Estimates in corona theorems for some subalgebras of H

c 2007 by Institut Mittag-Leffler. All rights reserved

Estimates in corona theorems for some subalgebras of H

Amol Sasane and Sergei Treil

Abstract.Ifnis a non-negative integer, then denote by∂⁻ⁿH^∞the space of all complex- valued functionsfdefined onDsuch thatf,f⁽¹⁾,f⁽²⁾,...,f⁽ⁿ⁾belong toH^∞, with the norm

f= n j=0

1

j!f^(j)∞.

We prove bounds on the solution in the corona problem for∂⁻ⁿH^∞. As corollaries, we obtain estimates in the corona theorem also for some other subalgebras of the Hardy spaceH^∞.

Notation We use the following notation:

:= equal by definition;

C the complex plane;

D the unit disk,D:={z∈C:|z|<1}; D the closed unit disk,D:={z∈C:|z|<1}; T the unit circle,T:=∂D={z∈C:|z|=1}; dm normalized Lebesgue measure onT,m(T)=1;

∂, ∂ derivatives with respect toz andz, respectively: ∂:=¹₂(∂/∂x+i∂/∂y) and∂:=¹₂(∂/∂x−i∂/∂y);

∆ the Laplacian, ∆:=4∂∂;

· , · When dealing with vector-valued functions with values in a Hilbert space (H, ·,· ), we use · for the norm in H induced by the inner product ·,· . We will use the symbol · (usually with a subscript) for the norm in the function space; thus for a vector-valued functionf, the symbolf∞denotes itsL^∞norm, which is the essential supremum

A. Sasane is supported by the Nuffield Grant NAL/32420.

S. Treil is supported by the NSF grant DMS-0501065.

of f(z) over z in the domain of definition off. On the other hand, the symbol f stands for the scalar-valued function whose value at a pointz is the norm of the vectorf(z);

·, ·,·^∗ IfM is a matrix (possibly infinite), thenM denotes the transpose of M. The complex conjugate ofM is denoted by M, andM^∗:=(M); H^∞ the space of bounded holomorphic functions onDwith the supremum

norm;

H^p the Hardy space, i.e. the space of analytic functions f on D such thatfp:=sup_0≤r<1

T f(rζ) dm(ζ)<∞; we will also use the vector- valued Hardy spaces H^p(E) of functions with values in a Hilbert (or Banach) space E;

A the space of bounded holomorphic functions onDwith continuous ex- tensions toTequipped with the supremum norm.

1. Introduction

This paper is devoted to estimates in the corona problem in some smooth subalgebras of the algebraH^∞of bounded analytic functions in the unit discD.

The main motivation for studying this problem comes from the idea of “vis- ibility” or “δ-visibility” of the spectrum, introduced by Nikolski [5].

Let us recall the main definitions. Let A be a commutative unital Banach algebra continuously embedded into the spaceC(X) of all continuous functions on a Hausdorff topological spaceX,A⊂C(X). The point evaluationsδ_x, (x∈X), given by

δ_x(f) =f(x), f∈ A,

are multiplicative linear functionals on A. Hence if A distinguishes points of X, then we can identifyXwith a subset of the maximal ideal space ofA(the spectrum M(A) ofA), that is,X⊂M(A).

Definition1.1. Let 0<δ≤1. The spectrum of A is said to be (δ, m)-visible (from X) if there exists a constantC(m) such that for any vectorf=(f₁, ..., f_m)∈ A^m satisfying

x∈Xinf m k=1

|f_k(x)|²≥δ²>0 (1.1)

and the normalizing condition

f²:=

m k=1

f_k²_A≤1,

the Bezout equation

g·f:=

m k=1

g_kf_k=e (1.2)

has a solutiong=(g₁, ..., g_m)∈A^mwith g=

_m

k=1

g_k²_{A 1/2}

≤C(m).

The spectrum is calledcompletely δ-visible if it is (δ, m)-visible for allm≥1 and the constantsC(m) can be chosen in such a way that sup_m≥1C(m)<∞.

This is a norm refinement of the usual corona problem for Banach algebras, and the motivations for the consideration of this problem can be found in Nikolski [5].

