WEAK TYPE (1, 1) INEQUALITIES FOR DISCRETE ROUGH MAXIMAL ...

arXiv:1305.0575v2 [math.CA] 10 Apr 2014

WEAK TYPE (1, 1) INEQUALITIES FOR DISCRETE ROUGH MAXIMAL FUNCTIONS MARIUSZ MIREK

Abstract. The aim of this paper is to show that the discrete maximal function X 1 f (x − n) , for x ∈ Z, Mh f (x) = sup N∈N |Nh ∩ [1, N ]| n∈Nh ∩[1,N]

is of weak type (1, 1), where Nh = {n ∈ N : ∃m∈N n = ⌊h(m)⌋} for an appropriate function h. As a consequence we also obtain pointwise ergodic theorem along the set Nh .

1. Introduction and statement of results Recently, Urban and Zienkiewicz in [20] proved that the maximal function (1.1)

N 1 X f (x − an ) , for x ∈ Z, N ∈N N n=1

Mf (x) = sup

is of weak type (1, 1), for an = ⌊nc ⌋ where 1 < c < 1.001 giving a negative answer for Rosenblatt– Wierdl’s conjecture – for more details and the historical background we refer to [19] page 74. Not long afterwards, LaVictoire [12] and Christ [7] provided some new examples of sequences (an )n∈N having Banach density 0, for which maximal function Mf is of weak type of (1, 1). The main aim of this article is to study maximal functions X 1 f (x − n) , for x ∈ Z, (1.2) Mh f (x) = sup N ∈N |Nh ∩ [1, N ]| n∈Nh ∩[1,N ]

defined along subsets of integers Nh of the form

(1.3)

Nh = {n ∈ N : ∃m∈N n = ⌊h(m)⌋},

where h is an appropriate function, see Definition 1.4. We are going to consider such functions h for which Mh f is of weak type (1, 1) – see Theorem 1.7 below. Our motivation to study maximal functions (1.2) for arithmetic sets defined in (1.3) is that: on the one hand, we were inspired by the series of papers of Bourgain [2], [3] and [4] where he proved ℓp (Z) – boundedness (p > 1) of ergodic averages modeled on integer valued polynomials and the recent results of Buczolich and Mauldin [6] and LaVictoire [13]. They showed that the pointwise convergence of ergodic averages along p(n) = nk for k ≥ 2 fails on L1 . On the other hand, we did not know (apart from the example given in [20]) any considerable examples of sequences (given by a concrete formula) for which Mf is of weak type of (1, 1). Similar problems were studied in [1] in the context of Lp – boundedness (p > 1) of ergodic averaging operators, but the case of p = 1 remained unresolved until the results of [5], [20], [12] and [7]. Here we will make the first attempt at characterizing a class of functions h for which maximal function in (1.2) is of weak type (1, 1). The author was supported by NCN grant DEC–2012/05/D/ST1/00053. 1

2

M.MIREK

The sets Nh , considered as subsets of the set of prime numbers P, have great importance in analytic number theory. Namely, in 1953 Piatetski–Shapiro established an asymptotic formula for Pγ = {p ∈ P : ∃n∈N p = ⌊n1/γ ⌋} = Nx1/γ ∩ P, of fixed type γ < 1 (γ is sufficiently close to 1). More precisely, it was shown in [17] that xγ as x → ∞, |Pγ ∩ [1, x]| ∼ log x for every γ ∈ (11/12, 1). Recently, the author [14] proved ℓp (Z) – boundedness (p > 1) of maximal functions modeled on subsets of primes of the form Nh ∩ P, for h as in Definition 1.4. In [14] we have also obtained related pointwise ergodic theorems and showed that the ternary Goldbach problem has a solution in the primes belonging to Nh ∩ P. On the other hand, in [15], we proved a counterpart of Roth theorem for the Piatetski–Shapiro primes. Throughout the paper we will use the convention that C > 0 stands for a large positive constant whose value may change from line to line. For two quantities A > 0 and B > 0 we say that A . B (A & B) if there exists an absolute constant C > 0 such that A ≤ CB (A ≥ CB). If A . B and A & B hold simultaneously then we will shortly write that A ≃ B. We will also write A .δ B (A &δ B) to indicate that the constant C > 0 depends on some δ > 0. Definition 1.4. Let c ∈ (1, 2) and Fc be the family of all functions h : [x0 , ∞) 7→ [1, ∞) (for some x0 ≥ 1) satisfying (i) h ∈ C 3 ([x0 , ∞)) and h′ (x) > 0,

h′′ (x) > 0, for every x ≥ x0 .

