Variational estimates for discrete operators modeled on multi-dimensional polynomial subsets of primes
We prove the extensions of Birkhoff’s and Cotlar’s ergodic theorems to multi-dimensional polynomial subsets of prime numbers . We deduce them from -boundedness of -variational seminorms for the corresponding discrete operators of Radon type, where and .
Let be a -finite measure space with invertible commuting and measure preserving transformations . Let denote a polynomial mapping such that each is a polynomial on having integer coefficients without a constant term. Let be an open bounded convex subset in containing the origin such that for some and all ,
where for , we have set
In this paper we consider the following averages
where , denotes the set of prime numbers, and
One of the results of this article establishes the following theorem.
Assume that . For every there exists such that
for -almost all .
While working with averaging operators we encounter additional problem, namely, the sums over prime numbers are very irregular. To overcome this, we introduce weighted averaging operators,
Then the pointwise convergence of is deduced from the properties of , 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 be a Calderón–Zygmund kernel satisfying the differential inequality
for all with , and the cancellation condition
for every . Then the truncated discrete singular operator is defined by
The logarithmic weights in and 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 .
Assume that . For every there exists such that
for -almost all .
The classical approach to the pointwise convergence in proceeds in two steps. Namely, one needs to show boundedness of the corresponding maximal function reducing the problem to showing the convergence on some dense class of functions. However, finding such a class may be a difficult task. This is the case of one dimensional averages along studied by Bourgain in . To overcome this issue he introduced the oscillation seminorm defined for a given lacunary sequence and a sequence of complex numbers as
Then the pointwise convergence of is reduced to showing that
while tends to infinity. However, in place of the oscillation seminorm, we investigate -variational seminorm. Let us recall that -variational seminorm of a sequence is defined by
In fact, -variational seminorm controls as well as the maximal function. Indeed, for any , by Hölder’s inequality we have
Moreover, for any ,
Nevertheless, the main motivation to study boundedness of -variational seminorm is the following observation: if for any then the sequence converges. Therefore, we can deduce Theorem A and Theorem B from the following result.
For every there is such that for all and all ,
The constant is independent of the coefficients of the polynomial mapping .
The variational estimates for discrete averaging operators have been the subject of many papers, see [9, 11, 13, 15, 16, 27]. In , Krause studied the case and has obtained the inequality (1.6) for and . On the other hand, Zorin-Kranich in  for the same case obtained (1.6) for all but for in some vicinity of . Only recently in  the variational estimates have been established in the full range of parameters, that is and , covering the case . In , Zorin-Kranich has proved (1.6) also for the averaging operators modeled on prime numbers, that is when with a polynomial . It is worth mentioning that the variational estimates for discrete operators are based on a priori estimates for their continuous counterparts developed in , see also [13, Appendix].
The variational estimates for discrete singular operators have been studied in [4, 13, 16]. In , the authors obtained the inequality (1.7), for the truncated Hilbert transform modeled on prime numbers, which corresponds to and a polynomial . In fact, discrete singular operators of Radon type required a new approach. An important milestone has been laid by Ionescu and Wainger in . Ultimately, the complete development of the discrete singular operators of Radon type has been obtained in .
Concerning pointwise ergodic theorems over prime numbers, there are some results using oscillation seminorms. In , Bourgain has shown pointwise convergence for the averages along prime numbers for functions from . Then his result was extended to all , , by Wierld in , see also [2, Section 9]. Not long afterwards, Nair in  has proved Theorem A for , , and any integer-valued polynomial. Nair also studied ergodic averages for functions in for , however, [19, Lemma 14] contains an error. In fact, the estimates on the multipliers are insufficient to show that the sum considered at the end of the proof has bounds independent of . Lastly, the extension of Cotlar’s ergodic theorem to prime numbers has been established in , see also .
In view of the Calderón transference principle, while proving Theorem C, we may work with the model dynamical system, namely, with the counting measure and the shift operators. Let us denote by and , the corresponding operators, namely,
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 the discrete Fourier multiplier corresponding to . In view of (2.4), we can split the -variation into two parts: long variations and short variations, and study them separately.
To deal with long variations, we adopt the partition of unity constructed in , that is
for some parameters and where each projector is supported by a finite union of disjoint cubes centered at rational points belonging to . In this way, we distinguish the part of the multipliers 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 estimates for multipliers of Ionescu–Wainger type. By the triangle inequality, to control the first part it is enough to show
Next, by the circle method of Hardy and Littlewood we find the asymptotic of the multiplier . Here we encounter the main difference from . Namely, for sufficiently close to the rational point we have
provided that , where is the Gaussian sum and is an integral version of . The limitation on the size of the denominator is a consequence of the fact that for a larger the Siegel–Walfish 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 (1.10). 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 much harder. For this reason we directly work with , which led to cleaner arguments. 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 (1.10), we divide the variation into two parts: and , where . For the large scales , we transfer a priori estimates on -norm for -variation of the related continuous multipliers. Since the Gaussian sums satisfies for some , we gain a decay on . Consequently, by interpolation the norm of -variation for large scales is bounded by provided that is sufficiently large. In the case of small scales , the estimate on is obtained with a help of the numerical inequality (2.3). We again show that norm is bounded by . Because of the weaker asymptotic (1.10), to obtain bounds for -variations over small scales required a new approach. We further divided the index set into dyadic blocks, then on each block we constructed a good approximation to the multiplier giving bounds on norm independent of the block. At the cost of additional factor of , we control norm of -variation. Again, by interpolation combined with a choice of large enough we can make the norm bounded by .
