Convergence and almost sure properties in Hardy spaces of Dirichlet series
aa r X i v : . [ m a t h . F A ] J a n CONVERGENCE AND ALMOST SURE PROPERTIES IN HARDYSPACES OF DIRICHLET SERIES
FR´ED´ERIC BAYART
Abstract.
Given a frequency λ , we study general Dirichlet series P a n e − λ n s . First, wegive a new condition on λ which ensures that a somewhere convergent Dirichlet seriesdefining a bounded holomorphic function in the right half-plane converges uniformly inthis half-plane, improving classical results of Bohr and Landau. Then, following recentworks of Defant and Schoolmann, we investigate Hardy spaces of these Dirichlet series.We get general results on almost sure convergence which have an harmonic analysisflavour. Nevertheless, we also exhibit examples showing that it seems hard to get generalresults on these spaces as spaces of holomorphic functions. Introduction λ -Dirichlet series and their convergence. A general Dirichlet series is a series P n a n e − λ n s where ( a n ) ⊂ C N , s ∈ C and λ = ( λ n ) is an increasing sequence of nonnegativereal numbers tending to + ∞ , called a frequency . We shall denote by D ( λ ) the space ofall formal λ -Dirichlet series. The two most natural examples are ( λ n ) = ( n ) which givesrise to power series and ( λ n ) = (log n ), the case of ordinary Dirichlet series. A proeminentproblem at the beginning of the twentieth century was the study of the convergence ofthese series, starting from the following theorem of Bohr [5] on ordinary Dirichlet series: let D = P n a n n − s be a somewhere convergent ordinary Dirichlet series having a holomorphicand bounded extension to the half-plane C . Then D converges uniformly on each half-place C ε for all ε >
0. Here, C θ means the right half-plane [Re s > θ ].An important effort was done to extend this result to general frequencies. Two suffi-cient conditions were isolated firstly by Bohr [5] and then by Landau [19]. Following theterminology of [21], we say that a frequency satisfies (BC) provided(BC) ∃ ℓ > , ∃ C > , ∀ n ∈ N , λ n +1 − λ n ≥ Ce − ℓλ n . A frequency λ satisfies (LC) provided(LC) ∀ δ > , ∃ C > , ∀ n ∈ N , λ n +1 − λ n ≥ Ce − e δλn . Of course, (BC) is a stronger condition than (LC), and Landau has shown that anyfrequency satisfying (LC) verifies Bohr’s theorem: all Dirichlet series D = P n a n e − λ n s Date : January 11, 2021.2020
Mathematics Subject Classification.
Key words and phrases.
Dirichlet series, Hardy spaces, abscissa of convergence, Helson theorem.The author was partially supported by the grant ANR-17-CE40-0021 of the French National ResearchAgency ANR (project Front). belonging to D ext ∞ ( λ ), the space of somewhere convergent λ -Dirichlet series which allow aholomorphic and bounded extension to C , converge uniformly to C ε , for all ε > D ∞ ( λ ) the subspace of D ext ∞ ( λ )of Dirichet series which converge on C (in general, D ∞ ( λ ) can be a proper subspaceof D ext ∞ ( λ ) if λ does not satisfy Bohr’s theorem) and define S N : D ext ∞ ( λ ) → D ∞ ( λ ), P + ∞ a n e − λ n s P N a n e − λ n s the N -th partial sum operator. In [21], a thorough studyof the norm of S N is done (the case of ordinary Dirichlet series was settled in [1]), leadingto consequences on the existence of a Montel’s theorem in D ext ∞ ( λ ) or on the completenessof D ∞ ( λ ).Our first main result is an extension of the results of [5, 19, 21]: we provide a new classof frequencies, that we will call (NC) (see Definition 2.2) such that Bohr’s theorem, andall its consequences, are true. Like (BC) or (LC), this class of frequencies quantifies howfast λ goes to + ∞ and how close its terms are, but in a less demanding way since (NC) isstrictly weaker than (LC).Our method of proof also differs from that of [21]. In [21], the estimation of k S N k is basedon Riesz means of λ -Dirichlet polynomials: recall that for a sequence of complex numbers( c n ) and for k >
0, the finite sum R λ,kx ( X n c n ) := X λ n
0. I. Schoolmann uses approx-imation of D by R λ,kx ( D ) for a suitable choice of k to deduce its result on k S N k . Ouralternate approach is based on mollifiers and on a formula of convolution due to Saksmanfor ordinary Dirichlet series.1.2. Hardy spaces of λ -Dirichlet series: Banach spaces and harmonic analysis. Our second approach deals with Hardy spaces of Dirichlet series. For ordinary Dirichletseries, they have been introduced and studied in [15] (see also [1]) and this has caused animportant renew of interest for this subject. The general case has been introduced andinvestigated very recently in [10, 12, 11]. For p ∈ [1 , + ∞ ), the Hardy space H p ( λ ) may bedefined as follows: given a λ -Dirichlet polynomial D = P Nn =1 a n e − λ n s , define its H p -normby k D k pp := lim T → + ∞ T Z T − T | D ( it ) | p dt. Then H p ( λ ) is the completion of the set of λ -Dirichlet polynomials for this norm. However,this internal description is often not sufficient to get the main properties of H p ( λ ) and weneed a group approach. Let G be a compact abelian group and let β : ( R , +) → G bea continuous homomorphism with dense range. Then we say that ( G, β ) is a λ -Dirichletgroup provided, for all n ∈ N , there exists h λ n ∈ ˆ G such that h λ n ◦ β = e − iλ n · . The space H λp ( G ), p ∈ [1 , + ∞ ] is then defined as the subspace of L p ( G ) of functions f such that ENERAL DIRICHLET SERIES 3 supp( ˆ f ) ⊂ { h λ n : n ∈ N } . Now define the Bohr map B by B : H λp ( G ) → D ( λ ) f X ˆ f ( h λ n ) e − λ n s and set H p ( λ ) = B ( H λp ( G )) with kB f k p := k f k p . Then it has been shown in [10] that • given a frequency λ , there always exists a λ -Dirichlet group ( G, β ); • the Hardy space H p ( λ ) does not depend on the choosen λ -Dirichlet group; • when p = + ∞ , it coincides with H p ( λ ) defined internally.Our second aim in this paper is solve some of the problems on the spaces H p ( λ ) raisedin [9] and to exhibit new properties of them. In particular, we investigate properties of H p ( λ ) coming from functional analysis and harmonic analysis.As an example, let us discuss a famous therorem of Helson [17] which ensures, in ourterminology, that if λ satisfies (BC) and D = P n a n e − λ n s belongs to H ( λ ), then for almostall homomorphisms ω : ( R , +) → T , the Dirichlet series P n a n ω ( λ n ) e − λ n s converges on C . This has been extended to the Hardy spaces H p ((log n )) for p ≥ H p ( λ ) or, equivalenty - via the Bohr transform - of H λp ( G ). When λ satisfies (BC), this has been done in [12], adding moreover the maximal inequality. Theorem A (Defant-Schoolmann) . Let λ satisfy (BC), let ( G, β ) be a λ -Dirichlet group.For every u > , there exists a constant C := C ( u, λ ) such that, for all ≤ p ≤ + ∞ andfor all f ∈ H λp ( G ) , (1) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup σ ≥ u sup N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X n =1 ˆ f ( h λ n ) e − σλ n h λ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ≤ C k f k p . In particular, for every u > , P + ∞ ˆ f ( h λ n ) e − uλ n h λ n converges almost everywhere on G . When λ satisfies (LC), the almost everywhere statement is known to be true, as well asthe maximal inequality for p > p . When p = 1,it is valid if we replace the L ( G )-norm by the weak L ( G )-norm. We shall prove thatinequality (1) remains true even on the weaker assumption that λ satisfies (NC), even for p = 1, and with a constant independent of p . Our approach, which seems less technicalthan that of [12], is based again on a version of Saksman’s convolution formula and on aCarleson-Hunt type maximal inequality of independent interest.1.3. H p ( λ ) as a Banach space of holomorphic functions. The results announced inthe previous section indicate that the spaces H p ( λ ) seem to behave well if we look at theiralmost sure properties. The classical case H p ((log n )) was also investigated as a Banachspace of holomorphic function. Even in that case, it is a nontrivial problem to determinethe optimal half-plane of convergence of elements in H p ( λ ), namely to compute σ H p ( λ ) := inf { σ ∈ R : σ c ( D ) ≤ σ for all D ∈ H p ( λ ) } FR´ED´ERIC BAYART where, for a Dirichlet series D ∈ D ( λ ), σ c ( D ) := inf { σ ∈ R : D converges on C σ } . Thishas been settled in [1], using that σ H ((log n )) = 1 / T σ ( P n a n e − λs ) = P n a n e − σλ n e − λ n s maps H p ((log n )) into H q ((log n )) forall p, q ∈ [1 , + ∞ ) and all σ >
0. The argument is based on multiplicativity (namely onthe fact that the natural λ -Dirichlet group for (log n ) is the infinite polytorus T ∞ ) and ona hypercontractive estimate for the Poisson kernel acting on the Hardy spaces H p ( T ) ofthe disk.We will show that there is no hope to get such a result for general frequencies λ evenif they satisfy (BC). For instance, if we will be able to prove that for all frequencies σ H ( λ ) ≤ σ H ( λ ) , we will nevertheless point out that, even if we assume (BC), this isoptimal and in particulat that we may have σ H ( λ ) > σ H ( λ ) . We will also exhibit asequence λ , which still satisfies (BC), such that T σ maps boundedly H ( λ ) into H k ( λ ) ifand only if σ ≥ ( k − / k . In particular, it seems very hard to compute σ H p ( λ ) in thegeneral case and the behaviour of H p ( λ ) as a space of holomorphic function seems moredifficult to predict if we assume only growth and separation conditions on λ .1.4. Notations.