The classical corona theorem for the algebra H^∞, see [1], says that if the functionsf_k∈H^∞=H^∞(D) satisfy

1≥^m

k=1

|f_k(z)|²≥δ²>0 for allz∈D, (1.3)

then the Bezout equation

m k=1

g_kf_k= 1 (1.4)

has a solutiong₁,g₂, ...,g_m, and moreover the solution satisfies the estimates m

k=1

|g_k(z)|²≤C(δ, m)² for allz∈D.

Later refinements obtained independently by Rosenblum [7] and Tolokonnikov [11], got the estimate independent of m and allowed the case m=∞, see Appendix 3 of [6] for a modern treatment.

Note that having estimates that are independent ofmin the corona theorem in fact gives us something slightly more than the completeδ-visibility of the spectrum of H^∞, since the normalizing condition in (1.3) is weaker than the corresponding normalizing condition in Definition 1.1.

On the other hand there are many algebras with invisible spectrum. For ex- ample, for the Wiener algebraW of analytic functions

f= ∞ k=0

fˆ(k)z^k, such that fW:=

|f(k)ˆ |<∞,

the corona theorem holds trivially, that is, the unit disc Dis dense in the maximal ideal space M(W), but it is in general impossible to control the norms of the solutions of the Bezout equation: the algebraW is not even (δ,1)-visible for smallδ.

It is general understanding among experts that the estimates hold for local norms, and may (generally) fail for non-local norms, for example for norms given in terms of Fourier coefficients.

In this article, we study the following subalgebras ofH^∞. Let us recall that A denotes the disc algebra of all bounded analytic functions continuous up to the boundary, A=H^∞∩C(T).

Definition1.2. For a positive integer ndefine the following algebras:

(1)∂⁻ⁿH^∞ is the set of analytic functionsf defined onDsuch thatf, f, ..., f⁽ⁿ⁾ belong toH^∞.

(2) ∂⁻ⁿA is the set of analytic functions f defined on D such that f, f, ..., f⁽ⁿ⁾ belong to the disk algebraA.

(3) More generally, ifS be an open subset ofT, then∂⁻ⁿA_S is the set of all analytic functions f defined on Dsuch thatf,f,...,f⁽ⁿ⁾belong toA_S, whereA_S denotes the class of functions defined on the disk that are holomorphic and bounded in Dand extend continuously toS.

The above spaces are Banach algebras with the norm given by f=

n j=0

1

j!f^(j)∞.

The factor 1/j! is chosen so that the norm satisfies the estimatef g≤f g.(¹) For a Hilbert space H, one can consider the H-valued spaces A(H), where A is one of the spaces∂⁻ⁿH^∞, ∂⁻ⁿA and ∂⁻ⁿA_S defined above. Namely, for an analyticH-valued functionf we define its norm as

f= n j=0

1

j!f^(j)∞, (1.5)

where the norm is understood as theL^∞norm of the vector-valued function with val- ues inH. For example, ifH=²(orH=C^m), then forf={f_k}^∞_k=1=(f₁, f₂, ..., f_k, ...),

f^(j)_∞= ess sup

z∈T f^(j)(z) = ess sup

z∈T

_∞

k=1

|f_k^(j)(z)|² _1/2

.

(¹) In the definition of Banach algebra it is usually required that the norm satisfies the estimate fg≤f g. However, in a unital Banach algebra, if one is given a norm which only satisfies a weaker inequality fg≤Cf g(so the multiplication is continuous), there is a standard way to replace the norm by an equivalent one satisfying the inequality with C=1.

Namely, the new norm of an elementf is defined as the operator norm of multiplication byf. It is an easy exercise to show that the new norm is equivalent to the original one; one needs the fact that the algebra is unital to get one of the estimates.

We prove in this paper that the corona theorem with estimates holds for all these algebras, and that the estimates do not depend on the number of functionsf_k. This fact implies completeδ-visibility of the spectrum for allδ >0.

One of the motivations for studying these algebras comes from control theory.

