On the Hardy--Littlewood majorant problem for arithmetic sets
aa r X i v : . [ m a t h . C A ] M a y ON THE HARDY–LITTLEWOOD MAJORANT PROBLEMFOR ARITHMETIC SETS
BEN KRAUSE, MARIUSZ MIREK, AND BARTOSZ TROJAN
Abstract.
The aim of this paper is to exhibit a wide class of sparse deterministic sets, B ⊆ N , so thatlim sup N →∞ N − | B ∩ [1 , N ] | = 0 , for which the Hardy–Littlewood majorant property holds:sup | a n |≤ (cid:13)(cid:13)(cid:13) X n ∈ B ∩ [1 ,N ] a n e πinξ (cid:13)(cid:13)(cid:13) L p ( T , d ξ ) ≤ C p (cid:13)(cid:13)(cid:13) X n ∈ B ∩ [1 ,N ] e πinξ (cid:13)(cid:13)(cid:13) L p ( T , d ξ ) , where p ≥ p B is sufficiently large, the implicit constant C p is independent of N , and the supremum istaken over all complex sequences ( a n : n ∈ N ) such that | a n | ≤ Introduction
In 1937, Hardy and Littlewood [7] conjectured that for each p ≥ C p > A ⊂ N and every sequence ( a n : n ∈ A ) of complex numbers satisfying sup n ∈ A | a n | ≤ (cid:13)(cid:13)(cid:13) X n ∈ A a n e πinξ (cid:13)(cid:13)(cid:13) L p ( T , d ξ ) ≤ C p (cid:13)(cid:13)(cid:13) X n ∈ A e πinξ (cid:13)(cid:13)(cid:13) L p ( T , d ξ ) . (1)This conjecture, known as the Hardy–Littlewood majorant problem , was suggested by a simple observation,based on Parseval’s identity, which implies that C p = 1 for every even integer p ≥
2. It was also noticedby Hardy and Littlewood that C >
1. In 1962, Boas [2] showed that C p > p
6∈ { k : k ∈ N } .Finally, in early seventies Bachelis [1] disproved the Hardy–Littlewood conjecture showing unboundednessof C p for every p
6∈ { k : k ∈ N } as | A | → ∞ .Although inequality (1) fails to hold in general, recently some attention has been paid to quantify thisfailure. To do so, for N ∈ N we consider C p ( N ) = sup A ⊆{ ,...,N } C p ( A, N )where for A ⊆ { , . . . , N } we have set C p ( A, N ) = sup | a n |≤ (cid:13)(cid:13)(cid:13) X n ∈ A a n e πinξ (cid:13)(cid:13)(cid:13) L p ( T , d ξ ) · (cid:13)(cid:13)(cid:13) X n ∈ A e πinξ (cid:13)(cid:13)(cid:13) − L p ( T , d ξ ) . It was proven in [10] that for every p ∈ (2 ,
4) there is a constant
C > C p ( N ) ≥ C log N log log N .
Consequently, the Hardy–Littlewood majorant problem was reformulated to a slightly weaker statement.Namely, it was conjectured that for every p ≥ ε > C p,ε > N ∈ N C p ( N ) ≤ C p,ε N ε . (2) It is worth mentioning that (2) implies the restriction conjecture for the Fourier transform on R d , i.e.that for every p > d/ ( d −
1) there exists a constant C p,d > (cid:13)(cid:13) d f d σ (cid:13)(cid:13) L p ( R d ) ≤ C p,d k f k L ∞ ( S d − , d σ ) (3)where σ is the spherical measure on the unit sphere S d − in R d . In [10] it was stated that for suitablesets A the inequality (1) may be treated as a restatement of (3). However, Mockenhaupt and Schlag [11]disproved (2) by showing that for all p > η > C > C p ( N ) ≥ CN η . For p = 3 the same result was obtained by Green and Ruzsa [5].In view of the restriction conjecture one may ask whether there are sets A ⊆ { , . . . , N } such that forevery p ≥ ε > C p,ε > C p ( A, N ) ≤ C p,ε N ε . (4)The question above has been extensively studied by Mockenhaupt and Schlag in [11] where the authorsproved that for every ̺ ∈ (0 ,