Throughout this work, we shall use the following notations. For λ afrequency and D ∈ D ( λ ), the abscissa of absolute convergence of D and the abscissa ofuniform convergence of D are defined by σ a ( D ) := inf { σ ∈ R : D converges absolutely on C σ } σ u ( D ) := inf { σ ∈ R : D converges uniformly on C σ } . We set L ( λ ) := lim sup N → + ∞ log Nλ N = sup D ∈D ( λ ) σ a ( D ) − σ c ( D ) . Given (
G, β ) a λ -Dirichlet group we shall denote by Pol λ ( G ) the set of polynomials withspectrum in λ , namely finite sums P nk =1 a k h λ k with λ k ∈ G for each k = 1 , . . . , n . Weshall also use the following result: for all f : G → C measurable, for almost all ω ∈ G ,the function f ω := f ( ωβ ( · )) : R → C is measurable. If additionally f ∈ L ∞ ( G ), then foralmost all ω ∈ G , f ω ∈ L ∞ ( R ) with k f ω k ∞ ≤ k f k ∞ . Moreover, if f ∈ L ( G ), then f ω islocally integrable for almost all ω ∈ G , and for g ∈ L ( R ), the convolution g ⋆ f ω ( t ) := Z R f ( ωβ ( t − y )) g ( y ) dy is almost everywhere defined on R and measurable (see [10, Lemma 3.11]).2. Preliminaries
A new class of frequencies.
We introduce our new condition, more general than(LC), under which most of our results will be satisfied. We first reformulate (LC).
Lemma 2.1.
A frequency λ satisfies (LC) if and only if there exists C > such that, forall δ > , for all n ∈ N , log (cid:18) λ n +1 + λ n λ n +1 − λ n (cid:19) ≤ Ce δλ n . ENERAL DIRICHLET SERIES 5
Proof.
Assume first that λ satisfies (LC) and let δ >
0. Then there exists
C > n ∈ N , λ n +1 − λ n ≥ Ce − e δ λn . Let n ∈ N and set ξ n = λ n + Ce − e δ λn . Sincethe function x ( x + λ n ) / ( x − λ n ) is decreasing on ( λ n , + ∞ ), one gets λ n +1 + λ n λ n +1 − λ n ≤ ξ n + λ n ξ n − λ n ≤ C − (cid:18) λ n + Ce − e δ λn (cid:19) e e δ λn ≤ C ′ e e δλn . The converse implication is easier and left to the reader. (cid:3)
The main idea to introduce (NC) is to allow to compare the position of λ n with λ m forsome m > n and not only with λ n +1 . Definition 2.2.
We say that a frequency λ satisfies (NC) if, for all δ >
0, there exists
C > n ≥
1, there exists m > n such that(NC) log (cid:18) λ m + λ n λ m − λ n (cid:19) + ( m − n ) ≤ Ce δλ n . Condition (NC) provides a nontrivial extension of (LC).
Example 2.3.
Let λ be defined by λ n + k = n + ke − e n for k = 0 , . . . , n −
1. Then L ( λ ) = + ∞ , λ satisfies (NC) and λ is not the finite union of frequencies satisfying (LC). Proof.
Let δ > n ∈ N , k ∈ { , . . . , n − } , then provided n is large enoughlog (cid:18) λ n +1 + λ n + k λ n +1 − λ n + k (cid:19) + (2 n +1 − n − k ) ≤ log (cid:0) n + 1) (cid:1) + 2 n ≤ Ce δn ≤ Ce δλ n + k for some C >
0. Moreover, if λ was the finite union of λ , . . . , λ p , each λ j satisfying (LC),then at least one of the λ j , say λ , will contain an infinite number of consecutive terms λ m = λ n + k , λ m +1 = λ n + k ′ with 1 ≤ k ′ − k ≤ p and k, k ′ ∈ { , . . . , n − } . For these m ,log (cid:18) λ m +1 + λ m λ m +1 − λ m (cid:19) ≥ e n − log p ≥ Ce λ m / contradicting that λ satisfies (LC). (cid:3) Saksman’s vertical convolution formula.
Saksman’s vertical convolution for-mula was introduced to express weighted sums of ordinary Dirichlet series using an integral.It says essentially that if D = P n a n n − s is an ordinary Dirichlet series and ψ is in L withˆ ψ compactly supported, then + ∞ X n =1 a n ˆ ψ (log n ) n − s = Z R D ( s + it ) ψ ( t ) dt FR´ED´ERIC BAYART with a sense that has to be made precise. It was used in [4] for Dirichlet series in H andin [20] for Dirichlet series in H ∞ . We shall extend it to general Dirichlet series and wewill use it as a much more flexible substitute of Perron’s formula. Theorem 2.4.
Let ψ ∈ L ( R ) be such that ˆ ψ is compactly supported and let λ be afrequency with ( G, β ) an associated λ -Dirichlet group.(a) Let D = P n a n e − λ n s ∈ D ext ∞ ( λ ) with bounded and holomorphic extension to C denotedby f . Then for all s ∈ C with ℜ e ( s ) > + ∞ X n =1 a n ˆ ψ ( λ n ) e − λ n s = Z R f ( s + it ) ψ ( t ) dt. (b) Let f = P n a n h λ n ∈ H λ ( G ) . Then for almost all ω ∈ G , + ∞ X n =1 a n ˆ ψ ( λ n ) h λ n ( ω ) = Z R f ω ( t ) ψ ( t ) dt. In the sequel, for D = P n a n e − λ n s ∈ D ext ∞ ( λ ), respectively for f = P n a n h λ n in H λ ( G ),and for ψ ∈ L ( R ) compactly supported, we shall denote R ψ ( D ) := X n a n ˆ ψ ( λ n ) e − λ n s R ψ ( f ) := X n a n ˆ ψ ( λ n ) h λ n . Proof. (a) Observe first that the equality is true provided D is a Dirichlet polynomial andthat the two members of the equality define an analytic function on C . Assume firstthat L ( λ ) < + ∞ . Then σ a ( D ) < + ∞ and for s > σ a ( D ), the formula is true just byexchanging the sum and the integral. We conclude by analytic continuation.When L ( λ ) = + ∞ , the proof is more difficult. We use (see [14, Theorem 41 p. 53] or[18, Theorem 22]) that there exist a half-plane C θ , θ >
0, and a sequence of λ -Dirichletpolynomials D j = P + ∞ n =1 a jn e − λ n s such that ( a jn ) tends to a n as j tends to + ∞ for any n and ( D j ) converges uniformly to f on C θ . Since each D j is a Dirichlet polynomial, weknow that for all s ∈ C θ and all j ∈ N , + ∞ X n =1 a jn ˆ ψ ( λ n ) e − λ n s = Z R D j ( s + it ) ψ ( t ) dt. Letting j to + ∞ in the previous inequality for a fixed s ∈ C θ , since the sum on the lefthandside is finite ( ˆ ψ has compact support), and by uniform convergence, we get the resulton C θ . We conclude again by analytic continuation.(b) When f ∈ Pol λ ( G ), the equality follows immediately by interverting a finite sum and ENERAL DIRICHLET SERIES 7 an integral, and the definition of the objects that come into play: R ψ ( f )( ω ) = Z R X n a n e − itλ n ψ ( t ) h λ n ( ω ) dt = Z R X n a n h λ n ( β ( t )) h λ n ( ω ) ψ ( t ) dt = Z R f ω ( t ) ψ ( t ) dt (here the equality is valid for all ω ∈ G ). Let now f ∈ H λ ( G ). Then Z G Z R | f ω ( t ) ψ ( t ) | dt ≤ k f k k ψ k . Therefore, for almost all ω ∈ G , the function t f ω ( t ) ψ ( t ) belongs to L ( R ) and theoperator S ψ : H λ ( G ) → L ( G, L ( R )), f [ ω f ω ( · ) ψ ( · )] is continuous. If ( f n ) is asequence in Pol λ ( G ) tending to f ∈ H λ ( G ), then there exists a sequence ( n k ) such that,for a.e. ω ∈ G , S ψ ( f n k )( ω ) → S ψ ( f )( ω ) in L ( R ) R ψ ( f n k )( ω ) → R ψ ( f )( ω )(recall that the sum defining R ψ is finite). Since R ψ ( f n k )( ω ) = R R S ψ ( f n k )( ω ) for all k andall ω ∈ G , we get the conclusion by taking the limit. (cid:3) Remark 2.5.
