arXiv:1405.7205v1 [math.FA] 28 May 2014
Multipliers of Dirichlet series and monomial series expansions of holomorphic functions in infinitely many variables Frédéric Bayart ∗ Andreas Defant† Leonhard Frerick‡ Manuel Maestre§ Pablo Sevilla-Peris ¶ Abstract P Let H ∞ be the set of all ordinary Dirichlet series D = n a n n −s representing bounded holomorphic functions on the right half plane. A multiplicative sequence (b n ) of complex numbers is said to be an ℓ1 -multiplier for P H ∞ whenever n |a n b n | < ∞ for every D ∈ H ∞ . We study the problem of describing such sequences (b n ) in terms of the asymptotic decay of the subsequence (b p j ), where p j denotes the j th prime number. Given a multiplicative sequence b = (b n ) we prove (among other results): b is an ℓ1 1 Pn ∗2 multiplier for H ∞ provided |b p j | < 1 for all j and limn logn j =1 b p j < 1, and conversely, if b is an ℓ1 -multiplier for H ∞ , then |b p j | < 1 for all j and 1 Pn ∗ ∗2 limn logn j =1 b p j ≤ 1 (here b stands for the decreasing rearrangement of b). Following an ingenious idea of Harald Bohr it turns out that this problem is intimately related with the question of characterizing those sequences z in the infinite dimensional polydisk D∞ (the open unit ball of ℓ∞ ) for which every bounded and holomorphic function f on D∞ has P ∂ f (0) an absolutely convergent monomial series expansion α αα! z α . Moreover, we study analogous problems in Hardy spaces of Dirichlet series and Hardy spaces of functions on the infinite dimensional polytorus T∞ . ∗
Laboratoire de Mathématiques Université Blaise Pascal Campus des Cézeaux, F-63177 Aubière Cedex (France) † Institut für Mathematik. Universität Oldenburg. D-26111 Oldenburg (Germany) ‡ Fachbereich IV - Mathematik, Universität Trier, D-54294 Trier (Germany) § Dep. Análisis Matemático. Fac. Matemáticas. Universidad de Valencia. 46100 Burjassot (Spain) ¶ IUMPA. Universitat Politècnica de València. 46022 València (Spain) The second, fourth and fifth authors were supported by MICINN and FEDER Project MTM2011-22417. The fourth author was also supported by PrometeoII/2013/013. The fifth author was also supported by project SP-UPV20120700. Mathematics Subject Classification (2010): 46E50, 42B30, 30B50, 46G25 Keywords: Dirichlet series, power series expansion; holomorphic function in infinitely many variables; Hardy spaces; multipliers; Bohr’s problem
1
Contents 1 Introduction
2
2 Monomial expansion of H ∞ -functions in infinitely many variable 2.1 Statement of the results . . . . . . . . . . . . . . . . . . . . . . . 2.2 The probabilistic device . . . . . . . . . . . . . . . . . . . . . . . 2.3 Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Proof of Theorem 2.1–lower inclusion . . . . . . . . . . . . . . . 2.5 Proof of Theorem 2.2–lower inclusion . . . . . . . . . . . . . . . 2.6 Dismissing candidates . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . .
. . . . . .
7 9 11 14 16 17 21
3 Series expansion of Hp -functions in infinitely many variables 22 3.1 The homogeneous case . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Representation of Hardy spaces . . . . . . . . . . . . . . . . . . . . 32 4 ℓ1 -multipliers of H p -Dirichlet series 34 4.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2 Bohr’s absolute convergence problem – old art in new light . . . . 37
1 Introduction Recall from [11] that the precise asymptotic order of the Sidon constant of all PN finite Dirichlet polynomials n=1 an n −s is given by PN
|an | ¯n=1 ¯ sup P N a 1 ,...,a N ∈C supt∈R ¯ n=1 a n n −i t ¯
=
p
e
p1 2
¡
N ¢p
1+o(1)
log N log log N
.
(1)
This result has its origin in fundamental works of Hilbert [21], Bohr [6], Toeplitz [28] and Bohnenblust-Hille [5], and it is the final outcome of a long series of results due to [2, 10, 23, 25, 26]. As usual we denote by H ∞ the vector space of all ordinary Dirichlet seP ries n an n −s representing bounded holomorphic functions on the right half plane (which together with the sup norm forms a Banach space). Applying (1) to dyadic blocks, it was proved in [11] (completing earlier results from [2]) that the supremum over all c > 0 for which ∞ X
n=1
|an |
ec
p
log n log log n
n
1 2
< ∞ for all
2
X n
an n −s ∈ H ∞
(2)
p equals 1/ 2. In other terms, all sequences p p p ¡ ¢ (b n ) = e (1/ 2−ε) log n log log n n −1/2 , 0 < ε < 1/ 2
are ℓ1 -multiplier of H ∞ in the sense that ∞ X
n=1
|an b n | < ∞ for all
X n
an n −s ∈ H ∞ .
(3)
Recall that a sequence (b n ) is said to be (completely) multiplicative whenever p b nm = b n b m for all n, m, and (3) obviously shows that the sequence (1/ n) is a multiplicative ℓ1 -multiplier of H ∞ . Clearly, there are more such multiplicative ℓ1 -multipliers of H ∞ . For example, it will turn out that all multiplicative sequences (b n ) with |b n | < 1 for all n and such that b p j = 0 for all but finitely many j have this property; here as usual p = (p j ) = {2, 3, 5 . . .} stands for the sequence of primes. In this article we intend to study the problem of describing all multiplicative ℓ1 -multipliers (b n ) of H ∞ in terms of the asymptotic decay of the subsequence (b p j ). Surprisingly, this question is intimately related with the following natural problem: Do C-valued holomorphic functions on the infinite dimensional polydisk B ℓ∞ (the open unit ball of the Banach space ℓ∞ of all bounded scalar sequences), like in finite dimensions, have a reasonable monomial series expansion? The crucial link is due to a genius observation of Harald Bohr from [6] which P we explain now: Denote by P the vector space of all formal power series α cα z α , P and by D the vector space of all Dirichlet series an n −s . By the fundamental theorem of arithmetics each n ∈ N has a unique prime decomposition n = p α = α α p 1 1 · · · p k k with a multiindex α ∈ N(0N) (i.e., α is a finite sequences of elements αk ∈ N0 ). Then the so-called Bohr transform B is a linear algebra homomorphism:
B : P −→ D ,
P
cα z α∈N(N) 0
α
P∞
n=1 a n n
−s
with a p α = cα .
(4)
Hilbert in [21] was among the very first who started a systematic study of the concept of analyticity for functions in infinitely many variables. According to Hilbert, an analytic function in infinitely many variables is a C-valued function defined on the infinite dimensional polydisk B ℓ∞ (see above) which has a pointwise convergent monomial series expansion: f (z) =
X
α∈N(N) 0
cα z α , z ∈ B ℓ∞ . 3
(5)
In [21] (see also [20, p. 65]) he gave the following criterion for a formal power P series α cα z α to generate such a function (i.e., to converge absolutely at each P point of B ℓ∞ ): Every k-dimensional section α∈Nk cα z α of the series is point0
wise convergent on Dk , and moreover ¯ ¯ X ¯ α¯ cα z ¯ < ∞ . sup sup ¯
(6)
k∈N z∈Dk α∈Nk 0
But this criterion is not correct as was later discovered by Toeplitz (see below (8)). Why? Today a holomorphic function f : B ℓ∞ → C is nothing else than a Fréchet differentiable function f : B ℓ∞ → C. As usual the Banach space of all bounded holomorphic f : B ℓ∞ → C endowed with the supremum norm will be denoted by H∞ (B ℓ∞ ). Important examples of such functions are bounded m-homogeneous polynomials P : ℓ∞ → C, restrictions of bounded m-linear forms on ℓ∞ × · · · × ℓ∞ to the diagonal. The vector space P (m ℓ∞ ) of all such P together with the norm kP k = supz∈Bℓ |P (z)| forms a closed subspace of H∞ (B ℓ∞ ). ∞ From the theory in finitely many variables it is well known that every holomorphic C-valued mapping f on the k-dimensional polydisk Dk has a monok mial series expansion which converges to f at ¡ every ¢ point of D . More precisely, for every such f there is a unique family cα ( f ) α∈Nk in C such that f (z) = 0 P α k α∈Nk0 c α ( f )z for every z ∈ D . The coefficients can be calculated through the Cauchy integral formula or partial derivatives: For each 0 < r < 1 and each α cα ( f ) =
1 ∂α f (0) = α! (2πi )k
Z
...
|z1 |=r
Z
|zk |=r
f (z) d z1 . . . d zk . z α+1
(7)
Clearly, every holomorphic function f : B ℓ∞ → C, whenever restricted to a finite dimensional section Dk = Dk × {0}, has an everywhere convergent power series P expansion α∈Nk cα(k) ( f )z α , z ∈ Dk . And from (7) we see that cα(k) ( f ) = cα(k+1) ( f ) 0 ¡ ¢ for α ∈ Nk0 ⊂ Nk+1 0 . Thus again there is a unique family c α ( f ) α∈N(N) in C such that at least for all k ∈ N and all z ∈ Dk X f (z) = cα ( f )z α .
0
α∈N(N) 0
This power series is called the monomial series expansion of f , and cα = cα ( f ) are its monomial coefficients; by definition they satisfy (7) whenever α ∈ Nk0 . At first one could expect that each f ∈ H∞ (B ℓ∞ ) has a monomial series expansion which again converges at every point and represents the function. But this is not the case: Just take a non-zero functional on ℓ∞ that is 0 on c0 (the 4
space of null sequences); by definition, its monomial series expansion is 0 and clearly does not represent the function. Moreover, since such a functional obviously satisfies (6), although it is not analytic (in Hilbert’s sense), the criterion of Hilbert turns out to be false. In order to avoid this example one could now try with the open unit ball B c0 of c0 instead of B ℓ∞ . But Hilbert’s criterion remains false: Note first that a simple extension argument (see e.g. [12, Lemma 2.2]) allows to identify all formal power series satisfying (6) with all bounded holomorphic functions on B c0 ; more precisely, each f ∈ H∞ (B c0 ) has a monomial series expansion as in (6), and conversely each power series satisfying (6) gives rise to a unique f ∈ H∞ (B c0 ) for which cα = cα ( f ) for all α. But then (6) does not imply (5) since by an example of Toeplitz from [28] there is a 2-homogeneous bounded polynomial P on c0 such that ∀ ε > 0 ∃ x ∈ ℓ4+ε :
X α
|cα (P )x α | = ∞ .