Namely, for a system (plant)Gwith coprime factorizationG=f₁/f₂, the construc- tion of a stabilizing feedback is equivalent to solving the Bezout equation

g₁f₁+g₂f₂≡1,

with the stabilizing controller given by −g₁/g₂. And assuming that the original plantG(more precisely, its coprime factorization) has some smoothness, we want to be able to construct the stabilizing controller with the same smoothness and to be sure that the smoothness of this stabilizer is controlled by the smoothness ofG.

Before proving the corona theorem with bounds for the subalgebras of H^∞ introduced above in Definition 1.2, we remark that the corona theorem itself (with- out the estimates) is trivial for them. Indeed it is easy to show (see Proposition 1.3 below) that the maximal ideal space of our algebras (for n∈N) is the closed unit disk. Then the well-known equivalence of the density of X in the maximal ideal space and the solvability of the Bezout equation (1.2) under the assumption (1.1) (withX=Din our case) gives the corona theorem for our algebras.

Proposition 1.3. Let A be one of the algebras ∂⁻ⁿH^∞, ∂⁻ⁿA, and ∂⁻ⁿA_S defined above, n≥1. The maximal ideal space ofA is the closed unit disk.

This proposition is definitely not new. It follows, for example from [12, The- orem 6.1]. This theorem says, in particular, that for any algebra of functions A satisfying the property

iff∈ Aandλ >f∞, λ∈C, then (f−λ)⁻¹∈ A, (GD)

its maximal ideal space coincides with the maximal ideal space of theL^∞-closure ofA.

The algebras we consider clearly satisfy the condition (GD), and theL^∞-closure of each algebra is the disc algebraA, whose maximal ideal space coincides with the closed unit discD.

For the convenience of the reader we present a (very simple) proof of Propos- ition 1.3.

Proof. Note that ∂⁻ⁿH^∞⊂A, and so point evaluation at a fixed λ∈Dgives a multiplicative linear functional on∂⁻ⁿA_S. We will show that every multiplicative linear functional arises in this manner.

LetL be a multiplicative linear functional and letλ:=L(z) (the value ofLon the functionf(z)≡z). Then clearlyL(f)=f(λ) for polynomialsf. We show that

for any polynomial f,

|L(f)| ≤ f_∞. (1.6)

This estimate immediately implies that|λ|≤1 (apply (1.6) to the functionf(z)≡z).

Since A⊂A, any functionf in A can be approximated by polynomials in the L^∞ norm. But (1.6) implies thatLis continuous in theL^∞norm, so formula (1.6) holds for allf∈A. Note that in this reasoning we do not need the density of polynomials in the norm of A(which happens only ifA=∂⁻ⁿA).

To prove (1.6) let us notice that iff∈Aand inf_z∈D|f(z)|>0, thenf is invertible inA. Indeed, sinceA⊂A, the condition inf_z∈D|f(z)|>0 implies thatf is invertible in A.

Differentiating 1/f, n times we get that all its derivatives up to the ordern are in the algebraH^∞orAorA_S, depending on the algebraAwe are considering.

Therefore, if 0∈/clos range(f)=range(f), then f is invertible in A, and so f does not belong to any proper ideal of A. Thus L(f)=0 for any maximal ideal (multiplicative linear functional) L. Replacing f by f−a, a∈C, we get that if a /∈range(f), then for any multiplicative linear functionalL,L(f)=a, that is,L(f)⊂ range(f). Thus|L(f)|≤f_∞, and (1.6) is proved.

Plan of the paper

In Section 2 we prove the corona theorem with estimates on the norm of the solution for the algebra ∂⁻ⁿH^∞, see Theorem 2.1. This result is stronger than the completeδ-visibility of the spectrum of∂⁻ⁿH^∞.

We will use this result to show that the corona theorem with the same estimates holds for the algebras∂⁻ⁿAand∂⁻ⁿA_S as well. That of course will imply that the spectra of these algebras are completelyδ-visible for allδ >0.

The estimates for the algebra ∂⁻ⁿA will be obtained from the estimates for

∂⁻ⁿH^∞ by a simple approximation argument. The same argument will be used to get the estimates for ∂⁻ⁿA_S, with the essential difference that the construction of the approximating functions is quite involved in this case: the reasoning “modulo the approximation” is very similar to the one for∂⁻ⁿA.