The statement of Theorem 2.4 remains true provided ψ is not compactlysupported but still satisfies P n | ˆ ψ ( λ n ) | < + ∞ . Remark 2.6.
To obtain Theorem 2.4, in both cases, we use the density of polynomialsfor a suitable topology. In H λ ( G ), this is trivial which is not the case in D ∞ ext ( λ ). Morespecifically we intend to use Theorem 2.4 to obtain results that do not seem easily reachableusing Riesz means. Therefore it is intesting to obtain a proof of Theorem 2.4 that do notuse Riesz means. This is the case if we use [18, Theorem 22]. We thank A. Defant and I.Schoolmann for pointing out to me this reference. Remark 2.7.
Part (b) of the vertical convolution formula is more precised than thestatement established and used in [4]. The equivalent statement in this context would bethat, for all f ∈ H λ ( G ), + ∞ X n =1 a n ˆ ψ ( n ) h λ n = Z R T t f ψ ( t ) dt, where T t : H λ ( G ) → H λ ( G ) , f f ( β ( t ) · ) is an onto isometry of H λ ( G ) and the righthandside denotes a vector-valued integral in H λ ( G ). We will need a pointwise statementin order to obtain maximal inequalities.2.3. Riesz means and Saksman’s vertical convolution formula.
We now show howthe results on ( λ, k )-Riesz means will follow from our results coming from Saksman’svertical convolution formula. This is a consequence of the following easy proposition.
Proposition 2.8.
Let α > . Then there exists an L ( R ) -function ψ such that, for all t ∈ R , ˆ ψ ( x ) = (1 − | x | ) α provided | x | < , ˆ ψ ( x ) = 0 otherwise. FR´ED´ERIC BAYART
Proof.
Define u ( x ) = (1 − | x | ) α [ − , ( x ). Then u is piecewise C , its derivative u ′ ( x ) = ± α (1 − | x | ) α − [ − , ( x ) belongs to L ( R ) and thus we know that, for all t = 0, ˆ u ( t ) = it b u ′ ( t ) . Now, it is easy to see that u ′ belongs to L ε ( R ) ∩ L ( R ) for some ε >
0. Hence, b u ′ belongs to L q ( R ) for some q < + ∞ . In particular, by H¨older’s inequality, ˆ u belongs to L , so that we may apply the inverse Fourier transform to get the statement. (cid:3) In the sequel, for λ a frequency, α >
0, (
G, β ) a λ -Dirichlet group and N >
0, we shall usethe following notations: R λ,αN ( f ) = X λ n ≤ N b f ( h λ n ) (cid:18) − λ n N (cid:19) α h λ n R λ,αN ( D ) = X λ n ≤ N a n (cid:18) − λ n N (cid:19) α e − λ n s where f ∈ H λ ( G ) and D = P n a n e − λ n s ∈ D ( λ ). Many of the results of [21, 12, 11]are based on a detailed study of these operators R λ,αN . We shall extend them via theconvolution formula to other operators R ψ , allowing better results with a different choiceof ψ . 3. Bohr’s theorem under (NC)
The case of D ext ∞ ( λ ) . In his study of Bohr’s theorem [21], I. Schoolmann used that,for all D = P n a n e − λ n s ∈ D ext ∞ ( λ ) with extension f , the sequence of its Riesz means oforder k R kx ( D ) = X λ n
0, as x → + ∞ . We now showthat we may replace the function ψ such that ˆ ψ ( t ) = (1 − | t | ) k [ − , ( t ) by any function L -function ψ such that ˆ ψ has compact support. Lemma 3.1.
Let λ be a frequency, let ψ ∈ L ( R ) be such that ˆ ψ has compact support andlet D = P n a n e − λ n s ∈ D ext ∞ ( λ ) with extension f . Then k R ψ ( D ) k ∞ ≤ k ψ k k f k ∞ . Moreover, if R R ψ = 1 ,denoting by ψ N ( · ) = N ψ ( N · ) , the sequence of λ -Dirichlet polyno-mials ( R ψ N ( D )) converges uniformly to f on each half-plane C ε , for all ε > .Proof. The inequality follows immediately from Theorem 2.4. The statement on uniformconvergence follows as well from this formula and from standard results on mollifiers,provided we know that f is uniformly continuous on C ε . Again, this can be deduced fromthe fact that on this half-plane, f is the uniform limit of λ -Dirichlet polynomials, whichare themselves uniformly continuous. (cid:3) We shall now apply this to a suitable choice of ψ in order to get good estimates of thenorm of the projection S N . ENERAL DIRICHLET SERIES 9
Theorem 3.2.
Let λ be a frequency. There exists C > such that, for all M > N ≥ , k S N k D ext ∞ ( λ ) →D ∞ ( λ ) ≤ C (cid:18) log (cid:18) λ M + λ N λ M − λ N (cid:19) + ( M − N − (cid:19) . Proof.
We set h = λ M − λ N . Let u be the function equal to 1 on [ − λ N , λ N ], to 0 on R \ ( − λ M , λ M ), and which is affine on ( − λ M , λ N ) and on ( λ N , λ M ). The function u maybe written u = [ − λ N − h,λ N + h ] ⋆ (cid:18) h [ − h,h ] (cid:19) . This formula allows us to compute the Fourier transform of u which is equal to b u ( t ) = 2 sin(( λ N + h ) t ) t · sin( ht ) ht which is an L function. Moreover k b u k ≤ Z + ∞ (cid:12)(cid:12)(cid:12)(cid:12) sin(( λ N + h ) t ) t (cid:12)(cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12)(cid:12) sin( ht ) ht (cid:12)(cid:12)(cid:12)(cid:12) dt ≤ Z + ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin (cid:16) λ N + hh x (cid:17) x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12)(cid:12) sin( x ) x (cid:12)(cid:12)(cid:12)(cid:12) dx ≤ Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin (cid:16) λ N + hh x (cid:17) x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx + 4 Z + ∞ x dx ≤ C log (cid:18) λ N + hh (cid:19) + 4 = C log (cid:18) λ M + λ N λ M − λ N (cid:19) + 4where we have used well-known estimates of the L -norm of the sinus cardinal function.We then applied Lemma 3.1 to ψ ∈ L defined by b ψ = u . By the Fourier inverse formula, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M − X n =1 a n b ψ ( λ n ) e − λ n s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ (cid:18) C log (cid:18) λ M + λ N λ M − λ N (cid:19) + 4 (cid:19) k f k ∞ . We get the conclusion by writing N X n =1 a n e − λ n s = M − X n =1 a n b ψ ( λ n ) e − λ n s − M − X n = N +1 a n b ψ ( λ n ) e − λ n s and by using that | a n | ≤ k f k ∞ (see [21, Corollary 3.9]) and k b ψ k ∞ ≤ (cid:3) From the Bohr-Cahen formula to compute the abscissa of uniform convergence of a λ -Dirichlet series, σ u ( D ) ≤ lim sup N log (cid:16) sup t ∈ R (cid:12)(cid:12)(cid:12)P Nn =1 a n e − λ n it (cid:12)(cid:12)(cid:12)(cid:17) λ N we get the following corollary. Corollary 3.3.
Let λ be a frequency satisfying (NC). Then λ satisfies Bohr’s theorem. Let us now compare Theorem 3.2 with the results of [21]. There it is shown that, for all N ≥ k ∈ (0 , k S N k D ext ∞ ( λ ) →D ∞ ( λ ) ≤ Ck (cid:18) λ N +1 λ N +1 − λ N (cid:19) /k . The right hand side is optimal for k = (cid:18) λN +1 λN +1 − λN (cid:19) which implies that k S N k D ext ∞ ( λ ) →D ∞ ( λ ) ≤ C log (cid:18) λ N +1 λ N +1 − λ N (cid:19) . Hence, we get the case M = N + 1 of Theorem 3.2.3.2. The case of H ∞ ( λ ) . So far, we have defined three spaces which are candidates forbeing the H ∞ -space of λ -Dirichlet series: D ∞ ( λ ), D ext ∞ ( λ ), and H ∞ ( λ ). We know that wealways have the canonical inclusion D ∞ ( λ ) ⊂ D ext ∞ ( λ ) ⊂ H ∞ ( λ ) (see [11, Theorem 2.17])and that, when λ satisfies Bohr’s theorem, the three spaces are equal. Observe also that H ∞ ( λ ) is the only space that is always complete.Thus, Theorem 3.2 does not always provide an answer for estimating the norm of S N asan operator on H ∞ ( λ ). Fortunately, the proof extends easily using the second (and easiestpart) of Theorem 2.4. Theorem 3.4.
