \(\phi (x)\equiv (\vartheta _\Omega (N_n))^{-1}\) for the \(\mathfrak {y}_t = \mathfrak {m}_{t}\) case and with \(\phi (x)=K(x)\) for the \(\mathfrak {y}_t = \mathfrak {n}_{t}\) case, noting the size condition (1.4) and the continuity condition (1.6).
-
Property 2.
Let A be the \(|\Gamma | \times |\Gamma |\) diagonal matrix with
$$\begin{aligned} (A v)_\gamma = |\gamma | v_\gamma . \end{aligned}$$(3.14)For any \(t>0\), we set \(t^A v: = \big (t^{|\gamma |} v_\gamma : \gamma \in \Gamma \big ).\) Then
$$\begin{aligned} \big |\Theta _{N_n}(\xi ) - \Theta _{N_{n-1}}(\xi )\big | \lesssim \min \big \{|N_n^A\xi |_\infty , |N_n^A \xi |_\infty ^{-1/|\Gamma |}\big \},\quad \text {for each }n\in \mathbb {N}. \end{aligned}$$In the \(\Theta _t = \Phi _t\) case, this follows from the mean value theorem and the standard van der Corput lemma. In the \(\Theta _t = \Psi _t\) case, this follows from the cancellation condition (1.5) and [27, Proposition B.2] (see [27, p. 21] for details).
-
Property 3.
For each \(\alpha > 0\), \(n \in \mathbb {N}\), and \(\xi \in \mathbb {T}^\Gamma \) satisfying
$$\begin{aligned} \bigg |\xi _\gamma - \frac{a_\gamma }{q} \bigg | \le N_n^{-|\gamma |} L\qquad \text {for all }\gamma \in \Gamma \end{aligned}$$with \(1 \le q \le L\), \(a\in A_q^\Gamma \), and \(1 \le L \le \exp \Big (c\sqrt{\log {N_n}}\Big ) (\log N_n)^{-\alpha }\), we have
$$\begin{aligned} \mathfrak {y}_{N_n}(\xi ) - \mathfrak {y}_{N_{n-1}}(\xi ) = G(a/q) \big (\Theta _{N_n}(\xi - a/q) - \Theta _{N_{n-1}}(\xi - a/q)\big ) + \mathcal {O}\big ((\log N_n)^{-\alpha }\big ), \end{aligned}$$for some constant \(c>0\) which is independent of \(n, \xi , a\) and q. In the \(\mathfrak {y}_t = \mathfrak {m}_t\), \(\Theta _t = \Phi _t\) case, this is [37, Property 6]. In the \(\mathfrak {y}_t = \mathfrak {n}_t\), \(\Theta _t = \Psi _t\) case, this follows from Property 1 alongside Lemma 7 with \(\Omega :=\Omega _{N_n}{\setminus }\Omega _{N_{n-1}}\) and \(\mathcal {K}(n,p):=K(n,p)\mathbbm {1}_{{\Omega }}\), noting the size condition (1.4) and the continuity condition (1.6). For details see [37, Lemmas 3 and 5].
