Holomorphic functions

II. Domains of convergence and other viewpoints 103

7.3. Holomorphic functions

Chapter 7. Sets of monomial convergence

In [32] it was furthermore shown that for 1< p <∞,q defined by ¹_q = ¹₂+_max{p,2}¹ and everyε >0

B`_p∩`q ⊂monH∞(B`_p)⊂B`_p∩`q+ε. (7·J) In the following we improve the lower inclusion and show in particular thatε= 0 is not possible. More precisely, we give necessary and sufficient conditions onα, β∈[0,∞) so that

1 k^α(log(k+ 2))^β

∈monH∞(B`_p). (7·K) Note that, by (7·I), a sufficient condition in the case p=∞ is given byα≥ ¹₂ and β >0. However, we do not know whetherα= ¹₂andβ = 0 is possible in this situation.

Moreover, by (7·H),

1 k(log(k+ 2))^β

∈monH_∞(B`1)

if and only ifβ >1. The following theorem collects results for the remaining cases:

Theorem 7.10. Let 1≤p≤ ∞and setσ:= 1−_min{p,2}¹ . (i) If1≤p≤2, then

1 k^σ(log(k+ 2))^θσ

·B`_p ⊂monH_∞(B`_p)

for every θ > ¹₂. In particular, (7·K) holds true for α =σ+¹_p = 1 and any β > ¹₂ 1 +¹_p

. (ii) If2≤p≤ ∞, then

1 k^σ(log(k+ 2))^θσ

·B`_p ⊂monH_∞(B`_p)

for everyθ >0. In particular, (7·K) holds true forα=σ+_p¹ =¹₂+¹_p and any β > ¹_p.

(iii) If

k^σ+¹^p(log(k+ 2))^β

∈monH_∞(B`_p), thenβ≥ ¹_p.

114

7.3. Holomorphic functions

Proof. We start with the proof of (i). Let 1 ≤ p ≤ 2 and θ > ¹₂. Recall that the sequenceq was defined byqk :=k log(k+ 2)^θ

. By Theorem 6.4 for anyf ∈H_∞(B_`_p) andu∈B_`_p

|cj(f) (q^−σ^u)j|

∞

N=1

j∈J(e^N) e^N−1<qj

1 qj^σ

|cj(f)uj|

≤

∞

N=1

1 e^(N−1)σ

j∈J(e^N)

|cj(f)uj|

≤

∞

N=1

e^(N−1)σe^{N σ}exp

−2σ q

θ−¹₂+o(1)p

NlogN kfk_B

<∞.

Hence,q^−σ^·^u^∈^mon^H∞(B`_p) for everyu∈B`_p, which had to be demonstrated.

We proceed to prove (ii). Let 2 ≤ p ≤ ∞ and recall that B_`_p = B_`_p ·B_`_∞. By Theorem 7.5,

mon

H∞(B`p)]∞·B`p⊂monH∞(B`p). We see at once that

H∞(B`p)

∞=H∞(B`∞). Therefore, it suffices to check that (ξ_k)_k = 1

k¹²(log(k+ 2))^θ

∈monH_∞(B_`_∞) for everyθ >0. We have

1 logn

k=1

k(log(k+ 2))^2θ ≤ 1 logn

Z n 3

t(logt)^2θdt+c

≤ (logn)^1−2θ+c logn with a universal constantc >0; therefore, by (7·I),ξ∈B⊂monH_∞(B_`_∞).

Finally, we get to (iii). In the case p = 1 the claim follows directly from the al-ready known result (7·H). The remaining cases 1 < p ≤ 2 and 2 ≤ p < ∞ will be proved separately. In the following denote by x the sequence in question (i.e.

xk =k^−σ−^p¹ log(k+ 2)−β

) and assumex∈monH∞(B`_p).

Chapter 7. Sets of monomial convergence

Let 1 < p≤2. By a closed graph argument we find a constant ˜c ≥1 such that for every f ∈H_∞(B`_p)

α∈N^(N)₀

|cα(f)x^α| ≤˜ckfk_B

`p.