Note that the results forn=0 are quite known. While we cannot give the exact reference, the fact that the estimates in the corona theorem for the disc algebra are the same as the estimates forH^∞is known to the specialists. The estimates in the corona theorem for the algebraA_S were considered by the first author, [8], although the equality of these estimates to the ones forH^∞was not mentioned there.

We should also mention that the corona theorem for various algebras of smooth functions was studied by Tolokonnikov [12]. In particular, the corona theorem

(without estimates) for the algebras considered in our paper follows from his results, see the remark immediately after Proposition 1.3 above. For some algebras of smooth functions he also obtained the corona theorem with estimates.

However, the estimates in the corona theorem for the algebras we are consid- ering do not follow from his results. Such estimates, which are the main goal of the present paper, are completely new. Also new is the fact that the estimates in all of the algebras we are considering are the same (for the samen), i.e. that they do not depend on continuity properties of the last derivative.

2. Estimates in the corona theorem for∂⁻ⁿH^∞

Theorem 2.1. Let n be a non-negative integer, and let A=∂⁻ⁿH^∞. There exists a constant C(δ, n)such that for anyf=(f₁, f₂, ..., f_k, ...)∈A(²)satisfying

0< δ≤ f(z) 2 for allz∈D, (2.1)

and

f_A(²₎≤1, (2.2)

there existsg=(g₁, g₂, ..., g_k, ...)∈A(²)such that ∞

k=1

g_k(z)f_k(z) = 1 for allz∈D, (2.3)

and

g_A(²₎≤C(δ, n).

(2.4)

Note that by considering sequencesf=(f₁, f₂, ..., f_n, ...) with finitely many non- zero entries, one can get the result aboutm-tuples as an elementary corollary.

2.1. Preliminaries for the proof

We want to introduce a different equivalent norm on the space∂⁻ⁿH^∞. Namely, for smooth functions on the circleTlet us consider the differential operatorD,

(Df)(e^it) :=−id dtf(e^it).

Define the spaceD⁻ⁿL^∞:={f∈L^∞|:D^kf∈L^∞, k=1,2, ..., n}. A natural norm on this class is given by

n k=0

f^(k)_∞. (2.5)

Of course, one can also define this space for functions with values in a Hilbert space H with inner product ·,· , and norm · . For our purposes it is more convenient to consider a different equivalent norm onD⁻ⁿL^∞,

f:= ˆf(0) +Dⁿf_∞, f∈D⁻ⁿL^∞, (2.6)

where ˆf(k), (k∈Z), denotes thekth Fourier coefficient off, fˆ(k) = 1

2π _π

π

f(e^it)e^−iktdt.

To show the equivalence of the two norms, let us notice that forζ∈[0,2π), f(e^iζ) = 1

2π _ζ+π

ζ−π

[f(e^iζ)−f(e^iθ)]dθ+ ˆf(0).

Since

f(e^iζ)−f(e^iθ) ≤ Df∞|θ−ζ|, we get by integrating this estimate

f_∞≤¹₄Df_∞+ ˆf(0) . (2.7)

AsDf(0)=0,Df_∞≤¹₄D²f_∞. Proceeding in a similar manner we get D^kf∞≤4^k−nDⁿf∞, k∈ {1, ..., n}, f∈D⁻ⁿL^∞,

so the norms of all derivatives can be estimated byDⁿf_∞and fˆ(0) . Therefore the norms (2.5) and (2.6) are equivalent.

Now we want to find the predual toD⁻ⁿL^∞. It is easy to see that if one writes an appropriate duality, thenD⁻ⁿL^∞is dual toL¹. Namely, it follows from the stan- dard L¹-L^∞ duality that any bounded linear functional on L¹ can be represented as

L(f) =fˆ(0),ˆg(0)+

Tf, Dⁿgdm, f∈L¹, (2.8)

where g is a function in D⁻ⁿL^∞. Moreover, the norm of L is comparable to the normg_D−nL^∞. Indeed, the functional Lcan be represented as

L(f) =

Tf, Fdm, f∈L¹,

whereF∈L^∞andF∞=L. LetD⁻¹denote the integration operator,D⁻¹e^int= (1/n)e^int,n=0. ThenD⁻ⁿ(F−F(0))+ F(0)=:g ∈D⁻ⁿL^∞with the normg_D⁻ⁿ_L^∞ comparable toF∞, which immediately implies the representation (2.8).