(8)
This means that there are functions f ∈ H∞ (B c0 ) that cannot be pointwise described by their monomial series expansions as in (5) which, at least at first glance, seems disillusioning. Indeed, this fact in infinite dimensions produces a sort of dilemma: There is no way to develop a complex analysis of functions in infinitely many variables which simultaneously handles phenomena on differentiability and analyticity (as it happens in finite dimensions). One of the main advances of this article is to give an almost complete description of what we call the set of monomial convergence of all bounded holomorphic functions on the open unit ball B ℓ∞ of ℓ∞ : o n X ¯ cα ( f )z α . mon H∞ (B ℓ∞ ) = z ∈ B ℓ∞ ¯ ∀ f ∈ H∞ (B ℓ∞ ) : f (z) =
(9)
α∈N(N) 0
We recall that the decreasing rearrangement of z ∈ ℓ∞ is given by ¯ © ª z n∗ = inf sup |z j | ¯ J ⊂ N , card(J ) < n , j ∈N\J
and use it to define the set
³ n ¯ B = z ∈ B ℓ∞ ¯ b(z) = lim sup n→∞
´1/2 o n 1 X z ∗2 < 1 . log n j =1 j
Then our main result is Theorem 2.2 that shows B ⊂ mon H∞ (B ℓ∞ ) ⊂ B . 5
(10)
As we intend to indicate in the following sections, this result has a long list of forerunners (due to various authors, see e.g. [5, 6, 7, 8, 14, 20, 21, 28]). In (16), (17), (19), (20), (21) as well as (47),(48), (50), (51) it will become clear that mon H∞ (B ℓ∞ ) was known to be very close to ℓ2 ∩B ℓ∞ . But (10) adds a new level of precision that enables us to extract much more precise information from monomial convergence of holomorphic functions on the infinite dimensional polydisk than before. This in particular gets clear if we finally return to the beginning of this introduction – let us return to the description of all multiplicative ℓ1 -multipliers of H ∞ using Bohr’s transform from (4). The following fact, essentially due to Bohr [6] and later rediscovered in [18, Lemma 2.3 and Theorem 3.1], is essential: The Bohr transform B induces an isometric algebra isomorphism from H∞ (B c0 ) onto H ∞ , H∞ (B c0 ) = H ∞ . (11) This identification in fact allows to identify the multiplicative ℓ1 -multipliers of H ∞ with the elements in mon H∞ (B ℓ∞ ): Given a sequence (b n ) ∈ CN , we have that (b n ) is an ℓ1 -multiplier for H ∞ ⇔ (b p k ) ∈ mon H∞ (B ℓ∞ ) . Observe that this way we may deduce from (2) that the sequence ¡ p ¢ 1/ p k ∈ mon H∞ (B ℓ∞ ) ;
(12)
this seems to be the first non-trivial example which distinguishes monH∞ (B ℓ∞ ) from ℓ2 ∩ B ℓ∞ . But note that this can also be seen using (10); indeed, (12) is a very particular case of the following reformulation of (10) (see Section 4) which is an almost complete characterization of all multiplicative ℓ1 -multipliers for H ∞ . For all multiplicative sequences (b n ) ∈ CN we have that • (b n ) is an ℓ1 -multiplier for H ∞ provided we have that |b p j | < 1 for all j ¡ ¢ and b (b p j ) < 1, and conversely, ¡ ¢ • if (b n ) is ℓ1 -multiplier for H ∞ , then |b p j | < 1 for all j and b (b p j ) ≤ 1.
In Section 3 we extend our concept on sets of monomial convergence to H p functions defined on the infinite dimension torus T∞ (see (32) for the precise definition); here T denotes the torus (the unit circle of C) and T∞ the infinite dimensional polytorus (the countable cartesian product of T). The Banach space H∞ (B c0 ) can be isometrically identified with the Banach space H∞ (T∞ ) of all L ∞ -functions f : T∞ → C with Fourier coefficients fˆ(α) = 0 for α ∈ Z(N) \ N(0N) ; this was proved in [8] (see also Proposition 3.5). For 1 ≤ p ≤ ∞ we define n o ¯X mon H p (T∞ ) = z ∈ CN ¯ | fˆ(α)z α | < ∞ for all f ∈ H p (T∞ ) . α
6
Then it is not hard to see that mon H∞ (T∞ ) = mon H∞ (B ℓ∞ ), but in contrast to (10) we have mon H p (T∞ ) = ℓ2 ∩ B ℓ∞ for all 1 ≤ p < ∞ . This way we extend and complement results of Cole and Gamelin from [8]. Finally, in Section 4 we use Bohr’s vision from (4) to interpret all these results on sets of monomial convergence of H p -functions in terms of multiplicative ℓ1 multipliers for H p -Dirichlet series (as was already described above for the case p = ∞).
2 Monomial expansion of H ∞ -functions in infinitely many variable Our definition of sets of monomial convergence (9) has its roots in Bohr’s seminal article [6], and the first systematic study of such sets was undertaken in [14]. Recall from the introduction that o n X ¯ cα ( f )z α , mon H∞ (B ℓ∞ ) = z ∈ B ℓ∞ ¯ ∀ f ∈ H∞ (B ℓ∞ ) : f (z) = α∈N(N) 0
and define similarly for m ∈ N o n X ¯ cα (P )z α . mon P (m ℓ∞ ) = z ∈ ℓ∞ ¯ ∀P ∈ P (m ℓ∞ ) : P (z) = α∈N(N) 0
Since we here consider functions f defined on B ℓ∞ as well as polynomials P defined on ℓ∞ , we clearly cannot avoid to define the preceding two sets as subsets of B ℓ∞ and ℓ∞ , respectively. Nevertheless we can give two slight reformulations which will be of particular importance when we translate our forthcoming results into terms of multipliers for Dirichlet series: o n X ¯ ¯ ¯ ¯cα ( f )z α ¯ < ∞ (13) mon H∞ (B ℓ∞ ) = z ∈ CN ¯ ∀ f ∈ H∞ (B ℓ∞ ) : α∈N(N) 0
n ¯ mon P (m ℓ∞ ) = z ∈ CN ¯ ∀P ∈ P (m ℓ∞ ) :
o X ¯ ¯ ¯cα (P )z α ¯ < ∞
(14)
α∈N(N) 0
The argument for these two equalities is short: Denote the set in (14) by U , and that in (13) by V . For z ∈ U it was shown in [14, p.29-30] that z ∈ c0 . Then an obvious continuity argument gives the equality in (14). Take now z ∈ V ⊂ U . 7
Considering bounded holomorphic functions on the open disk D, we see immediately that |z n | < 1 for all n. The equality in (13) again follows by continuity. In the above definitions we may replace ℓ∞ by c0 . Davie and Gamelin showed in [9, Theorem 5] that every function in H∞ (B c0 ) can be extended to a function in H∞ (B ℓ∞ ) with the same norm. Using this it was shown in [14, Remark 6.4] that we in fact have mon H∞ (B ℓ∞ ) = mon H∞ (B c0 ) and mon P (m ℓ∞ ) = mon P (m c0 ) .
(15)
Let us collect some more basic facts on sets of monomial convergence which in the following will be used without further reference: • If z ∈ mon H∞ (B ℓ∞ ), then every permutation of z is again in monH∞ (B ℓ∞ ); this was proved in [13, p. 550]. • We know from [14, p. 29-30] that mon H∞ (B ℓ∞ ) ⊂ c0 . Hence, the decreasing rearrangement z ∗ of any z ∈ mon H∞ (B ℓ∞ ) is a permutation of |z|. This implies that z ∈ mon H∞ (B ℓ∞ ) if and only if z ∗ ∈ mon H∞ (B ℓ∞ ) . • Let z ∈ mon H∞ (B ℓ∞ ) and x = (xn )n ∈ B ℓ∞ such that |xn | ≤ |z n | for all but finitely many n’s. Then x ∈ mon H∞ (B ℓ∞ ); this result is from [13, Lemma 2] and was inspired by [6, Satz VI] (see also Lemma 3.7). • Similar results hold for mon P (m ℓ∞ ). What was so far known on sets of monomial convergence? Bohr [6] proved ℓ2 ∩ B ℓ∞ ⊂ mon H∞ (B ℓ∞ ) ,
(16)
and Bohnenblust-Hille in [5] ℓ
2m m−1
⊂ mon P (m ℓ∞ ).
(17)
Moreover, these two results in a certain sense are optimal; to see this define ¯ ª © M := sup 1 ≤ p ≤ ∞ ¯ ℓp ∩ B ℓ∞ ⊂ mon H∞ (B ℓ∞ ) , ¯ © ª (18) Mm := sup 1 ≤ p ≤ ∞ ¯ ℓp ⊂ mon P (m ℓ∞ ) for m ∈ N .
These are two quantities which measure the size of both sets of convergence in terms of the largest possible slices ℓp ∩ B ℓ∞ included in them. The definition of M (at least implicitly) appears in [6], and (16) of course gives that M ≥ 2. The idea of graduating M through Mm appears first in Toeplitz’ article [28]; clearly the estimate M2 ≤ 4 is a reformulation of (8). After Bohr’s paper [6] the intensive search for the exact value of M and Mm was not succesful for more then 15 8
years. The final answer was given by Bohnenblust and Hille in [5], who were able to prove that 2m 1 Mm = and M = . (19) m −1 2 Their original proofs of the upper bounds are clever and ingenious. Using modern techniques of probabilistic nature, different from the original ones, they were improved in [14, Example 4.9 and Example 4.6]: ℓ2 ∩ B ℓ∞ ⊂ mon H∞ (B ℓ∞ ) ⊂
\
ℓ2+ε ,
(20)
ε>0
and mon P (m ℓ∞ ) ⊂ ℓ
2m m−1 ,∞
.
(21)
Recall that for 1 ≤ q < ∞ the Marcinkiewicz space ℓq,∞ consists of those sequences z for which supn z n∗ n 1/q < ∞ (and this supremum defines the norm of this Banach space). Clearly, ℓq,∞ ⊂ c0 , hence z ∗ = (|z σ(n) |) with σ some permutation of N. In Section 2.2 a simplified proof of (21) will be given.