From Corollary 3.24, we obtain a constantc≥1 such that for anyn, m∈N Xⁿ

k=1

|xk|m

≤c(nlogm)¹⁻^p¹m^m(1−^p¹⁾. Taking them^th root, we get

k=1

k(log(k+ 2))^β ≤c^m¹ (nlogm)^m¹⁽¹⁻¹^p⁾m¹⁻¹^p

for everyn, m∈Nwith a universal constantc≥1. The left-hand side of this equation is now asymptotically equivalent to (logn)^1−β and withm:=blognc the right-hand side is asymptotically equivalent to (logn)¹⁻^p¹ asn→ ∞. Hence,β ≥¹_p.

Now, let p ≥ 2. Define ξ := k⁻¹^q(log(k+ 2))⁻¹^q^−ε

k for some ε > 0 where q is determined by ¹_p+¹_q = ¹₂. Considerf ∈H_∞(B`₂) and setg:=f◦Dξ. By Hölder’s inequality,Dξ defines a bounded operator`p→`2. Therefore,g∈H∞(B`_p) and thus

|cj(f)| 1

j₁(log(j₁+ 2))^q¹^+β+ε· · · 1

j_m(log(j_m+ 2))¹^q^+β+ε

cj(f)ξj

cj(g)xj

<∞ as we assumedx∈monH∞(B`p). Hence, we have

k log(k+ 2)¹_q+β+ε

k∈monH∞(B`₂).

From what was already proven (the case p= 2), we obtain that ¹_q +β+ε≥ ¹₂ and thusβ+ε≥¹_p for every ε >0.

We are finally able to give an answer to our previously stated question: The upper inclusion (7·J) holdsnot true forε= 0.

Theorem 7.11. Let 1< p <∞and set ¹_q := ¹₂+_max{p,2}¹ . Then B`p∩`_q 6= monH_∞(B`p).

116

7.3. Holomorphic functions

Proof. Assume equality and let q ^:= ^k^log(k^{+ 2)}_k. By Theorem 7.10, this implies that the diagonal operator`p→`qinduced by the sequenceq^−σ^where^σ^{:= 1−}_min{p,2}¹ is well-defined and, by a closed graph argument, bounded. Hence,

^∞ X

k=1

qk^−σ

r¹_r

= sup

x∈B_`p

X^∞

k=1

x_kqk^−σ

q_q¹

=kD_q−σk<∞ where ¹_q :=¹_p +¹_r. Therefore, we haveq^−σ^∈^`r, but

∞

k=1

qk^−σr=

∞

k=1

klog(k+ 2) =∞, a contradiction.

Theorem 7.12. Let 1≤p≤ ∞, setσ:= 1−_min{p,2}¹ , and letp denote the sequence of primes. Then

p^−σ^·^B`_p⊂monH_∞(B`_p) and the exponentσis optimal.

Proof. We proceed analogously to the proof of Theorem 7.10, (i). By Theorem 6.10, for anyf ∈H_∞(B`_p) and every u∈B`_p

|cj(f) (p^−σ^u)^j^|

∞

N=1

j∈J(e^N) e^N−1<pj

1 pj^σ

|cj(f)uj|

≤

∞

N=1

1 e^(N^−1)σ

j∈J(e^N)

|cj(f)uj|

≤

∞

N=1

e^(N^−1)σe^{N σ}exp

−√

2σ+o(1)p

NlogN kfk_B

<∞.

Hence,p^−σ^·^u^∈^mon^H∞(B_`_p) for everyu∈B_`_p, which had to be demonstrated.

Analogously to the result (7·I) forp=∞a plausible conjecture could be

Bp⊂monH∞(B`_p)⊂Bp (7·L)

Chapter 7. Sets of monomial convergence

withB_pand B_pdefined by Bp =

x∈B`∞

lim sup

1 logn

k=1

x^∗_k

q <1

Bp =

x∈B`∞

lim sup

1 logn

k=1

x^∗_k

q ≤1

where ¹_q := ¹₂+_max{p,2}¹ . This conjecture (at least the lower inclusion) is false. Indeed, for sufficiently smallβ >0

ξ:=

k^p¹^+σ(log(k+ 2))^β

6∈monH_∞(B`_p) by Theorem 7.10; but

1 logn

k=1

|ξ_k^∗|^q= 1 logn

k=1

k(log(k+ 2))^βq ∼(logn)^−βq→0 as n→ ∞.