And finally, it is easy to see that ∂⁻ⁿH^∞=H^∞∩D⁻ⁿL^∞ and the norm · _D⁻ⁿ_L^∞ is equivalent to the norm in ∂⁻ⁿH^∞. Indeed, since D(e^ikt)=ke^ikt we conclude thatDf(z)=zf(z) for analytic polynomialsf=_N

k=0a_kz^k. Iterating the formulaDf(z)=zf(z) and using the fact that multiplication byzdoes not change the norm inL^∞(T) we get the estimate

D^kf∞≤C k j=1

f^(j)∞, k= 1,2, ..., n, which implies thatf_D⁻ⁿ_L^∞≤Cf_∂⁻ⁿ_H^∞.

To get the opposite inequality, we iterate the identity f(z)=z⁻¹DF(z), and since the multiplication byz⁻¹does not change theL^∞(T) norm we get the estimate

f^(k)_∞≤C k j=1

D^jf_∞, k= 1,2, ..., n.

Using standard approximation reasoning we get that the norms are equivalent for functionsf∈Hol(D), where Hol(D) is the set of all functions analytic in a neighbor- hood of the closed disc D. It is also easy to see that ∂⁻ⁿH^∞∩Hol(D)=Hol(D)=

D⁻ⁿL^∞∩H^∞∩Hol(D).

Finally, for both X=∂⁻ⁿH^∞ and X=D⁻ⁿL^∞∩H^∞ we have that f∈X if and only if sup{f_rX:0≤r<1}<∞, where f_r(z):=f(rz), and, moreover f_X= lim_r!1−f_rX.

Note that the operatorDis symmetric, namely, for smoothfandg, integration by parts or use of the Fourier series representations yields

TDf, gdm=

Tf, Dgdm.

(2.9)

Therefore, for smooth functionsf the duality (2.8) can be rewritten as L(f) =fˆ(0),ˆg(0)+

TDⁿf, gdm, f∈L¹. (2.10)

Remark 2.2. Given a Φ∈C^∞(D), there always exists a Ψ∈C^∞(D) such that

∂Ψ=Φ on some neighbourhood of D. Indeed, let O be open and let D⊂O. Let α∈C₀^∞(O) be such that α=1 on a neighbourhood ofD. Defining Ψ by

Ψ(z) =−1 π

R²

α(ζ)Φ(ζ)

ζ−z dx dy, z∈C, it can be seen that Ψ∈C^∞(C) and∂Ψ=Φ.

2.2. Setting up the∂-equation

We follow the standard way of setting up the∂-equations to solve the corona problem, as presented for example in [6]. We assume that we are given a column vectorf=(f₁, f₂, ..., f_m, ...) and we want to find a row vectorg=(g₁, g₂, ..., g_m, ...) satisfying

g·f= ∞ k=1

g_kf_k≡1.

We will use the standard linear algebra conventions, for example for a matrix A, A^∗=A. In particular, f^∗ is a row vector f^∗=(f₁, f₂, ..., f_m, ...). Also, for two vectors f, g∈² we will use the notation g·f for the “dot product”, g·f:=gf= _∞

k=1g_kf_k.

As usual, it is sufficient to prove the theorem under the additional assumption that f is holomorphic in a neighborhood ofD. Let 0<r<1, and setf_r(z)=f(rz), z∈D. Then f_r∈Hol(D), and we have f_r≤1, and f_r(z) ≥δfor allz∈D. If the statement of the theorem is true for f’s in Hol(D), then there exists a g_r∈Hol(D) such thatg_r(z)f_r(z)=1 for allz∈D, andg_r≤C(δ). If we chooser_k!1 such that g_r!guniformly on compact subsets ofD(which is possible by Montel’s theorem), then theg satisfies (2.3) and (2.4) of the theorem.