1) and p ≥ A ⊆ { , . . . , N } with cardinality N ̺ satisfying (4) with a large probability.The Hardy–Littlewood majorant property plays an important role in combinatorial problems. In [4]Green used a variant of the inequality (1) for the set of prime numbers P to deduce that every subset of P with non-vanishing relative upper-density contains at least one arithmetic progression of length three.Specifically, Green proved that for every p ≥ C p > N ∈ N C p ( P N , N ) ≤ C p where P N = P ∩ [1 , N ], the set of primes less than or equal to N . Generally speaking, in problems of thiskind it is critical to know whether the majorant property (1) holds for some p ∈ (2 ,
3) with the uniformconstant C p , independent of the cardinality of the set A (see [6, 12]).The present article is devoted to study a wide class of deterministic infinite sets A ⊆ N with vanishingBanach density, i.e. lim sup N →∞ | A ∩ [1 , N ] | N = 0 , and obeying the Hardy–Littlewood majorant property. In particular, we will be concerned with the sets A = (cid:8) ⌊ h ( n ) ⌋ : n ∈ N (cid:9) (5)where h is a regularly varying function of the form h ( x ) = xℓ ( x ), for a suitably chosen slowly varyingfunction ℓ , e.g. ℓ ( x ) = (log x ) B , or ℓ ( x ) = exp (cid:0) B (log x ) C (cid:1) , or ℓ ( x ) = l m ( x ) , where B > C ∈ (0 , l ( x ) = log x and l m +1 ( x ) = log( l m ( x )), for m ∈ N . We show that for every p ≥ C p > N ∈ N we have C p ( A N , N ) ≤ C p where A N = A ∩ [1 , N ]. We also consider the sets (5) with h ( x ) = x c ℓ ( x )for some c > p c > p > p c there exists a constant C c,p > N ∈ N C p ( A N , N ) ≤ C c,p . Moreover, lim c → p c = 2. N THE HARDY–LITTLEWOOD MAJORANT PROBLEM 3
Statement of the results.
Before we precisely formulate the main results we need to introducesome definitions.
Definition 1.1.
Let L be a family of slowly varying functions ℓ : [ x , ∞ ) → (0 , ∞ ) such that ℓ ( x ) = exp (cid:16) Z xx ϑ ( t ) t d t (cid:17) where ϑ ∈ C ([ x , ∞ )) is a real function satisfyinglim x →∞ ϑ ( x ) = 0 , lim x →∞ xϑ ′ ( x ) = 0 , lim x →∞ x ϑ ′′ ( x ) = 0 . We also distinguish a subfamily L of L . Definition 1.2.
Let L be a family of slowly varying functions ℓ : [ x , ∞ ) → (0 , ∞ ) such that ℓ ( x ) = exp (cid:16) Z xx ϑ ( t ) t d t (cid:17) where ϑ ∈ C ([ x , ∞ )) is positive decreasing real function satisfyinglim x →∞ ϑ ( x ) = 0 , lim x →∞ xϑ ′ ( x ) ϑ ( x ) = 0 , lim x →∞ x ϑ ′′ ( x ) ϑ ( x ) = 0 , and for every ε > C ε > ≤ C ε ϑ ( x ) x ε and lim x →∞ ℓ ( x ) = ∞ .Finally, we define the subfamily R c of regularly varying functions. Definition 1.3.
For every c ∈ (0 , \ { } let R c be a family of increasing, convex, regularly-varyingfunctions h : [ x , ∞ ) → [1 , ∞ ) of the form h ( x ) = x c L ( x )where L ∈ L . If c = 1 we impose that L ∈ L .We fix two functions h ∈ R c and h ∈ R c for c ∈ [1 ,
2) and c ∈ [1 , / ϕ and ϕ be theinverse of h and h , respectively. We consider a function ψ : [ x , ∞ ) → (0 , ∞ ) such that for all x ≥ x , ψ ( x ) ≤ / x → + ∞ ψ ( x ) ϕ ′ ( x ) = 1 , lim x → + ∞ ψ ′ ( x ) ϕ ′′ ( x ) = 1 , lim x → + ∞ ψ ′′ ( x ) ϕ ′′′ ( x ) = 1 . (6)Finally, we define two sets B + = (cid:8) n ∈ N : { ϕ ( n ) } < ψ ( n ) (cid:9) , B − = (cid:8) n ∈ N : {− ϕ ( n ) } < ψ ( n ) (cid:9) . Let us observe that if h = h = h is the inverse function ϕ and ψ ( x ) = ϕ ( x + 1) − ϕ ( x ) then B − = A .Indeed, we have the following chain of equivalences m ∈ A ⇐⇒ m = ⌊ h ( n ) ⌋ for some n ∈ N ⇐⇒ h ( n ) − < m ≤ h ( n ) < m + 1 ⇐⇒ ϕ ( m ) ≤ n < ϕ ( m + 1) , since ϕ is well-defined and monotonically increasing ⇐⇒ ≤ n − ϕ ( m ) < ϕ ( m + 1) − ϕ ( m ) = ψ ( m ) < / ⇐⇒ ≤ {− ϕ ( m ) } < ψ ( m ) ⇐⇒ m ∈ B − . The main result of this paper is the following theorem.