(ii) There exists a real valued function ϑ ∈ C 3 ([x0 , ∞)) and a constant Ch > 0 such that h(x) = Ch xc ℓh (x), where ℓh (x) = e

(1.5)

Rx

x0

ϑ(t) t dt

, for every x ≥ x0 ,

and (1.6)

lim ϑ(x) = 0,

x→∞

lim xϑ′ (x) = 0,

x→∞

lim x2 ϑ′′ (x) = 0

x→∞

lim x3 ϑ′′′ (x) = 0.

x→∞

Among the functions belonging to the family Fc are (up to multiplicative constant Ch > 0) h1 (x) = xc logA x, h2 (x) = xc eA log

B

x

, h3 (x) = xc lm (x),

where c ∈ (1, 2), A ∈ R, B ∈ (0, 1), C > 0, l1 (x) = log x and lm+1 (x) = log(lm (x)), for m ∈ N. From now on we will focus our attention on subsets of integers Nh defined in (1.3) with h ∈ Fc . Let δn (x) stands for Dirac’s delta, i.e. δn (x) = 1 if x = n, and δn (x) = 0 otherwise. Our main result is the following. Theorem 1.7. Assume that c ∈ (1, 30/29) and h ∈ Fc . Let η ∈ C ∞ (R) be a smooth cut–off function supported in (1/2, 4) such that η(x) = 1 for x ∈ [1, 2] and 0 ≤ η(x) ≤ 1 for x ∈ R. Define a maximal function Mh f (x) = sup |Kh,N ∗ f (x)| for x ∈ Z,

(1.8)

N ∈N

corresponding with the kernel (1.9)

Kh,N (x) =

Then

1 |Nh ∩ [1, N ]|

X

n∈Nh ∩[1,N ]

δn (x)η

n N

for x ∈ Z.

kMh f kℓ1,∞ (Z) . kf kℓ1 (Z) ,

(1.10) 1

for every f ∈ ℓ (Z). In particular Mh f is bounded on ℓp (Z) for every f ∈ ℓp (Z) and p > 1.

3

The boundedness of (1.8) implies the boundedness of (1.2) (possibly with a different constant) and vice versa. Therefore, it will cause no confusion if we use the same letter Mh f in the definitions (1.2) and (1.8). The proof of Theorem 1.7 (see Section 6) will be based on the concepts of [20]. In [20] the authors used a subtle version of Calder´ on–Zygmund decomposition, which was pioneered by Fefferman [9] and later on developed by Christ [8], to study maximal functions. Fefferman’s ideas turned out to be applicable to the discrete settings as it was shown in [20], and recently also in [12] and [7]. Heuristically speaking, the weak type (1, 1) bound of Mh f is obtained by considering the recalcitrant part of the Calder´ on–Zygmund decomposition in ℓ2 (see Lemma 6.6 e h,N ∗ f, gi, and and Theorem 6.1 in Section 6), using the fact that hKh,N ∗ f, Kh,N ∗ gi = hKh,N ∗ K 1 e decomposing Kh,N ∗ Kh,N (x) into several manageable pieces (a delta mass at 0, a slowly varying function GN (x), and a small error term EN (x), see Section 5) obtained by special Van der Corput estimates, (we refer to Section 3). As we mentioned before our motivations to study such maximal functions are derived in part by scant knowledge of the structure of functions h for which Mh f is of weak type (1, 1). The family Fc was studied in [14] to generate various thin subsets of primes in the context of pointwise ergodic theorems and it turned out to be a good candidate to improve qualitatively theorem from [20]. On the other hand the family Fc gives rise to renew the discussion initiated in [1] and sheds some new light on L1 – pointwise ergodic theorems which have not been brought up there. It is worth pointing out that the complexity of the family Fc causes some obstructions which did not occur in [20]. Namely, we had to completely change the method of approximation of the kernel e h,N (x) compared to the method form [20] and this is the novelty of this paper (see Section Kh,N ∗ K 3 and Section 5). Their approach is inadequate here since it leads us to study exponential sums with a complicated form of a phase function, and loosely speaking this is the reason why we prefer to consider Nh ∩ [1, N ] in (1.2) instead of {⌊h(m)⌋ : m ∈ [1, N ]}. Now we have to emphasize that our method does not settle the case when c = 1. It would be nice to know, for instance, if Mh f is of weak type (1, 1) for h(x) = x log x. We hope to return this matter at a future time. Although the argument as stated works only for 1 < c < 30/29, the obstacles involved in getting a similar result for 1 < c < 2 pale in comparison to the obstacles e h,N (x) no longer has any useful properties. Nevertheless, for c > 2, since at that point Kh,N ∗ K LaVictoire [12] and Christ [7] provided a certain wide class of sequences for which Mf from (1.1) is of weak type (1, 1). If it comes to ℓp (Z) – boundedness of Mh f for p > 1, one can conclude, thanks to Lemma 3.20, that it holds for all h ∈ Fc provided that c ∈ [1, 4/3). However, if c = 1 then the conditions in (1.6) from Definition 1.4 must be modified in the following way. Remark 1.11. If c = 1, then we additionally assume that ϑ(x) is positive, decreasing and for every ε > 0 x 1 (1.12) .ε xε , and lim = 0. x→∞ h(x) ϑ(x) Furthermore, (1.13)

