CYCLICITY OF NON VANISHING FUNCTIONS IN THE POLYDISC AND IN THE BALL

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CYCLICITY OF NON VANISHING FUNCTIONS IN THE POLYDISC AND IN THE BALL

Eric Amar, Pascal J. Thomas

To cite this version:

Eric Amar, Pascal J. Thomas. CYCLICITY OF NON VANISHING FUNCTIONS IN THE POLY- DISC AND IN THE BALL. Algebra i Analiz, Nauka, Leningradskoe otdelenie, 2019, 31 (5), pp.1-23.

�hal-01453468v2�

CYCLICITY OF NON VANISHING FUNCTIONS IN THE POLYDISC AND IN THE BALL

ERIC AMAR AND PASCAL J. THOMAS

Abstract. We use a special version of the Corona Theorem in several variables, valid when all but one of the data functions are smooth, to generalize to the polydisc and to the ball results ob- tained by El Fallah, Kellay and Seip about cyclicity of non vanish- ing bounded holomorphic functions in large enough Banach spaces of analytic functions determined either by weighted sums of powers of Taylor coefficients or by radially weighted integrals of powers of the modulus of the function.

1. Introduction

The Hardy space can be seen as a space of square integrable func- tions on the circle with vanishing Fourier coefficients for the negative integers, a space of holomorphic functions on the unit disk, or the space of complex valued series with square summable moduli, and the inter- action between those viewpoints has generated a long and rich history of works in harmonic analysis, complex function theory and operator theory.

The present work aims at generalizing one particular aspect of this to several complex variables: the study of cyclicity of some bounded holomorphic functions under the shift operator in large enough Banach spaces containing the Hardy space.

1.1. Definitions.

Definition 1. Let ω : N^d −→ (0,∞), where d ∈ N^∗, and p ≥ 1. We define the Banach space of power series in several variables

X_ω,p :=

(

f(z) := X

I∈N^d

a_Iz^I :kfk^p_X

ω,p := X

I∈N^d

|a_I| ω(I)

p

<∞ )

,

with the usual multiindex notation,z = (z₁, . . . , z_d)∈C^d,I = (i₁, . . . , i_d)∈ N^d, z^I :=zⁱ₁¹· · ·z_d^id.

We also write |I| :=i₁ +· · ·+i_d, I! :=i₁!· · ·i_d!. We say that ω is nondecreasing if for any I, J, ω(I+J)≥ω(J).

1

Recall that domains of convergence of power series are logarithmi- cally convex complete Reinhardt domains (for a definition and those terms and proofs, see e.g. [9], [8]). In what follows, we shall restrict our attention to the cases of the polydisc D^d :={z ∈C^d : max_1≤j≤d|z_j|<

1} and the unit ball B^d := {z ∈ C^d : P

1≤j≤d|zj|² < 1}. The letter Ω will stand for either one of those two domains, except in the more general Theorem 11.

Ifω(I) = 1 for anyI, then we obtain the Hardy spaceH²(D^d), which can also be described as the set of functions in the Nevanlinna class of the polydisc with boundary values (radial limits a.e.) on the torus (∂D)^d which are in L²((∂D)^d), and

kfk²_H2(D^d)= X

I∈N^d

|a_I|² = 1 (2π)^d

Z

(∂D)^d

|f|²dθ₁. . . dθ_d. The standard references for Hardy spaces on polydiscs is [11].

There is a Hardy space for B^d, which is most easily described as as the set of functions in the Nevanlinna class of the ball with boundary values (radial limits a.e.) on the sphere∂B^dwhich are inL²(∂B^d), and

kfk²_H2(B^d)= Z

∂B^d

|f|²dσ,

whereσ is the (2d−1)-real dimensional Lebesgue measure normalized so that σ(∂B^d) = 1. The standard reference for function theory on the unit ball is [12]. Lemma 12 gives a description of H²(B^d) in terms of the coefficients in the Taylor expansion.

Definition 2. We set ω₂^Dd(J) := 1, and ω₂^Bd(J) := 1

kz^Jk_H²_(B^d₎ =

(|J|+d−1)!

(d−1)!J! 1/2

.

We sometimes use the notation ω₂(J) (without superscript) when either of those quantities is meant.

We still have to understand in what sense a power series f can be understood as a function of z ∈ Ω. We will want to consider weights which satisfy the following relative monotonicity condition : there ex- ists a constant C_m ≥1 such that, for any I, J ∈N^d,

(1) Cmω(I+J)≥ω(J)ω^Ω₂(I).

One can check that ω=ω^B₂^d itself verifies condition (1).

When Ω = D^d, then ω^Ω₂(I) = 1 and if Cm = 1, we recover the usual monotonicity. We observe that for the polydisc, we can reduce ourselves to the case C_m = 1.

Lemma 3. Let Ω =D^d. If ω satisfies Condition (1), then Xω,p admits an equivalent norm given by the nondecreasing weight

ˆ

ω(I) := inf

J∈N^d

ω(I+J).

Proof. Since 0∈N^d and we have (1), 1 ≥ ^ω(I)_ω(I)^ˆ ≥C_m⁻¹, and ˆ

ω(I+K) = inf

J∈N^d

ω(I+K+J) = inf

J∈K+N^dω(I+J)≥ inf

J∈N^d

ω(I+J) = ˆω(I).

Since the new norm is equivalent to the original one, the problem is unchanged and there is no loss of generality in assuming that ω has been modified and made nondecreasing, and we shall do so henceforth.

Lemma 4. Let Ω =D^d or =B^d.