2.1 Statement of the results We already mentioned in (12) that the left inclusion in (20) is strict. The aim of this section is to show that our two sets of monomial convergence can be ‘squeezed’ in a much more drastic way. Our first theorem gives a complete description of mon P (m ℓ∞ ) and extends all results on this set mentioned so far. Theorem 2.1. Let m ∈ N. Then mon P (m ℓ∞ ) = ℓ
2m m−1 ,∞
,
and moreover there exists a constant C > 0 such that for every z ∈ ℓ 2m ,∞ and m−1 every P ∈ P (m ℓ∞ ) we have X |cα (P )z α | ≤ C m kzkm kP k . (22) |α|=m
In view of Bohr’s transform B from (4) this theorem can be seen as a sort of polynomial counterpart of a recent result on m- homogeneous Dirichlet series. P A Dirichlet series an n −s is called m-homogeneous whenever an = 0 for every Ω(n) 6= m; following standard notation, for each n ∈ N we write Ω(n) = |α| if n = m p α (this counts the prime divisors of n, according to their multiplicity). By H ∞ we denote the closed subspace of all m-homogeneous Dirichlet series in the 9
Banach space H ∞ . Then the restriction of the isometric algebra isomorphism B : H∞ (B c0 ) → H ∞ from (11) defines an isometric and linear bijection: m P (m c0 ) = H ∞ .
(23)
The following estimate due to Balasubramanian, Calado and Queffélec [2, Theorem 1.4] is a homogeneous counterpart of (2) and of Theorem 2.1: For each P m m ≥ 1 there exists C m > 0 such that for every an n −s ∈ H ∞ we have X n
|an |
(log n) n
m−1 2
m−1 2m
¯X ¯ ¯ ¯ ≤ C m sup ¯ an n i t ¯ , t∈R
(24)
n
and the parameter m−1 2 is optimal by [24, Theorem 3.1] (here, in contrast to (22), it seems unknown whether the constant C m is subexponential). At least philosophically holomorphic functions can be viewed as polynomials of degree m = ∞. Hence it is not surprising that the complete characterization of mon P (m ℓ∞ ) from Theorem 2.1 improves (16) and even the highly non-trivial fact from (12): With ¯ ª © p ℓ2,0 = z ∈ ℓ∞ ¯ lim z n∗ n = 0 n
we have
ℓ2 ∩ B ℓ∞ & ℓ2,0 ∩ B ℓ∞ ⊂ mon H∞ (B ℓ∞ ) ; (25) ¡ −1/2 ¢ note that by the prime number theorem we have p n ∈ ℓ2,0 ∩ B ℓ∞ while this sequence does not belong to ℓ2 . We sketch the proof of (25): Since B ℓ2,∞ ⊂ T m∈N B ℓ 2m ,∞ , by (22) and [14, Theorem 5.1] there exists an r > 0 such that m−1 ¡ ¢ r B ℓ2,∞ ⊂ mon H∞ (B ℓ∞ ). Then we conclude that prn n ∈ mon H∞ (B ℓ∞ ) which easily gives that z ∗ ∈ mon H∞ (B ℓ∞ ) for every z ∈ ℓ2,0 ∩ B ℓ∞ . By the general remarks on mon from the beginning of this section this completes the proof. Improving (25) considerably, the following theorem is our main result on monomial convergence of bounded holomorphic functions on the infinite dimensional polydisk. It can be seen as the power series counterpart of (2), and in Section 4 we will see that it gives far reaching information on the general theory of Dirichlet series. Theorem 2.2. For each z ∈ B ℓ∞ the following two statements hold: (a) If lim sup n→∞
n 1 X z ∗2 < 1, then z ∈ mon H∞ (B ℓ∞ ). logn j =1 j
10
n 1 X z ∗2 j ≤1; n→∞ logn j =1 moreover, here the converse implication is false.
(b) If z ∈ mon H∞ (B ℓ∞ ), then lim sup
In the remaining part of this section, we prove Theorems 2.1 and 2.2. To do so, we need some more notation: Given k, m ∈ N we consider the following sets of indices M (m, k) = {j = ( j 1 , . . . , j m ) | 1 ≤ j 1 , . . . , j m ≤ k} = {1, . . . , k}m
J (m, k) = {j ∈ M (m, k) | 1 ≤ j 1 ≤ · · · ≤ j m ≤ k} Λ(m, k) = {α ∈ Nk0 | |α| = α1 + · · · + αk = m} .
An equivalence relation is defined in M (m, k) as follows: i ∼ j if there is a permutation σ such that i σ(r ) = j r for all r . We write |i| for the cardinality of the equivalence class [i]. For each i ∈ M (m, k) there is a unique j ∈ J (m, k) such that i ∼ j. On the other hand, there is a one-to-one relation between J (m, k) and Λ(m, k): Given j, one can define α by doing αr = |{q | j q = r }|; conversely, for each α, we consider jα = (1, .α.1., 1, 2, .α.2., 2, . . . , k, .α.k., k). Note that |jα | = m! α! for every α ∈ Λ(m, k). Taking this correspondence into account, the monomial series expansion of a polynomial P ∈ P (m ℓk∞ ) can be expressed in different ways (we write cα = cα (P )) X X X c j 1 ... j m z j 1 · · · z j m . cj zj = cα z α = α∈Λ(m,k)
1≤j 1 ≤...≤j m ≤k
j∈J (m,k)
2.2 The probabilistic device The upper inclusions in Theorem 2.1 and Theorem 2.2 are based on the following probabilistic device known as the Kahane-Salem-Zygmund inequality (see e.g. [22, Chapter 6, Theorem 4]): There is a universal constant C KSZ > 0 such that for any m, n and any family (aα )α∈Λ(m,n) of complex numbers there exists a choice of signs εα = ±1 for which ¯ ¯ sup ¯
X
z∈Dn α∈Λ(m,n)
r ¯ X εα aα z ¯ ≤ C KSZ n logm |aα |2 . α¯
(26)
α
Let us start with the proof of the upper inclusion of Theorem 2.1. As we have already mentioned earlier (see (21)), this result is from [14], where it appears as a special case of a more general result proved through more sophisticated probabilistic argument. For the sake of completeness we here prefer to give a direct argument based on (26).
11
Proof of the upper inclusion in Theorem 2.1. Take z ∈ mon P (m ℓ∞ ). We show that the decreasing rearrangement r = z ∗ ∈ ℓ 2m ,∞ . Since r ∈ mon P (m ℓ∞ ), a m−1 straightforward closed graph argument (see also [14, Lemma 4.1]) shows that there is a constant C (z) > 0 such that for every Q ∈ P (m ℓ∞ ) we have X |cα (Q)r α | ≤ C (z) kQk . (27) α∈N(N) 0
By (26) for each n there are signs εα = ±1, α ∈ Λ(m, n) such that the m-homogeneous polynomial X m! P (u) = εα u α , u ∈ Cn α! α∈Λ(m,n)
satisfies
kP k ≤ C KSZ
s
n logm
X
α∈Λ(m,n)
But by the multinomial formula we have X
α∈Λ(m,n)
¯ ¯ ¯cα (P )¯2 =
X
α∈Λ(m,n)
³ m! ´2 α!
≤ m!
¯ ¯ ¯cα (P )¯2 .
(28)
m! = m!n m , α! α∈Λ(m,n) X
and hence we conclude from (27) and (28) (and another application of the multinomial formula) that ³X n
rj
j =1
´m
=
q m+1 m! α r ≤ C (z)C KSZ m! logm n 2 . α∈Λ(m,n) α! X
Finally, this shows that for all n we have rn ≤
n m+1 1 1 1 X r j ≤ C (z)C KSZ (m! log m) m n 2m −1 ≪ m−1 , n j =1 n 2m
the conclusion. A similar argument leads to the Proof of the upper inclusion in Theorem 2.2. Let us fix some z ∈ mon H∞ (B ℓ∞ ). Then z ∈ B c0 and without loss of generality we may assume that r = z is nonincreasing. Again a closed graph argument assures that there is C (z) > 0 such that for every f ∈ H∞ (B ℓ∞ ) X ° ° |cα ( f )|r α ≤ C (z)° f ° . α
12
For each m, n and aα = r α , α ∈ Λ(m, n) we choose signs εα according to (26), P and define f (u) = α∈Λ(m,n) εα r α u α , u ∈ Dn . Then the preceding estimate gives X
α∈Λ(m,n)
r 2α =
X
α∈Λ(m,n)
° ° |εα r α |r α ≤ C (z)° f °
³ ≤ C (z)C KSZ n log m
X
α∈Λ(m,n)
This implies ³
Now,
X
r 2α
α∈Λ(m,n)
´1
2
|r α |2
≤A
(r 12 + · · · + r n2 )m ≤ m!
´1 2
=A
q
n logm
³
X
α∈Λ(m,n)
r 2α
´1 2
.
q n log m . r 2α .
X
α∈Λ(m,n)
Using Stirling’s formula and taking the power 1/m, we get 1
1
1
1
r 12 + · · · + r n2 ≤ A m me −1 m 2m n m (log m) m . We then choose m = ⌊log n⌋ so that e −1 n 1/m → 1. This yields ¶ ¶ µµ log logn 1 2 2 , + o(1) r 1 + · · · + r n ≤ log n × exp 2 logn and we immediately deduce lim sup n→∞
n 1 X r j2 ≤ 1 . log n j =1
Moreover, the converse is false, since if we consider a decreasing sequence (r n ) satisfying, for large values of n, µ ¶ log log n 2 2 r 1 + · · · + r n = log n × exp , log n then lim sup n→∞
n 1 X r 2 ≤ 1 whereas (r n ) ∉ mon H∞ (B ℓ∞ ). log n j =1 j
Remark 2.3. The same argument gives also informations on the constant C appearing in (22). More precisely, if there exists A,C > 0 such that, for every z ∈ ℓ 2m ,∞ and for every P ∈ P (m ℓ∞ ), we have m−1
X
|α|=m
|cα (P )z α | ≤ AC m kzkm kP k, 13
(29)
then we claim that C ≥ e 1/2 . Indeed, provided (29) is satisfied, and arguing as in the proof of Theorem 2.2, we see that for any 0 < r 1 , . . . , r n , p q (r 12 + · · · + r n2 )m/2 ≤ AC m kr km C KSZ m! n log m. We choose r j =
1
j
m−1 2m
so that kr k = 1 and
r 12 + · · · + r n2
=
n X
1
j =1
j
1 1− m
≥
Zn 1
dx x
1 1− m
1
≥ mn m − m.