Let λ be a frequency. There exists C > such that, for all M > N ≥ , k S N k H ∞ ( λ ) →D ∞ ( λ ) ≤ C (cid:18) log (cid:18) λ M + λ N λ M − λ N (cid:19) + ( M − N − (cid:19) . Proof.
We do the proof in H λ ∞ ( G ). Let f = P n a n h λ n ∈ H λ ∞ ( G ). We pick the samefunction ψ and observe, that for almost all ω ∈ G , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ∞ X n =1 a n b ψ ( λ n ) h λ n ( ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R | f ω ( it ) ψ ( t ) | dt ≤ (cid:18) C log (cid:18) λ M + λ N λ M − λ N (cid:19) + 4 (cid:19) k f k ∞ and we conclude as above. (cid:3) Maximal inequalities in H λp ( G )4.1. Helson’s theorem under (NC).
In this section, we prove the following theorem,which improves the main result of [12] and answers an open question of [9].
Theorem 4.1.
Let λ satisfy (NC), let ( G, β ) be a λ -Dirichlet group. For every u > ,there exists a constant C := C ( u, λ ) such that, for all ≤ p ≤ + ∞ and for all f ∈ H λp ( G ) , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup σ ≥ u sup N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X n =1 ˆ f ( h λ n ) e − σλ n h λ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ≤ C k f k p . In particular, for every u > , P + ∞ ˆ f ( h λ n ) e − uλ n h λ n converges almost everywhere on G . ENERAL DIRICHLET SERIES 11
Let us explain the strategy for the proof. When p >
1, the almost everywhere convergenceis known to hold without any assumption on λ . This is a consequence of the Carleson-Hunt type result proved in [12]: for all frequencies λ , for all ( G, β ) a λ -Dirichlet group,for all p ∈ (1 , + ∞ ), there exists C ( p ) > f ∈ H λp ( G ),(2) (cid:18)Z G sup n | S n f ( ω ) | p dω (cid:19) /p ≤ C ( p ) k f k p where S n ( f ) = P nk =1 b f ( h λ k ) h λ k is the partial sum operator (the constant C ( p ) does noteven depend on λ ). We shall prove a variant of (2) under (NC), namely(3) (cid:18)Z G sup n e − δλ n | S n f ( ω ) | p dω (cid:19) /p ≤ C ( λ, δ ) k f k p valid for all p ≥
1, all δ > f ∈ H λp ( G ), with a constant C ( λ, δ ) independent of p .The proof of (3) will be done for p = 1 and for p = + ∞ and will be finished by interpo-lation. Unfortunately, it is in general false that [ H λp ( G ) , H λp ( G )] θ = H λp θ ( G ) (see [3]) andwe will use an auxiliary operator defined on the whole L ( G ).We begin by establishing several lemmas. First, we shall prove that we may requireadditional properties on a sequence satisfying (NC). Lemma 4.2.
Let λ be a frequency satisfying (NC). Then there exists a frequency λ ′ suchthat λ ⊂ λ ′ and, for all δ > , there exists C > such that, for all n ∈ N , there exists m > n with log( λ ′ m + λ ′ n ) ≤ Ce δλ ′ n (4) − log( λ ′ m − λ ′ n ) ≤ Ce δλ ′ n (5) m − n ≤ Ce δλ ′ n . (6) Proof.
We construct inductively λ ′ as follows. We set λ ′ = λ . Assume that the sequence λ ′ has been built until step n , namely that we have constructed λ ′ , . . . , λ ′ k n with λ ′ k n = λ n .If λ n +1 ≤ λ n + 1, then we set λ ′ k n +1 = λ n +1 and k n +1 = k n + 1. Otherwise, we includeas many terms λ ′ k n +1 , . . . , λ ′ k n +1 as necessary so that, for all j = k n + 1 , . . . , k n +1 − / ≤ λ ′ j +1 − λ ′ j ≤ λ ′ k n +1 = λ n +1 . Namely, we add terms in the sequence λ whenthere is a gap greater than 1 between two successive terms, and the difference betweentwo consecutive terms of λ ′ is now less than 1.Let us show that the sequence λ ′ satisfies the above conclusion. Let λ ′ n be any term ofthe sequence λ ′ . If λ ′ n does not belong to λ , then we have just to consider m = n + 1.Otherwise, if λ ′ n = λ k for some k ≤ n , there exists l > k such thatlog (cid:18) λ l + λ k λ l − λ k (cid:19) ≤ Ce δλ k (7) l − k ≤ Ce δλ k . Set m = n + ( l − k ) and observe that we havelog( λ ′ m + λ ′ n ) ≤ log (cid:16) Ce δλ ′ n + 2 λ ′ n (cid:17) ≤ C ′ e δλ ′ n . If there is no gap between λ k and λ l , then λ ′ m = λ l and (7) implies − log( λ ′ m − λ ′ n ) = − log( λ l − λ k ) ≤ Ce δλ k = Ce δλ ′ n . If there is a gap between λ k and λ l , then λ ′ m − λ ′ n ≥ /
2, and (5) holds trivially. (cid:3)
In the sequel, when we will pick a frequency λ satisfying (NC), we will in fact assume thatit satisfies the stronger properties given by Lemma 4.2.For a > h >
0, we shall denote by ψ a,h the function defined by ψ a,h ( t ) = sin(( a + h ) t ) t × sin( ht ) ht . The estimation of the L -norm of ψ a,h was a crucial point in order to apply Saksman’sconvolution formula during the proof of Theorem 3.2. In order to obtain our maximalestimates, we will need a similar inequality allowing now a and h to vary. Lemma 4.3.
Let a : R → (0 , + ∞ ) and h : R → (0 , + ∞ ) be two measurable functions.Assume that there exists κ > such that a ( t ) + h ( t ) ≤ κ and h ( t ) ≥ κ − for all t ∈ R .Then Z R (cid:12)(cid:12) ψ a ( t ) ,h ( t ) ( t ) (cid:12)(cid:12) dt ≤ κ. Proof.
It suffices to observe that • when 0 < | t | ≤ κ − , then (cid:12)(cid:12) ψ a ( t ) ,h ( t ) ( t ) (cid:12)(cid:12) ≤ | a ( t ) + h ( t ) | × ≤ κ. • when κ − ≤ | t | ≤ κ , then (cid:12)(cid:12) ψ a ( t ) ,h ( t ) ( t ) (cid:12)(cid:12) ≤ | t | × | t | . • when | t | ≥ κ , then (cid:12)(cid:12) ψ a ( t ) ,h ( t ) ( t ) (cid:12)(cid:12) ≤ h ( t ) t ≤ κt . (cid:3) We now fix a frequency λ satisfying (NC) and δ >
0. Let
C > m : N → N be suchthat m ( n ) > n for all n ∈ N and (4), (5), (6) are satisfied for m = m ( n ). For n ∈ N , weshall denote by h n = ( λ m ( n ) − λ n ) / φ n the function φ n = ψ λ n ,h n . Let us recallthat R φ n is defined on H λ ( G ) by(8) R φ n ( f ) = X k b f ( h λ k ) c φ n ( λ k ) h λ k . By the vertical convolution formula, we also know that R φ n is given by, for a.e. ω ∈ G ,(9) R φ n ( f )( ω ) = Z R f ( ωβ ( t )) φ n ( t ) dt. Now the right hand side of the previous equality is well-defined for all functions in L ( G ).Thus we will think at R φ n as the operator on L ( G ) defined by (9), keeping in mind thatit also verifies (8) for f ∈ H λ ( G ). In this context, we shall prove the following maximalinequality on R φ n : ENERAL DIRICHLET SERIES 13
Lemma 4.4.
For all δ > , there exists C > such that, for all p ∈ [1 , + ∞ ] , for all N ∈ N , for all f ∈ L p ( G ) , Z G sup n ≤ N | R φ n f ( ω ) | p dω ! /p ≤ Ce δλ N k f k p . Proof.