BEN KRAUSE, MARIUSZ MIREK, AND BARTOSZ TROJAN
Theorem 1.
Assume that c ∈ [1 , and c = 1 . Then for every p ≥ there exists a constant C p > such that for every N ∈ N and any sequence of complex numbers ( a n : n ∈ N ) satisfying sup n ∈ N | a n | ≤ we have (cid:13)(cid:13)(cid:13) X n ∈ B ± N a n e πinξ (cid:13)(cid:13)(cid:13) L p ( T , d ξ ) ≤ C p (cid:13)(cid:13)(cid:13) X n ∈ B ± N e πinξ (cid:13)(cid:13)(cid:13) L p ( T , d ξ ) (7) where B ± N = B ± ∩ [1 , N ] . We observe that by the Hausdorff–Young inequality for every p ≥ (cid:13)(cid:13)(cid:13) X n ∈ B ± N a n e πinξ (cid:13)(cid:13)(cid:13) L p ( T , d ξ ) ≤ | B ± N | /p ′ . Moreover, by integrating over frequencies | ξ | ≤ / (100 N ), we have the following lower bound (cid:13)(cid:13)(cid:13) X n ∈ B ± N e πinξ (cid:13)(cid:13)(cid:13) L p ( T , d ξ ) & | B ± N | N − /p . These inequalities combined together yield (cid:13)(cid:13)(cid:13) X n ∈ B ± N a n e πinξ (cid:13)(cid:13)(cid:13) L p ( T , d ξ ) . | B ± N | /p N − /p (cid:13)(cid:13)(cid:13) X n ∈ B ± N e πinξ (cid:13)(cid:13)(cid:13) L p ( T , d ξ ) . (8)By Proposition 2.1 for c = 1, we have | B ± N | ∼ ϕ ( N ) where ϕ ( N ) = N L ϕ ( N ) for some slowly varyingfunction L ϕ ∈ L . Therefore, applying inequality (8), we obtain C p ( B ± N , N ) . L ϕ ( N ) /p . N ε for any ε >
0. Hence, the main difficulty in proving Theorem 1 is to show that the constant in (7) isindependent of N .Next, we would like to relax the hypothesis in Theorem 1 to allow any c ∈ [1 , / p . Let us introducecheck p ( c , c ) = 2 /c − /c + 61 /c + 3 /c − − /c /c + 3 /c − . We observe that if c ∈ [1 ,
2) and c ∈ [1 , /
5) then 1 < c + c , thus12 − /c /c + 3 /c − ≥ . Also notice that lim c → p ( c , c ) = 2 . The extended version of Theorem 1 has the following form.
Theorem 2.
Assume that c ∈ [1 , and c ∈ [1 , / . Then for every p ≥ p ( c , c ) there exists aconstant C p > such that for every N ∈ N and any sequence of complex numbers ( a n : n ∈ N ) satisfying sup n ∈ N | a n | ≤ we have (cid:13)(cid:13)(cid:13) X n ∈ B ± N a n e πinξ (cid:13)(cid:13)(cid:13) L p ( T , d ξ ) ≤ C p (cid:13)(cid:13)(cid:13) X n ∈ B ± N e πinξ (cid:13)(cid:13)(cid:13) L p ( T , d ξ ) where B ± N = B ± ∩ [1 , N ] . N THE HARDY–LITTLEWOOD MAJORANT PROBLEM 5
We were inspired to study Hardy–Littlewood majorant property by the paper of Mockenhaupt andSchlag [11] where the authors considered sparse random subsets of the integers. The desire to betterunderstand structure of deterministic sets which satisfy (1) was our principal motivation.Before turning to the arguments, let us begin with some preliminary remarks. The heart of the matterlies in proving our Proposition 3.1, which can be though of as a restriction estimate for our sets B ± N .We accomplish this using a Tomas–Stein T T ∗ argument, which forces us to estimate certain exponentialsums, see Section 3 below. These estimates are quite delicate, and lead to the technical restriction on therange of L p spaces which we are able to handle; in particular, we do not yet know how to extend Theorem2 to the full regime 2 < p < p ( c , c ). Finally, it is worth calling attention to the explicit constructionof the sets B ± N for which the full strength of the Hardy–Littlewood property holds. To the best of theauthors knowledge it is the first treatment where such a wide family of subsets of the integers satisfiesproperty (1). 2. Some properties of the sets B ± As it has been observed, when c ∈ [1 ,
2) and c ∈ [1 , /
5) we have 1 < c + c , or equivalently3(1 − γ ) + (1 − γ ) < , where γ = 1 /c and γ = 1 /c . Under this assumption, we prove the asymptotic formula for thecardinality of sets B ± . Proposition 2.1.