If ω verifies the relative monotonicity condition (1) and (2) logω^Ω₂(I)≤logω(I)≤logω^Ω₂(I) +o(|I|),

for any f ∈ X_ω,p, the series defining f converges on Ω, and the map X_ω,p 3 f 7→ f(z) is continuous with respect to the norm k · k_Xω,p. In particular, Xω,p can be seen as a subset of the space H(Ω) of holomor- phic functions on Ω.

Furthermore, there is no larger domain on which every f ∈X_ω,p has to be holomorphic.

This lemma will be proved in Section 2.

In the ball case, consider λ a probability measure on [0,1).

Definition 5. The radially weighted Bergman space associated to λis B=B^p(λ) = B^p(λ)(B^d)

:=

f ∈ H(B^d) :kfk^p_p :=

Z 1 0

Z

∂B^d

|f(rζ)|^pdσ(ζ)dλ(r)<∞

.

Typical examples are provided by dλ(r) = c_α(1−r²)^αr^2d−1dr, where α >−1 andc_αis an appropriate normalizing constant; they correspond to a weight c_α(1−P

1≤j≤d|z_j|²)^α, z ∈B^d.

Let λ be a probability measure on [0,1)^d, the elements of which are denoted r := (r1, . . . , rd). The torus (∂D)^d is endowed with its normalized Haar measure denoted by dθ.

Definition 6. The weighted Bergman space associated to λ is B=B^p(λ) = B^p(λ)(D^d)

:=

f ∈ H(D^d) :kfk^p_p :=

Z

[0,1)^d

Z

T^d

f(r₁e^iθ1, . . . , r_de^iθd)

pdθdλ(r)<∞

.

Let H^∞(Ω) stand for the set of bounded holomorphic functions on Ω. In each case, the conditions onλ ensure thatH^∞(Ω)⊂ B^p(λ).

Note that the norms of the monomials are given by moments of the measureλ. In the case where Ω =D^d,

kz^Ik^p_p = Z

[0,1)^d

r^pIdλ(r),

so that log(kz^Ik⁻¹_p ) is a concave function of I.

When p= 2, B²(λ) is a Hilbert space and the monomialsz^I form an orthogonal system. Notice that in X_ω,2, kz^Ik_ω,2 =ω(I)⁻¹, so that

B²(λ)(D^d) = X_ω,2 with ω(I) = Z

[0,1)^d

r^2Idλ(r) ^−1/2

.

In the case where Ω =B^d, kz^Ik^p_p =

Z 1 0

r^p|I|dλ(r) Z

∂B^d

|ζ^I|^pdσ(ζ)

.

When p = 2, since the surface measuredσ on ∂B^d desintegrates as an integral of Haar measures on tori, the monomials (z^J) again form an orthogonal system, and in this case

kz^Ik²₂ = Z 1

0

r^2|I|dλ(r)

ω₂^Bd(J)⁻², so that

B²(λ)(B^d) =X_ω,2 with ω(I) = Z 1

0

r^2|I|dλ(r) ^−1/2

ω₂^Bd(J).

In general, whenever we consider a spaceX, we define the correspond- ing weight by ω(J) := 1/

z^J X.

1.2. Main results. Let X be a Banach space as above, defined by power series or as a weighted Bergman space.

Definition 7. We say that a function f ∈ X is cyclic if for any g ∈ X, there exists a sequence of holomorphic polynomials (P_n) such that lim_n→∞kg−P_nfk_X = 0.

Note that using the word “cyclic” is a slight abuse of language, since for d ≥ 2 we are not iterating a single operator, but taking composi- tions of the multiplication operators by each of the coordinate functions z1, . . . , zd. It is, however, a straightforward generalization of the usual notion of cyclicity under the shift operator f(z)7→zf(z).

By Lemma 4 in the case of power series spaces, or by the mean value inequality in the case of Bergman spaces, the point evaluations are

continuous, therefore any cyclic f must verify that f(z) 6= 0 for any z ∈Ω.

Definition 8. (1) When Ω =D^d, for any k ∈N, let 1

˜ ω(k) :=

d

X

j=1

kz_j^kk_X =

d

X

j=1

kz^kejk_X,

where (e_j) stands for the elementary multiindices of N^d: e₁ = (1,0, . . . ,0),e₂ = (0,1,0, . . . ,0), etc, soke_j = (0, . . . ,0, k,0, . . . ,0), with k in the j-th place.

(2) When Ω = B^d, for any k ∈N, let 1

˜

ω(k) := X

J, |J|=k

p_J ω(J),

where p_J is the multinomial coefficient, p_J := ^|J|!_J!. When X =X_ω,p and Ω = D^d, notice that

d⁻¹ min

1≤j≤dω(ke_j)≤ω(k)˜ ≤ min

1≤j≤dω(ke_j).

Note that if ω satisfies (2) then log ˜ω(k) =o(k), but the converse does not hold when d >1.

Here are two interesting special cases of our results.

Theorem 9. Let Ω :=D^d or B^d.

Suppose that lim_k→∞ω(k) =˜ ∞, and ω satisfies (1) and (2), and that

(3) X

k≥1

log ˜ω(k) k

2

=∞.

Let U ∈H^∞(Ω), verifying U(z)6= 0 for any z∈Ω.

• (i) If d≥1, p≥1 and X =B^p(λ), then U is cyclic in X.

• (ii) IfΩ = D^d and X =Xω,2, then U is cyclic in X.

When we demand a growth condition of a slightly stronger nature on ˜ω, we can expand the range of spaces where the result applies.