118

Chapter 8. Interfaces with D IRICHLET series

The investigations of this thesis are closely linked to the theory of Dirichlet series as we already mentioned in our introduction. In this chapter we want to point out this connection in more detail. An ordinaryDirichletseries is a series of the form

D(s) =

∞

n=1

a_n 1 n^s

with complex coefficients (a_n)_nand a complex variables. Such a series is conditional, uniform, and absolute convergent on half-planes

[Re> σ] :={s∈C| Res > σ}.

For a Dirichlet series D, we define the abscissa of conditional convergence σc(D) as the infimum over allσ∈Rsuch that D converges conditionally on [Re> σ]. The abscissae of uniform and absolute convergence are defined analogously and denoted byσa(D) and σu(D) respectively. Clearly we have σc(D)≤σu(D)≤σa(D) for any DirichletseriesD.

On its half-plane of uniform convergence any Dirichlet series D converges to a holomorphic function f : [Re > σ_u(D)] → C. By σ_b(D) we want to denote the abscissa of boundedness, which is defined as the infimum over allσ∈Rsuch thatf can be extended to a bounded holomorphic function on [Re > σ]. An outstanding result ofBohr [18] shows thatσ_u(D) =σ_b(D) for anyDirichletseries D.

By H∞ we want to denote the linear space of all Dirichlet series converging to a bounded holomorphic function on the half-plane [Re>0];H_∞forms aBanachspace when endowed with the supremum norm on [Re>0].

Chapter 8. Interfaces withDirichletseries

σc σu σa

conditional convergence uniform convergence

absolute convergence

Figure 8.1.: Abscissae of convergence.

8.1. The B OHR transform — connecting D IRICHLET

series and power series

In his paper [18],Bohrintroduced an algebra isomorphism between the set of formal power series in infinitely many variables and the set of all ordinaryDirichletseries.

By the fundamental theorem of arithmetics we have a correspondence between the natural numbers and the set of all multi-indicesN⁽0^N⁾: n=p^α^whereⁿ^and^α^determine each other uniquely.

What we call today theBohr transform is then the algebra homomorphism B:P→D, X

α∈N^(N)0

cαz^α7→X

n∈N

ann^−s wherea_p^α :=cα.

A natural question might be: Which spaces on the side of power series correspond to which spaces on the side of ordinaryDirichlet series? Do we have isomorphisms or even isometries?

Hedenmalm,Lindqvist, andSeip[41] first proved that theBohrtransform defines an isometry between H_∞(Bc₀) andH_∞. For an alternative proof see the upcoming book [31].

Proposition 8.1 (cf. Section 2.2 of [41]). The Bohr transform defines a bijective isometry

B:H_∞(Bc₀)→ H_∞.

120

8.1. TheBohrtransform — connecting Dirichletseries and power series

In addition to the spaceH∞as the image ofH_∞(B_c₀) under theBohrtransform we can construct further examples of spaces ofDirichlet series. Form∈Ndefine

H^m_∞:=B P(^mc₀) . We easily check forD =P

nn^−s ∈ H^m_∞ that a_n 6= 0 only if n has exactlym prime factors (counting with multiplicity). Such Dirichlet series are called m–homoge-neous.

Let 1 ≤ p ≤ ∞ and denote by m the normalized product measure on the infinite dimensional polytorusT^∞. For a functionf ∈Lp(T^∞) we define theFourier coeffi-cient ˆf(α) withα∈Z⁽^N⁾by

fˆ(α) :=

Tⁿ

f(ω)ω^−αdm(ω) =hf, z^αi_L

p(T^∞),L_p0(T^∞). The so-calledHardy spaces are then defined as

Hp(T^∞) :=

f ∈Lp(T^∞)

∀α∈Z⁽^N⁾\N⁽0^N⁾: ˆf(α) = 0 .