Hence, C≥
1 (AC KSZ )
1 m
×
1 (log m)
1 2m
×
1 (m!)
1 2m
µ
× m−
m n
1 m
¶1
2
.
Letting n to infinity and then m to infinity, and using lim
m→+∞
m 1
(m!) m
= e,
we get the claim. We will see later that (29) is satisfied with C any constant greater than (2e)1/2 .
2.3 Tools The proof of Theorem 2.1 and Theorem 2.2–(a) share some similarities. They need several lemmas. The first one is a Khinchine-Steinhaus type inequality for m-homogeneous polynomials on the n-dimensional torus Tn (see [3] and also [29]). Following [27] and [30] m n and m will denote the product of the normalized Lebesgue measure respectively on Tn and T∞ (i.e. the unique rotation invariant Haar measures). Lemma 2.4. Let 1 ≤ r ≤ s < ∞ . Then for every m-homogeneous polynomial P ∈ P (m Cn ) we have r m ³Z ´1/s ´1/r ³Z ¯ ¯s ¯ ¯ s ¯P (w)¯ d m n (w) ¯P (w)¯r d m n (w) ≤ . r Tn Tn
The second lemma needed for the proof of Lemma 2.6 is the following hypercontractive Bohnenblust-Hille inequality for m-homogeneous polynomials on the n-dimensional torus. This was recently shown in [4], improving a result from [11].
14
Lemma 2.5. For every κ > 1 there is a constant C (κ) > 0 such that for every mP homogeneous polynomial P = |α|=m cα z α , z ∈ Cn we have à ! m+1 2m X ° 2m |cα | m+1 ≤ C (κ) κm kP ° . α=Λ(m,n)
We are now ready to give the main technical tool.
Lemma 2.6. Let n ≥ 1, let m ≥ p ≥ 1 and let κ > 1. There exists C (κ) > 0 such that, for any P ∈ P (m ℓn∞ ) with coefficients (cj )j , we have
1×
2
2p p+1
X X |c(i,j) |2 j∈J (p,n) i∈J (m−p,n) i m−p ≤j 1
Proof. Let us start by denoting
p+1 2p
· µ ¶¸ 1 m ≤ C (κ) κ 1 + kP k . p
1×
2
2p p+1
X X |c(i,j) |2 H := j∈J (p,n) i∈J (m−p,n) i m−p ≤j 1
p+1 2p
.
Let L be the symmetric m-linear form associated to P , whose coefficients ai 1 ,...,i m = L(e i 1 , . . . , e i m ) satisfy, for i ∈ J (m, n), ci ai = . |i|
We fix j ∈ J (p, n), and note that for any i ∈ J (m − p, n)
|(i, j)| ≤ m(m − 1) · · ·(m − p + 1)|i| . Then X
i∈J (m−p,n) i m−p ≤j 1
|c(i,j) |2 ≤
X
i∈J (m−p,n)
|(i, j)|2 |a(i,j) |2 ≤ m 2p
We now apply Lemma 2.4 with the exponent polynomial z 7→ L(z, . . . , z, e j 1 , . . . , e j p ) to get ¶ µ X 1 m 2 2p 1+ |c(i,j) | ≤m p i∈J (m−p,n) i m−p ≤j 1
ÃZ ¯ ¯ × ¯
X
Tn i∈J (m−p,n)
2p p+1
X
i∈J (m−p,n)
|i|2 |a(i,j) |2 .
to the (m − p)-homogeneous
! p+1 ×2 2p 2p ¯ p+1 ¯ . |i|a(i,j) w i 1 · · · w i m−p ¯ d m n (w)
15
We then sum over j ∈ J (p, n). This yields H
2p p+1
≤m
(2p)2 2(p+1)
µ
1 1+ p
¶m×
2p p+1
×
Z
X
Tn j∈J (p,n)
¯ ¯ 2p ¯L(w, . . . , w, e j , . . . , e j )¯ p+1 d m n (w) . 1 p
For each fixed w ∈ Tn we apply Lemma 2.5 with 1 < κ0 < κ to the p-homogeneous polynomial z 7→ L(w, . . . , w, z, . . . , z): H
2p p+1
¶ ¸m× 2p 2p p+1 1 1+ ≤ C (κ0 )m κ0 sup |L(w, . . . , w, z, . . . , z)| p+1 p w,z∈Tn 2p ¶ ¸m× ·µ (2p)2 p 2p p+1 1 ′ 2(p+1) κ0 ≤ C (κ0 )m m p+1 kP k p+1 , 1+ p (2p)2 2(p+1)
·µ
where in the last estimate we have used an inequality from Harris [17, Theorem 1].
2.4 Proof of Theorem 2.1–lower inclusion m−1
Let z ∈ ℓ 2m ,∞ , so that supn z n∗ n 2m = kzk < ∞. Let us fix n ≥ 1 and let us conm−1 sider P ∈ P (m ℓn∞ ) with coefficients (cj )j . Using the Cauchy-Schwarz inequality, we may write X X ∗ X |cj |z j∗ ≤ zj |c( j ,i) |z i∗1 · · · z i∗m−1 j∈J (m,n)
j ≥1
≤
X
j ≥1
i 1 ≤···≤i m−1 ≤j
z ∗j
Ã
X
i 1 ≤···≤i m−1 ≤j
|c( j ,i) |2
!1/2 Ã
X
i 1 ≤···≤i m−1 ≤j
z i∗2 · · · z i∗2 1 m−1
Now, X
i 1 ≤···≤i m−1 ≤j
z i∗2 · · · z i∗2 ≤ 1 m−1
X
kzk2(m−1)
m−1
i 1 ≤···≤i m−1 ≤j
For k ≤ m and u ≤ v , we have X
u≤v
1 u
k 1− m
≤
Zv 0
1 u
k 1− m
16
du =
m−1
m i 1 m · · · i m−1
m k/m v . k
.
!1/2
.
By applying the above inequality for k = 1, . . . , m − 1, an easy induction yields X
i 1 ≤···≤i m−1 ≤j
z i∗2 · · · z i∗2 1 m−1
≤ ≤
j X
iX m−1
j X
iX m−1
i m−1 =1 i m−2 =1
i 1 =1 i
...
i m−1 =1 i m−2 =1
≤ ...
≤ kzk2(m−1)
kzk2(m−1)
i2 X
...
i3 X
1 1 1− m 1− 1 1− m i 2 m · · · i m−1 1 2(m−1)
m
kzk
i 2 =1 i
2 1− m
2
1 1− m
i3
1−
1
m · · · i m−1
m m−1 m−1 j m (m − 1)!
≤ e m−1 kzk2(m−1) j
m−1 m
.
We then deduce that X
j∈J (m,n)
|cj |z j∗ ≤ e
m−1 2
kzkm
X
Ã
X
j ≥1 i 1 ≤···≤i m−1 ≤j
|c( j ,i) |2
!1/2
≤ C m kzkm kP k
where the conclusion comes from Lemma 2.6 with p = 1. This shows that z ∗ ∈ mon P (m ℓ∞ ), and hence the conclusion follows by the general properties of sets of monomial convergence (given at the beginning of this section).
2.5 Proof of Theorem 2.2–lower inclusion The proof of Theorem 2.2–(a) is technically more demanding and needs further lemmas. Lemma 2.7. Let n ≥ 1, p > 1 and ρ > 0, and take 0 < r i < ρ for i = 1, . . . , n. Then for any sequence (ci )i∈Sm≥p J (m,n) of nonnegative real numbers we have
j1 X Y ci r i 1 . . . r i m ≤ r j 1 · · · r j p j∈J (p,n) m=p i∈J (m,n) l =1 ∞ X
X
1−
2p 1/2 p−1
1 ³ ´2 rl ρ
X X X 2(m−p) 2 ρ c × (i,j) j∈J (p,n) m≥p i∈J (m−p,n)
17
1× 2
i m−p ≤j 1
p−1 2p
2p p+1
p+1 2p
.
Proof. We begin by writing ∞ X
X
m=p i∈J (p,n)
ci r i 1 . . . r i m =
X
j∈J (p,n)
r j1 · · · r j p
X
X
m≥p i∈J (m−p,n) i m−p ≤j 1
ρ (m−p) c(i,j) ρ −(m−p) r i 1 · · · r i m−p .
We apply the Cauchy-Schwarz inequality (inside) to get: ∞ X
X
m=p i∈J (m,n)
ci r i 1 . . . r i m ≤
X
j∈J (p,n)
X r j1 · · · r j p X
m≥p i∈J (m−p,n) i m−p ≤j 1
X
j∈J (p,n)
X
m≥p i∈J (m−p,n) i m−p ≤j 1
X × ≤
1/2
r j1 · · · r j p
X ×
X
1/2
r i21 · · · r i2m−p ρ −2(m−p) j1 Y
1/2
1 ³ ´2
l =1 1 − r l ρ
m≥p i∈J (m−p,n) i m−p ≤j 1
2 ρ 2(m−p) c(i,j)
1/2
2 ρ 2(m−p) c(i,j)
.