lim ϑ(x) = 0,

x→∞

xϑ′ (x) = 0, x→∞ ϑ(x) lim

x2 ϑ′′ (x) = 0, x→∞ ϑ(x) lim

x3 ϑ′′′ (x) = 0. x→∞ ϑ(x) lim

On the one hand, our approach supplies one more different method to the techniques developed in [1] which permits us to treat with Lp – boundedness for ergodic averages. On the other hand, it is worth noting that some of the Lp results for p > 1 are new, as there are some h ∈ Fc which do not belong to any Hardy field and are thus not covered by the results of [1]. 1where K e h,N (x) = Kh,N (−x)

4

M.MIREK

Theorem 1.7 is the main ingredient in the following. Theorem 1.14. Assume that c ∈ (1, 30/29) and h ∈ Fc . Let (X, B(X), µ, T ) be a dynamical system, where µ is a σ–finite measure and T is an invertible and measure preserving transformation on X. Then for every f ∈ Lp (X, µ) where p ≥ 1, the ergodic averages X 1 (1.15) f (T n x) for x ∈ X, Ah,N f (x) = |Nh ∩ [1, N ]| n∈Nh ∩[1,N ]

converges µ–almost everywhere on X. The paper is organized as follows. In Section 2 we give the necessary properties of function h ∈ Fc and its inverse ϕ. In Section 3 we estimate some exponential sums which allow us to e h,N (x) in the penultimate section. Assuming momentarily Theorem decompose the kernel Kh,N ∗ K 1.7 (its proof has been postponed to the last section), we prove Theorem 1.14 in Section 4. Despite the fact that Theorem 1.7 works only for c ∈ (1, 30/29) we decided to formulate the results in Section 2 and Section 3 also for c = 1 (see Remark 1.11), mainly due to new examples of functions in the family F1 . Acknowledgements I would like to thank Jacek Zienkiewicz for years of discussions on related problems. The author is grateful to the referee for careful reading of the manuscript and useful remarks that led to the improvement of the presentation. 2. Basic properties of functions h and ϕ In this section we gather all necessary properties of function h ∈ Fc and its inverse ϕ and we follow the notation used in Section 2 from [15]. Lemma 2.1. Assume that c ∈ [1, 2) and h ∈ Fc . Then for every i = 1, 2, 3 there exists a function ϑi : [x0 , ∞) 7→ R such that (2.2)

xh(i) (x) = h(i−1) (x)(αi + ϑi (x)), for every x ≥ x0 ,

where αi = c − i + 1, ϑ1 (x) = ϑ(x), (2.3)

ϑi (x) = ϑi−1 (x) +

xϑ′i−1 (x) , for i = 2, 3 and αi−1 + ϑi−1 (x)

lim ϑi (x) = 0, for i = 1, 2, 3.

x→∞

If c = 1, then there exist constants 0 < c1 ≤ c2 and a function ̺ : [x0 , ∞) 7→ [c1 , c2 ], such that (2.4)

xϑ′2 (x) = 0. x→∞ ϑ2 (x)

ϑ2 (x) = ϑ(x)̺(x), for every x ≥ x0 and lim

In particular (2.2) with i = 2 reduces to (2.5)

xh′′ (x) = h′ (x)ϑ(x)̺(x), for every x ≥ x0 .

The cases for i = 1, 3 remain unchanged. Proof. For the proof we refer to [15].



Lemma 2.6. Assume that c ∈ [1, 2), h ∈ Fc , γ = 1/c and let ϕ : [h(x0 ), ∞) 7→ [x0 , ∞) be its inverse. Then there exists a function θ : [h(x0 ), ∞) 7→ R such that xϕ′ (x) = ϕ(x)(γ + θ(x)) and (2.7)

ϕ(x) = xγ ℓϕ (x),

where

ℓϕ (x) = e

Rx

h(x0 )

θ(t) t dt+D

,

5

for every x ≥ h(x0 ), where D = log(x0 /h(x0 )γ ) and limx→∞ θ(x) = 0. Moreover, θ(x) =

(2.8)

1 ϑ(ϕ(x)) −γ =− . (c + ϑ(ϕ(x))) c(c + ϑ(ϕ(x)))

Additionally, for every ε > 0 lim x−ε L(x) = 0,

(2.9)

x→∞

and

lim xε L(x) = ∞,

x→∞

where L(x) = ℓh (x) or L(x) = ℓϕ (x). In particular, for every ε > 0 ϕ(x) = 0. x is increasing for every δ < c, (if c = 1, even δ ≤ 1 is allowed) and for every xγ−ε .ε ϕ(x),

(2.10) Finally, x 7→ xϕ(x)−δ x ≥ h(x0 ) we have

and

lim

x→∞

ϕ(x) ≃ ϕ(2x), and ϕ′ (x) ≃ ϕ′ (2x).