We suppose therefore thatf∈Hol(D) and (2.1) holds.

Define the row vector

ϕ= f^∗ f ².

Then ϕ∈C^∞(D), and ϕf≡1 in a neighbourhood of D. So ϕ solves the Bezout equationϕf≡1, but it is not analytic inD. Note that

∂ϕ=(f)^∗

f ²−(f)^∗f f ⁴ f^∗. If we find a matrix Ψ solving the ∂-equation

∂Ψ =ϕ∂ϕ=: Φ, then

g:=ϕ+f(Ψ−Ψ) will be analytic inD, since

∂g=∂ϕ+f(∂Ψ−∂Ψ) =∂ϕ+f((∂ϕ)ϕ−ϕ∂ϕ)

=∂ϕ+((∂ϕ)f)ϕ−∂ϕ= ((∂ϕ)f)ϕ= (∂(ϕf))ϕ= 0

where the equalities come from the fact that ∂f=0, ∂Ψ=ϕ∂ϕ and ϕf≡1.

Moreover, since the matrix Ξ=Ψ−Ψ is antisymmetric (Ξ=−Ξ), we have f(Ψ−Ψ)f=0, sogf=ϕf≡1.

2.3. Estimates of the solution of the ∂-equation from the boundedness ofL

Let us see what we need to get the estimate of the norm of the solution. Since Dⁿ(Ξf)=_n

k=0 n k

(D^kΞ)D^n−kf, the estimates ess sup

ζ∈T Ψ^(k)(ζ) ≤C <∞, k= 1,2, ..., n,

where · denotes the operator norm of a matrix, imply that the solution g is in the spaceD⁻ⁿL^∞(²). As the solutiongwe get is analytic, that is exactly what we need.

Since the operator norm of a matrix is dominated by the Hilbert–Schmidt norm

· S2, it is sufficient to estimate the Hilbert–Schmidt norms of the derivatives, that is, to estimate the norm of the solution Ψ in the spaceD⁻ⁿL^∞(S2). Note that the spaceS₂ of Hilbert–Schmidt operators (matrices) is a Hilbert space with the inner product A, B_S2:=trAB^∗=trB^∗A, so all the previous discussions about norms and duality for the spaceD⁻ⁿL^∞do apply here.

We estimate the norm of the solution of the∂-equation by duality. Let Ψ₀ be any smooth solution of the∂-equation

∂Ψ = Φ :=ϕ∂ϕ= f f ²

(f)^∗

f ²−(f)^∗f f ⁴ f^∗

. (2.11)

Define the linear functionalLonH₀¹(S₂):=zH¹(S₂), L(h) =

T

tr{(Dⁿh)Ψ₀}dm=

TDⁿh,Ψ^∗₀S2dm.

Note that the above expression is well defined on a dense subspace of smooth func- tions inH₀¹(S2), for example on the subspaceX₀=H₀¹(S2)∩Hol(D,S2).

If we prove thatLis a bounded functional on H₀¹(S2), it can be extended by the Hahn–Banach theorem to a bounded functional on the whole space L¹(S₂).

That means, according to our discussions of duality, see (2.8) and (2.10), that there exists a function Ψ∈D⁻ⁿL^∞,Ψ_D⁻ⁿ_L^∞L, such that

L(h) =

T

tr{(Dⁿh)Ψ₀}dm=

T

tr{(Dⁿh)Ψ}dm for allh∈X₀.

Note that ˆh(0)=0 for h∈X₀, so the term corresponding to fˆ(0),g(0)ˆ from (2.8) and (2.10) disappears.

Since

Ttr{(Dⁿh)(Ψ−Ψ₀)}dm=0 on a dense setX inH₀¹, the function Ψ−Ψ₀ is analytic in D, so Ψ solves the ∂-equation∂Ψ=Φ.

2.4. Estimates of the functional L

To estimateL(h), we use Green’s formula,

T

u dm−u(0) =2 π

D

(∂∂u(z)) log 1

|z|dx dy (G)

which holds for C²-smooth functionsu in the closed discD(recall that ∂∂=¹₄∆).