We conclude by applying Hölder’s inequality with the couple of conjugate ex2p 2p ponents p+1 , p−1 . The strategy now will be to bound each factor in the preceding lemma. The first factor will be controlled by the condition given in Theorem 2.2. Lemma 2.8. Fix p > 1, 0 < α < ρ, and let (r n )n∈N be a nonincreasing sequence of nonnegative real numbers satisfying, for all n ≥ 1, ½ rn < ρ 1 2 (r + · · · + r n2 ) ≤ α2 . log(n+1) 1 Then the sequence
j1 X Y r j 1 · · · r j p j∈J (p,n) l =1
18
1−
2p 1/2 p−1
1 ³ ´2 rl ρ
n
is bounded. Proof. It is enough to prove that p p−1 2p j1 ∞ Y X 1 p−1 H := rj ³ ´2 1 r j 1 =1 l =1 1 − l ρ
2p
X
j 1 ≤j 2 ≤···≤j p
(r j 2 · · · r j p ) p−1
is finite. We first consider the last (r n )n is nonincreasing, it is psum. Because p plain that, for any n ≥ 1, r n ≤ α log(n + 1)/ n. We will use that there is a constant A p ≥ 1 such for all a ∈ N we have p
p
X (log(k + 1)) p−1
≤ Ap
p
k p−1
k≥a
This implies X
2p
j 1 ≤j 2 ≤···≤j p
(r j 2 · · · r j p ) p−1 ≤
≪
Ã
X p
p
k p−1
k=j 1
1
.
a p−1 2p
j 2 ,··· ,j p ≥j 1
∞ (log(k + 1)) p−1 X
1 + (log a) p−1
(r j 2 · · · r j p ) p−1
!p−1
≪
¡
p
1 + (log j 1 ) p−1 j1
¢p−1
≪
1 + (log j 1 )p , j1
where the constant in the last inequality only depends on α and p. Furthermore, !! à à j1 j1 Y X r l2 1 ³ ´2 = exp − log 1 − 2 . ρ l =1 1 − r l l =1 ρ Let ε > 0 be such that α2 (1 + ε) < ρ 2 . Since (r n )n goes to zero, there exists some A > 0 such that à ! j1 j1 r 2 X X r l2 α2 l − log 1 − 2 ≤ A + (1 + ε) ≤ A + (1 + ε) log j 1 2 ρ ρ2 l =1 l =1 ρ
for any j 1 ≥ 1 (use again that limx→0 j1 Y
−log(1−x) x
= 1). This yields
1 δ ³ ´2 ≪ j 1
l =1 1 − r l ρ
for some δ < 1. Hence, H≪
p
X (log( j 1 + 1)) p−1 (1 + log j 1 )p
j 1 ≥1
p
1+(1−δ) p−1
( j1)
The last sum is convergent and this completes the proof. 19
.
Finally, we are ready to give the Proof of Theorem 2.2–(a). Take z ∈ B ℓ∞ such that A := lim sup n→∞
n 1 X z ∗2 < 1 . log n j =1 j
We write for simplicity r n for z n∗ , and we are going to show that r ∈ mon H∞ (B ℓ∞ ) (see the preliminaries). Choose A < α < ρ < 1. Moreover, we know that changing a finite number of terms does not change the property r ∈ mon H∞ (B ℓ∞ ) (see again [13, Lemma 2]), hence we may assume that for all n ≥ 1 ½ rn < ρ 1 2 2 2 (r + · · · + r n) ≤ α . log(n+1) 1 ³ ´ Now we choose p > 1 and κ > 1 such that κρ 1 + p1 < 1, and consider for each fixed f ∈ H∞ (B ℓ∞ ) and for each n the decomposition X
α∈Nn
Since
|cα |r α =
p−1 X
X
m=1 j∈J (m,n)
|cj |r j 1 . . . r j m +
r∈ ℓ
2k k−1 ,∞
∞ X
X
m=p j∈J (m,n)
|cj |r j 1 . . . r j m
for all k ,
we deduce from Theorem 2.1 (here in fact only the weaker version from (19) is needed) that the first summand is bounded by a constant independent of n. Moreover, by Lemmas 2.7 and 2.8, the second summand can be majorized as follows: p+1 1 2p 2p 2 × p+1
∞ X
X
m=p j∈J (m,n)
|cj |r j 1 . . . r j m
X X X 2m 2 ρ |c | ≪ (i,j) j∈J (p,n) m≥p i∈J (m−p,n) i m−p ≤j 1
.
We then apply Minkowki’s inequality and Lemma 2.6. Using the Taylor series P expansion f = m≥0 P m we get ∞ X
X
m=p j∈J (m,n)
|cj |r j 1 . . . r j m
1×
2
X X |c(i,j) |2 ≪ ρm j∈J (p,n) i∈J (m−p,n) i m−p ≤j 1
· µ ¶ ¸m X 1 ρ 1+ ≪ κ kP m k. p m≥p
2p p+1
p+1 2p
This yields the conclusion, since by the Cauchy inequalities we have that kP mk ≤ k f k. 20
2.6 Dismissing candidates A natural question seems to be whether or not there is a sequence space X (i.e., a vector space X of complex sequences) such that X ∩B ℓ∞ = mon H∞ (B ℓ∞ ). The first natural candidate to do that job was ℓ2 (see again (16), (19), and (20)). But, as we already have seen in (12), the sequence (p n−1/2 )n belongs to mon H∞ (B ℓ∞ ) although it is not in ℓ2 . The three other natural candidates are the spaces ℓ2,0 , ℓ2,∞ and ℓ2,log : n o ℓ2,0 = z ∈ ℓ∞ | lim z n∗ n 1/2 = 0 n n o ℓ2,∞ = z ∈ ℓ∞ | ∃c ∀n : z n∗ ≤ c p1n q o n log n ∗ . ℓ2,log = z ∈ ℓ∞ | ∃c ∀n : z n ≤ c n
Theorem 2.2 shows that neither ℓ2,0 nor ℓ2,log are the proper spaces since we have ¯ á ℓ2,log ∩ B ℓ ℓ2,0 ∩ B ℓ∞ á B ⊂ mon H∞ (B ℓ∞ ) ⊂ B (30) ∞ ¯ from (10)). We prove this: Note first (recall the definition of B ( ³ c ´ ∈ mon H∞ (B ℓ∞ ) for c < 1 p n n∈N ∉ mon H∞ (B ℓ∞ ) for 1 < c
since lim sup n→∞
(31)
n 1 1 X = 1. log n j =1 j
Now, (31) immediately gives ℓ2,0 ∩ B ℓ∞ á B. The last inclusion in (30) follows P from the fact that lim supn log1 n n1 z ∗2 j < ∞ obviously implies that z ∈ ℓ2,log . On the other hand, s n ³ log j ´2 n log 3 1 X 1 X ≥ lim sup = log 3 > 1 lim sup j n→∞ log n j =1 j n→∞ log n j =1 gives
³q
´
logn n n
¯ and shows that this inclusion is also strict. 6∈ B
In view of (31) the following interesting problem remains open: ³ 1 ´ ∈ mon H∞ (B ℓ∞ ) ? p n n In fact, Theorem 2.2, even proves that there is no sequence space X at all for which mon H∞ (B ℓ∞ ) = X ∩ B ℓ∞ : Indeed, assume that such an X exists. By 21
(31) we have that ( 21 n −1/2 )n≥9 ∈ mon H∞ (B ℓ∞ ), and ¡therefore ¢ by assumption ( 23 n −1/2 )n≥9 ∈ X ∩B ℓ∞ . But then, again by assumption, 32 n −1/2 n≥9 ∈ mon H∞ (B ℓ∞ ), a contradiction to (31). Finally, we compare ℓ2,∞ ∩B ℓ∞ with mon H∞ (B ℓ∞ ). Again by (31) we see that B ℓ2,∞ ⊂ mon H∞ (B ℓ∞ ) , and moreover that there are sequences in ℓ2,∞ ∩B ℓ∞ that do not belong to mon H∞ (B ℓ∞ ). But it also can be shown that mon H∞ (B ℓ∞ ) * ℓ2,∞ ; the proof is now slightly more complicated: Take a strictly increasing sequence ¡ ¢ of non-negative integers (n k )k with n 1 > 1, satisfying that the sequence k+1 nk k is strictly decreasing and ∞ k +1 X < 1; k=1 n k 2
(take for example n k = a k (k+1) for a ∈ N big enough). Now we define q 1 1 ≤ j ≤ n1 n rj = q 1 k+1 n < j ≤ n , k = 1, 2, . . . . k k+1 n k+1
The sequence (r n ) is decreasing to 0. Clearly, n k r n2k = k for all k. Thus (r n ) does not belong to ℓ2,∞ . But for n > n 1 , if n k < n ≤ n k+1 and limk condition satisfied by the above example), then
k+1 log nk
= 0 (a
n1 n n h+1 X nX X ¢ 1 X 1 k−1 1 ¡X + r j2 = r j2 + r j2 log n j =1 log n j =1 n 1 h=1 j =nh +1 j =nk +1
≤
k−1 X n h+1 − n h ¢ n k+1 − n k 1 ¡ (h + 1) + (k + 1) 1+ log n n h+1 n k+1 h=1
≤ Hence lim sup n→∞
k−1 ∞ h +1 X h +1 X k +1 k +1 1 + + < + . log n 1 h=1 log n h+1 log n k h=1 n h logn k
n 1 X r 2 < 1, and therefore (r n )n ∈ mon H∞ (B ℓ∞ ). log n j =1 j
3 Series expansion of H p -functions in infinitely many variables We draw now our attention to functions on T∞ , the infinite dimensional polytorus. We recall that m denotes the product of the normalized Lebesgue measure on T∞ . Given a function f ∈ L p (T∞ ), its Fourier coefficients ( fˆ(α))α∈Z(N) 22
R α α are defined by fˆ(α) = T∞ f (w)w −α d m(w) = 〈 f , w α 〉 where w α = w 1 1 . . . w n n if α = (α1 . . . αn , 0, . . .) for w ∈ T∞ , and the bracket 〈·, ·〉 refers to the duality between L p (T∞ ) and L q (T∞ ) for 1/p + 1/q = 1. With this, for 1 ≤ p ≤ ∞ the Hardy spaces are defined as n o ¯ H p (T∞ ) = f ∈ L p (T∞ ) ¯ fˆ(α) = 0 , ∀α ∈ Z(N) \ N(0N) . (32) We will also consider, for each m, the following closed subspace n o ¯ H pm (T∞ ) = f ∈ H p (T∞ ) ¯ fˆ(α) 6= 0 ⇒ |α| = m
of L p (T∞ ). By [8, Section 9] this is the completion of the m-homogeneous trigonometric polynomials (functions on T∞ that are finite sums of the form P α |α|=m c α w ). It is important to note that H qm (T∞ ) = H pm (T∞ ) , 1 ≤ p, q < ∞ and m ∈ N ;
(33)
this was first observed in by [8, 9.1 Theorem] (here it also follows from Lemma 2.4 and a density argument). In analogy to (13) and (14) we define for every for 1 ≤ p ≤ ∞ and m ∈ N the following two sets of monomial convergence: n o ¯X mon H p (T∞ ) = z ∈ CN ¯ | fˆ(α)z α | < ∞ for all f ∈ H p (T∞ ) α n o ¯X m ∞ N¯ mon H p (T ) = z ∈ C | fˆ(α)z α | < ∞ for all f ∈ H pm (T∞ ) . α
Obviously both sets are increasing in p. In sections 3.1 and 3.2 we will prove that mon H∞ (T∞ ) = mon H∞ (B c0 ) = mon H∞ (B ℓ∞ ) (34) m mon H∞ (T∞ ) = mon P (m c0 ) = mon P (m ℓ∞ ) ,
(35)
which by Theorem 2.1 and Theorem 2.2 then in particular implies that mon H p (T∞ ) ⊂ B and mon H pm (T∞ ) ⊂ ℓ m−1 ,∞ . 2m
But we are going to see in this section that a much more precise description is possible.