(2.11)

Proof. For the proof we refer to [15].



The next lemma will be very important in the sequel. Lemma 2.12. Assume that h ∈ Fc and let ϕ : [h(x0 ), ∞) 7→ [x0 , ∞) be its inverse. Then p ∈ Nh ⇐⇒ ⌊−ϕ(p)⌋ − ⌊−ϕ(p + 1)⌋ = 1,

(2.13)

for all sufficiently large p ∈ Nh . Proof. For the proof we refer to [15].



We finish this section by proving Lemma 2.14. Lemma 2.14. Assume that c ∈ [1, 2), h ∈ Fc , γ = 1/c and let ϕ : [h(x0 ), ∞) 7→ [x0 , ∞) be its inverse. Then for every i = 1, 2, 3, there exists a function θi : [h(x0 ), ∞) 7→ R such that xϕ(i) (x) = ϕ(i−1) (x)(βi + θi (x)), for every x ≥ h(x0 ),

(2.15)

where βi = γ − i + 1, limx→∞ θi (x) = 0 and limx→∞ xθi′ (x) = 0. If c = 1, then there exists a positive function σ : [h(x0 ), ∞) 7→ (0, ∞) and a function τ : [h(x0 ), ∞) 7→ R such that (2.15) with i = 2 reduces to xθ2′ (x) xϕ′′ (x) = ϕ′ (x)σ(x)τ (x), for every x ≥ h(x0 ) and lim (2.16) = 0. x→∞ θ2 (x) The cases for i = 1, 3 remain unchanged. Moreover, σ(x) is decreasing, limx→∞ σ(x) = 0, σ(2x) ≃ σ(x), and σ(x)−1 .ε xε , for every ε > 0. Finally, there are constants 0 < c3 ≤ c4 such that c3 ≤ −τ (x) ≤ c4 for every x ≥ h(x0 ). Proof. The proof is based on simple computations. However, for the convenience of the reader we shall present the details. In fact, (2.15) for i = 1 with θ1 (x) = θ(x), has been shown in Lemma 2.6. Arguing likewise in the proof of Lemma 2.1 we obtain (2.15) for i = 2, 3. More precisely, θ1 (x) = θ(x) = −

(2.17) and θ1′ (x) = θ′ (x) = (2.18)



1 c+ϑ(ϕ(x))

θ2 (x) = θ(x) +

−γ

′

ϑ(ϕ(x)) 1 = − γ, c(c + ϑ(ϕ(x))) c + ϑ(ϕ(x)) ′

′

(ϕ(x))ϕ (x) = − ϑ(c+ϑ(ϕ(x))) 2 . Thus

1 ϑ′ (ϕ(x))ϕ(x) xθ′ (x) = Θ2 (ϕ(x)), = −γ− γ + θ(x) c + ϑ(ϕ(x)) (c + ϑ(ϕ(x)))2

6

M.MIREK 1 c+ϑ(x)

ϑ′ (x)x (c+ϑ(x))2

and θ2′ (x) = Θ′2 (ϕ(x))ϕ′ (x) with  ′ 1 (ϑ′′ (x)x + 2ϑ′ (x))(c + ϑ(x)) − 2ϑ′ (x)2 x ϑ′ (x)x Θ′2 (x) = = − , −γ− c + ϑ(x) (c + ϑ(x))2 (c + ϑ(x))3

where Θ2 (x) =

xθ2′ (x) =

−γ −

(ϑ′′ (ϕ(x))ϕ(x)2 + 2ϑ′ (ϕ(x))ϕ(x))(c + ϑ(ϕ(x))) − 2ϑ′ (ϕ(x))2 ϕ(x)2 Θ′2 (ϕ(x))ϕ(x) . =− c + ϑ(ϕ(x)) (c + ϑ(ϕ(x)))4

Finally, (2.19)

θ3 (x) = θ(x) +

xθ′ (x) xθ2′ (x) xθ2′ (x) + = θ2 (x) + , γ + θ(x) γ − 1 + θ2 (x) γ − 1 + θ2 (x)

Therefore, θ3 (x) can be rewritten as θ3 (x) = Θ3 (ϕ(x)) where (ϑ′′ (x)x2 +2ϑ′ (x)x)(c+ϑ(x))−2ϑ′(x)2 x2 (c+ϑ(x))4 ϑ′ (x)x 1 c+ϑ(x) − 1 − (c+ϑ(x))2 2 ′