Theorem 10. LetX=X_ω,p, withp≥2, orX =B^p(λ). Iflimk→∞ω(k) =˜

∞, and ω satisfies (1), (2) and

(4) lim sup

k

log ˜ω(k)

√k =∞, then any zero-free U ∈H^∞(Ω) is cyclic in X.

1.3. Previous results. Many results have been proved for the case d = 1, and even more for p= 2. In one dimension, Ω =D and ω = ˜ω of course. When furthermorep= 2,X_ω,2 has a norm equivalent to that of a Bergman space if and only if logω(n) is a concave function of n [3, Theorem A.2 and Proposition 4.1].

In his seminal monograph [10], N. K. Nikolski proved that ifωis non- decreasing, limk→∞ω(k) = ∞, logω(k) = o(k), logω(n) is a concave function of n and

(5) X

k≥1

log ˜ω(k) k^3/2 =∞, then any zero-free f ∈H^∞(D) is cyclic in X_ω,2.

Our main inspiration comes from [5], where O. El Fallah, K. Kellay and K. Seip show, still ford= 1 andp= 2, that (3), with no condition of concavity, is enough to imply cyclicity of any nonvanishing bounded function. Even though (3) is a stronger condition than (5), the concav- ity condition means that there exist weights to which the new result applies while Nikolski’s cannot [5, Remark 2].

The novelty in the present work is of course that we have several variables, and exponentsp6= 2. We also notice that it is not necessary to make use of the inner-outer factorization: the much easier Harnack inequality suffices.

1.4. A Corona-like Theorem. As in [5], our main tool is a ver- sion of the Corona Theorem. In full generality, this is still a vexingly open question in several variables, be it in the ball or the polydisc.

However, following an earlier result of Cegrell [4], a simpler proof [1]

gives a Corona-type result in the special case where most of the given generating functions are smooth. That result is enough to yield the required estimates in this instance. For Ω a bounded domain inC^d, let A¹(Ω) :=H(Ω)∩ C¹(Ω).

Theorem 11. Let Ω be a bounded pseudoconvex domain in C^d, such that the equation ∂u¯ = ω, 1 ≤ q ≤ n, admits a solution u = Sqω ∈ L^∞_(0,q−1)(Ω) when ∂ω¯ = 0, ω ∈L^∞_(0,q)(Ω), with the bounds :

kuk_∞ ≤E_qkωk_∞.

There exists a constantC =C(d,Ω)such that if N ∈N, N ≥2, and if f_j ∈A¹(Ω), 1≤j ≤N −1, f_N ∈H^∞(Ω), verify

sup

z∈Ω

1≤j≤Nmax |f_j(z)| ≤1, inf

z∈Ω N

X

j=1

|f_j(z)| ≥δ >0,

then there existg1, . . . , gN ∈H^∞(Ω) such thatPN

j=1fj(z)gj(z) = 1and for 1≤j ≤d,

1≤j≤Nmax kg_jk∞ ≤C(d,Ω)N^4d+2max1≤j≤N−1k∇fjk^d_∞ δ^2d+1 .

Note that the polydisc and the ball verify the hypotheses of the theorem.

1.5. Structure of the paper. First we clarify the easy relationship between weights and domains of convergence in Section 2. Then we gather some preliminary results and a first reduction of the problem in Section 3. Theorem 11 is proved in Section 6, and used in the proofs of the two main theorems. The relatively easy proof of Theorem 10 is given in Section 4. Theorem 9 will follow from a more general and more technical result, Theorem 19, which is stated and proved in Section 5.

2. Domains of convergence

Lemma 12. Let f be holomorphic on the unit ball B^d, represented by the Taylor expansion f(z) = P

Ja_Jz^J. Then f ∈H²(B^d) if and only if X

J∈N^d

|aJ| ω₂^Bd(J)

2

<∞, where (ω₂^Bd(J))⁻² = (d−1)!J!

(|J|+d−1)!. Proof. The surface measure dσ on ∂B^d desintegrates as an integral of Haar measures on tori, so the monomials (z^J) form an orthogonal system in H²(B^d), which is a basis since the polynomials are dense in the space. Thenkfk²_H2(B^d) =P

J|a_J|²kz^Jk²_H2(B^d). The explicit value of kz^Jk_H²_(B^d₎= (ω^B₂^d(J))⁻¹ (Definition 2) can be found in [12, p. 12].

As an immediate consequence of this Lemma and of the remarks before Definition 2, if X_ω,p is as in Definition 1 and p ≥ 2, and if for allJ,ω₂^Ω(J)≤ω(J), thenH²(Ω) ⊂X_ω,p.

Definition 13. Forz ∈C^d, let|z|_Dd := max_1≤j≤d|z_j|,|z|²

B^d :=P

1≤j≤d|z_j|². In each case, Ω ={z :|z|Ω <1}.

Proof of Lemma 4. The last statement follows from the fact that (1) implies that H²(Ω)⊂X_ω,p, as in the remark before the Definition.

To prove the convergence of the series, write f(z) = P

Ja_Jz^J and take a point z such that|z|_Ω =ρ <1. Since X_ω,p ⊂Xω,∞,|a_J| ω(J) and it will be enough to prove the convergence of P

Jω(J)|z^J|.

In the case where Ω =D^d, then

log(ω(J)|z^J|)≤ |J|logρ+o(|J|)≤ −η|J|

for some η > 0 when |J| is large enough, so the general term is domi- nated by the general term of a convergent geometric (multi)series.

In the case where Ω =B^d, for anyk ∈N,

|z|^2k_Ω = X

J:|J|=k

|J|!