It is well known that these areBanachspaces when endowed with theLp norm. For m∈Ndefine furthermore

H_p^m(T^∞) :=

f ∈Hp(T^∞)

fˆ(α) = 0 if|α| 6=m .

From [21] we know that H_p^m(T^∞) is the completion of them–homogeneous trigono-metric polynomials in H_p(T^∞). By means of the Bohr transform, applied on the Fourier series expansion, we define now the Banach spaces (transferring the re-spective topology)

Hp:=B H_p(T^∞) and

H^m_p :=B H_p^m(T^∞) . ForX =`p where 1≤p <∞andX =c0 define moreover

H_∞[X] :=B H_∞(BX) and

H^m_∞[X] :=B H_∞(P(^mX) . With this obviouslyH∞[c0] =H∞and H^m_∞[c0] =H^m_∞.

Chapter 8. Interfaces withDirichletseries

8.2. Multipliers on spaces of D IRICHLET series

LetDdenote a set ofDirichletseries. We call a sequence (bn)nof complex numbers an`_p–multiplier forDif

(anbn)n

p=Xⁿ

n=1

|anbn|^p_p¹

<∞ for allP

na_nn^−s∈ D. In [9], Bayart,Defant,Frerick,Maestre, and Sevilla-Perisconduct an profound research about the set of`1–multipliers forH_∞,H^m_∞, and the spacesHp,H^m_p . Using results presented in Chapter 7 and the fact that theBohr transform defines an bijective isometryH_∞(B_c₀)→ H_∞ they find (among others):

Theorem 8.2 (cf. Theorem 4.2 in [9]). Let (bn)n be a completely multiplicative sequence of complex numbers, that is bnm=bnbmfor any n, m∈N. Then:

(i) If b_p_k

<1 for everyk∈Nand lim sup

n→∞

1 logn

k=1

b^∗_p_k2

<1, then is(bn)n an `1–multiplier forH_∞.

(ii) If(bn)n is an`1–multiplier for H∞, then b_p_k

<1 for allk∈Nand lim sup

n→∞

1 logn

k=1

b^∗_p_k2

≤1.

In particular,(n⁻¹²)nis an`₁–multiplier forH_∞and(n⁻¹²^+ε)nisnotan`₁–multiplier forH_∞ for everyε >0.

In their proof they use the following evident connection, which follows directly from the definition.

Lemma 8.3. Let X a Banachsequence space andb= (b_n)_n be a completely multi-plicative sequence with

b_p_k

<1 for every k∈N. Then:

(i) bis an `1–multiplier forH_∞[X] if and only if(b_p_k)k∈monH_∞(BX).

(ii) bis an `1–multiplier forH^m_∞[X] if and only if(b_p_k)k∈monP(^mX).

122

8.2. Multipliers on spaces of Dirichletseries

From Theorem 7.6, (7·J), and the preceding lemma we get the following characteri-zation for the`1–multipliers forH_∞[`p] and H^m_∞[`p]:

Theorem 8.4. Letb= (b_n)n be a completely multiplicative sequence of complex num-bers with

b_p_k

<1 for every k∈Nand let1≤p <∞. Then:

(i) b is an`1–multiplier for H_∞[`1] if and only if(b_p_k)k∈`1. (ii) In the case1< p <∞ andqdefined by ¹_q =¹₂+_max{p,2}¹ :

(1) If(b_p_k)k∈B`_p∩`q, then isb is an`1–multiplier for H∞[`p].

Conversely:

(2) Ifbis an`₁–multiplier forH∞[`_p], then(b_p_k)_k ∈B_`_p∩`_q+εfor everyε >0.

Furthermore, form∈Nwe have:

(iii) b is an`₁–multiplier for H_∞^m[`₁] if and only if(b_p_k)_k∈`₁. (iv) In the case that1< p <2:

(1) If(b_p_k)k ∈`_(mp⁰₎⁰_−ε,∞for someε >0, then isban`1–multiplier forH^m_∞[`p].