ϑ′ (x)x 1 − −γ− Θ3 (x) = c + ϑ(x) (c + ϑ(x))2

(ϑ′′ (x)x + 2ϑ (x)x) (c + ϑ(x))2 − (c + ϑ(x))3 − ϑ′ (x)x(c + ϑ(x)) 2ϑ′ (x)2 x2 + , (c + ϑ(x))3 − (c + ϑ(x))4 − ϑ′ (x)x(c + ϑ(x))2 = Θ2 (x) −

and θ3′ (x) = Θ′3 (ϕ(x))ϕ′ (x) where ϑ′′′ (x)x2 + 4ϑ′′ (x)x + 2ϑ′ (x) (c + ϑ(x))2 (1 − c − ϑ(x)) − ϑ′ (x)x(c + ϑ(x)) (ϑ′′ (x)x2 + 2ϑ′ (x)x)((c + ϑ(x))ϑ′ (x) − 3(c + ϑ(x))2 ϑ′ (x) − ϑ′′ (x)x(c + ϑ(x)) − ϑ′ (x)2 x) + (((c + ϑ(x))2 (1 − c − ϑ(x)) − ϑ′ (x)x(c + ϑ(x)))2 4ϑ′ (x)x(ϑ′′ (x)x + ϑ′ (x)) + 3 (c + ϑ(x)) (1 − c − ϑ(x)) − ϑ′ (x)x(c + ϑ(x))2 2ϑ′ (x)2 x2 (2(c + ϑ(x))2 ϑ′ (x) − 4(c + ϑ(x))3 ϑ′ (x) − ϑ′′ (x)x(c + ϑ(x))2 − 2ϑ′ (x)2 x(c + ϑ(x))) − . ((c + ϑ(x))3 (1 − c − ϑ(x)) − ϑ′ (x)x(c + ϑ(x))2 )2

(2.20) Θ′3 (x) = Θ′2 (x) −

These computations and (1.6) yield limx→∞ θi (x) = 0 and limx→∞ xθi′ (x) = 0 for every i = 1, 2, 3. The proof will be completed, if we elaborate the case c = 1. We know that xϕ′′ (x) = ϕ′ (x)θ2 (x), with   1 ϑ′ (ϕ(x))ϕ(x) ϑ′ (ϕ(x))ϕ(x) ϑ(ϕ(x)) . = ϑ(ϕ(x)) − − − θ2 (x) = − 1 + ϑ(ϕ(x)) (1 + ϑ(ϕ(x)))2 1 + ϑ(ϕ(x)) ϑ(ϕ(x))(1 + ϑ(ϕ(x)))2 Therefore (2.16) is proved with σ(x) = ϑ(ϕ(x)) and   1 ϑ′ (ϕ(x))ϕ(x) τ (x) = − . + 1 + ϑ(ϕ(x)) ϑ(ϕ(x))(1 + ϑ(ϕ(x)))2 In order to show that σ(2x) ≃ σ(x) it is enough to prove that ϑ(2x) ≃ ϑ(x). Notice that for some ξx ∈ (0, 1) we have ϑ(2x) (x + ξx x)ϑ′ (x + ξx x) ϑ(x + ξx x) (x + ξx x)ϑ′ (x + ξx x) x −−−→ 0, ≤ ϑ(x) − 1 = x + ξx x x→∞ ϑ(x + ξx x) ϑ(x) ϑ(x + ξx x) since ϑ(x) is decreasing. It is easy to see that

σ(x)−1 . xε , for every ε > 0,

7

since ϑ(x)−1 .ε xε for every ε > 0 and by (2.10). Furthermore, there exist 0 < c3 ≤ c4 such that c3 ≤ −τ (x) ≤ c4 for every x ≥ h(x0 ), by (1.13). The only what is left is to verify that xθ ′ (x) limx→∞ θ22(x) = 0 and limx→∞ xθ3′ (x) = 0. Indeed, xθ2′ (x) lim = lim x→∞ θ2 (x) x→∞

(ϑ′′ (ϕ(x))ϕ(x)2 +2ϑ′ (ϕ(x))ϕ(x))(1+ϑ(ϕ(x)))−2ϑ′(ϕ(x))2 ϕ(x)2 ϑ(ϕ(x))(1+ϑ(ϕ(x)))4 ϑ′ (ϕ(x))ϕ(x) 1 1+ϑ(ϕ(x)) + ϑ(ϕ(x))(1+ϑ(ϕ(x)))2

= 0.

In order to show that limx→∞ xθ3′ (x) = 0 it suffices to prove that lim xθ3′ (x) = lim

x→∞

x→∞

Θ3 (ϕ(x))ϕ(x) = 0, 1 + ϑ(ϕ(x))

but this follows from (1.13) and (2.20), since limx→∞ xΘ′3 (x) = 0. This completes the proof.