J! |z^2J|.

First consider only sums of terms with all powers even:

X

J:|J|=k

ω(2J)|z^2J| ≤ sup

J:|J|=k

ω(2J)J!

|J|!

X

J:|J|=k

|J|!

J! |z^2J|

≤ sup

J:|J|=k

ω(2J)J!

|J|!

ρ^2k,

so using (2), we need to estimate ^ω2(2J)_(|J|!)^2(J!)2 ². Stirling’s formula implies that for any n ∈N,

log(n!) = n(logn−1) +o(n), so we have, for |J|=k,

log

ω₂(2J)²(J!)² (|J|!)²

= log

(2k+d−1)!(J!)² (d−1)!(2J)!(k!)²

=

= (2k+d−1) (log(2k+d−1)−1)−2k(logk−1) + 2

d

X

i=1

j_i(logj_i−1)−

d

X

i=1

2j_i(log(2j_i)−1) +o(k)

= 2klog 2−2

d

X

i=1

jilog 2 + 2k

log(k+d−1

2 )−logk

+o(k)

= 0 +O(1) +o(k) =o(k).

This proves that P

J:|J|=kω(2J)|z^2J| is dominated by the general term of a convergent geometric series for k large enough.

Now consider a generalJ such that|J|=k: thenJ = 2J⁰+K, with j_i⁰ = 2 [ji/2], andK ∈ {0,1}^d. Let 2J⁰⁰ :=J +K. Condition (1) shows thatω(J)ω(2J⁰)ω(2J⁰⁰),|z^J| ≤ |z^2J0|, and eachJ⁰ corresponds to at most 2^d different multi-indicesJ. SoP

J:|J|=kω(J)|z^J|is dominated by the general term of a convergent geometric series fork large enough.

Continuity of the evaluation map follows, for instance, from the Dom- inated Convergence Theorem applied to the series.

Lemma 14. Suppose that H^∞(Ω) is a multiplier space for X; or that X =X_ω,p, with p ≥2 and ω verifying (1). Let g ∈H^∞(Ω), K ∈N^d.

Then

kz^Kgk_Xω,p ≤ C_m

ω(K)kgk_H^∞_(Ω).

Observe that in the special case K = 0, X =X_ω,p, we get back the fact that H²(Ω)⊂X_ω,p.

Proof. Under the first assumption, we immediately have kz^Kgk^p_X ≤ kz^Kk^p_Xω,pkgk_H^∞_(Ω) = 1

ω(K)kgk_H^∞_(Ω).

Under the second assumption, by scaling we may assume 1 =kgk_H²_(Ω) ≤ kgk_H^∞_(Ω). Let g(z) =P

Ja_Jz^J. Then sup_{J ω}^|aJ|2

2(J)² ≤1. Then kz^Kgk^p_Xω,p =X

J

|a_J|^p ω(J +K)^p ≤

sup

J

ω₂(J) ω(J+K)

p

X

J

|a_J|^p ω₂(J)^p

≤ C_m^p ω(K)^p

X

J

|a_J|²

ω₂(J)² = C_m^p

ω(K)^p ≤ C_m^p

ω(K)^pkgk^p_H∞(Ω). 3. Auxiliary results

3.1. Multiplier property.

Definition 15. We shall say that H^∞(Ω) is a multiplier algebra for X if there exists Cm >0 such that

∀f ∈X,∀g ∈H^∞(Ω), gf ∈X and kgfk_X ≤C_mkgk_∞kfk_X. Notice that, since constants are inX, this implies thatH^∞(Ω)⊂X.

It is immediate thatH^∞(Ω) is a multiplier algebra for eachB^p(λ)(Ω), with C_m = 1. In the case of X_ω,p, writing ω^Ω_∞(I) := kz^Ik⁻¹_L∞(Ω), an obvious necessary condition is that

(6) C_mω(I+J)≥ω^Ω_∞(I)ω(J),

but sufficient conditions are not so easy to state in general.

Observe that (6) is very similar to (1). In fact,ω_∞^Dd(I) =ω₂^Dd(I) = 1 for all I, and one can show that

ω₂^Bd(I)≥ω^B_∞^d(I)≥ω₂^Bd(I)−O(log|I|),

by an appropriate minoration of |z^I| on a strip of ∂B^d of width com- parable to |I| around its maximum modulus set (we omit the details;

this can provide an alternate proof of Lemma 4 without recourse to Stirling’s formula).

3.2. Some tools. Our first technical tool is a bound from below for the modulus of a zero-free bounded holomorphic function.

Forz ∈Ω, z^∗ :=z/|z|_Ω ∈∂Ω, where |z|_Ω is as in Definition 13.

Lemma 16. LetU be a zero-free holomorphic function on Ωsuch that kUk_∞≤1, and z ∈Ω. Let c² := log_|U(0)|¹ . Supposek ≥4c². Then, for Ω =D^d,

|U(z)|+|z₁|^k+· · ·+|z_d|^k ≥e^−2c

√ k, and for Ω =B^d,

|U(z)|+ X

|J|=k

|f_J(z)| ≥e^−2c

√ 2k,

where f_J(z) :=p_Jz^2J = ^|J|!_J!z^2J.

Proof. The conclusion is obvious ifz = 0. If not, define a holomorphic function on D byf_z^∗(ζ) :=U(ζz^∗). Then

• kf_z^∗k_∞≤1;

• f_z^∗(0) =U(0);

• ∀ζ ∈D, f_z^∗(ζ)6= 0;

• f_z^∗(|z|_Ω) =U(z).