3.3 Parameters discussion
Let \(p\in (1,\infty )\) be fixed and let \(\chi \in (0,1/10)\). Fix \(\tau \) with \(0< \tau < 1-\min (2,p)^{-1}\) and let \(N_n:=\lfloor 2^{n^\tau } \rfloor \) for \(n\in \mathbb {N}\). If \(p \in (1,2)\), fix \(p_0\) such that \(1< p_0 < p\). If instead \(p \in (2,\infty )\), fix \(p_0 > p\). If \(p=2\), the discussion is moot since all the interpolation arguments in the article become unnecessary. We choose \(\rho \) with
so that interpolation of the estimates
yields
Property 1 gives us a corresponding \(\beta _\rho \). We fix a choice of \(\beta > \beta _\rho \) and then fix a choice of \(u\in \mathbb {N}\) with \(u>|\Gamma |\beta \). We also have the value of \(\delta \) coming from the Gaussian sum estimate (3.5). With these fixed, we choose the value of \(\varrho \) in Theorem 5 to be
4 Proof of Theorem 4
By the monotone convergence theorem and standard density arguments it is enough to prove that
holds for every finite subset \(\mathbb {I}\subset \mathbb {R}_+\) with the implicit constant independent of the set \(\mathbb {I}\). We start by splitting (cf. [16, Lemma 1.3]) into long oscillations/jumps and short variations along the subexponential sequence \(N_n\):
4.1 Short variations
By using the arguments from [28, Section 3.1], the estimate for the short variations will follow from the estimate
Let \(t_1<t_2<\cdots <t_{J(n)}\) be a sequence of elements of \([N_n,N_{n+1})\cap \mathbb {I}\). Since the number of elements in \([N_n,N_{n+1})\cap \mathbb {I}\) is finite, it is easy to see that
for any \(n\in \mathbb {N}_0.\) Moreover, we have
This follows from the monotonicity of the sets \(\Omega _t\) and having \(\vartheta _\Omega (t)\approx t^k\) by the prime number theorem in the \(M_t^{k',k''} = A_t^{\mathcal {Q}, k',k''}\) case or the size condition (1.4) in the \(M_t^{k',k''} = H_t^{\mathcal {Q}, k',k''}\) case. By [37, Eq. 4.10], the right hand side of (4.2) is bounded by \(n^{\tau -1}\Vert f\Vert _{\ell ^1(\mathbb {Z}^\Gamma )}\), proving (4.1).
4.2 Long oscillations/jumps and the circle method
Let \(\eta :\mathbb {R}^\Gamma \rightarrow [0,1]\) be a smooth function with
For \(N\in \mathbb {R}_+\), we define the scaling notation
where A is the matrix given in (3.14) and \(\textrm{Id}\) is the \(|\Gamma | \times |\Gamma |\) identity matrix. For dyadic integers \(s \in 2^{u\mathbb {N}}\), we define the annuli sets of fractions by
where the \(\Sigma _{\le \cdot }\) are the sets of Ionescu–Wainger fractions as in Theorem 5. For \(t\ge 2^u\), we set \(F(t):=\max \{s \in 2^{u\mathbb {N}}: s \le t\}\). We define
and, for \(s \in 2^{u\mathbb {N}}\), we define the annuli functions
By (4.3), we have the telescoping property
Note that \(\eta _{j^\tau }(\xi )\) satisfies the hypothesis about the support for \(\mathfrak {m}\) in Theorem 5 since \(\frac{1}{8 |\Gamma |}2^{-j^{\tau }+j^{\tau \chi }} \le e^{-j^{2 \tau u \varrho }}\) provided that \(\varrho \le \chi /(10 u)\). Using the \(\Xi _{\le j^{\tau u}}\) functions, we bound the long oscillations/jumps by
These terms correspond to major and minor arcs respectively.
4.3 Minor arcs
Since \(V_1\) controls the oscillation/jump seminorms, we have
It then suffices to show that
for some \(\varepsilon > 0\). This uses Property 1 and follows from the proof of [37, Eqs. (5.8), (5.9)] with only small changes due to our differing scaling in the definition of \(\eta _N(\xi )\). We omit the details.