Conversely:

(2) Ifbis an `1–multiplier forH^m_∞[`p], then(b_p_k)k∈`(mp⁰)⁰,∞. (v) In the case2≤p <∞:

(1) If(b_p_k)_k∈` 2m

m−1,∞·`_p, then isb an`₁–multiplier for H_∞^m[`_p].

Conversely:

(2) Ifbis an `₁–multiplier forH^m_∞[`_p], then(b_p_k)_k∈` _m−1

2m +¹_p⁻¹

,∞.

The analysis of the underlying results of this summarizing theorem, in particular Theorem 6.10, brings another interesting finding to light: Let (an)n be a sequence of complex numbers. We verify easily that

sup

(b_n)_n∈B_`p N

n=1

|anbn|= sup

(b_n)_n∈B_`p

n=1

anbn

(8·A)

Chapter 8. Interfaces withDirichletseries

for every N ∈ N. Furthermore, a quick calculation shows that for any completely multiplicative sequence (bn)n of complex numbers

(b_n)n ∈`_p⇔(b_p_k)k∈`_p and∀k: b_p_k

<1. (8·B) Having (8·A) and (8·B) in mind it is peculiar that there exist sequences (an)n of complex numbers such that

sup

(b_n)_nmult.

(b_p_k)k∈B_`p N

n=1

|anbn|> sup

(b_n)_nmult.

(b_p_k)k∈B_`p

n=1

anbn

Indeed, in the proof of Proposition 3.5 we constructed a polynomial P =PN k=1ckz^k such thatc_k=±1 and|P(x)| ≤√

2Nfor everyx∈T. Witha₂k=c_k fork= 1, . . . , N and an = 0 otherwise the right-hand side is bounded by√

2N whereas the left-hand side evaluates toN.

From Theorem 6.10 we obtain the following curious inequality:

Theorem 8.5. Let (an)n be a sequence of complex numbers. For 1 ≤ p ≤ ∞ set σ:= 1−_min{p,2}¹ . For anyN ∈Nthen

sup

(b_n)_nmult.

(b_p_k)k∈B_`p N

n=1

|anbn| ≤N^σexp

−√

2σ+o(1)p

logNlog logN sup

(b_n)_nmult.

(b_p_k)k∈B_`p

n=1

anbn

We conclude this chapter by interpreting the results ofBohrandBohnenblustand Hille in this new fashion. The result ofBohr, namely

S:= sup

σa(D)−σu(D)

D a Dirichletseries ≤¹₂,

is equivalent to the fact that (n⁻¹²^−ε)n is for every ε > 0 an `1–multiplier for H_∞. Conversely,S ≥ ¹₂, which was proved by Bohnenblustand Hille, is equivalent to the fact that (n⁻¹²^+ε)_n is for anyε >0 not an`₁–multiplier forH_∞. Both statements can hence be concluded from Theorem 8.2.

BohnenblustandHille showed in their proof of the lower boundS≥ ¹₂ that S_m:= sup

σ_a(D)−σ_u(D)

Da m–homogeneousDirichletseries = ^m−1_2m . This is equivalent to two of the statements in Theorem 8.4: (n⁻^m−1^2m ^−ε)n is for every ε >0 an`₁–multiplier forH^m_∞and (n⁻^m−1^2m ^+ε)_n is for everyε >0 not an`₁–multiplier forH^m_∞.

124

Chapter 9. B ^OHR radii

Already in 1913,Bohrwas aware that the absolute convergence of Dirichletseries is closely related to the absolute convergence of power series in infinitely many vari-ables; we introduced theBohr transform, which relates these facts, in the previous chapter.

A reasonable strategy to tackle the convergence of power series is to consider finite dimensional sections. We define then^th Bohr radius as

Kn:= supn

0≤r≤1

∀f ∈H_∞(B`ⁿ_∞) : sup

x∈rB_`n

∞

α∈Nⁿ₀

cα(f)x^α

≤ kfk_B

`n∞

o .