We start with the case p = 1. It is enough to prove it for f ∈ C ( G ). Define n : G → { , . . . , N } , ω n ( ω ) by n ( ω ) = inf ( l ∈ { , . . . , N } : | R φ l f ( ω ) | = sup n ≤ N | R φ n f ( ω ) | ) . The function n is measurable and Z G sup n ≤ N | R φ n ( f )( ω ) | dω = Z G | R φ n ( ω ) ( f )( ω ) | dω ≤ Z R Z G | f ( ωβ ( t )) | · | ψ λ n ( ω ) ,h n ( ω ) ( t ) | dωdt. In the inner integral we do the change of variables ω ′ = ωβ ( t ) so that Z G sup n ≤ N | R φ n ( f )( ω ) | dω ≤ Z R Z G | f ( ω ′ ) | · | ψ λ n ( ω ′· β ( t ) − ,h n ( ω ′· β ( t ) − ( t ) | dωdt ≤ Z G | f ( ω ′ ) | Z R | ψ λ n ( ω ′· β ( t ) − ,h n ( ω ′· β ( t ) − ( t ) | dtdω. We now use Lemma 4.3 together with (4) and (5). This yields Z G sup n ≤ N | R φ n ( f )( ω ) | dω ≤ C Z G | f ( ω ′ ) | e δλ N dω ′ = Ce δλ N k f k . We then do the case p = + ∞ . Let f ∈ L ∞ ( G ). Thensup ω ∈ G sup n ≤ N | R φ n f ( ω ) | = sup n ≤ N sup ω ∈ G Z R | f ( ωβ ( t )) | · | ψ λ n ,h n ( t ) | dt ≤ sup n ≤ N k ψ λ n ,h n k k f k ∞ ≤ Ce δλ N k f k ∞ . We then conclude by interpolation. (cid:3)
We deduce from the above work a weighted Carleson-Hunt maximal inequality for H λ ( G )-functions, which seems interesting for itself when p = 1 (for p ∈ (1 , + ∞ ), an unweightedCarleson-Hunt inequality is true, the point here is that the constant does not depend on p ). This statement was inspired by [2] where a similar result in the (much easier) caseof H ( T ) was essential to do a multifractal analysis of the divergence of Fourier series offunctions of H ( T ). Theorem 4.5.
Let λ satisfying (NC). For all δ > there exists C > such that, for all N ∈ N , for all p ≥ , for all f ∈ H λp ( G ) , Z G sup n ≤ N | S n f ( ω ) | p dω ! /p ≤ Ce δλ N . Proof.
We argue as in the proof of Theorem 3.2, namely we write for a fixed n ∈ N , | S n f ( ω ) | ≤ | R φ n f ( ω ) | + m ( n ) − n ≤ | R φ ( n ) f ( ω ) | + Ce δλ n . Therefore, sup n ≤ N | S n f ( ω ) | ≤ sup n ≤ N | R φ n ( f )( ω ) | + Ce δλ N and we conclude by taking the L p ( G )-norm. (cid:3) We are now ready for the proof of Theorem 4.1.
Proof of Theorem 4.1.
We first proceed with the case p ∈ [1 , + ∞ ). We may assume that f ∈ P ol λ ( G ). Let δ = u/
3. For σ ≥ u , using Lemma 3.4 of [12], we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X n =1 b f ( h λ n ) e − σλ n h λ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ C ( u ) p sup n ≤ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − δλ n n X k =1 b f ( h λ k ) h λ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p . Hence, sup σ>u sup N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X n =1 b f ( h λ n ) e − σλ n h λ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ C ( u ) p sup N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − δλ N N X n =1 b f ( h λ n ) h λ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p . For ω ∈ G , we define n ( ω ) = inf ( l ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − δλ l l X n =1 b f ( h λ n ) h λ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = sup N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − δλ N N X n =1 b f ( h λ n ) h λ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)) which is measurable. For k ≥
0, we set A k = { n : λ n ∈ [ k, k + 1) } , G k = { ω ∈ G : n ( ω ) ∈ A k } ,I ( σ ) = Z G sup σ>u sup N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X n =1 b f ( h λ n ) e − σλ n h λ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dω. ENERAL DIRICHLET SERIES 15
We can write I ( σ ) ≤ C ( u ) p X k ≥ Z G k sup N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − δλ N N X n =1 b f ( h λ n ) h λ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dω ≤ C ( u ) p X k ≥ Z G k sup N ∈ A k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − δλ N N X n =1 b f ( h λ n ) h λ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dω ≤ C ( u ) p X k ≥ Z G k e − δpk sup N ∈ A k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X n =1 b f ( h λ n ) h λ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dω ≤ C ( u, λ ) p X k ≥ e − δpk e δp ( k +1) k f k pp ≤ C ( u, λ ) p k f k pp . As for the proof of Lemma 4.4, the proof is easier for p = ∞ and is left to the reader. (cid:3) If we analyze the previous proof carefully, we observe that we have obtained the following(slightly stronger) variant of Theorem 4.5.
Corollary 4.6.
Let λ satisfying (NC). For all δ > , there exists C > such that, for all p ≥ , for all f ∈ H λp ( G ) , (cid:18)Z G sup N (cid:12)(cid:12)(cid:12)(cid:12) S N f ( ω ) e δλ N (cid:12)(cid:12)(cid:12)(cid:12) p dω (cid:19) /p ≤ C ( δ ) k f k p . When λ satisfies (BC), it is possible to improve this inequality. Proposition 4.7.
Let λ satisfy (BC). For all α > , there exists C > such that, for all p ≥ , for all f ∈ H λp ( G ) , (cid:18)Z G sup N (cid:12)(cid:12)(cid:12)(cid:12) S N f ( ω ) λ αN (cid:12)(cid:12)(cid:12)(cid:12) p dω (cid:19) /p ≤ C ( α ) k f k p . Proof.
We just sketch the proof. If λ satisfies (BC), then we know that there exists C > n ∈ N , log( λ n +1 − λ n ) ≥ − Cλ n . Adding terms if necessary, we can alsoassume that log( λ n +1 + λ n ) ≤ Cλ n . Arguing exactly as in the proof of Theorem 4.5, wecan prove the existence of C > f ∈ H λp ( G ), for all n ∈ N , Z G sup n ≤ N | S n f ( ω ) | p dω ≤ Cλ pN . Let now α >
1, fix f ∈ Pol λ ( G ) and define, for ω ∈ G , n ( ω ) = inf ( l ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ − αl l X n =1 b f ( h λ n ) h λ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = sup N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ − αN N X n =1 b f ( h λ n ) h λ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)) A k = { n : λ n ∈ [2 k , k +1 ) } G k = { w : n ( ω ) ∈ A k } . Then Z G sup N (cid:12)(cid:12)(cid:12)(cid:12) S N f ( ω ) λ αN (cid:12)(cid:12)(cid:12)(cid:12) p dω = X k Z G k sup N (cid:12)(cid:12)(cid:12)(cid:12) S N f ( ω ) λ αN (cid:12)(cid:12)(cid:12)(cid:12) p dω = X k Z G k sup N ∈ A k (cid:12)(cid:12)(cid:12)(cid:12) S N f ( ω ) λ αN (cid:12)(cid:12)(cid:12)(cid:12) p dω ≤ X k − pkα Z G sup λ N ≤ k +1 | S N f ( ω ) | p dω ≤ C X k − pkα p ( k +1) k f k pp . (cid:3) Question 4.8.
We know that λ satisfies Bohr’s theorem if and only if for all δ > , thereexists C > such that, for all f ∈ H λ ∞ ( G ) , for all N ≥ , (10) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup n ≤ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X k =1 b f ( h λ k ) h λ k ( ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( G ) ≤ Ce δλ N k f k ∞ . We have shown that if λ satisfies (NC), then it satisfies the previous inequality. To provethat Helson’s theorem is satisfied (and even to prove that the relevant maximal inequalityholds true), it is sufficient to prove that, for all δ > , there exists C > such that, forall f ∈ H λ ( G ) , for all N ≥ , (11) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup n ≤ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X k =1 b f ( h λ k ) h λ k ( ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( G ) ≤ Ce δλ N k f k . Again we have shown that if λ satisfies (NC), then (11) is true. It seems natural to askwhether (11) always follows from (10) or, equivalently, if any frequency λ satifying Bohr’stheorem also satisfies Helson’s theorem. Inequalities in H ( λ ) have already been deducedfor their vector-valued counterpart in H ∞ ( λ ) in [10] . At first glance, it seems that thisargument cannot be applied here. Failure of Helson’s theorem for p = 1 . Since for p >
1, for any frequency λ , forany ( G, β ) a λ -Dirichlet group, for any g ∈ H λp ( G ), the series P + ∞ n =1 ˆ f ( h λ n ) h λ n convergesalmost everywhere on G (this follows from the Carleson-Hunt theorem of [12]), it is naturalto ask whether Theorem 4.1 remains true without any assumption on λ . We show thatthis is not the case. Theorem 4.9.
There exists a frequency λ , a λ -Dirichlet group ( G, β ) and f ∈ H λ ( G ) such that, for all u > , the series P + ∞ n =1 ˆ f ( h λ n ) e − uλ n h λ n diverges almost everywhere on G . As we might guess, the proof will use the results of Kolmogorov on a.e. divergent Fourierseries in L ( T ) (see for instance [22]). Lemma 4.10.