3.1 The homogeneous case The homogeneous case can be solved completely.
23
Theorem 3.1. mon H pm (T∞ ) =
(
for 1 ≤ p < ∞
ℓ2 ℓ
2m m−1 ,∞
for p = ∞.
Moreover, there is C > 0 such that if z ∈ mon H pm (T∞ ) and f ∈ H pm (T∞ ), then X | fˆ(α)z α | ≤ C m kzkm k f kp , (36) |α|=m
where kzk is the norm in the corresponding sequence space (here 1 ≤ C ≤ 1 ≤ p ≤ 2 and C = 1 for 2 ≤ p < ∞).
p 2 for
Again we prepare the proof with some lemmas of independent interest. We deal with two separate situations: p = ∞ and p = 2 (covering the case for arbitrary 1 ≤ p < ∞). The first case will follow from Theorem 2.1, after showing that m H∞ (T∞ ) can be identified with P (m c0 ). The basic idea here is, given a polynomial on c0 , extend it to ℓ∞ and then restrict it to T∞ . Let us very briefly recall how m-homogeneous polynomials on a Banach space X can be extended to its bidual (see [16, Section 6] or [15, Proposition 1.53]). First of all, every m-linear mapping A : X ×· · ·× X → C has a unique extension (called the Arens extension) A˜ : X ∗∗ × · · · × X ∗∗ → C such that for all j = 1, . . . , n, all xk ∈ X and z k ∈ X ∗∗ , ˜ 1 , . . . , x j −1 , z, z j +1 , . . . , z m ) is weak∗ the mapping that to z ∈ X ∗∗ associates A(x continuous. Now, given P ∈ P (m X ), we take its associated symmetric m-linear ˜ . . . , z). form A and define its Aron–Berner extension P˜ ∈ P (m X ∗∗ ) by P˜ (z) = A(z, By [9, Theorem 3] we have sup |P (x)| = sup |P˜ (z)| .
x∈B X
(37)
z∈B X ∗∗
Hence, the operator AB : P (m X ) → P (m X ∗∗ ) , AB(P ) = P˜ is a linear isometry. Lemma 3.2. The mapping m ψ : P (m c0 ) → H∞ (T∞ ) , ψ(P )(w) = AB(P )(w)
is a surjective isometry. Proof. Let us note first that, by the very definition of the Aron–Berner extension, for each α ∈ N(0N) , the monomial x ∈ c0 7→ x α is extended to the monomial P z ∈ ℓ∞ 7→ z α . Then the set of finite sums of the type |α| cα x α is bijectively and isometrically mapped onto the set of m-homogeneous trigonometric polynomials. By [15, Propositions 1.59 and 2.8] the monomials on c0 with |α| = m generate a dense subspace of P (m c0 ). On the other hand, by [8, Section 9] the m trigonometric polynomials are dense in H∞ (T∞ ). This gives the result. 24
To deal with the case 1 ≤ p < ∞ we need the following lemma. Lemma 3.3. mon H pm (T∞ ) ⊂ mon H pm−1 (T∞ ) Proof. Let 0 6= z ∈ mon H pm (T∞ ) and f ∈ mon H pm−1 (T∞ ). We choose z i 0 6= 0 and define f˜(w) = w i 0 f (w). Let us see that f˜ ∈ H pm (T∞ ); indeed, take a sequence ( f n )n of (m −1)-homogeneous trigonometric polynomials that converges in the P space L p (T∞ ) to f . Each f n is a finite sum of the type |α|=m−1 cα(n) w α . We define for w ∈ T∞ X αi +1 α α cα(n) w 1 1 · · · w i 0 · · · w k k . f˜n (w) = w i 0 f n (w) = 0
|α|=m−1
Clearly f˜n is an m-homogeneous trigonometric polynomial. Moreover
´ 1 ³Z ´1 p p p p |w i 0 f n (w) − w i 0 f (w)| d m(w) = |w i 0 | | f n (w) − f (w)| d m(w) ∞ ∞ T T ´1 ³Z p | f n (w) − f (w)|p d m(w) . ≤
³Z
p
T∞
The last term converges to 0, hence ( f˜n )n converges in L p (T∞ ) to f˜ and f˜ ∈ H pm (T∞ ). We compute now the Fourier coefficients: fˆ˜(α) =
Z
T∞
Z −α ˜ f (w)w d m(w) = w i 0 f (w)w −α d m(w) ∞ T Z −αi +1 −α −α f (w)w 1 1 · · · w i 0 · · · w n n d m(w) = 0 ∞ ZT −(αi −1) −α −α f (w)w 1 1 · · · w i 0 · · · w n n d m(w) = 0
T∞
That is fˆ˜(α) =
= fˆ(α1 , . . . , αi 0 − 1, . . . , αn ) .
(
fˆ(β) 0
if α = (β1 , . . . , βi 0 + 1, . . . , βn ) otherwise
and this gives X β
| fˆ(β)z β | =
1 X ˆ | f (β)z β | |z i 0 | |z i 0 | β 1 X ˆ˜ 1 X ˆ βi +1 β β | f (β)z 1 1 · · · z i 0 · · · z n n | = | f (α)z α | < ∞ . = 0 |z i 0 | β |z i 0 | α
Hence z ∈ mon H pm−1 (T∞ ). 25
Finally, we are ready to give the Proof of Theorem 3.1. The case p = ∞ follows from Theorem 2.1 and Lemma 3.2. Assume that 1 ≤ p < ∞, and observe that by (33) it suffices to handle the case p = 2. If z ∈ ℓ2 and f ∈ H2m (T∞ ), then we apply the Cauchy-Schwarz inequality and the binomial formula to get X
α∈N(N) 0 |α|=m
| fˆ(α)z α | ≤
³ X
α∈N(N) 0 |α|=m
| fˆ(α)|2
´1 ³ X 2
2α
α∈N(N) 0 |α|=m
|z|
´1 2
≤ kzkm 2 k f k2 < ∞ ;
(38)
this implies z ∈ mon H2m (T∞ ). Let us now fix z ∈ mon H21 (T∞ ). By a closedgraph argument, there is c z > 0 such that for every f ∈ H21 (T∞ ) the inequality P∞ ˆ We fix y ∈ ℓ2 , and define for each N the function n=1 | f (n)z n | ≤ c z k f k2 holds. PN f N : T∞ → C , f N (w) = n=1 w n y n . Clearly fˆ(n) = y n for n = 1, . . . , N . Hence f ∈ H21 (T∞ ), and as a consequence we have N X
n=1
| fˆ(n)z n | ≤ c z
³X N
n=1
2
|y n |
´1
2
≤ c z kyk2 < ∞ .
P ˆ This holds for every N , hence ∞ n=1 | f (n)z n | ≤ c z kyk2 and, since this holds for every y ∈ ℓ2 , we obtain z ∈ ℓ2 . This gives ℓ2 ⊂ mon H2m (T∞ ) ⊂ mon H21 (T∞ ) ⊂ ℓ2 . Finally, (36) follows immediately from Theorem 2.1 and Lemma 3.2 for the case p = ∞. Moreover, (38) gives (36) for 2 ≤ p < ∞ with C = 1, and (38) p combined with Lemma 2.4 (s = 2 and r = 1) give the inequality with C ≤ 2 whenever 1 ≤ p < 2.
3.2 The general case We address now our main goal of describing mon H p (T∞ ). There are three significant cases: p = 1, p = 2, and p = ∞. The description of mon H p (T∞ ) for 1 ≤ p < ∞ will follow from the cases p = 1 and p = 2, showing that these two coincide. Theorem 3.4. (a) B ⊂ mon H∞ (T∞ ) ⊂ B (b) mon H p (T∞ ) = ℓ2 ∩ B ℓ∞ for 1 ≤ p < ∞. 26
Again we prepare the proof (which will be given after Lemma 3.7) by some independently interesting observations. Part (a) is an immediate consequence of Theorem 2.2 and the fact that mon H∞ (T∞ ) = mon H∞ (B c0 ) = mon H∞ (B ℓ∞ ) , which we already mentioned without proof in (34): For the second equality see (15) whereas the proof of mon H∞ (T∞ ) = mon H∞ (B c0 ) is a consequence of the following theorem due to Cole and Gamelin [8, 11.2 Theorem] (see also [18, Lemma 2.3]). For the sake of completeness we include an elementary direct proof; the statement about the inverse mapping seems to be new. Proposition 3.5. There exists a unique surjective isometry φ : H∞ (T∞ ) → H∞ (B c0 )
such that for every f ∈ H∞ (T∞ ) and every α ∈ N0(N) we have ¡ ¢ cα φ( f ) = fˆ(α) .
m Moreover, when restricted to H∞ (T∞ ), the mapping ψ defined in Proposition 3.2 and φ are inverse to each other.