For every ǫ > | B ± N | = ϕ ( N ) (cid:0) O (cid:0) N − ǫ (cid:1)(cid:1) . (9)From now on we only work with the sets B + because all the results remain valid for B − with similarproofs. To simplify the notation we write B = B + = { n ∈ N : { ϕ ( n ) } < ψ ( n ) } . We need the following working characterizations of the sets B . Lemma 2.2. n ∈ B if and only if ⌊ ϕ ( n ) ⌋ − ⌊ ϕ ( n ) − ψ ( n ) ⌋ = 1 .Proof. We begin with the forward implication; it suffices to show that if n ∈ B , the integer ⌊ ϕ ( n ) ⌋ − ⌊ ϕ ( n ) − ψ ( n ) ⌋ belongs to (0 , / n ∈ B then 0 ≤ ϕ ( n ) − ⌊ ϕ ( n ) ⌋ < ψ ( n ), thus − ϕ ( n ) ≤ −⌊ ϕ ( n ) ⌋ < ψ ( n ) − ϕ ( n )if and only if ϕ ( n ) ≥ ⌊ ϕ ( n ) ⌋ > ϕ ( n ) − ψ ( n ) , from where it follows that ⌊ ϕ ( n ) ⌋ − ⌊ ϕ ( n ) − ψ ( n ) ⌋ > { ϕ ( n ) − ψ ( n ) } ≥ . In view of ⌊ ϕ ( n ) − ψ ( n ) ⌋ ≥ ϕ ( n ) − ψ ( n ) −
1, we obtain ⌊ ϕ ( n ) ⌋ − ⌊ ϕ ( n ) − ψ ( n ) ⌋ ≤ ⌊ ϕ ( n ) ⌋ − ϕ ( n ) + ψ ( n ) + 1 ≤ ψ ( n ) + 1 < / . We now turn to the reverse implication; if ⌊ ϕ ( n ) ⌋ = 1 + ⌊ ϕ ( n ) − ψ ( n ) ⌋ , we have0 ≤ ϕ ( n ) − ⌊ ϕ ( n ) ⌋ = ϕ ( n ) − − ⌊ ϕ ( n ) − ψ ( n ) ⌋ < ϕ ( n ) − ψ ( n ) − ϕ ( n ) = ψ ( n ) . Consequently, we get { ϕ ( n ) } < ψ ( n ), as desired. (cid:3) BEN KRAUSE, MARIUSZ MIREK, AND BARTOSZ TROJAN
Our next task is to show that for every δ ≥ − γ ) + (1 − γ ) + 6 δ < δ ′ > X n ∈ B N e πiξn = N X n =1 ψ ( n ) e πiξn + O (cid:0) ϕ ( N ) N − δ − δ ′ (cid:1) (10)where the implied constant is independent of ξ and N . Let us observe that the asymptotic formula (9)follows from (10) by taking ξ = 0. Indeed, we have | B N | = X n ∈ B N N X n =1 ψ ( n ) + O (cid:0) ϕ ( N ) N − ε (cid:1) and summation by parts yields1 ϕ ( N ) N X n =1 ψ ( n ) = N ψ ( N ) ϕ ( N ) − ϕ ( N ) Z N xψ ′ ( x ) d x = 1 ϕ ( N ) Z N ψ ( x ) dx = 1 + o (1) . (11)Although, for the proof of (9) we only needed (10) with ξ = 0, the more general version will be used inour future works.For the proof of (10), let us introduce the “sawtooth” function Φ( x ) = { x } − /
2. Notice that ⌊ ϕ ( n ) ⌋ − ⌊ ϕ ( n ) − ψ ( n ) ⌋ = ψ ( n ) + Φ (cid:0) ϕ ( n ) − ψ ( n ) (cid:1) − Φ (cid:0) ϕ ( n ) (cid:1) . With this in mind, we may write X n ∈ B N e πiξn = N X n =1 ψ ( n ) e πiξn + N X n =1 (cid:0) Φ (cid:0) ϕ ( n ) − ψ ( n ) (cid:1) − Φ (cid:0) ϕ ( n ) (cid:1)(cid:1) e πiξn . (12)The second sum we absorb into an error term of the order O (cid:0) ϕ ( N ) N − ε (cid:1) . To do so, see [8], we expandΦ into its Fourier series, i.e.