Let
A, δ > . There exists P ∈ H ( T ) a polynomial and E ⊂ T measurablesuch that ENERAL DIRICHLET SERIES 17 • k P k ≤ δ . • m T ( E ) ≥ − δ (here, m T denotes the Lebesgue measure on T ). • for all z ∈ E , there exists n ( z ) ∈ N such that | S n ( z ) P ( z ) | ≥ A .Proof. By induction on j ≥
1, we construct a sequence of holomorphic polynomials ( P j )with deg( P j ) = d j , two sequences of positive real numbers ( µ j ) and ( ε j ) and a sequence( E j ) of measurable subsets of T such that the following properties are true for each j :(a) m T ( E j ) ≥ − − j (b) k P j k ≤ − j (c) µ j > µ j − + d j − ε j − (d) the real numbers 2 π, µ , . . . , µ j , ε , . . . , ε j are Q -independent(e) for each z ∈ E j , there exists an integer n j ( z ) such that, for all u ∈ [0 , j ], (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n j ( z ) X k =0 c P j ( k ) e − ukε j z k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ je jµ j . Let us proceed with the construction. We choose for µ j any real number such that µ j >µ j − + d j − ε j − and the real numbers 2 π, µ , . . . , µ j , ε , . . . , ε j − are independent over Q (when j = 1, we simply choose µ > π with (2 π, µ ) independent over Q ). We then applyLemma 4.10 with A = ( j + 1) e jµ j and δ = 2 − j to get a polynomial P j with degree d j anda subset E j ⊂ T satisfying (a) and (b). Since the functions ( u, z ) P nk =0 b P j ( k ) e − ukε z k ,for 0 ≤ n ≤ d j , converge uniformly on [0 , j ] × T to ( u, z ) S n P ( z ) as ε →
0, we maychoose ε j a sufficiently small positive real number such that (d) and (e) are satisfied.Define now λ = { µ j + kε j : j ≥ , ≤ k ≤ d j } , G = Q + ∞ j =1 T endowed with thecanonical product structure and define β : ( R , +) → G, t (cid:0) ( e − itµ j , e − itε j ) (cid:1) j . By (d)and Kronecker’s theorem, the homomorphism β has dense range. Moreover, let λ n ∈ λ .Then λ n = µ j + kε j for some j ≥ ≤ k ≤ d j . Write an element ω ∈ G as Q + ∞ l =1 ( w l , z l ) and define h λ n ( ω ) = w j z kj . Then h λ n ◦ β ( t ) = e − it ( µ j + kε j ) = e − iλ n t so that ( G, β ) is a λ -Dirichlet group. Now, define f j ( ω ) = w j P j ( z j ) = d j X k =0 c P j ( k ) h µ j + kε j ( ω ) . We get k f j k H λ ( G ) = k P j k H ( T ) so that the series f = P j ≥ f j converges in H λ ( G ). Let usalso define F j = { ω ∈ G : z j ∈ E j } . Then m G ( F j ) = m T ( E j ) ≥ − − j (here, m G denotes the Haar measure on G ). Thus, if weset F = T j ≥ S j ≥ j F j , then m G ( F ) = 1. Pick now ω ∈ F . We may find j as large as wewant such that ω ∈ F j . The construction of P j ensures that there exists 0 ≤ n j ( z j ) ≤ d j such that, for all u ∈ [0 , j ], (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n j ( z j ) X k =0 c P j ( k ) e − u ( µ j + kε j ) z kj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ j. Setting N , resp. M , such that λ N = µ j − + d j − ε j − , resp. λ M = µ j + n j ( z j ) ε j , theprevious inequality translates into (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M X n = N +1 ˆ f ( h λ n ) e − uλ n h λ n ( ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ j. This easily yields the a.e. divergence of P + ∞ n =1 b f ( h λ n ) e − uλ n h λ n . (cid:3) Maximal inequalities for mollifiers.
Since ( S n f ) does not necessarily convergepointwise or even in norm for all functions in H λ ( G ), Defant and Schoolmann looked in [11]for a substitute by changing the summation method. They succeeded by choosing Rieszmeans. Precisely they showed (see [11, Theorem 2.1]), through a maximal inequality, thatfor all frequencies λ , for all f ∈ H λ ( G ), for all α >
0, the sequence ( R λ,αN ( f )( ω )) convergesto f ( ω ) for almost all ω ∈ G . We extend this to a large class of mollifiers. Theorem 4.11.
Let λ be a frequency, ( G, β ) a λ -Dirichlet group. Let ψ ∈ L ( R ) be acontinuous function (except at a finite number of points) such that b ψ has compact supportand there exists a nonincreasing function g ∈ L (0 , + ∞ ) such that | ψ ( x ) | ≤ g ( | x | ) for all x ∈ R . For N ≥ , define ψ N ( · ) = ψ ( · /N ) . Then R max ,ψ ( f ) := sup N | R ψ N ( f ) | defines a bounded sublinear operator from H λ ( G ) into L , ∞ ( G ) . Moreover, if R ψ = 1 ,then for all f ∈ H λ ( G ) , R ψ N ( f )( ω ) converges for almost every ω ∈ G to f ( ω ) .Proof. Again, the key point is the vertical convolution formula. Indeed, we know that fora.e. ω ∈ G , R ψ N ( f )( ω ) = f ω ⋆ ψ N (0) . For those ω , using [13, Theorem 2.1.10 and Remark 2.1.11],sup N | R ψ N ( f )( ω ) | ≤ sup N | f ω | ⋆ ψ N (0) ≤ k g k M f ( ω )where M ( f )( ω ) = sup T > T R T − T | f ω ( t ) | dt is the appropriate Hardy-Littlewood maximaloperator. Since M maps H λ ( G ) into L , ∞ ( G ) by [11, Theorem 2.10], we can concludeabout the first assertion of the theorem. The result on a.e. convergence is then a standardcorollary of it, using that it is clearly true for polynomials since b ψ (0) = 1. (cid:3) Remark 4.12.
We can replace the assumption that ψ is compactly supported by theassumption that, for all N ≥ P n | b ψ ( λ n /N ) | < + ∞ .This last theorem covers many examples. For instance, for all 0 ≤ a < b , we may choosethe function ψ ∈ L ( R ) such that ˆ ψ = 1 on [ − a, a ], ˆ ψ = 0 on ( −∞ , − b ) ∪ ( b, + ∞ ) and ˆ ψ is affine on ( − b, − a ) and on ( a, b ). As already observed during the proof of Theorem 3.2,the function ψ is given by ψ ( x ) = C ( a, b ) sin (cid:0) a + b x (cid:1) sin (cid:0) b − a x (cid:1) x which clearly satisfies the assumptions of Theorem 4.11. This is also the case for ψ ( x ) = e −| x | or ψ ( x ) = e − x , provided the frequency λ satisfies P | b ψ ( λ n /N ) | < + ∞ for all N ≥ α > ψ ∈ L ( R ) that satisfies b ψ ( t ) = (1 − | t | ) α [ − , ( t )verifies the assumptions of Theorem 4.11. Let x >
0. We already have observed that ψ ( x ) = 1 ix F (cid:0) ± α (1 − | t | ) α − [ − , (cid:1) ( x ) . Fix β > | β ( α − | < x ≥
1. Then | ψ ( x ) | ≤ αx (cid:12)(cid:12)(cid:12)(cid:12)Z (1 − t ) α − e itx dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ αx (cid:12)(cid:12)(cid:12)(cid:12)Z u α − e − iux du (cid:12)(cid:12)(cid:12)(cid:12) . We split the integral into two parts. First, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z x − β u α − e − iux du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ α x − αβ . Second, integrating by parts, Z x − β u α − e − iux du = − ix (cid:2) u α − e − iux (cid:3) x − β + α − ix Z x − β u α − e − iux du so that (cid:12)(cid:12)(cid:12)(cid:12)Z x − β u α − e − iux du (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) x + 1 x α − β (cid:19) . Our choice of β guarantees that there is δ > C > x ≥ | ψ ( x ) | ≤ Cx δ . This shows that the assumptions of Theorem 4.11 are satisfied with g ( x ) = Cx δ , x ≥ k ψ k ∞ , C ) , x ∈ [0 , . Remark 4.13.
Lemma 4.4 and Theorem 4.11 are of course very close. The latter one istrue for all frequencies λ , but we start from a fixed function ψ and it does not cover fullythe case p = 1. Lemma 4.4 adapts at each step the L -function to the frequency λ and tothe function f . The price to pay is that we lose some factor e δλ N and that we cannot usegeneral results on the Hardy-Littlewood maximal function. Horizontal translations
In this section, we investigate the boundedness from H p ( λ ) into H q ( λ ), for q > p , of thehorizontal translation map T σ ( P n a n e − λ n s ) = P n a n e − σλ n e − λ n s . We are interested inthis map to determine the exact value of σ H p ( λ ) = inf { σ ∈ R : σ c ( D ) ≤ σ for all D ∈ H p ( λ ) } since it is easy to prove, using the Cauchy-Schwarz inequality, that σ H ( λ ) = L ( λ ) / λ = (log n ), σ H p ( λ ) = 1 / p ∈ [1 , + ∞ ). In the general case, it isalways possible to majorize σ H ( λ ) if we know σ H ( λ ) . Proposition 5.1.