The Harnack inequality applied to the positive harmonic function log |f_z^∗|⁻¹ shows that

|f_z^∗(ζ)| ≥exp

−1 +|ζ|

1− |ζ|log 1

|U(0)|

≥exp

− 2

1− |ζ|log 1

|U(0)|

.

The computation implicit at the beginning of the proof of [5, Lemma 3] shows that inf_D|f_z^∗(ζ)|+|ζ|^k ≥e^−2c

√

k as soon as k ≥4c²; applying this to ζ =|z|_Ω, we find

|U(z)|+|z|^k_Ω ≥f_z^∗(|z|_Ω) +|z|^k_Ω ≥e^−2c

√ k.

In the case where Ω =D^d, this yields

|U(z)|+|z₁|^k+· · ·+|z_d|^k ≥ |U(z)|+ ( max

1≤j≤d|z_j|)^k≥e^−2c

√ k.

In the case where Ω =B^d, substituting 2k for k, we obtain

|U(z)|+ (|z1|²+· · ·+|zd|²)^k=|U(z)|+ X

|J|=k

|fJ(z)| ≥e^−2c

√2k

.

Lemma 17. If X =Xω,p or B^p(λ)from Definitions 1 or 6 or 5 respec- tively, then the space of polynomials C[Z] :=C[Z₁, . . . , Z_d] is dense in X.

Proof. By construction, the polynomials are dense inXω,p. Forf ∈ B^p(λ)(D^d), r= (r₁, . . . , r_d)∈[0,∞)^d, let

m_r,p(f) :=

Z

T^d

f(r₁e^iθ1, . . . , r_de^iθd)

pdθ

denote the mean of |f|^p on the torus T(r) of multiradius r. Since

|f|^p is plurisubharmonic, this is an increasing function with respect to each component of r. In particular, if we set for any γ ∈ (0,1), f_γ(z) := f(γz), m_r,p(f_γ)≤m_r,p(f) for each r.

We claim that limγ→1kf −f_γkB^p(λ) = 0. Indeed, kf −f_γkB^p(λ) = R

[0,1)^dF_γ(r)dλ(r), where F_γ(r) :=

Z

T^d

f(r₁e^iθ1, . . . , r_de^iθd)−f_γ(r₁e^iθ1, . . . , r_de^iθd)

pdθ.

Since |f−f_γ|^p ≤C_p(|f|^p+|f_γ|^p),

F_γ(r)≤C_p(m_r,p(f) +m_r,p(f_γ))≤2C_pm_r,p(f)∈L¹(dλ).

Sincefγ →f uniformly on the torusT(r) for eachrasγ →1,Fγ(r)→ 0 for each r, and we can apply Lebesgue’s Dominated Convergence theorem.

For each γ ∈(0,1), f_γ is holomorphic on a larger polydisc, so can be uniformly approximated by truncating its Taylor series.

When Ω =B^d, we can perform an analogous (and simpler) argument.

3.3. First reduction. We begin by showing that it is enough to obtain a relaxed version of the conclusion.

Lemma 18. Let U ∈H^∞(Ω) be a non-vanishing function.

If either:

• (i) H^∞(Ω) is a multiplier algebra for X,

• or (ii) X =X_ω,p, p≥2 and (1) is satisfied, and if there exists a sequence (f_n)⊂H^∞(Ω) such that

n→∞lim k1−f_nUk_X = 0, then U is cyclic in X.

Proof. By Lemma 17, it is enough to show that we can approximate any polynomial P.

Let us show that it is enough to prove that for anyε >0, there exists Q∈C[Z] such thatk1−QUk_X ≤ε.

LetP(z) :=P

|J|≤Na_Jz^J, then

kP −P QUk_X =kP(1−QU)k_X ≤ kPk_∞k1−QUk_X

in the case of assumption (i), and kP(1−QU)k_X ≤ X

|J|≤N

|a_J|kz^J(1−QU)k_X ≤C_m



 X

|J|≤N

|a_J| ω₂^Ω(J)



k1−QUk_X,

in the case of assumption (ii), and each upper bound can be made arbitrarily small by choosing Q.

In the case of assumption (i), let us then show that the constant function 1 can be approximated. Let ε > 0. Take f ∈ H^∞(D^d) such that k1−f Uk_X < ε/2. By Lemma 17, we can choose Q∈ C[Z] such that kf −QkX ≤ _C ¹

mkUk∞

ε

2, whereCm is as in Definition 15. Then k1−QUk_X ≤ k1−f Uk_X+kU(f−Q)k_X < ε

2+C_mkUk∞kf−Qk_X ≤ε.

In the case of assumption (ii), again take f so that k1−f Uk_ω,p is small, then because H²(Ω)⊂X_ω,p,

kf U −QUk_ω,p ≤Ckf U −QUk_H² ≤CkUk∞kf −Qk_H²,

and this last quantity can be made arbitrarily small by taking Q a

Taylor expansion of f for instance.

4. Proof of Theorem 10 Proof of Theorem 10.

Observe that if X = B^p(λ), then H^∞(Ω) is a multiplier algebra, so Lemma 14 always applies here.

Case 1: Ω =D^d.

Letc² :=−log |U(0)|and B >2c(2d+ 1). By the hypothesis of the theorem, there exists a strictly increasing sequence (nk)k≥1 such that for all k, log ˜ω(nk)≥B√

nk.

By Lemma 16 and Theorem 11, we get g_j ∈ H^∞(D^d), for j = 1, . . . , d+ 1, such that

(7) g_d+1U+g₁z₁^nk +· · ·+g_dz_d^nk = 1, and

(8) ∀j = 1, . . . , d+ 1, kg_jk_∞≤C(d)n^d_ke^{2c(2d+1)√nk}. Set f_k :=g_d+1, we get, using Lemma 14,

(9)

k1−f_kUk_X ≤

d

X

j=1

g_jz_j^nk

X ≤C_m

d

X

j=1

kg_jk_∞

ω(n_ke_j) ≤C_mC(d)n^d_ke^{2c(2d+1)√nk}

˜

ω(n_k) .

By the choice of B, this tends to 0 as k → ∞. It only remains to apply lemma 18 to conclude.

Case 2: Ω =B^d.

Letc, (n_k) be as above and B >2c√

2(2d+ 1).

By Theorem 11, we will getg₀, g_J ∈H^∞(D^d), for|J|=n_k, such that

(10) g₀U+ X

|J|=n_k

g_Jf_J ≡1, where f_J is as in Lemma 16.

We need to estimate the size of the g_J, g₀. First P

|J|=n_k|f_J(z)|=|z|²ⁿ_Ω^k <1.

The number of terms in the Bezout equation is N =N(d, k) = #

J ∈N^d:|J|=nk = (nk+d−1)!

n_k!(d−1)! ≤n^d−1_k . We also needk∇f_Jk_∞. We have

∂

∂z_ifJ =pJ2ji

z^2J

z_i ⇒ ∇fJ =pJz^2J(2j₁

z₁ , ...,2j_d

z_d ) = 2fJ(z)(j₁ z₁, ..., j_d

z_d).

If we set ˜J := (max(0,2j₁−1), ...,max(0,2j_d−1)) and

˜

z_i :=z₁· · ·zi−1zˆ_iz_i+1· · ·z_d, where ˆz_i is omitted, then

∇f_J(z) = 2p_Jz^J˜(j₁z˜₁, ..., j_dz˜_d).

So we get, because |˜z_i| ≤1 in the ball,

|∇f_J(z)| ≤2p_J z^J˜

d

X

i=1

j_i|˜z_i| ≤2p_J z^J˜

|J|.

But if we write J⁰ := ((j₁ −1)₊, . . . ,(j_d−1)₊), then |z^J˜| ≤ |z^2J0| and P

|J⁰|=n_k−dp_J⁰|z^2J0|<1. Furthermore,

p_J ≤ n_k(n_k−1)· · ·(n_k−d+ 1) j1· · ·jd

p_J⁰ ≤n^d_kp_J⁰. All together then, k∇f_J(z)k∞≤C(d)n^d+1_k .

By Lemma 16 (in the case of the ball), δ = inf

z∈B^d



|U(z)|+ X

|J|=n_k

|fJ(z)|



≥e^−2c

√2nk.

Gathering the estimates, we get

(11) kg_Jk_∞≤C(d)N(d, k)^4d+2e^2c(2d+1)

√2nk ≤C(d)n^5d_k ²e^2c(2d+1)

√2nk.

Then let fk =g0 (at thenk step) (12) k1−f_kUk_X ≤ X

|J|=n_k

kg_Jf_Jk_X ≤C_m X

|J|=n_k

p_Jkz^2Jk_Xkg_Jk∞

≤C_mC(d)n^5d_k ²e^2c(2d+1)

√2nk

˜

ω(n_k) ,

and we finish as before.

5. Proof of Theorem 9 5.1. Main intermediate result.

Theorem 19. Let X be a Banach space as in Definitions 1, 5 or 6.

Suppose that H^∞(D^d) is a multiplier algebra for X. Suppose also that lim_k→∞ω(k) =˜ ∞, that log ˜ω(k) = o(k), and that conditions (1), (2) and (3) hold.

Then any U ∈ H^∞(D^d), verifying U(z)6= 0 for any z ∈D^d is cyclic in X.

Proof. Now we need to distinguish two cases according to the growth of ω(k).

Case 1: sup_k ^{log ˜√ω(k)}

k =∞.

Then Theorem 10 applies.

Case 2: sup_k^{log ˜√ω(k)}

k = B < ∞. To deal with this more delicate case, we shall need the full power of the proof scheme in [5]. Since our Corona-like estimates are slightly different from those in dimension 1, we first need a refined version of [5, Lemma 1].

Lemma 20. Letω˜ be as in Theorem 19. Let C₀ >0. Then there exists a strictly increasing sequence (n_k)k≥1 such that

X

k≥1

(log ˜ω(n_k))² n_k =∞,

and for all k, log ˜ω(n_k+1)≥2 log ˜ω(n_k) and log ˜ω(n_k)≥C₀logn_k. The last condition is the only novelty with respect to [5, Lemma 1].

Proof. First notice that there exists an infinite set E ⊂ N^∗ such that for all n ∈ E, log ˜ω(n) ≥ C₀logn. Indeed, if not, for n large enough, we would have

log ˜ω(n)≤C₀logn≤n^1/4, and (3) would be violated.

Now let n0 = 1 and ifnj is defined, let

n⁰_j+1 := min{n > nj : log ˜ω(n)≥2 log ˜ω(nj)},

n_j+1 := min{n > n_j, n∈E : log ˜ω(n)≥2 log ˜ω(n_j)}. Obviously, n_j < n⁰_j+1 ≤n_j+1. We claim that

S:=X

j≥0 nj−1

X

k=n⁰_j

log ˜ω(k) k

2

<∞.