Proof. First of all, let us note that in the finite dimensional setting the result is true: It is a well known fact (see e.g. [27, 3.4.4 exercise (c)]) that for¡ each¢n there exists an isometric bijection φn : H∞ (Tn ) → H∞ (Dn ) such that cα φ( f ) = f˜(α) for every f ∈ H∞ (Tn ) and every α ∈ Nn0 . Take now f ∈ H∞ (T∞ ) and fix n ∈ N; since we can consider T∞ = Tn ×T∞ , we write w = (w 1 , . . . , w n , w˜ n ) ∈ T∞ . Then we define f n : Tn → C by Z f n (w 1 , . . . w n ) = f (w 1 , . . . , w n , w˜ n )d m(w˜ n ) . T∞
By the Fubini theorem f n is well defined a.e. and Z Z ³Z ´ f (w)d m(w) = f (w 1 , . . . , w n , w˜ n )d m(w˜ n ) d m n (w 1 , . . . , w n ) , T∞
Tn
T∞
hence f n ∈ L ∞ (Tn ). Moreover, for α ∈ Zn we have, again by Fubini Z fˆn (α) = f (w)w −α d m(w) = fˆ(α) . Tn ×T∞
Thus fˆn (α) = fˆ(α) = 0 for every α ∈ Zn \Nn0 and f n ∈ H∞ (Tn ). Obviously k f n k∞ ≤ k f k∞ since the measure is a probability. We take g n = φn ( f n ) ∈ H∞ (Dn ). We have kg n k∞ = k f n k∞ ≤ k f k∞ and X X fˆ(α)z α fˆn (α)z α = g n (z) = α∈Nn0
α∈Nn0
27
for every z ∈ Dn . Since this holds for every n we can define g : D(N) → C by P g (z) = α∈Nn0 fˆ(α)z α . We have kg k∞ = supn kg n k∞ ≤ k f k∞ . By [12, Lemma 2.2] there exists a unique extension g˜ ∈ H∞ (B c ) with cα (g˜ ) = fˆ(α) and kg˜ k∞ = 0
kg k∞ ≤ k f k∞ . Setting φ( f ) = g˜ we have that φ : H∞ (T∞ ) → H∞ (B c0 ) is well ∞ defined k f k∞ and ¡ ¢and such that for every(Nf) ∈ H∞ (T ) we have kφ( f )k∞ ≤ ∞ cα φ( f ) = f˜(α) for every α ∈ N0 . On the other hand if f ∈ L ∞ (T ) is such that fˆ(α) = 0 for all α then f = 0. Hence φ is injective. Let us see that it is also surjective and moreover an isometry. Fix g ∈ H∞ (B c0 ) and consider g n its restriction to the first n variables. Clearly g n ∈ H∞ (Dn ) and kg n k∞ ≤ kg k∞ . Using again [27, 3.4.4 exercise (c)] we can choose f n ∈ H∞ (Tn ) such that k f n k∞ = kg n k∞ and cα (g n ) = fˆn (α) for all α ∈ Nn0 . Since cα (g n ) = cα (g ) we have fˆn (α) = cα (g ). We define now f˜n ∈ H∞ (T∞ ) by f˜n (w) = f n (w 1 , . . . , w n ) for w ∈ T∞ . Then the sequence ( f˜n )∞ n=1 is contained in the closed ∞ ball in L ∞(T ) centered at 0 and with radius kg k∞. Since this ball is w ∗-compact and metrizable, there is a subsequence ( f˜nk )k that w ∗ -converges to some f ∈ L ∞ (T∞ ) with k f k∞ ≤ kg k∞ . Moreover, fˆ(α) = 〈 f , w α 〉 = limk→∞ 〈 f˜nk , w α 〉 = limk→∞ fˆ˜n (α) for every α ∈ Z(N) and this implies f ∈ H∞ (T∞ ). Let us see that 0
k
φ( f ) = g , which shows that φ is onto; indeed, if α = (α1 , . . . , αn0 , 0, . . .) then for n k ≥ n 0 we have Z Z −α α f nk (w)w −α d m nk (w) = fˆnk (α) = cα (g ) . f˜nk (w)w d m(w) = 〈 f˜nk , w 〉 = T∞
Tn k
Hence fˆ(α) = cα (g ) for all α ∈ N(0N) . Furthermore, since k f k∞ ≤ kg k∞ = kφ( f )k∞ we also get that φ is an isometry. Let us fix P ∈ P (m c0 ) and show that φ−1(P )(w) = P˜ (w) for every w ∈ T∞ . We choose (J k )k a sequence of finite families of multiindexes included in {α : α ∈ N(0N) : |α| = m} and such that the sequence P k = P α α∈J k c α,k x converges uniformly to P on the unit ball of c 0 . Since each J k is finite, we have X φ−1 (P k )(w) = cα,k w α = P˜k (w) , α∈J k
for every w ∈ T∞ . The linearity of the AB operator and (37) give that kP˜ − P˜k k = kP − P k k = kφ−1 (P ) − φ−1 (P k )k converges to 0 and complete the proof. Observe that this argument actually works to prove that φ−1 (g )(w) = g˜ (w) for every w ∈ T∞ and every function g in the completion of the space of all polynomials on c0 .
We handle now the case p = 2 of part (b) of Theorem 3.4 where slightly more can be said (for the proof of Theorem 3.4 this will not be needed). Here, since H2 (T∞ ) is a Hilbert space with the orthonormal basis {w α }α , we have k f k2 = ¢ ¡P 2 1/2 ˆ which simplifies the problem a lot. α | f (α)| 28
Proposition 3.6. We have mon H2 (T∞ ) = ℓ2 ∩ B ℓ∞ , and for each z ∈ ℓ2 ∩ ℓ∞ and f ∈ H2 (T∞ ), X
α∈N(N) 0
Moreover, the constant
| fˆ(α)z α | ≤
¢1/2 1 n 1−|zn |2
¡Q
´1 1 2 k f k2 . 2 1 − |z | n n=1
³Y ∞
(39)
is optimal.
Proof. The fact that ℓ2∩B ℓ∞ ⊂ mon H2 (T∞ ) follows by using the Cauchy-Schwarz inequality in a similar way as in (38): X
α∈N(N) 0
| fˆ(α)z α | ≤
³ X
α∈N(N) 0
| fˆ(α)|2
´1 ³ X 2
α∈N(N) 0
|z|2α
´1 2
= k f k2
´1 1 2 < ∞. 2 n=1 1 − |z n |
³Y ∞
On the other hand, since H21 (T∞ ) ⊂ H2 (T∞ ) we have that mon H2 (T∞ ) is a subset of mon H21 (T∞ ) and Theorem 3.1 gives the conclusion. To see that the constant in the inequality is optimal, let us fix z in mon H2 (T∞ ) and take c > 0 such that X | fˆ(α)z α | ≤ ck f k2 . α∈N(N) 0
P For each n ∈ N we consider the function f n (w) = α∈Nn0 z α w α that clearly satisfies f z ∈ H2 (T∞ ) and fˆz (α) = z α for every α ∈ Nn (and 0 otherwise). Hence 0
X
α∈Nn0
|z α |2 =
X
α∈Nn0
| fˆz (α)z α | ≤ ck f k2 = c
This gives c≥ for every n. Hence c ≥
³ X
α∈Nn0
α 2
|z |
´1 2
¢1/2 1 n=1 1−|zn |2
¡ Q∞
³ X
α∈Nn0
|z α |2
´1 2
.
´1 1 2 = 2 n=1 1 − |z n | ³Y n
and the proof is completed.
In order to extend this result to the general case 1 ≤ p < ∞ we need another important lemma – an H p –version of [6, Satz VI] (see also [13, Lemma 2]). Lemma 3.7. Let z ∈ mon H p (T∞ ) and x = (xn )n ∈ B ℓ∞ such that |xn | ≤ |z n | for all but finitely many n’s. Then x ∈ mon H p (T∞ ). 29
Proof. We follow [13, Lemma 2] and choose r ∈ N such that |xn | ≤ |z n | for all n > r . We also take a > 1 such that |z n | < a1 for n = 1, . . . , r . Let f ∈ H p (T∞ ) with k f kp ≤ 1. We fix n 1 , . . . , n r ∈ N and define for each u ∈ T∞ , Z −n −n f (w 1 , . . . , w r , u 1 , . . .)w 1 1 · · · w r r d m r (w 1 , . . . , w r ) . f n1 ,...,nr (u) = Tr
Let us see that f n1 ,...,nr ∈ H p (T∞ ); indeed, using Hölder inequality we have ³Z
´ p1 | f n1 ,...,nr (u)| d m(u) p
T∞
¯p ´ p1 ³Z ¯Z ¯ ¯ −nr −n1 f (w 1 , . . . , w r , u 1 , . . .)w 1 · · · w r d m r (w 1 , . . . , w r )¯ d m(u) = ¯ Tr T∞ ´ ´ p1 ³Z ³Z p | f (w 1 , . . . , w r , u 1 , . . .)| d m r (w 1 , . . . , w r ) d m(u) = k f kp . ≤ T∞
Tr
Hence f n1 ,...,nr ∈ L p (T∞ ) and k f n1 ,...,nr kp ≤ k f kp ≤ 1. Now we have, for every multi index α = (α1 , . . . , αk , 0, . . .) fˆn1 ,...,nr (α) Z f n1 ,...,nr (u)u −α d m(u) = T∞ ³ Z Z f (w , . . . , w , u , . . . , u ) ´ 1 r 1 k = d m (w , . . . , w )d m (u , . . . , u , 0, . . .) r 1 r k 1 k αk n1 nr α1 T∞ Tr w 1 · · · w r u 1 · · · u k = fˆ(n 1 , . . . , n r , α1 , . . . , αk , 0, . . .) . Therefore fˆn1 ,...,nr (α) =
(
fˆ(n 1 , . . . , n r , α1 , . . . , αk , 0, . . .) 0
if α = (0, .r. ., 0, α1 , . . . αk , 0, . . .)
otherwise
and this implies f n1 ,...,nr ∈ H p (T∞ ). Now, using (41) (below) and doing exactly P the same calculations as in [13, Lemma 2] we conclude α | fˆ(α)x α | < ∞ and x belongs to mon H p (T∞ ). Finally, we are ready for the Proof of Theorem 3.4–(b). Lower inclusion: Let us remark first that mon H1 (T∞ ) ⊂ mon H p (T∞ ) since H p (T∞ ) ⊂ H1 (T∞ ). Then to get the lower bound it is enough to show that ℓ2 ∩ B ℓ∞ ⊂ mon H1 (T∞ ). As a first step we show that there exists 0 < r < 30
p 1 such that r B ℓ2 ∩ B ℓ∞ ⊂ mon H1 (T∞ ). Let r < 1/ 2 and choose f ∈ H1 (T∞ ) and z ∈ r B ℓ2 ∩ B ℓ∞ . Then z = r y for some y ∈ B ℓ2 . By [8, 9.2 Theorem] there exists a projection P m : H1 (T∞ ) → H1m (T∞ ) such that kP m g k1 ≤ kg k1 for every g ∈ H1 (T∞ ). We write f m = P m ( f ) and we have fˆm (α) = fˆ(α) if |α| = m and 0 otherwise. Then X α
| fˆ(α)z α | =
∞ X X
m=0 |α|=m
| fˆ(α)(r y)α | = ≤
∞ X
m=0
∞ X X
m=0 |α|=m
| fˆm (α)(r y)α |
∞ X p p (r 2)m k f k1 < ∞ , r m ( 2)m k f m k1 ≤ m=0
where in the first inequality we used that y ∈ ℓ2 and (36), and in the second one that the projection is a contraction. Take now some z ∈ ℓ2 ∩ B ℓ∞ . Then ¡ P∞ ¢ 2 1/2 < r for some n 0 , and we define n=n0 |z n | x = (0, . . . , 0, z n0 , z n0 +1 , . . .) ∈ r B ℓ2 ∩ B ℓ∞ .