4.4 Introduction to major arcs
Using the annuli multipliers (4.4) and Proposition (3), we bound the major arcs term by
It then suffices to show for large \(s\in 2^{u\mathbb {N}}\) that
for some \(\varepsilon > 0\) since \(\sum _{s \in 2^{u\mathbb {N}}} s^{-\varepsilon } < \infty \). Let \(\kappa _s:=s^{2 \lfloor \varrho \rfloor }\). By splitting the left hand side of (4.5) at \(n \approx 2^{\kappa _s}\) into small and large scales, it suffices to prove that
and
For the small scales (4.6), we will use the Rademacher–Menshov inequality (2.5) and Theorem 5. For the large scales (4.7), we will use the Magyar–Stein–Wainger sampling principle from [23, Proposition 2.1] and its counterpart for the jump inequality from [26, Theorem 1.7]. We first establish an approximation lemma to replace our discrete multipliers with continuous counterparts. Let
and
Lemma 8
Let \(M \in \mathbb {N}\), \(\alpha ' > 0\), and \(S_M:=\lfloor 2^{M^\tau - 3\,M^{\tau \chi }}\rfloor \). For \(j\in \mathbb {N}\) with \(s^{1/(\tau u)} \le j\) and \(M \le j \le 2M\), we have
and
Proof
For (4.10), since the \(\eta _{j^{\tau }}(\xi - a/q)\) bump functions in the definitions of \(\Xi _j^s\) and \(v_j^s\) have disjoint supports for distinct fractions a/q, it suffices to prove for a fixed \(a/q \in \Sigma _s\) that
with the implied constant independent of the choice of a/q. Using the definition of \(\Sigma _s\), property (i) from Theorem 5, \(s \le j^{\tau u}\), and \(\varrho \le \chi /(10 u)\), we have \(q \le e^{s^{\varrho }} \le e^{j^{\tau u \rho }}\le 2^{j^{\tau \chi }} =: L_1.\) On the support of \(\eta _{j^\tau }(\xi - a/q)\), we have \(|\xi _\gamma - a_\gamma /q| \lesssim N_j^{-|\gamma |} L_1\) for all \(\gamma \in \Gamma \). Moreover, we have
The estimate (4.10) then follows from Property 3 with \(\alpha = \alpha '\) and \(L = L_1\).
For (4.11), we use (4.10) and are reduced to showing that
Fixing \(\xi \) in the support of \(\eta _{j^\tau }(\xi - a/q)\), we have
for all \(\gamma \in \Gamma \). By the triangle inequality, we have
For the first term, we use the estimate (3.5), the mean value theorem, and Property 2 to obtain
For the second term, we use that
and \(q \le 2^{j^{\tau \chi }} \le 2^{(2\,M)^{\tau \chi }} = L_2 \le \exp \left( \sqrt{ \log S_M}\right) (\log S_M)^{-\alpha ' / \chi }.\) Hence, we may apply [37, Lemma 3] with \(\alpha = \alpha '/\chi \), \(N = S_M\), and \(L = L_2\) to obtain
This completes the proof of (4.11). \(\square \)
4.5 Small scales
Using that \(V_2\) dominates oscillations/jumps, splitting \([s^{1/u},2^{\kappa _s+1}]\) into dyadic intervals, and preparing via the triangle inequality to use (4.11), we bound the left hand side of (4.6) by
For Error Term 1, it will suffice to show that
for some \(\varepsilon ' > 0\) since we would then bound it by
using that \(V_1\) dominates \(V_2\). We note by Theorem 5 that
and, by (4.11) with \(\alpha ' = \rho \), that
Interpolation of the above inequalities yields (4.12).
For Main Term 1, we apply the Rademacher–Menshov inequality (2.5) to bound it by
where j is taken over \(j \ge 0\) such that \(I_{i,j}^M:= [j2^i,(j+1)2^i] \cap [M^{1/\tau }, (2\,M)^{1/\tau }] \ne \emptyset \). Let \(\tilde{\eta }_{N}(\xi ):=\eta _N(\xi /2)\). Then \(\tilde{\eta }_{N} \eta _{k^\tau } = \eta _{k^\tau }\) for \(k^\tau \ge N\) due to the nesting supports. This lets us write
for \(k \in I_{i,j}^M\) since then \(k \ge M^{1/\tau }\). We have for any \(p\in (1,\infty )\) that
since, by Theorem 5, the above estimate is a consequence of its continuous counterpart
The above square function estimate follows by appealing to Property 2 and arguments from Littlewood–Paley theory. We refer to [27] for more details, see also [28, Theorem 4.3, p. 42]. Thus,
using the uniform \(\ell ^{p}\)-boundedness of the averaging operators. We get an improved bound on \(\ell ^2\). To do this, we show that
for \(M \in 2^\mathbb {N}\cap [s^{1/u},2^{\kappa _s}]\). Since the bump functions in the sum have disjoint supports, it suffices to prove for a fixed \(a/q \in \Sigma _s\) that
with the implied constant independent of the choice of a/q. On the support of \(\tilde{\eta }_M(\xi - a/q)\), we have
We follow the same arguments as in the proof of (4.11), choosing \(\alpha ' = \delta u /\tau \), to show that
For any \(\xi \in \mathbb {T}^\Gamma \) and \(a/q \in \Sigma _s\), we have
using that \(|G(a/q)| \lesssim q^{-\delta } \lesssim s^{-\delta }\) since \(q \ge s/2^u\) by the construction of \(\Sigma _s\). Hence,
Interpolation of (4.13) with (4.14) then gives that
since \(8\varrho \le \delta /(\rho \tau )\). Thus, we may dominate Main Term 1 by
since \(\kappa _s \le s^{2\varrho }\), concluding the proof of (4.6).