Accepting the claim, the proof finishes as in [5]:

X

k≥1

log ˜ω(k) k

2

≤S+X

j≥0 n⁰_j+1−1

X

k=nj

log ˜ω(k) k

2

≤S+X

j≥0

4 (log ˜ω(n_j))²

n⁰_j+1−1

X

k=nj

1

k² ≤S+ 4X

j≥0

(log ˜ω(n_j))² n_j−1 , so the last sum must diverge.

We now prove the claim. If n⁰_j ≤k < n_j, then n /∈ E, so for j large enough and n⁰_j ≤k < n_j, log ˜ω(k)≤k^1/4, thus

(13)

nj−1

X

k=n⁰_j

log ˜ω(k) k

2

≤ X

k≥n⁰_j

1

k^3/2 ≤ 2 pn⁰_j−1.

The definition of n_j implies that ˜ω(n_j) ≥ C₀2^j, and n⁰_j+1 ∈/ E (if it is distinct from n_j+1) so

C₀logn⁰_j+1 >log ˜ω(n⁰_j+1)≥2 log ˜ω(n_j)≥C₀2^j,

and the series with general term the last expression in (13) must con-

verge.

We follow the proof of [5, Theorem 1], with a couple of wrinkles.

ChooseA := max(2,logC(d)) whereC(d) is the constant in (8) when Ω =D^d(resp. (11) when Ω =B^d). Then forc² =−log|U(0)|as above, letC₁² := (8√

2(2d+ 1)Ac)²+B². We choose C₀ ≥C₁/c when Ω =D^d (resp. C₀ ≥ ^5d2C1

2√

2(2d+1)c when Ω = B^d), and define the sequence (n_j) as in Lemma 20 above. For any given j₀ ∈N, let α²_j := (^{log ˜ω(nj}0+j))²

nj0+j , N := min

( M :

M

X

j=1

α²_j ≥(8√

2(2d+ 1)Ac)² )

,

λ_j :=α_j PN

i=1α²_i−1/2

.

Notice that for anyj, α_j ≤B by the hypothesis of Case 2, and that

N

X

i=1

α_i² ≤

N−1

X

i=1

α²_i +α²_N ≤(8√

2(2d+ 1)Ac)²+B² =C₁²,

so that λ_j ≥α_j/C₁. Clearly, λ_j ≤α_j/(8√

2(2d+ 1)Ac).

We write U_j := U^λ2j, so that U = QN

j=1U_j. As above, choose f_j :=

g_d+1 satisfying (7) and (8), but with U_j instead of U andn_j0+j instead of n_k. The quantity c must then be replaced by cλ_j.

When Ω =D^d, the bound (8) can be rewritten (14) kfjk∞ ≤exp 2c(2d+ 1)λj

√nj0+j+dlognj0+j + logC(d) .

Notice that cλ_j√

n_j0+j ≥ c

C₁ log ˜ω(n_j0+j)≥ cC₀

C₁ logn_j0+j ≥logn_j0+j, by our choice of C0, so that

(15) kf_jk∞ ≤expA 2c(2d+ 1)λ_j√

n_j0+j+ 1 .

We finish as in [5]. Let f :=QN

j=1f_j. Since

1−f U = 1−

N

Y

j=1

f_jU_j =

N

X

k=1

(1−U_kf_k)

k−1

Y

j=1

f_jU_j,

k1−f Uk_X ≤C_m

N

X

k=1

k1−U_kf_kk_X

k−1

Y

j=1

kf_jU_jk∞,

which by (9) becomes

(16) ≤C_m

N

X

k=1

C(d)n^d_j

0+ke^{2c(2d+1)√nj0+k}

˜

ω(n_j0+k)

k−1

Y

j=1

kf_jk_∞

≤C_m

N

X

k=1

1

˜

ω(n_j0+k)exp A

k

X

j=1

(2c(2d+ 1)λ_j√

n_j0+j + 1)

! ,

and using the growth of log ˜ω(nj) obtained in Lemma 20,

≤Cm N

X

k=1

1

˜

ω(n_j0+k)exp Ak+

k

X

j=1

1

4log ˜ω(nj0+j)

!

≤Cm N

X

k=1

exp

Ak− 1

2log ˜ω(nj0+k)

.

Now choosej₀ such that log ˜ω(n_j0)≥A, the sum above has terms with better than geometric decrease, so is bounded by ˜ω(n_j0+1)^−1/2, which can be made arbitrarily small by choosing j₀ large enough.

When Ω =B^d, we need to make the changes indicated at the begin- ning of the argument, and replace the bound (14) by the following:

kf_jk∞≤exp 2c√

2(2d+ 1)λ_j√

n_j0+j + 5d²logn_j0+j+ logC(d) .

Then the choice (for Ω =B^d) ofC₀implies that 2c√

2(2d+1)λ_j√

n_j0+j ≥ 5d²logn_j0+j, and this leads again to (15). In the succession of majora- tions that follow, (16) becomes

≤C_m

N

X

k=1

C(d)n^5d_j ²

0+ke^2c

√2(2d+1)√ n_j0+k

˜

ω(nj0+k)

k−1

Y

j=1

kf_jk∞

≤C_m

N

X

k=1

1

˜

ω(n_j0+k)exp A

k

X

j=1

(2c(2d+ 1)λ_j√

n_j0+j + 1)

! ,

and the proof concludes in the same way.

5.2. Proof of Theorem 9. We now obtain cyclicity results as soon as we can prove that H^∞(D^d) is a multiplier algebra on the space X. As remarked after Definition 15, this is always the case when X =B^p(λ).