Bohr’s power series theorem states that K1 = ¹₃ and Bayart, Pellegrino, and Seoane-Sepúlveda[11] recently proved, using ideas of [27], that

n→∞lim Kn

qlogn n

= 1.

We introduce a more general definition: For aReinhardt domain R ⊂`_∞ and an index set Λ⊂N⁽0^N⁾define

K(R; Λ) := supn

0≤r≤1

∀f ∈H_∞(R) : sup

x∈rR

α∈Λ

c_α(f)x^α

≤ kfk_Ro . We call K(R; Λ) theBohr radius of the Reinhardt domain R with respect to Λ.

With this we have clearlyK(B_`n

∞;N⁽0^N⁾) =K_n. In the `_p case we have by results of Dineenand Timoney[38]; Boasand Khavinson [16]; Aizenberg[1]; Boas[15];

andDefantandFrerick[25, 26] the following theorem:

Chapter 9. Bohrradii

Theorem 9.1 (cf. Theorem 3 inBoas[15] and Theorem 1.1 in [26]). Let1≤p≤ ∞ and set σ:= 1−_min{p,2}¹ . There exists a constantsc≥1 such that

c⁻¹ logn

n σ

≤K B`ⁿ_p;N⁽0^N⁾

≤c logn

n σ

for every n∈N.

The upper estimate is due toBoas[15] (see also [28]); the proof uses a probabilistic argument. In [26] a proof of the lower estimate can be found which uses local Ba-nachspace theory and symmetric tensor products. Using Theorem 4.1, or rather its corollary, we want to give a simplified proof of the lower estimate, which moreover covers a wider range of Banachsequence spaces:

Theorem 9.2. Let 1≤p≤ ∞and setσ:= 1−_min{p,2}¹ . For any Banachsequence space with p–exhaustible unit ball there exists a constantsc≥1 such that

c⁻¹ logn

n ^σ

≤K BX_n;N⁽0^N⁾

for every n∈N.

The proof is based on the following lemma:

Lemma 9.3. Let1≤p≤ ∞and setσ:= 1−_min{p,2}¹ . There exists a constantsc≥1 such that for any index setΛ and every n∈N

K(B_X_n; Λ)≥ c sup

|Λ(n, m)^∗|^m^σ

whereΛ(n, m) := Λ∩Λ(n, m). Moreover, we havec≥ _3e¹2.

We will at first give the proof of Theorem 9.2. Afterwards, we give the proof of this lemma.

Proof of Theorem 9.2. Take the full index set Λ =N⁽0^N⁾. Obviously Λ(n, m) =Λ(n, m) and Λ(n, m)^∗=Λ(n, m−1); thus, by Lemma 4.8,

|Λ(n, m)^∗|=|Λ(n, m−1)|=

(m−1) +n−1 m−1

≤e^m−1

1 + n m−1

^m−1 .

126

Distinguishing the two casesn≤m−1 andn≥mwe have now

|Λ(n, m)^∗| ≤e^m−1

1 + n m−1

m−1

≤

(2e ifn≤m−1 and

2e_m−1ⁿ m−1

ifn≥m.

From Lemma 9.3, we hence obtain K(B_X_n; Λ)≥ c

sup

|Λ(n, m)^∗|^m^σ ≥c min

2e−σ

,inf

m 2e_m−1ⁿ −^m−1_m σ .

It remains to find a lower bound of the infimum. Leth_n(m) := ^m−1_m logn−log(m−1) . By differentiation we find thathn attains its maximum atM =W(ⁿ_e) + 1 whereW denotes theLambert W function; that is the inverse function ofx7→xe^xon (0,∞).