As explained x ∈ mon H1 (T∞ ), and hence Lemma 3.7 implies as desired z ∈ mon H1 (T∞ ). Upper inclusion: Again by 2.4 we have H p1 (T∞ ) = H21 (T∞ ) with equivalent norms. This, together with Theorem 3.1, gives mon H p (T∞ ) ⊂ mon H p1 (T∞ ) = mon H21 (T∞ ) ⊂ ℓ2 ∩ B ℓ∞ . Remark 3.8. Denote by P fin the space of all trigonometric polynomials on CN P (all finite sums α∈J cα z α ). For each z ∈ ℓ∞ the evaluation mapping δz : P fin → C , δz ( f ) = f (z)
is clearly well defined. One of the main problems considered in [8] is to determine for which z’s the evaluation mapping δz extends continuously to the whole space H p (T∞ ), 1 ≤ p < ∞. This can be reformulated as to describe the following set ¯ © ª z ∈ ℓ∞ ¯ ∃c z > 0 ∀ f ∈ P fin : | f (z)| ≤ c z k f kp . Since for each f ∈ P fin and every α we have fˆ(α) = cα , the previous set can be written as ¯X ¯ o n ¯ ¯ α¯ ¯ ˆ (40) z ∈ ℓ∞ ∃c z > 0 ∀ f ∈ P fin : ¯ f (α)z ¯ ≤ c z k f kp . α
In [8, 8.1 Theorem] it is shown that for 1 ≤ p < ∞ the set in (40) is exactly ℓ2 ∩ B ℓ∞ . By a closed-graph argument, for each 1 ≤ p < ∞ a sequence z belongs 31
to the set mon H p (T∞ ) if and only if there exists c z > 0 such that for every f ∈ H p (T∞ ) ´1 ³Z ¯ X ¯ α ¯ f (w)¯p d m(w) p . ˆ | f (α)z | ≤ c z (41) T∞
α∈N(N) 0
This implies
n o X ¯ mon H p (T∞ ) = z ∈ ℓ∞ ¯ ∃c z > 0 ∀ f ∈ P fin : | fˆ(α)z α | ≤ c z k f kp .
(42)
α
In view of (42) we have that mon H p (T∞ ) is contained in the set in (40). Then the upper inclusion in Theorem 3.4-(b) follows from [8, 8.1 Theorem]. The proof we presented here is independent from that in [8]. But the lower inclusion in Theorem 3.4-(b) is stronger than the result in [8].
3.3 Representation of Hardy spaces We have seen in Proposition 3.5 how, like in the finitely dimensional case, the Hardy space H∞ (T∞ ) can be represented as a space of holomorphic functions on c0 . In [8, 10.1 Theorem] it is proved that every element of H p (T∞ ) can be represented by an holomorphic function of bounded type on B ℓ∞ ∩ ℓ2. A characterization of the holomorphic functions coming from elements of H p (T∞ ) can be given for 1 ≤ p < ∞, in terms of the following Banach space H p (B ℓ∞ ∩ ℓ2 ) of all holomorphic functions g : B ℓ∞ ∩ ℓ2 → C (here B ℓ∞ ∩ ℓ2 is considered as a complete Reinhardt domain in ℓ2 ) for which kg kHp (Bℓ∞ ∩ℓ2 ) = sup sup
n∈N 0 0 the sequences p k 2 B . By (2) this infimum from (51) is attained – a result which in our setting can alternatively be deduced from Theorem 4.2-(bii) since (p k )−1/2 ∈ B. • For Bohr strips of m-homogeneous Dirichlet series we by (50) have that S m = m−1 . Again this result can be reformulated into a result on ℓ1 -multi2m m pliers for H ∞ of the type (1/n σ ), and hence it can be easily deduced from the more general Theorem 4.2-(aii). 37
• Let now 1 ≤ p < ∞. It is known that each H p -Dirichlet series D has an absolut convergence abscissa σa (D) ≤ 1/2, and that this estimate is optimal: sup σa (D) =
D∈H p
1 . 2
(52)
This is an H p -analog of (47) (or equivalently (48)) which can be found (implicitly) in [3] and (explicitly) in [2, Theorem 1.1]. After the following reformulation in terms of ℓ1 -multipliers for H p : © ¯ ª 1 inf σ ¯ (1/n σ ) is an ℓ1 − multiplier for H p = , 2
(53)
we obtain (52) as an immediate consequence of Theorem 4.2-(bi). Note that here in contrast with (51) the infimum in (53) is not attained since (p k−1/2 )k ∉ ℓ2 (see also [2] where this was observed for the first time). • Similarly we obtain supD∈H ∞m σa (D) = m−1 2m as a consequence of Theorem 4.2-(aii), and observe that here the infimum corresponding to (53) is attained (see also (24)).
References [1] A. Aleman, J.F. Olsen, and E. Saksman. Fatou type theorems on the infinite dimensional polydisk, preprint. [2] R. Balasubramanian, B. Calado, and H. Queffélec. The Bohr inequality for ordinary Dirichlet series. Studia Math., 175(3):285–304, 2006. [3] F. Bayart. Hardy spaces of Dirichlet series and their composition operators. Monatsh. Math., 136(3):203–236, 2002. [4] F. Bayart, D. Pellegrino, and J. B. Seoane-Sepúlveda. The Bohr radius of the p n-dimensional polydisk is equivalent to (log n)/n. Preprint 2014.
[5] H. F. Bohnenblust and E. Hille. On the absolute convergence of Dirichlet series. Ann. of Math. (2), 32(3):600–622, 1931. [6] H. Bohr. Über die Bedeutung der Potenzreihen unendlich vieler Variablen Pa in der Theorie der Dirichlet–schen Reihen nns . Nachr. Ges. Wiss. Göttingen, Math. Phys. Kl., pages 441–488, 1913. [7] H. Bohr. Über die gleichmäßige Konvergenz Dirichletscher Reihen. J. Reine Angew. Math., 143:203–211, 1913. 38
[8] B. J. Cole and T. W. Gamelin. Representing measures and Hardy spaces for the infinite polydisk algebra. Proc. London Math. Soc. (3), 53(1):112–142, 1986. [9] A. M. Davie and T. W. Gamelin. A theorem on polynomial-star approximation. Proc. Amer. Math. Soc., 106(2):351–356, 1989. [10] R. de la Bretèche. Sur l’ordre de grandeur des polynômes de Dirichlet. Acta Arith., 134(2):141–148, 2008. [11] A. Defant, L. Frerick, J. Ortega-Cerdà, M. Ounaïes, and K. Seip. The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive. Ann. of Math. (2), 174(1):485–497, 2011. [12] A. Defant, D. García, and M. Maestre. New strips of convergence for Dirichlet series. Publ. Mat., 54(2):369–388, 2010. [13] A. Defant, D. García, M. Maestre, and D. Pérez-García. Bohr’s strip for vector valued Dirichlet series. Math. Ann., 342(3):533–555, 2008. [14] A. Defant, M. Maestre, and C. Prengel. Domains of convergence for monomial expansions of holomorphic functions in infinitely many variables. J. Reine Angew. Math., 634:13–49, 2009. [15] S. Dineen. Complex analysis on infinite-dimensional spaces. Springer Monographs in Mathematics. Springer-Verlag London Ltd., London, 1999. [16] K. Floret. Natural norms on symmetric tensor products of normed spaces. Note Mat., 17:153–188 (1999), 1997. [17] L. A. Harris. Bounds on the derivatives of holomorphic functions of vectors. In Analyse fonctionnelle et applications (Comptes Rendus Colloq. Analyse, Inst. Mat., Univ. Federal Rio de Janeiro, Rio de Janeiro, 1972), pages 145–163. Actualités Aci. Indust., No. 1367. Hermann, Paris, 1975. [18] H. Hedenmalm, P. Lindqvist, and K. Seip. A Hilbert space of Dirichlet series and systems of dilated functions in L 2 (0, 1). Duke Math. J., 86(1):1–37, 1997. [19] H. Helson, and D. Lowdenslager, Prediction theory and Fourier series in several variables. Acta Math., 99:165–202, 1958. [20] D. Hibert. Gesammelte Abhandlungen (Band 3). Verlag von Julius Springer, Berlin, 1935.
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[21] D. Hilbert. Wesen und Ziele einer Analysis der unendlichvielen unabhängigen Variablen. Rend. del Circolo mat. di Palermo, 27:59–74, 1909. [22] J.-P. Kahane. Some random series of functions, volume 5 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 1985. [23] S. V. Konyagin and H. Queffélec. The translation 21 in the theory of Dirichlet series. Real Anal. Exchange, 27(1):155–175, 2001/02. [24] B. Maurizi and H. Queffélec. Some remarks on the algebra of bounded Dirichlet series. J. Fourier Anal. Appl., 16(5):676–692, 2010. [25] H. Queffélec. H. Bohr’s vision of ordinary Dirichlet series; old and new results. J. Anal., 3:43–60, 1995. [26] H. Queffélec and M. Queffélec. Diophantine approximation and Dirichlet series. HRI Lecture Notes Series, New Delhi 2013. [27] W. Rudin. Function theory in polydisks. W. A. Benjamin, Inc., New YorkAmsterdam, 1969. [28] O. Toeplitz. Über eine bei den Dirichletschen Reihen auftretende Aufgabe aus der Theorie der Potenzreihen von unendlichvielen Veränderlichen. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, pages 417–432, 1913. [29] F. B. Weissler. Logarithmic Sobolev inequalities and hypercontractive estimates on the circle. J. Funct. Anal., 37(2):218–234, 1980. [30] P. Wojtaszczyk. Banach spaces for analysts, volume 25 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1991.
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