3. Estimates for some exponential sums The aim of this section is to establish Lemma 3.6 and 3.8 which will be essential for us and will be applied repeatedly in the sequel. Both proofs are based on Van der Corput’s type estimates. In this section we will assume that c ∈ [1, 4/3), γ = 1/c, h ∈ Fc and ϕ is the inverse function to h. Lemma 3.1 (Van der Corput). Assume that a, b ∈ R and a < b. Let F ∈ C 2 ([a, b]) be a real valued function and let I be a subinterval of [a, b]. If there exists λ > 0 and r ≥ 1 such that λ . |F ′′ (x)| . rλ, for every x ∈ I, then

X e2πiF (k) . r|I|λ1/2 + λ−1/2 . k∈I

Proof of Lemma 3.1 can be found in [11], see Corollary 8.13, page 208. Lemma 3.6 is a rather straightforward application of Lemma 3.1, whereas the estimate given in Lemma 3.8 is more involved and its proof will explore brilliant ideas from [20]. Throughout the paper, we will use the following version of summation by parts. Lemma 3.2. Let u(n) and Pg(n) be arithmetic functions and a, b ∈ Z such that 0 ≤ a < b. Define the sum function Ua (t) = a+1≤n≤t u(n), for any t ≥ a + 1. Then (3.3)

b X

u(n)g(n) = Ua (b)g(b) −

n=a+1

b−1 X

Ua (n)(g(n + 1) − g(n)).

n=a+1

Let x and y be real numbers such that 0 ≤ y < x. If g ∈ C 1 ([y, x]), then Z x X (3.4) u(n)g(n) = U⌊y⌋ (x)g(x) − U⌊y⌋ (t)g ′ (t)dt. y 1, 2 ′′ t ϕ (t) = (3.5) ϕ(t)(γ + θ1 (t))σ(t)τ (t), if c = 1. Lemma 3.6. Assume that N ≥ 1, x ∈ Z, α ∈ [0, 1], m ∈ Z \ {0}, l ≥ 1 and p, q ∈ {0, 1}. If N1,x = max{N/2, N/2 − x}, N2,x = min{4N, 4N − x} then X
−1/2 2πi(αln+mϕ(n+px+q)) 1/2 e (3.7) . . |m| N ϕ(N )σ(N ) N1,x 1 (see Section 2) σ is constantly equal to 1. Proof. We shall apply Lemma 3.1 to the exponential sum in (3.7). We can assume, without loss of generality, that m > 0 and let F (t) = αlt + mϕ(t + px + q) for t ∈ (N1,x , N2,x ]. According to (3.5) we see that |F ′′ (t)| = |mϕ′′ (t + px + q)| ≃

mϕ(N )σ(N ) mϕ(t + px + q)σ(t + px + q) ≃ , 2 (t + px + q) N2

since p, q ∈ {0, 1}, N/2 < t + px + q ≤ 5N , ϕ(2x) ≃ ϕ(x) and σ(2x) ≃ σ(x). One can think that σ is constantly equal to 1, when c > 1 (see Section 2). Now by Lemma 3.1 we obtain X N m1/2 ϕ(N )1/2 σ(N )1/2 2πi(αln+mϕ(n+px+q)) + 1/2 e . N · N m ϕ(N )1/2 σ(N )1/2 ′ N1,x 0 be a small enough real number whose precise value will be specified later. If |A(t)| ≥ ε′ ma1 N a2 −2 σ(N )−a3 ϕ(N )κ−a4 for every t ∈ (N1,x , N2,x ], then |F ′′ (t)| = |ϕ′′ (t)||A(t)| & ε′ ma1 N a2 −4 σ(N )1−a3 ϕ(N )1+κ−a4 . Assume now that there is some t0 ∈ (N1,x , N2,x ] such that |A(t0 )| ≤ ε′ ma1 N a2 −2 σ(N )−a3 ϕ(N )κ−a4 . By the mean value theorem there is ξ ∈ (0, 1) such that |A(t) − A(t0 )| = |t − t0 ||A′ (ξt,t0 )|, where ξt,t0 = t + ξ(t0 − t), if t0 ≥ t and ξt,t0 = t0 + ξ(t − t0 ), if t0 < t. In both cases ξt,t0 ≃ N and ξt,t0 + x ≃ N , since 1 ≤ ϕ(N )κ ≤ x ≤ 4N . Thus it is enough to estimate |A′ (t)| from below for any t ≃ N . Indeed, again by the mean value theorem, we see that for some ξx ∈ (0, 1) we have ϕ′′′ (t + x)ϕ′′ (t) − ϕ′′ (t + x)ϕ′′′ (t) ϕ′′ (t)2   ′′ ϕ (t + x) γ − 2 + θ3 (t + x) γ − 2 + θ3 (t) = m2 − ϕ′′ (t) t+x t   ′′ ϕ (t + x) 2 − γ − θ3 (t + ξx x) + (t + ξx x)θ3′ (t + ξx x) . = xm2 ϕ′′ (t) (t + ξx x)2