Applying this formula to u=tr{(Dⁿh)Ψ}, whereDⁿhin the disc is defined as the harmonic (analytic) extension from the boundary, we get

L(h) =

T

tr{(Dⁿh)Ψ}dm=2 π

D

(∂∂tr{(Dⁿh)Ψ}) log 1

|z|dx dy

= 2 π

D

(∂tr{(Dⁿh)Φ}) log 1

|z|dx dy=2

π(I₁+I₂), where we have used that Dⁿh(0)=0,∂(Dⁿh)=0 and∂Ψ=Φ, and let

I₁:=

D

tr{(Dⁿh)∂Φ}log 1

|z|dx dy and I₂:=

D

tr{(∂Dⁿh)Φ}log 1

|z|dx dy.

To estimate the integralsI₁ andI₂we would like to move the derivatives to Φ.

To do this, let us extend the operatorDto the whole disc as follows:

Dw(re^iθ) =−id

dθw(re^iθ).

Then Dzⁿ=nzⁿ and Dzⁿ=−nzⁿ for n≥0, and so for holomorphic w, Dw(z)=

zw(z) andDw=−zw(z).

Note that if we treatDⁿhas the “extended” operatorDⁿ applied to the func- tion in the disc, we get the same result as before, when we definedDⁿhin the disc as the harmonic (analytic) extension from the boundary.

2.4.1. Estimates ofI₁.Using the symmetry ofD, see (2.9), we get I₁=

D

tr{(Dⁿh)∂Φ}log 1

|z|dx dy=

DDⁿh, ∂Φ^∗S2log 1

|z| dx dy

=

Dh, Dⁿ∂Φ^∗_S2log 1

|z| dx dy, where the last equality can be seen as follows: we write the integral in polar co- ordinates, then, in the integral with respect to dθ we apply the formula (2.9) and

finally we go back todx dy. Note that we used the inner product notation, because the symmetry of the operatorDis more transparent and is easier to write this way.

Applying the operatorD ntimes, we get that Dⁿ∂Φ^∗ can be represented as a sum of terms of form

a product of analytic and antianalytic factors f ^2r

(2.12)

where (up to the transpose) the antianalytic factors can be only of the form (f^(j))^∗ and the analytic ones can be only of the form f^(l), j, l=0,1, ..., n+1. Moreover, if one looks at the derivatives of the maximal possible order k=n+1, each term of form (2.12) can have at most one factorf^(k) and at most one factor (f^(k))^∗ (it can have both f^(k) and (f^(k))^∗). Indeed, the direct computations show that the function∂Φ^∗ clearly is represented as such a sum, with the maximal order of each derivative being 1. Each differentiation D preserves the form, and increases the maximal order of the derivative at most(²) by 1.

The terms in the decomposition (2.12) ofDⁿ∂Φ^∗ containing both factorsf^(k) and (f^(k))^∗ of maximal possible orderk=n+1 can be estimated by C f^{(n+1) 2}₂. Note thatf⁽ⁿ⁾∈H^∞(²).

It is well known (see Section 2.5 below for all necessary information about Car- leson measures) that for a bounded analytic functionFwith values in a Hilbert space the measure F(z) ²log(1/|z|)dx dyis Carleson, with the Carleson norm estimated by CF²_∞. Thus we can conclude that the measure f⁽ⁿ⁺¹⁾ 2log(1/|z|)dx dy is Carleson. Therefore

D

h(z) _S2 f^{(n+1) 2}2log 1

|z|dx dy≤Ch_H¹_(S2) so the terms ofI₁ containing bothf⁽ⁿ⁺¹⁾ and (f⁽ⁿ⁺¹⁾)^∗ are estimated.

The terms in the decomposition (2.12) ofDⁿ∂Φ^∗containing only the derivatives of orderk<n+1 are bounded, so the corresponding terms inI₁are easily estimated, because the measure log(1/|z|)dx dyis trivially Carleson.