For short variations we again remove the highly oscillating part of a multiplier with a help of Weyl–Vinogradov’s inequality and estimates for multipliers of Ionescu–Wainger type. The complementary part of the multiplier is decomposed by a family of projectors , reducing the problem to showing that
Let us briefly describe the structure of the article. In Section 2.1 we collect basic properties of the variational seminorm. In Section 2.2, we show how to deduce Theorem A from -variational estimates (1.6). Then we present the lifting procedure, which allows us to replace any polynomial mapping by a canonical one . In the next section, we describe multipliers of Ionescu–Wainger type whose norm estimates are essential to our argument. In Section 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 and , respectively. Finally, to get completely unified approach to the variational estimates for the averaging operators and truncated singular operators, at the beginning of Section 5, we list the properties shared by them which are sufficient to prove Theorem C. In next two sections based on this list we show the estimates on long and short variations.
Throughout the whole article, we write () if there is an absolute constant such that (). Moreover, stand for a large positive constant whose value may vary from occurrence to occurrence. If and hold simultaneously then we write . Lastly, we write () to indicate that the constant depends on some . Let . For a vector , we set and denote the set of dyadic numbers. Given a subset and , we set .
2.1. Variational norm
Let . For a sequence , , we define -variational seminorm by
The function is non-decreasing and satisfies
for any . Therefore,
Finally, for any increasing sequence , we have
The following lemma is essential in studying variational seminorm.
[15, Lemma 1] If then for any sequence of complex numbers
We split a variational seminorm into two parts long variations , and short variations . They are defined for a sequence by formulas
To estimate short variations we are going to use the following lemma.
[13, Lemma 2.2] Let , . Then for any integer and every strictly increasing sequence of integers with and , and each , we have
Moreover, if and for each , then
The implied constants are independent of , , and the sequence .
Let us observe that, if and is a sequence of functions from , and then
To see this, let
Then . If , then by Lemma 2, we get
2.2. Pointwise ergodic theorems
Let be a -finite space. Suppose we are given invertible commuting and measure preserving transformations . Let be a polynomial mapping such that each is a polynomial on with integral coefficients without a constant term. We fix , an open convex subset of containing the origin such that for some and all ,
where for , we have set
Because sums over prime numbers are irregular it is more convenient to work with weighted operators (1.3) instead of averages (1.2). In this section we show how to deduce the pointwise ergodic theorem (Theorem A) from a priori -variational estimates for .
Let and . Suppose that there is such that for all ,
Then there is such that for all ,
and the averages converges for -almost all .
Let us fix . For each and , we set
For , by the partial summation we obtain
where we have used the trivial estimate
which is the consequence of (2.7) and the prime number theorem. Observe that
thus by repeated application of (2.9), we arrive at the conclusion that
because the prime number theorem implies that . In particular, by taking and in (2.10) we get
Hence, for any and ,
Next, if then we can write
Hence, while proving -almost everywhere convergence of the averages for , we may assume that the function is bounded. By (2.11), for , we can write
Therefore, the convergence of implies the convergence of to the same limit. ∎
2.3. Lifting lemma
For the polynomial mapping , let us define
It is convenient to work with the set
equipped with the lexicographic order. Then each can be expressed as
for some . The cardinality of the set is denoted by . We identify with . Let be a diagonal matrix such that for all and ,
For , we set
Finally, we introduce the canonical polynomial mapping,
by setting . Now, if we define to be the linear transformation such that for ,
then . The following lemma allows us to reduce the problems to studying the canonical polynomial mappings.
[12, Lemma 2.1] Let be any of the operators or . Suppose that for some and ,
In the rest of the article by and we denote the averaging and the truncated discrete singular operator for the canonical polynomial mapping , that is and .
2.4. Ionescu–Wainger type multipliers
Let denote the Fourier transform on , that is for any ,
If , then we set
To simplify the notation, by we denote the inverse Fourier transform on as well as the inverse Fourier transform on the -dimensional torus identified with . We also fix , a smooth function such that , and
We additionally assume that is a convolution of two non-negative smooth functions with supports contained inside .
For any , we set and where . We define
wherein for we have set
Notice that . For , let us define
Lastly, we set
Given a multiplier on , such that for each there is such that for all ,
For each , its discrete counterpart is given by the formula
3. Trigonometric sums
3.1. Weyl–Vinogradov sum
We say that a subset of integers is polynomially regular, if for all , there are and a constant so that for any integer , and any polynomial of the form
for some coprime integers and , such that , and
for all and .
Let us check that is polynomially regular. We write
Set and with . Then