A′ (t) = m2

10

M.MIREK

Therefore, there is a universal constant C > 0 such that for sufficiently large N ∈ N, by Lemma 2.14, we have ′′   ϕ (t + x) |θ3 (t + ξx x)| + |(t + ξx x)θ3′ (t + ξx x)| 2−γ ′ − |A (t)| ≥ x ϕ′′ (t) (t + ξx x)2 (t + ξx x)2 ′′ ϕ (t + x) ϕ(N )κ 2−γ ≥ x ≥ C , ϕ′′ (t) 2(t + ξx x)2 N2

since (t + ξx x) ≃ N and |θ3 (t + ξx x)| + |(t + ξx x)θ3′ (t + ξx x)| ≤ (2 − γ)/2 for sufficiently large N ∈ N. This implies that, if |t − t0 | ≥ N0 = ma1 N a2 σ(N )−a3 ϕ(N )−a4 , then |A(t) − A(t0 )| = |t − t0 ||A′ (ξt,t0 )| ≥ Cma1 N a2 −2 σ(N )−a3 ϕ(N )κ−a4 . Finally, taking ε′ = C/2, we obtain that |A(t)| & ma1 N a2 −2 σ(N )−a3 ϕ(N )κ−a4 , for every |t − t0 | ≥ N0 = ma1 N a2 σ(N )−a3 ϕ(N )−a4 as desired and the proof of Lemma 3.8 is completed.  Now we have some refinements of Lemma 3.6 and Lemma 3.8. Corollary 3.12. Assume that N ≥ 1, x ∈ Z, α ∈ [0, 1], m1 , m2 ∈ Z \ {0}, l ≥ 1 and p, q ∈ {0, 1}. Let N1,x = max{N/2, N/2 − x}, N2,x = min{4N, 4N − x}. Then X 2πi(αln+m1 ϕ(n+px+q)) x (3.13) e Fm1 (n) N1,x 0 satisfy 4(1 − γ) + 6χ < 1, then there exists ε > 0 such that for every N ∈ N and for every α ∈ [0, 1] X X (3.21) e2πiαn + O(N 1−χ−ε ). ϕ′ (n)−1 e2πiαn = n∈Nh ∩[1,N ]

n∈[1,N ]

The implied constant is independent of α and N .

12

M.MIREK

Proof. According to Lemma 2.1 (we may assume that it holds for all n ∈ Nh ) and the definition of function Φ(x) = {x} − 1/2 we obtain X

X

ϕ′ (n)−1 e2πiαn =

n∈Nh ∩[1,N ]

X

=

n∈[1,N ] ′

−1

ϕ (n)

n∈[1,N ]

X

+

n∈[1,N ]

X

=


ϕ′ (n)−1 ⌊−ϕ(n)⌋ − ⌊−ϕ(n + 1)⌋ e2πiαn


ϕ(n + 1) − ϕ(n) e2πiαn


ϕ′ (n)−1 Φ(−ϕ(n + 1)) − Φ(−ϕ(n)) e2πiαn e2πiαn +

X

n∈[1,N ]

n∈[1,N ]


ϕ′ (n)−1 Φ(−ϕ(n + 1)) − Φ(−ϕ(n)) e2πiαn + O(log N ).

The proof will completed if we show that

(3.22)

sup

X

X

1 2πim

P ∈[1,N ] P 0. Then there exist a finite constant C > 0, a set of indexes B ⊆ N ∪ {0} × Z and functions g and (bs,j )(s,j)∈B such that X X bs,j = g + bs , f =g+ (s,j)∈B

s≥0

and

• kgkℓ∞ (Z) ≤ λ, • bs,j is supported on the dyadic cube Qs,j = [j2s , (j + 1)2s ) ∩ Z,

20

M.MIREK

• for any fixed s ≥ 0

X

bs =

bs,j ,

j∈Z: (s,j)∈B

• {Qs,j : (s, j) ∈ B} is a disjoint collection, • kbs,j kℓ1 (Z) ≤ λ|Qs,j | = λ2s , • The constant C > 0 is independent of λ > 0 and f . Moreover, X Ckf kℓ1 (Z) |Qs,j | ≤ . λ (s,j)∈B

Note that we have not assumed a cancellation condition for bs,j . However, instead of that we make further modifications of bs,j . Namely, we split bs as follows bs = bns + Bsn + gsn , where (in the sequel we will use the following convenient notational convention [f ]Q = bns (x) = bs (x)1{x∈Z: |bs (x)|>λdn } (x), hns (x) = bs (x) − bns (x) = bs (x)1{x∈Z: |bs (x)|≤λdn } (x), Bsn (x) = hns (x) − gsn (x), where

• • • • for any fixed s ≥ 0

gsn (x) =

X

1 |Q|

R

Q

f ),

[hns ]Qs,j 1Qs,j (x).