So we obtain Theorem 9 (i).

When X = X_ω,2 and ω is relatively nondecreasing, then Lemma 3 reduces us to the nondecreasing case, where the multiplication opera- tors by each z_j are commuting contractions on a Hilbert space. Von Neumann’s inequality was generalized by Ando in the case of two con- tractions, and to an arbitrary number of weighted shifts by Michael Hartz [6]: this is precisely our situation. It implies that for any poly- nomial f, and thus for anyf ∈H^∞(D^d), kf gk_X ≤ kfk_∞kgk_X. So we obtain Theorem 9 (ii).

6. Proof of the Corona theorem with smooth data We begin by constructing a partition of unity which exploits the smoothness of the data.

Because of the corona hypothesis, and f_j is continuous up to the boundary of Ω, for 1≤j ≤N −1, we have that g(z) :=PN−1

j=1 |f_j(z)|

is continuous in Ω, and even Lipschitz with a constant controlled by max1≤j≤N−1k∇fjk∞.

Set

U_N⁰ :={z ∈Ω :¯ g(z)< N −1

4N δ}and U_N :={z ∈Ω :¯ g(z)< N−1 2N δ}, and

U_j :={z ∈Ω :¯ |f_j|> δ

5N}and U_j⁰ :={z∈Ω :¯ |f_j|> δ 4N}.

Then U_j⁰ bU_j,1≤j ≤N.

Lemma 21. There exist C₁ >0 and χ_j ∈ C_c^∞(U_j), j = 1, ..., N, such that for z ∈Ω, 0≤χ_j ≤1, PN

j=1χ_j(z) = 1, and

χ_j f_j

≤ C₁N

δ , k∇χ_jk_∞≤ C₁N²

δ max

1≤i≤N−1k∇f_ik_∞, j = 1, . . . , N,

1≤j≤Nmax sup

z∈Ω

|∇χj(z)|

|f_j(z)| ≤C₁N³

δ² max

1≤i≤N−1k∇f_ik∞, where C₁ is an absolute constant.

Proof. We can construct a functionψN ∈ C_c^∞(UN) such that 0 ≤ψN ≤ 1 and ψ_N ≡ 1 on U_N⁰ , with k∇ψ_Nk∞ ≤ ^C_δ, for instance by composing

|g| with an appropriate smooth one-variable function.

We have

O := ¯U_N⁰ ∪

N−1

[

j=1

U_j⁰ ⊃Ω,¯ because for z /∈SN−1

j=1 U_j⁰, then

∀j = 1, ..., N −1, |f_j(z)| ≤ δ 4N ⇒

N−1

X

j=1

|f_j(z)| ≤ N −1

4N δ⇒z ∈U¯_N⁰ . Now we construct a partition of unity{χj}j=1,...,N subordinated to{Uj} in the usual way: we take a nonnegative functionψ_j ∈ C_c^∞(U_j) such that ψ_j ≤ 1 everywhere and ψ_j ≡ 1 on U_j⁰, with k∇ψ_jk∞ ≤ C^N_δk∇f_jk∞. We set

χ_j := ψ_j PN

k=1ψ_k.

Since PN

k=1ψk ≥ 1, we have 0≤ χj ≤ 1, χj ∈ C_c^∞(Uj) and χ1+· · ·+ χ_N = 1 on ¯Ω and

k∇χ_jk∞≤CN²

δ max

1≤i≤N−1k∇f_ik∞. This yields a partition of unity such that^χ_f^j

j ∈ C^∞( ¯Ω) for 1≤j ≤N and for j ≤N −1,

χj

fj

≤ ^5N_δ , because supp χ_j ⊂U_j, where |f_j| > _5N^δ and χ_j ≤1.

For j = N on the other hand, we have suppχ_N ⊂ U_N and, by the corona hypothesis,

z ∈U_N ⇒ |f_N(z)| ≥δ−g(z) =δ−N −1

2N δ= N + 1 2N δ hence

χN

f_N

≤ 2N

(N + 1)δ ≤ 5N δ . An analogous reasoning yields the bound on ^|∇χ_|f ^j|

j| , 1≤j ≤N.

Proof of Theorem 11. We shall now go through the Koszul complex method, introduced in this context by H¨ormander [7], to obtain the explicit bounds we need. We follow the notations of [2].

Let∧^k(C^N) be the exterior algebra onC^N,lete_j, j = 1, ..., N,be the canonical basis of ∧¹(C^N), and e_α :=e_α1 ∧ · · · ∧e_αk, α_j ∈ {1, . . . , N}, the associated basis of ∧^k(C^N).

Let L^k_r be the space of bounded and infinitely differentiable differ- ential forms in Ω of type (0, r) with values in ∧^k(C^N). The norm on these spaces is defined to be the maximum of the uniform norms of the coefficients.

We define two linear operators on L^k_r.

∀ω∈L^k_r, R_f(ω) :=ω∧

N

X

j=1

χ_j

f_je_j ∈L^k+1_r . We see that kR_fωk ≤C_fkωk, with

(17) C_f :=N sup

1≤j≤N,z∈Ω

χ_j(z) f_j(z) .

The operator d_f : L^k+1_r −→ L^k_r is defined by induction and linearity.

For ω ∈ L⁰_r, d_fω = 0. To define the operator on L¹_r, set d_f(e_j) := f_j and extend by linearity.

To define d_f onL^k+1_r , fore_α∈ ∧^k(C^N), 1≤j ≤d, set d_f(e_α∧e_j) :=f_je_α−d_f(e_α)∧e_j ∈L^k_r.