Therefore, with an absolute constantc≥1 n

m−1 ^m−1_m

exphn(m)≤exphn(M) = n

W(ⁿ_e)

W(n e) W(n

e)+1

≤c n logn for anym∈NasW(x) = logx−log logx+o(1). Together, we obtain

K(B_X_n; Λ)≥c logn

n ^σ

We proceed with the proof of Lemma 9.3. For this purpose we need adaptations of Lemma 2.1 and Theorem 2.2 in [28]:

Proposition 9.4(cf. Lemma 2.1 in [28]). For each Banachsequence space X, any set of indicesΛ, and any n, m∈N

K BX_n; Λ(n, m)

= 1

χmon P(^Λ(n,m)Xn) .

Proof. LetP ∈ P(^Λ(n,m)Xn) and (θα)α∈T^Λ(n,m). Then withkn:=K BX_n; Λ(n, m)

α∈Λ(n,m)

θαcα(P)z^α _B

Xn ≤ sup

x∈B_Xn

α∈Λ(n,m)

|cα(P)x^α|

= sup

x∈kn·B_Xn

α∈Λ(n,m)

|cα(P) _k^x

^α

| ≤ 1 k_n^mkPk_B

Xn.

Chapter 9. Bohrradii

This yields the upper estimate ofK B_X_n; Λ(n, m)

. On the other hand, sup

x∈B_Xn

α∈Λ(n,m)

|cα(P)x^α|=

α∈Λ(n,m)

|cα(P)|z^α _B

Xn ≤χ_mon P(^Λ(n,m)X_n) kPk_B

and thus

α∈Λ(n,m)

|c_α(P)x^α| ≤ kPk_B

for anyx∈X_n withkxk^m≤

χ_mon P(^Λ(n,m)X_n)⁻¹ .

Proposition 9.5 (cf. Theorem 2.2 in [28]). Let X be a Banachsequence space and R⊂X a Reinhardt domain. Then for any set of indicesΛ and any n∈N

K Rn; Λ

≥1 3 inf

m K Rn; Λ(n, m) . Proof. For simplicity we writekn:= infmK Rn; Λ(n, m)

forn∈N. Letf ∈H_∞(Rn) withkfk_R

n≤1, fixx∈R_n, and define g:

ξ∈C

|ξ| ≤1 →C, ξ7→ X

α∈Nⁿ₀

c_α(f)ξ^|α|x^α.

Clearly|g(ξ)| ≤1 for|ξ| ≤1 and thus Re(1−e^iθg)≥1 forθso that e^iθc0(f) =|c0(f)|.

By Carathéodory’sinequality for any m≥1

α∈Nⁿ0

|α|=m

cα(f)x^α

≤2 Re 1−e^iθc0(f)

= 2 1− |c0(f)|

Hence, for anym∈N X

α∈Λ(n,m)

c_α(f) x^k₃ⁿα

≤2 1− |c0(f)| 1 3^m and thus

α∈Λ

|cα(f)x^α|=|c0(f)|+

∞

m=1

α∈Λ(n,m)

|cα(f)x^α| ≤ |c0(f)|+ 2 1− |c0(f)|

∞

m=1

1 3^m = 1 for anyx∈Rn withkxk_X ≤^k₃ⁿ.

Altogether, we have

K R_n; Λ

≥kn

3 = 1 3 inf

m K R_n; Λ(n, m) .

128

Proof of Lemma 9.3. We now simply have to combine the previous propositions:

K(B_X_n; Λ)≥ 1 3 inf

m K B_X_n; Λ(n, m)

= inf

1 3

χmon P(^Λ(n,m)Xn) which is, by Corollary 4.2,

≥inf

1 3e²

|Λ(n, m)^∗|^m^σ .

Chapter 10. Outlook — where to continue

To conclude this thesis we want give a short outlook and draw the readers attention to some questions remaining unanswered.

Regarding Part I: In Part I we investigated the unconditional basis constant of the monomials in certain spaces of polynomials. Although we were able to establish an abstract inequality to get upper bounds, we presented only one vigorous application;

namely the case that the index set is generated by an increasing sequence.

In this setting the asymmetric reduction method displays its full strength. However, in the setting presented in Theorem 5.24 the asymmetric reduction has no advantage over the symmetric reduction method. Are there other relevant examples of index sets, different from index sets generated by increasing sequences, for which the asymmetric reduction method shows its full potential?