Φ( x ) = X < | m |≤ M πim e − πimx + O (cid:18) min (cid:26) , M k x k (cid:27)(cid:19) , for some M > k x k = min {| x − n | : n ∈ Z } is the distance of x ∈ R to the nearest integer. Next,we expand min (cid:26) , M k x k (cid:27) = X m ∈ Z b m e πimx (13)where | b m | . min (cid:26) log MM , | m | , M | m | (cid:27) . (14)We split the second sum in (12) into three parts, I = X < | m |≤ M πim N X n =1 e πi ( nξ − mϕ ( n )) (cid:0) e πimψ ( n ) − (cid:1) ,I = O (cid:18) N X n =1 min (cid:26) , M k ϕ ( n ) − ψ ( n ) k (cid:27) (cid:19) ,I = O (cid:18) N X n =1 min (cid:26) , M k ϕ ( n ) k (cid:27) (cid:19) . N THE HARDY–LITTLEWOOD MAJORANT PROBLEM 7
Now, our aim is to show that each part I , I and I is O (cid:0) ϕ ( N ) N − ε (cid:1) . In the proof we use the estimatesfor the following trigonometric sums: for m ∈ Z \ { } , l ∈ { , } and X ≤ X ′ ≤ X we consider X X ≤ n ≤ X ′ ≤ X e πi ( ξn + m ( ϕ ( n ) − lψ ( n ))) By [9, Lemma 2.14], if c = 1 then there is a positive decreasing real function σ satisfying σ (2 x ) ≃ σ ( x )and σ ( x ) & x − ε for any ε >
0, such that(15) ϕ ′′ ( x ) ≃ ϕ ( x ) σ ( x ) x . We set σ ≡ c >
1. Similarly, by [9, Lemma 2.14] for ϕ ′′′ we obtain(16) ϕ ′′′ ( x ) ≃ ϕ ( x ) x . Therefore, by (6) we may write | ψ ′′ ( x ) | ≃ | ϕ ′′′ ( x ) | ≃ ϕ ( x ) x . Since 1 /c ≤ < /c , we get ϕ ( x ) xσ ( x ) ϕ ( x ) = o (1) , thus | ψ ′′ ( x ) | = o (cid:16) σ ( x ) ϕ ( x ) x (cid:17) . Let F ( x ) = ξx + m ( ϕ ( x ) − lψ ( x )). By (15) and (16), for any X ≤ x ≤ X ′ ≤ X we may write | F ′′ ( x ) | = | m | · | ϕ ′′ ( x ) − lψ ′′ ( x ) | ≃ | m | σ ( X ) ϕ ( X ) X . Therefore, the Van der Corput lemma (see [3, Theorem 2.2]) yields (cid:12)(cid:12)(cid:12) X
X There is a positive decreasing real function σ satisfying σ (2 x ) ≃ σ ( x ) and σ ( x ) & x − ε ,for any ε > , such that for every m ∈ Z \ { } , l ∈ { , } , and N ≥ we have (cid:12)(cid:12)(cid:12) X ≤ n ≤ N e πi ( ξn + m ( ϕ ( n ) − lψ ( n ))) (cid:12)(cid:12)(cid:12) . | m | / N (log N ) (cid:0) σ ( N ) ϕ ( N ) (cid:1) − / . If c > then σ ≡ . The implied constant is independent of m , N and ξ . Next, we return to bounding I , I and I . BEN KRAUSE, MARIUSZ MIREK, AND BARTOSZ TROJAN The estimate for I . Let S ( x ) = X x ≤ n ≤ x ′ < x e πi ( nξ − mϕ ( n )) and φ m ( x ) = e πimψ ( x ) − 1. We observe that | φ m ( x ) | . mx − ϕ ( x )and | φ ′ m ( x ) | . mx − ϕ ( x ) . Applying to the inner sum in I summation by parts together with (17) we obtain (cid:12)(cid:12)(cid:12) N X n =1 e πi ( nξ − mϕ ( n )) φ m ( n ) (cid:12)(cid:12)(cid:12) ≤ (log N ) sup X ∈ [1 ,N ] (cid:16) | S (2 X ) | · | φ m (2 X ) | + | S ( X ) | · | φ m ( X ) | + Z XX | S ( x ) | · | φ ′ m ( x ) | d x (cid:17) . (log N ) sup X ∈ [1 ,N ] m / ϕ ( X ) (cid:0) σ ( X ) ϕ ( X ) (cid:1) − / ≤ m / ϕ ( N )(log N ) (cid:0) σ ( N ) ϕ ( N ) (cid:1) − / . Therefore, | I | . M X m =1 m / (log N ) ϕ ( N ) (cid:0) σ ( N ) ϕ ( N ) (cid:1) − / . M / (log N ) ϕ ( N ) (cid:0) σ ( N ) ϕ ( N ) (cid:1) − / . The estimates for I and I . We only treat I because I can be handled by a similar reasoning.By (13), (14) and Lemma 2.3 we have N X n =1 min (cid:26) , M k ϕ ( n ) − ψ ( n ) k (cid:27) ≤ X m ∈ Z | b m | (cid:12)(cid:12)(cid:12)(cid:12) N X n =1 e πim ( ϕ ( n ) − ψ ( n )) (cid:12)(cid:12)(cid:12)(cid:12) . N (log M ) M + (cid:18) X < | m |≤ M log MM + X | m | >M M | m | (cid:19) | m | / (log N ) ϕ ( N ) (cid:0) σ ( N ) ϕ ( N ) (cid:1) − / . N (log M ) M + M / (log M )(log N ) ϕ ( N ) (cid:0) σ ( N ) ϕ ( N ) (cid:1) − / . Concluding remarks. Based on Subsections 2.1 and 2.2 we get | I | + | I | + | I | . N log MM + M / (log M )(log N ) ϕ ( N ) (cid:0) σ ( N ) ϕ ( N ) (cid:1) − / . Therefore, by taking M = N δ (log N ) ϕ ( N ) − , we conclude | I | + | I | + | I | . ϕ ( N ) N − δ (cid:16) N / δ/ (log N ) σ ( N ) − / ϕ ( N ) − / ϕ ( N ) − / (cid:17) . ϕ ( N ) N − δ (cid:16) N / δ/ − γ / − γ / (cid:17) which is bounded by a constant multiple of ϕ ( N ) N − δ since 3(1 − γ ) + (1 − γ ) + 6 δ < N THE HARDY–LITTLEWOOD MAJORANT PROBLEM 9 Proof of Theorem 1 Let F ( f )( ξ ) = X n ∈ Z f ( n ) e πiξn denote the Fourier transform on Z , and ˆ f ( n ) = Z T f ( ξ ) e − πiξn d ξ denote the Fourier transform on T . For any measure space X , let C ( X ) be the space of all continuousfunctions on X . For N ∈ N we introduce on Z a measure µ N defined µ N ( x ) = N − X n ∈ B N ψ ( n ) − δ n ( x ) . Let T N : C ( B N ) → C ( T ) be the linear operator given by T N ( f ) = F (cid:0) f µ N (cid:1) . We are going to prove the following proposition. Proposition 3.1. For each (18) p ≥ − /c /c + 3 /c − there is a constant C p > such that for all N ∈ N and f ∈ L (cid:0) B N , µ N (cid:1) k T N f k L p ( T ) ≤ C p N − /p k f k L ( B N ,µ N ) . Before embarking on the proof we show the following. Lemma 3.2. For every δ > satisfying (1 − γ ) + 3(1 − γ ) + 6 δ < there is δ ′ > such that X n ∈ B N ψ ( n ) − e πiξn = N X n =1 e πiξn + O (cid:0) N − δ − δ ′ (cid:1) . The implied constant is independent of ξ and N .Proof. For N ∈ N and ξ ∈ T we set S N ( ξ ) = X k ∈ B N e πiξk . Then, by the summation by parts we have X n ∈ B N ψ ( n ) − e πiξn = N X n =1 ψ ( n ) − (cid:0) S n ( ξ ) − S n − ( ξ ) (cid:1) = ψ ( N + 1) − S N ( ξ ) + N X n =1 (cid:0) ψ ( n ) − − ψ ( n + 1) − (cid:1) S n ( ξ ) . (19)Similarly, we may write(20) N X n =1 e πiξn = ψ ( N + 1) − N X n =1 ψ ( n ) e πiξn + N X n =1 (cid:0) ψ ( n ) − − ψ ( n + 1) − (cid:1) n X k =1 ψ ( k ) e πiξk . Thus, subtracting (20) from (19) we may estimate (cid:12)(cid:12)(cid:12) X n ∈ B N ψ ( n ) − e πiξn − N X n =1 e πiξn (cid:12)(cid:12)(cid:12) ≤ N X n =1 (cid:12)(cid:12) ψ ( n ) − − ψ ( n + 1) − (cid:12)(cid:12) · (cid:12)(cid:12)(cid:12) S n ( ξ ) − n X k =1 ψ ( k ) e πiξk (cid:12)(cid:12)(cid:12) + ψ ( N + 1) − (cid:12)(cid:12)(cid:12) S N ( ξ ) − N X k =1 ψ ( k ) e πiξk (cid:12)(cid:12)(cid:12) . By (10), for any δ > − γ ) + (1 − γ ) + 6 δ < δ ′ > n ∈ N (cid:12)(cid:12)(cid:12) S n ( ξ ) − n X k =1 ψ ( k ) e πiξk (cid:12)(cid:12)(cid:12) ≤ Cϕ ( n ) n − δ − δ ′ . Using (6) together with [9, Lemma 2.14] we obtain ψ ′ ( n ) . ϕ ′′ ( n ) . ϕ ( n ) n . Therefore, again by (6) and the monotonicity of ϕ we get (cid:12)(cid:12) ψ ( n ) − − ψ ( n + 1) − (cid:12)(cid:12) . sup t ∈ [ n,n +1] (cid:12)(cid:12) ψ ( t ) − ψ ′ ( t ) (cid:12)(cid:12) . ϕ ( n ) − . Hence, N X n =1 (cid:12)(cid:12) ψ ( n ) − − ψ ( n + 1) − (cid:12)(cid:12) · (cid:12)(cid:12)(cid:12) S n ( ξ ) − n X k =1 ψ ( k ) e πiξk (cid:12)(cid:12)(cid:12) . N X n =1 n − δ − δ ′ . N − δ − δ ′ . (cid:3) Proof of Proposition 3.1. The T T ∗ argument will be critical in the proof. Firstly, let us calculate T ∗ N .By Plancherel’s theorem we have h T N f, g i L ( T ) = Z T F (cid:0) f µ N (cid:1) ( ξ ) g ( ξ ) d ξ = X n ∈ Z f ( n )ˆ g ( n ) µ N ( n ) = h f, T ∗ N g i L ( B N ,µ N ) . Therefore, the adjoint operator T ∗ N : C ( T ) ∗ → C ( B N ) ∗ = C ( B N ) is given by T ∗ N ( g ) = ˆ g · B N , and consequently, T N T ∗ N : C ( T ) ∗ → C ( T ) ∗ may be written as T N T ∗ N f = f ∗ F ( µ N ) . Let us observe that it is enough to show k T N T ∗ N k L p ′ ( T ) → L p ( T ) ≤ C p N − /p . (21)Indeed, for f ∈ L (cid:0) B N , µ N (cid:1) and g ∈ L p ′ ( T ) we have |h T N f, g i L ( T ) | = |h f, T ∗ N g i L ( B N ,µ N ) | ≤ k f k L ( B N ,µ N ) k T ∗ N g k L ( B N ,µ N ) and since k T ∗ N g k L ( B N ,µ N ) = h T N T ∗ N g, g i L ( T ) ≤ k T N T ∗ N k L p ′ ( T ) → L p ( T ) k g k L p ′ ( T ) , we obtain k T N f k L p ( T ) ≤ k T N T ∗ N k / L p ′ ( T ) → L p ( T ) k f k L ( B N ,µ N ) . For the proof of (21), for N ∈ N , let us introduce an auxiliary measure ν N on Z and the correspondinglinear operator S N : C ( N N ) → C ( T ), by setting ν N ( x ) = N − X n ∈ N N δ n ( x ) , N THE HARDY–LITTLEWOOD MAJORANT PROBLEM 11 and S N f = F ( f ν N ) . Here, N N := N ∩ [1 , N ]. Reasoning similar to the above applied to the operator S N leads to S N S ∗ N f = f ∗ F ( ν N ) . Since L p ( T ) can be embedded into C ( T ) ∗ for any p ≥ T N T ∗ N and S N S ∗ N as mappings on L p ( T ) spaces. Next, we write (cid:13)(cid:13) T N T ∗ N f (cid:13)(cid:13) L p ( T ) = (cid:13)(cid:13) f ∗ F ( µ N ) (cid:13)(cid:13) L p ( T ) ≤ (cid:13)(cid:13) f ∗ F ( ν N ) (cid:13)(cid:13) L p ( T ) + (cid:13)(cid:13) f ∗ F ( µ N − ν N ) (cid:13)(cid:13) L