Let λ be a frequency. Then σ H ( λ ) ≤ σ H ( λ ) .Proof. Let ε >
0. It is sufficient to prove that, for all f = P j a j e − λ j s belonging to H ( λ ),for all σ > σ H ( λ ) + ε = L ( λ ) + ε , + ∞ X j =1 | a j | e − λ j σ < + ∞ . Let J ≥ j ≥ J , log( j ) /λ j ≤ L ( λ ) + ε . Then X j | a j | e − λ j σ ≤ ( J − k f k + + ∞ X j = J k f k e − σL ( λ )+ ε log( j ) < + ∞ by our assumption on σ . (cid:3) It turns out that, even if we put strong growth and separation conditions on λ , we cannotgo further. Theorem 5.2.
There exists a frequency λ satisfying (BC) such that σ H ( λ ) = 2 σ H ( λ ) and σ H ( λ ) > .Proof. For n ≥
2, let δ n ∈ (2 − n − , − n ] such that (2 π, n, δ n ) are Z -independent. We set λ n + k = n + kδ n for n ≥ , k = 0 , . . . , n − . It is easy to check that L ( λ ) = log(2) so that σ H ( λ ) = (log 2) /
2. Moreover, it is also easyto check that λ satisfies (BC). Indeed, for n ≥ k = 0 , . . . , n − λ n + k +1 − λ n + k = δ n ≥
12 2 − n ≥ Ce − (log 2) λ n + k and similarly λ n +1 − λ n +1 − ≥ − n ≥ Ce − (log 2) λ n +1 − . Pick now any σ > σ H ( λ ) . By the principle of uniform boundedness, there exists C > D = P j a j e − λ j s belonging to H ( λ ), for all N ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X j =2 a j e − λ j σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C k D k Let n ≥ D = P n − k =0 e − λ n + k s . Then(13) n − X k =0 e − λ n + k σ ≥ n e − σ ( n +1) ≥ C e (log 2 − σ ) n . On the other hand, set λ ′ = { n + kδ n : k ≥ } and observe that, using the internaldescription of the norm of H , k D k H ( λ ) = k D k H ( λ ′ ) . We shall compute k D k H ( λ ′ ) using Fourier analysis. Indeed, since (2 π, n, δ n ) are Z -independent, the map β : R → T , t ( e − itn , e − itδ n ) has dense range, so that ( T , β ) isa λ ′ -Dirichlet group. Therefore,(14) k D k H ( λ ′ ) = Z T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − X k =0 z z k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dz dz ≤ C n by the classical estimate of the norm of the Dirichlet kernel. Hence, (12), (13) and (14)imply that, for all σ > σ H ( λ ) , σ ≥ log(2). This yields σ H ( λ ) ≥ σ H ( λ ) . (cid:3) In view of the previous results, it seems natural to study how arithmetical properties ofthe frequency λ can influence the values of σ for which T σ : H p ( λ ) → H q ( λ ), p < q , isbounded. We concentrate on the case p = 2 and q = 2 k , k ≥
1, because we can computethe norms using the coefficients. We define λ ∗ λ as { λ l + λ k : l, k ≥ } and λ ∗ k = λ ∗ · · · ∗ λ (with k factors). Definition 5.3.
Let λ be a frequency and k ≥
1. Write λ ∗ k = ( µ l ) where the sequence( µ l ) is increasing. We set A ( λ, k ) = lim sup l → + ∞ log (card { ( n , . . . , n k ) : λ n + · · · + λ n k = µ l } )2 µ l . Proposition 5.4.
Let λ be a frequency and k ≥ . Then for σ > A ( λ, k ) , T σ mapsboundedly H ( λ ) into H k ( λ ) .Proof. We shall prove a slightly stronger statement : if σ >
C > µ > e − µσ card { ( n , . . . , n k ) : λ n + · · · + λ n k = µ } ≤ C, then T σ maps boundedly H ( λ ) into H k ( λ ). Indeed, let D = P n a n e − λ n s belonging to H ( λ ). We write T σ ( D ) k = P l b µ l e − µ l s where b µ l = X λ n + ··· + λ nk = µ l a n · · · a n k e − µ l σ . We just need to prove that the sequence ( b µ l ) is square summable, namely that for allsquare summable sequences ( c µ l ), P l b µ l c µ l is convergent, namely that X n ,...,n k a n · · · a n k e − ( λ n + ··· + λ nk ) σ c λ n + ··· + λ nk is convergent . By the Cauchy-Schwarz inequality, since ( a n ) is square summable, it is sufficient to provethat X n ,...,n k e − λ n σ · · · e − λ nk σ | c λ n + ··· + λ nk | < + ∞ . Rewriting this X l | c µ l | e − µ l σ card { ( n , . . . , n k ) : λ n + · · · + λ n k = µ l } < + ∞ this follows from the assumption. (cid:3) Corollary 5.5.
Let λ be a frequency, k ≥ and σ > ( k − L ( λ ) / . Then T σ maps H ( λ ) into H k ( λ ) .Proof. Let ε >
0. There exists N ≥ n ≥ N , log( n ) /λ n ≤ L ( λ ) + ε . Let µ ∈ ( λ ∗ ) k and n , . . . , n k be such that λ n + · · · + λ n k = µ . Then each λ n i is smaller than µ so that either n i ≤ N or n i ≤ exp( µ ( L ( λ ) + ε )). Since the knowledge of n , . . . , n k − determines the value of n k , we havecard { ( n , . . . , n k ) : λ n + · · · + λ n k = µ } ≤ N k − exp (cid:0) ( k − µ ( L ( λ ) + ε ) (cid:1) . Taking the logarithm and letting µ to + ∞ , we find A ( λ, k ) ≤ ( k − L ( λ )+ ε )2 , hence theinequality A ( λ, k ) ≤ ( k − L ( λ )2 since ε is arbitrary. (cid:3) Corollary 5.6.
Let λ be a frequency such that L ( λ ) = 0 . Then T σ maps boundedly H ( λ ) into H q ( λ ) for all q ≥ . Question 5.7.
Let p ≥ and let λ be a frequency. Does T σ maps H ( λ ) into H p ( λ ) assoon as σ > ( p − L ( λ )4 ? Example 5.8.
Let λ = (log n ). Then for all k ≥ A ( λ, k ) = 0. Proof.
We first observe that ( λ ∗ ) k = λ . Pick now log n ∈ λ . We want to know the cardinalnumber of { ( n , . . . , n k ) ∈ N : n × · · · × n k = n } . Decompose n into a product of primenumbers, n = p α · · · p α r r . Then each n k writes p α ( k )1 · · · p α r ( k ) r with α j (1)+ · · · + α j ( r ) = α j ,1 ≤ j ≤ r . Hence, ( α j (1) , · · · α j ( r )) is a weak composition of α j into k parts which can bedone in (cid:0) α i + k − k − (cid:1) ways. In total, there are r Y i =1 (cid:18) α i + k − k − (cid:19) ≤ r Y i =1 ( α i + k ) k ways to write n as a product of k factors. Thus, A ( λ, k ) ≤ lim sup n = Q ri =1 p αii → + ∞ P ri =1 k log( α i + k )2 P ri =1 α i log( p i ) = 0 . (cid:3) We finish this section by exhibiting a frequency λ satisfying (BC) and such that, for all k ≥ T σ maps H ( λ ) into H k ( λ ) if and only if σ ≥ A ( λ, k ) = k − k . We begin with twocombinatorial lemmas. ENERAL DIRICHLET SERIES 23
Lemma 5.9.
Let b, c > , let n ∈ N and let λ j = b + jc , j ≥ . For all k ∈ N , there exist γ k ∈ (0 , and δ k > such that, for all n ≥ k , for all ℓ ∈ [( k − γ k ) n, ( k + γ k ) n ] ∩ N , card n ( j , . . . , j k ) ∈ { , . . . , n } k : λ j + · · · + λ j k = kb + ℓc o ≥ δ k n k − . Proof.
We define the sequences ( γ k ) and ( δ k ) by γ k = 2 − ( k − and δ = 1, δ k +1 = δ k · γ k +1 .We proceed by induction over k . The case k = 1 is trivial. Assume that the result hasbeen proved for k and let us prove it for k + 1. Let n ≥ k +1 . Choose j k +1 any integer in[(1 − γ k +1 ) n, (1 + γ k +1 ) n ] and ℓ ∈ [( k + 1 − γ k +1 ) n, ( k + 1 + γ k +1 ) n ] ∩ N . Then(15) λ j + · · · + λ j k +1 = ( k + 1) b + ℓc ⇐⇒ λ j + · · · + λ j k = kb + ( ℓ − j k +1 ) c. Now, | ℓ − j k +1 − kn | ≤ γ k +1 n = γ k n so that there exist at least δ k n k − choices of ( j , . . . , j k ) such that (15) is true, j k +1 beingfixed. Now, there are 2 ⌊ γ k +1 n ⌋ + 1 choices of j k +1 and since γ k +1 n − ≥ γ k +1 n/ γ k +1 n ≥
2, we get the result. (cid:3)
Lemma 5.10.