4.6 Large scales
Since \(V_1\) dominates \(\mathcal {S}_{\mathbb {Z}^\Gamma }^p\), we may bound the left hand side of (4.7) by
For Error Term 2, it will suffice to show that
for some \(\varepsilon ' > 0\) since we would then bound it by
We have
by (4.10) with \(\alpha ' = \rho \). We also have
simply by the triangle inequality and property (i) from Theorem 5. Consequently (4.15) follows by interpolation.
For Main Term 2, we define
and
Let \(Q_s:=\textrm{lcm}(q: a/q \in \Sigma _s).\) By property (iv) from Theorem 5, we have \(Q_s \le 3^s\). The function \(\omega _n^s\) is supported on \([-\frac{1}{4Q_s}, \frac{1}{4Q_s}]\) for large \(s\in 2^{u\mathbb {N}}\) since, on the support of \(\eta _{2^{\kappa _s}}\), we have \(|\xi _\gamma | \le 2^{-2^{-\kappa _s}+2^{\kappa _s\chi }} \le (4Q_s)^{-1}\) for all \(\gamma \in \Gamma \) and large s. We also have
Therefore, it suffices to prove
and
for some \(\varepsilon > 0\).
By the Magyar–Stein–Wainger sampling principle [23, Proposition 2.1] for the oscillation seminorm or the sampling principle for the jumps [26, Theorem 1.7], (4.16) follows from
To prove (4.18), we use that the \(\omega _n^s\) functions are almost telescoping. We define
Then (4.18) follows from
since the error term is bounded by
using Property 2 and interpolation.
On the other hand, due to translation invariance of \(\mathcal {S}_{\mathbb {R}^\Gamma }^p\), (4.19) follows from
For the jump inequality, the estimate (4.20) was proven in [27, Theorem 1.22, Theorem 1.30] for both \(\Phi _t\) and \(\Psi _t\). For the oscillation inequality, (4.20) was proven in [21, Eq. 3.38] for \(\Phi _t\) and in [34, Theorem 1.9] for \(\Psi _t\). This concludes the proof of (4.16).
For (4.17), we note by (3.5) that
On \(\ell ^{p_0}\), we start by splitting
where \(J_s = \lfloor 2^{2^{\kappa _s} - 3 \cdot 2^{\kappa _s \chi }} \rfloor \). By Theorem 5, we have
Let \(p_{00} \in (1,\infty )\). Then
by property (i) from Theorem 5. Therefore, it suffices to show that
since interpolating (4.24) with (4.23) for an appropriate choice of \(p_{00}\) gives
combining this with (4.22) gives
and interpolating the above inequality with (4.21) completes the proof of (4.17) and, thereby, that of (4.7). The proof of (4.24) will proceed similarly as in the proof of (4.11). For \(\xi \) in the support of \(\tilde{\eta }_{2^{\kappa _s}}(\xi - a/q)\), we have
for all \(\gamma \in \Gamma \). By the triangle inequality, we have
For the first term, we use (3.5) and the mean value theorem to write
For the second term, we use that
and \(q \le L_3 \le \exp \Big (\sqrt{ \log J_s}\Big ) (\log J_s)^{-1}\) to apply [37, Lemma 3] with \(\alpha = 1\), \(N = J_s\), and \(L = L_3\). This gives
completing the proof.