j∈Z: (s,j)∈B

The task now is to show that Mf (x) is of weak type (1, 1). Since X X X f =g+ gsn + bns + Bsn , s≥0

s≥0

s≥0

we observe that

 X  {x ∈ Z : |Mf (x)| > 4Cλ} ⊆ {x ∈ Z : sup Kn ∗ g + gsn (x) > Cλ} n∈N

s≥0

X  ∪ {x ∈ Z : sup Kn ∗ bns (x) > Cλ} n∈N

s≥0

∪ {x ∈ Z : sup Kn ∗ n∈N

 s(n)−1 s=0

 Bsn (x) > Cλ}

∞   X Bsn (x) > Cλ} = S1 ∪ S2 ∪ S3 ∪ S4 , ∪ {x ∈ Z : sup Kn ∗ n∈N

s

X

s=s(n)

where s(n) = min{s ∈ N : 2 ≥ Dn }. We shall deal with each set separately.

6.1. Step 1. Estimates for |S1 |. If C > 0 is sufficiently large then X  |S1 | ≤ |{x ∈ Z : sup |Kn ∗ g(x)| + sup Kn ∗ gsn (x) > Cλ}| = 0, n∈N

n∈N

s≥0

≤ [|bs |]Qs,j ≤ |Qs,j |−1 kbs,j kℓ1 (Z) ≤ λ, ≤ ≤ since kgkℓ∞ (Z) ≤ λ and which in turn implies that X X X 1Qs,j (x) ≤ λ. |[hns ]Qs,j |1Qs,j (x) ≤ λ gsn (x) ≤ |[hns ]Qs,j |

s≥0

[|hns |]Qs,j

(s,j)∈B

[|bns |]Qs,j

(s,j)∈B

21

6.2. Step 2. Estimates for |S2 |. X  XX |S2 | ≤ |{x ∈ Z : sup Kn ∗ |{x ∈ Z : Kn ∗ |bns (x)| > 0}| bns (x) > Cλ}| ≤ n∈N

≤

XX

|supp Kn | · |{x ∈ Z :

n∈N s≥0

≤

XX

dn

s≥0 n∈N

=

XX

n∈N s≥0

s≥0

X

|bns (x)|

> 0}| .

XX

dn · |{x ∈ Z : |bs (x)| > λdn }|

n∈N s≥0

|{x ∈ Z : λdk < |bs (x)| ≤ λdk+1 }|

k≥n

|{x ∈ Z : λdk < |bs (x)| ≤ λdk+1 }| ·

k X

dn

n=1

s≥0 k∈N



kf kℓ1 (Z) 1X 1 XX , λdk |{x ∈ Z : λdk < |bs (x)| ≤ λdk+1 }| . kbs kℓ1 (Z) . . λ λ λ s≥0 k∈N

s≥0

as desired.

6.3. Step 3. Estimates for |S4 |. It remains to show that

|S4 | ≤

Ckf kℓ1 (Z) for every s ≥ s(n), λ

which will follow from the definition of s(n) = min{s ∈ N : 2s ≥ Dn }. Indeed, supp Kn ⊆ [0, Dn ] ⊆ [0, 2s ] since s ≥ s(n). Thus supp Kn ∗ Bsn ⊆ supp Kn + supp Bsn ⊆ [0, 2s ] +

[

Qs,k ⊆

k∈Z

[

3Qs,k ,

k∈Z

where 3Q denotes the unique cube with the same center as Q and side length equal to 3 times of the side length of Q. Therefore, S4 ⊆

[

[

{x ∈ Z : |Kn ∗ Bsn (x)| > 0} ⊆

[

[

[

3Qs,k ⊆

n∈N s≥s(n) k∈Z

n∈N s≥s(n)

and consequently

|S4 | ≤

X

(s,k)∈B

|3Qs,k | .

X

(s,k)∈B

|Qs,k | ≤

Ckf kℓ1 (Z) . λ

[

(s,k)∈B

3Qs,k ,

22

M.MIREK

6.4. Step 4. Estimates for |S3 |. What is left is to estimate S3 . For this purpose we will proceed as follows. Notice that by Lemma 6.6 we obtain  s(n)−1  X n λ |{x ∈ Z : sup Kn ∗ Bs (x) > Cλ}| 2

n∈N

s=0

s(n)−1 X n = λ |{x ∈ Z : sup (x) > Cλ}| Kn ∗ Bs(n)−1−s 2

n∈N

.

X

x∈Z

≤

s=0

s(n)−1

2 X s(n)−1

2 X

X n n sup Kn ∗ Bs(n)−1−s (x) . Kn ∗ Bs(n)−1−s

2

n∈N

s=0

X X s(n)−1 X

2

Kn ∗B n s(n)−1−s ℓ2 (Z) +2

n∈N

s=0

X

n∈N 0≤s2