Let ( b n ) and ( c n ) be two sequences of positive real numbers such that thesequences ( b , . . . , b N , c , . . . , c N ) are Z -independent for all N ≥ , n + 1 ≤ exp( b n ) and nc n ≤ for each n ∈ N . Define a sequence ( λ n ) by λ m + j = b m + jc m , m ≥ , j = 0 , . . . , m . Then for all k > there exists C k > such that, for all µ > , card n ( n , . . . , n k ) ∈ N k : λ n + · · · + λ n k = µ o ≤ C k exp (cid:18) ( k − µk (cid:19) . Proof. If µ can be written µ = λ n + · · · + λ n k for some sequence ( n , . . . , n k ), it can beuniquely written(16) µ = α b r + β c r + · · · + α l b r l + β l c r l with 1 ≤ l ≤ k , r < r < · · · < r l , α i ≥ α + · · · + α l = k and 0 ≤ β i ≤ α i r i . We willfirst estimate card (cid:8) ( n , . . . , n k ) ∈ N k : λ n + · · · + λ n k = µ (cid:9) by a quantity depending on k , l , α i , r i and β i . In view of the definition of the sequence λ and of (16), we are reducedto estimate the number of 2 k -tuples ( m , . . . , m k , j , . . . , j k ) such that for all s = 1 , . . . , k ,0 ≤ j s ≤ m s and, for all i = 1 , . . . , l , • there are α i elements in m , . . . , m k which are equal to r i ; • if φ i (1) , . . . , φ i ( α i ) are the indices of these elements, then(17) j φ i (1) + · · · + j φ i ( α i ) = β i . We first choose the values of m , . . . , m k . We choose the α indices in { , . . . , k } such thatthe corresponding m i are equal to r . We then do the same for the α elements equal to r and so on until k − m i are fixed and equal to r l ). Thus the number ofchoices for m , . . . , m k is equal to (cid:18) kα (cid:19) × (cid:18) k − α α (cid:19) × · · · × (cid:18) k − ( α + · · · + α l − ) α l − (cid:19) . Because l ≤ k and α + · · · + α l = k , this number can be bounded from above bysome number depending only on k . The integers m , . . . , m k having been fixed, we nowchoose the integers j , . . . , j k . For each i ∈ { , . . . , l } , (17) implies that there are at most (2 r i + 1) α i − choices for the values of j φ i (1) , . . . , j φ i ( α i ) : indeed, each j φ i ( t ) belongs to { , . . . , r i } and the last one is fixed when we know the values of the first α i − n ( n , . . . , n k ) ∈ N k : µ = λ n + · · · + λ n k o ≤ C k l Y i =1 (2 r i + 1) α i − ≤ C k exp l X i =1 ( α i − b r i ! whre the last inequality follows from the assumption 2 n + 1 ≤ exp( b n ) for all n ∈ N . Nowwe have P li =1 α i b r i ≤ µ and k l X i =1 b r i ≥ l X i =1 α i b r i = µ − l X i =1 β i c r i ≥ µ − l X i =1 α i r i c r i ≥ µ − k l X i =1 r i c r i ≥ µ − k since nc n ≤ l ≤ k . This implies that l X i =1 ( α i − b r i ≤ µ − µk + 2 k = ( k − µk + 2 k, hence the result. (cid:3) Theorem 5.11.
There exists a frequency ( λ n ) satisfying (BC) such that for all k ≥ , T σ maps H ( λ ) into H k ( λ ) if and only if σ ≥ k − k = A ( λ, k ) .Proof. Let ( b n ) and ( c n ) be two sequences of positive real numbers such that • for all n ≥
1, log(2 n + 1) ≤ b n ≤ log(2 n + 2); • for all n ≥
1, ( b n +1 − b n ) / n ≤ c n ≤ ( b n +1 − b n ) / n ; • for all N ≥
1, the sequences ( b , . . . , b N , c , . . . , c N ) are Z -independent.We then define λ by λ m + j = b m + jc m , m ≥ j = 0 , . . . , m . We may argue as inthe proof of Theorem 5.2 to show that the frequency λ satisfies (BC). Using Proposition5.4 (look at the first sentence of the proof) and Lemma 5.10, we get easily that T σ maps H ( λ ) into H k ( λ ) for σ ≥ k − k and also that A ( λ, k ) ≤ k − k .Conversely, assume that T σ maps H ( λ ) into H k ( λ ) (boundedness is automatic by theclosed graph theorem). Let us consider D [ m ] = P mj =0 e − λ m j s for m ≥
1, so that k D [ m ] k = √ m + 1. Write λ ∗ k as the increasing sequence ( µ l ) and observe that( T σ D [ m ] ) k = X l a µ l e − µ l σ e − µ l s where a µ l = card (cid:8) ( j , . . . , j k ) ∈ { , . . . , m } k : µ l = kb m + ( j + · · · + j k ) c m (cid:9) . Lemma5.9 tells us that, for m sufficiently large, there is at least γ k m terms of the sequence ( µ l )so that a µ l ≥ δ k m k − . Furthermore, for those µ l , µ l ≤ k ( b m + 2 mc m ) ≤ k log m + A k . ENERAL DIRICHLET SERIES 25
In particular, log a µ l µ l ≥ ( k −
1) log m + log δ k k log m + A k )which shows that A ( λ, k ) ≥ k − k . Furthermore, k T σ D [ m ] k k = k ( T σ D [ m ] ) k k /k ≥ C k (cid:16) m · m k − · m − σk (cid:17) / k ≥ C k m k + k − k − σ . Therefore, the boundedness of T σ from H ( λ ) into H k ( λ ) implies that m k + k − k − σ ≤ C ′ k √ m + 1for all sufficiently large m , which itself yields σ ≥ k − k . (cid:3) Question 5.12.
Let p ≥ . Is it true that, for the previous sequence ( λ n ) , T σ maps H ( λ ) into H p ( λ ) if and only if σ ≥ p − p ? Other results
Norm of the projection in H ( λ ) . In [10], Defant and Schoolmann have shown, us-ing a vector-valued argument, that for all frequencies λ and for all N ≥ k S N k H ( λ ) →H ( λ ) ≤k S N k H ∞ ( λ ) →H ∞ ( λ ) . We provide a different approach to estimate k S N k H ( λ ) →H ( λ ) , inspiredby [6]. Proposition 6.1.
Let λ be a frequency. The, for all N ≥ , k S N k H →H ≤ C log(Λ N ) where Λ N = sup (cid:13)(cid:13)(cid:13)P Nn =1 a n e − λ n s (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)P Nn =1 a n e − λ n s (cid:13)(cid:13)(cid:13) : a , . . . , a N ∈ C . Proof.
We work in H λ ( G ) where ( G, β ) is a λ -Dirichlet group. Let g ∈ L ( G ). Then, for ε ∈ (0 , Z | g | = Z | g | ε ε | g | − ε ε ≤ (cid:18)Z | g | (cid:19) ε ε (cid:18)Z | g | − ε (cid:19) ε where we have applied H¨older’s inequality for (1 + ε ) /ε and 1 + ε . Applying this to S N f ,where f ∈ H λ ( G ), we get k S N f k ≤ k S N f k ε ε k S N f k − ε ε − ε . We now use a result of Helson [16], saying that k S N f k − ε ≤ Aε k f k where the constant A is absolute. Therefore, assuming k f k = 1, after some simplifica-tions, we get k S N f k ≤ (cid:18) Aε (cid:19) exp (cid:18) ε − ε log(Λ N ) (cid:19) . We conclude by choosing ε = 1 / log(Λ N ). (cid:3) Corollary 6.2.
There exists
C > such that, for all frequency λ , k S N k H ( λ ) →H ( λ ) ≤ C log( N ) .Proof. We get immediately that Λ N ≤ √ N by using the Cauchy-Schwarz inequality andthe fact that | a n | ≤ k D k for all D = P Nn =1 a n e − λ n s . (cid:3) This last corollary has an interest provided we are unable to prove that k S N k H ∞ ( λ ) →H ∞ ( λ ) is less than C log( N ). The best known estimation on k S N k H ∞ ( λ ) →H ∞ ( λ ) is given by The-orem 3.4 and indeed it provides worst estimations for some sequences λ . Indeed, pick thesequence λ defined in Example 2.3. Let N = 2 n for some n and pick M > N . Then, if
M < n +1 , then log (cid:18) λ M + λ N λ M − λ N (cid:19) ≥ e n whereas, if M ≥ n +1 , then M − N − ≥ n − . A variant of Proposition 6.1 was already used in the classical case λ = (log n ) to provethat k S N k H →H ≤ C log N log log N . A precise solution to the problem on how large can be Λ N in this case can be found in [8]. References [1] F. Bayart,
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