5 Remarks
As a simple consequence of our results, we can prove the convergence of the Wiener–Wintner type averages. This result is probably known, but we have not found anything like this in the literature in the presented generality.
Let \((X,\mu )\) be a measure space endowed with a measure preserving transformation \(T:X\rightarrow X\) and let
be a polynomial with real coefficients. Moreover, let \(P:\mathbb {Z}\rightarrow \mathbb {Z}\) be a polynomial with integer coefficients such that \(P(0)=0\). For \(p\in (1,\infty )\), the Wiener–Wintner type averages
converge \(\mu \)-almost everywhere for any \(f\in L^p(X,\mu )\). According to Assani [1, p. 179], the convergence of the averages (5.1) in the case when \(\deg P\ge 2\) is known only for \(R\equiv 0\). However, in [10, Theorem 1.9], the authors have established the convergence in the case when \(P(n)=n^2\) and \(R(n)=\theta n\) for any \(\theta \in \mathbb {R}.\)
Let us show how to deduce the convergence of the averages (5.1) from Corollary 2. Clearly, we may assume that \(R(0)=0\). We consider the measure space \((Y,\nu )\) where \(Y:=X\times \mathbb {T}\), \(\nu :=\mu \times \lambda \), and \(\lambda \) is the normalized Lebesgue measure on \(\mathbb {T}\). We equip the space \((Y,\nu )\) with the family of measure preserving commuting transformations \(S_1, S_2,\ldots ,S_m,S_{m+1}:X\times \mathbb {T}\rightarrow X\times \mathbb {T}\) where, for \(j=1,\ldots ,m\), we put \(S_j:=\textrm{Id}\times D_j\) with
being a rotation on \(\mathbb {T}\), and \(S_{m+1}:=T\times \textrm{Id}.\) We consider the following polynomial mapping
By Corollary 2, we know that the averages
converge \(\nu \)-almost everywhere for any \(h\in L^p(Y,\nu )\). If, for \(f\in L^p(X,\mu )\), we consider the function \(h(y):=f(x)\xi \), then we see that the convergence of the averages (5.1) follows from the convergence of the averages (5.2).
The procedure described above can be extended to obtain that, for \(\mathcal {P}\) being a polynomial mapping of the form (1.1) and \(\mathcal {R}:\mathbb {R}^k\rightarrow \mathbb {R}\) being a polynomial with real coefficients, the averages
and
converge \(\mu \)-almost everywhere for any \(f\in L^p(X,\mu )\) with \(p\in (1,\infty )\). Moreover, we can deduce that the analogue of Corollary 2 holds for \(\mathcal {A}_{t}^{\mathcal {P},\mathcal {R},k',k''}\) and \(\mathcal {H}_t^{\mathcal {P},\mathcal {R},k',k''}\).
Unfortunately, we are not able to prove the Wiener–Wintner theorem for the averages \(\mathcal {A}_{t}^{\mathcal {P},\mathcal {R},k',k''}\) and \(\mathcal {H}_t^{\mathcal {P},\mathcal {R},k',k''}\). In our case, that would mean showing that, for any \(M\in \mathbb {N}\), there is a subset of X of full measure on which the convergence holds regardless of the choice of polynomial \(\mathcal {R}\) with \(\deg \mathcal {R}\le M\).
It is an interesting question whenever the Wiener–Wintner theorem can be somehow deduced from the inequality
This question is motivated by the fact that the constant in (5.3) depends only on the degree of \(\mathcal {R}\) and not its coefficients. We hope to investigate this problem in the near future.
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The authors were partially supported by the National Science Foundation (NSF) Grant DMS-2154712. The second author was supported by the Basque Government through the BERC 2022–2025 program and by the Ministry of Science, Innovation and Universities: BCAM Severo Ochoa accreditation SEV-2017-0718.
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Mehlhop, N., Słomian, W. Oscillation and jump inequalities for the polynomial ergodic averages along multi-dimensional subsets of primes. Math. Ann. 388, 2807–2842 (2024). https://doi.org/10.1007/s00208-023-02597-8
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DOI: https://doi.org/10.1007/s00208-023-02597-8