In Theorem 6.11 we tried to give an idea of the optimality of Theorem 6.4 and 6.10.

However, we were only able to prove that the exponent ofxis optimal; different from χmon P(^J(x)`∞)

with J(x) generated by the sequence of primes no precise lower bound are known. Are the estimates in Theorem 6.4 and 6.10 optimal?

Regarding Part II: In the latter part we used Theorem 6.4 to investigate the sets of monomial convergence and`₁–multipliers for sets of Dirichlet series. It seems that only index sets generated by increasing sequences yield results useful in this context.

Is there another choice of index sets more reasonable for this purpose?

Chapter 10. Outlook — where to continue

Regarding domains of monomial convergence: Theorem 7.3 yields a useful tool to prove that a certain sequence lies in the domain of monomial convergence forP(^m`_∞) orH∞(B`∞). It is perfectly reasonable to expect that this result holds (possibly with additional assumptions) also true for`_p(or even anyBanachsequence space) instead of `_∞. However, the issues one stumbles upon trying to adapt the proof in the `_∞ case seem indissoluble. Attempts to use the trick of Theorem 7.5 result in contrasting assumptions preventing success. It remains open to prove (or disprove) Theorem 7.3 for any Banachsequence space instead of`_∞.

In (7·L) we presented a plausible conjecture for an approximation of monH_∞(B_`_p).

Unfortunately we were instantly able to reveal a flaw. If we modify the setsBp and Bp a little we can bypass this shortcoming. We conjecture forp≥2

B_p⊂monH_∞(B_`_p)⊂B_p withBpand Bpdefined by

B_p:=

x∈B_`_∞

lim sup

1 (logn)^p+2^p

k=1

x^∗_k

q<1

B_p:=

x∈B`∞

lim sup

1 (logn)^p+2^p

k=1

x^∗_k

q≤1

where ¹_q := ¹₂+¹_p.

For`1,Lempert[45] proved that the domain of monomial convergence forH∞(B`₁) coincides with the whole domain of holomorphy, i.e. monH_∞(B_`₁) = B_`₁. It is unclear if there exist otherBanach sequence spacesX for which this is the case. Prove or disprove:

monH_∞(BX) = BX ⇒ X =`1. This is equivalent to the implication

infn K(BX_n;N⁽0^N⁾)>0 ⇒ X=`1.

By Proposition 9.4, this is on the other hand equivalent to the implication

∃c≥1∀m, n∈N:χmon P(^mXn)

≤c^m ⇒ X =`1.

132

RegardingDirichlet series: Theorem 6.10 in the casep=∞reads in the setting of Dirichletseries as: For anyDirichletpolynomial D=P

n≤xann^−s∈ H_∞ X

n≤x

|an| ≤√ xexp

−^√¹

2+o(1)p

logxlog logx sup

t∈R

n≤x

ann^−it .

Here a reasonable question might be: Can we obtain an analogous result for Dirich-letseries inHp?

In Chapter 8, we investigated`p–multipliers of sets ofDirichletseries and obtained results for multiplicative`₁–multipliers. It remains open to investigate on one hand non-multiplicative`1–multipliers and on the other hand to identify `p–multipliers at all.

We furthermore defined the spacesH∞[`_p] as the image ofH_∞(B`p) under theBohr transform and proved conditions on multiplicative sequences (bn)nto be`1–multipliers.

However, we didn’t investigate how theDirichletseries inH∞[`p] andH^m_∞[`p] look like.

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Im Dokument Unconditionality in spaces of holomorphic functions (Seite 123-153)

II. Domains of convergence and other viewpoints 103

7.3. Holomorphic functions

Chapter 8.

Interfaces with D IRICHLET series

8.1. The B OHR transform — connecting D IRICHLET

series and power series

8.2. Multipliers on spaces of D IRICHLET series

Chapter 9.

B OHR radii

Chapter 10.

Outlook — where to continue

Bibliography

B ^OHR radii