p ( T ) . We are going to show that for each p satisfying (18) there is C p > (cid:13)(cid:13) f ∗ F ( ν N ) (cid:13)(cid:13) L p ( T ) ≤ C p N − /p k f k L p ′ ( T ) , (22) (cid:13)(cid:13) f ∗ F ( µ N − ν N ) (cid:13)(cid:13) L p ( T ) ≤ C p N − /p k f k L p ′ ( T ) (23)for all f ∈ L p ′ ( T ). We start by proving (22) for p = 2. By Plancherel’s theorem we have (cid:13)(cid:13) f ∗ F ( ν N ) (cid:13)(cid:13) L ( T ) = (cid:13)(cid:13) ˆ f ν N (cid:13)(cid:13) ℓ ( Z ) ≤ k ν N k ℓ ∞ ( Z ) k f k L ( T ) ≤ N − k f k L ( T ) . On the other hand, for p = ∞ we may write k f ∗ F ( ν N ) k L ∞ ( T ) ≤ kF ( ν N ) k L ∞ ( T ) k f k L ( T ) ≤ k f k L ( T ) . Therefore, for p ≥ (cid:13)(cid:13) f ∗ F ( µ N − ν N ) (cid:13)(cid:13) L ( T ) = (cid:13)(cid:13) ˆ f (cid:0) µ N − ν N (cid:1)(cid:13)(cid:13) L ( T ) ≤ (cid:13)(cid:13) µ N − ν N (cid:13)(cid:13) ℓ ∞ ( Z ) k f k L ( T ) ≤ ϕ ( N ) − k f k L ( T ) . Secondly, for p = ∞ we get k f ∗ F ( µ N − ν N ) k L ∞ ( T ) ≤ kF ( µ N ) − F ( ν N ) k L ∞ ( T ) k f k L ( T ) ≤ N − δ − δ ′ k f k L ( T ) where in the last estimate we have used Lemma 3.2. Thus, again by Riesz–Thorin interpolation theorem,for p ≥ (cid:13)(cid:13) f ∗ F ( µ N − ν N ) (cid:13)(cid:13) L p ( T ) ≤ (cid:13)(cid:13) µ N − ν N (cid:13)(cid:13) /pℓ ∞ ( Z ) · (cid:13)(cid:13) F ( µ N − ν N ) (cid:13)(cid:13) − /pL ∞ ( T ) · k f k L p ′ ( T ) . ϕ ( N ) − /p N − ( δ + δ ′ )(1 − /p ) k f k L p ′ ( T ) . Let us recall that for any ε > ϕ ( N ) & ε N γ − ε Therefore, for the inequality (23) to hold true, we need to have ε > p > − γ − ε ) /p − ( δ + δ ′ )(1 − /p ) ≤ − /p. Hence, ε ≤ − (1 − γ ) + ( δ + δ ′ )( p − / . Because the right hand side has to be positive, we obtain the condition( δ + δ ′ )( p − / − (1 − γ ) > , which is equivalent to p > − γ ) / ( δ + δ ′ ) . Since 3(1 − γ ) + (1 − γ ) + 6 δ < p ≥ − γ ) γ + 3 γ − . (cid:3) Next, we show Theorem 1 and Theorem 2. Proof of Theorem 1 and Theorem 2. Let ( a n : n ∈ N ) be a sequence of complex numbers such thatsup n ∈ N | a n | ≤ 1. Using Proposition 3.1 with f ( n ) = a n ψ ( n ) we get Z T (cid:12)(cid:12)(cid:12) X n ∈ B N f ( n ) ψ ( n ) − e πiξn (cid:12)(cid:12)(cid:12) p d ξ . p N p/ − (cid:16) X n ∈ B N | f ( n ) | ψ ( n ) − (cid:17) p/ , thus, by (11), Z T (cid:12)(cid:12)(cid:12) X n ∈ B N a n e πiξn (cid:12)(cid:12)(cid:12) p d ξ . p N p/ − (cid:16) X n ∈ B N ψ ( n ) (cid:19) p/ . p N − ϕ ( N ) p . Finally, we may estimate Z T (cid:12)(cid:12)(cid:12) X n ∈ B N e πiξn (cid:12)(cid:12)(cid:12) p d ξ & Z | ξ |≤ / (100 N ) (cid:12)(cid:12)(cid:12) X n ∈ B N e πiξn (cid:12)(cid:12)(cid:12) p d ξ & N − ϕ ( N ) p . 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