Finally, the terms in (2.12) containing only one of the factorsf⁽ⁿ⁺¹⁾or (f⁽ⁿ⁺¹⁾)^∗ can be estimated byC f⁽ⁿ⁺¹⁾ 2, and since by the Cauchy–Schwarz inequality

D

h(z) _S2 f⁽ⁿ⁺¹⁾(z) 2log 1

|z|dx dy

≤

D

h(z) _S2 f⁽ⁿ⁺¹⁾(z) ²2log 1

|z|dx dy

_1/2

D

h(z) _S2log 1

|z|dx dy _1/2

≤Ch_H¹_(S2)

(²) It can be shown by more careful analysis, that no cancellation happens, and the maximal order of the derivative increasesexactlyby 1, but we do not need this for the proof: we only need that it cannot increase by more than 1.

(as we discussed above, the measures in both integrals in the second line are Car- leson), so the corresponding terms inI₁ are also easily estimated.(³)

2.4.2. Estimates of I₂.Let us now estimate I₂. By trivial estimates we have for

|z|<¹₂,

|tr{(∂Dⁿh)Φ}| ≤Ch_H¹_(S2),

so we need only estimate the integralI₂, where one integrates over ¹₂≤|z|<1.

Indeed, the derivatives ofhcan be estimated by standard estimates for power series, if one recalls that ˆh(k) _S2≤h_H¹_(S2). We also have Φ(z) ≤C f(z) , and using similar reasoning with power series one can show that f(z) ≤Cfor|z|<¹₂. Note that for analyticf we have∂f=z⁻¹Df, and so we can replace∂Dⁿhby z⁻¹Dⁿ⁺¹hinI₂. Thus

I₂ =

1/2≤|z|<1

tr{(∂Dⁿh)Φ}log 1

|z|dx dy

=

1/2<|z|<1z⁻¹Dⁿ⁺¹h,Φ^∗_S2log 1

|z|dx dy.

Using the symmetry ofD we get as in the case ofI₁ I₂ =

1/2<|z|<1Dh, Dⁿ((z)⁻¹Φ^∗)_S2log 1

|z|dx dy

=

1/2<|z|<1z⁻¹h(z), Dⁿ((z)⁻¹Φ^∗)S2log 1

|z|dx dy.

Applying the operator D repeatedly to (z)⁻¹Φ^∗, we get the representation of Dⁿ((z)⁻¹Φ^∗) as the sum of terms of form (2.12), with slight differences. Namely, the analytic factors, as in the case ofI₁ can be of the formf^(l),l=1,2, ..., n+1, and the antianalytic factors (and this is the difference to the case ofI₁) can only be of the form (f^(j))^∗, j=1,2, ..., nor (z)⁻,≥1. And again, any term containing the derivativef⁽ⁿ⁺¹⁾of the highest possible order can contain it only once.

We notice that (z)⁻¹Φ^∗has such a representation withn=0, and each differen- tiation preserves the form of the decomposition and increases the maximal possible order of the derivatives f^(l)and (f^(j))^∗ by at most 1.

To estimate I₂, let h₁ be a scalar-valued outer function in H² such that

|h₁(ζ)|²= h(ζ) a.e. onT. Thenh∈H¹(S₂) can be represented ash=h₁h₂, where h₁∈H² (scalar),h₂∈H²(S₂), andh₁²_H2=h₂²_H2(S2)=h_H¹_(S2).

(³) A careful analysis ofDⁿ∂Φ^∗can show that the terms containing only one derivative of the maximal order are impossible here, but the above reasoning is significantly simpler than the careful analysis of derivatives.

Sinceh=h₁h₂+h₁h₂, we can estimates the terms ofI₂ containing the deriva- tivef⁽ⁿ⁺¹⁾ of the highest possible order by

D

h(z) _S2 f⁽ⁿ⁺¹⁾ 2log 1

|z|dx dy

≤

D

(|h₁| h₂ +|h₁| h₂ ) f⁽ⁿ⁺¹⁾ 2log 1

|z|dx dy.

Since, as we discussed, when treatingI₁, the measure f⁽ⁿ⁺¹⁾(z) 2log(1/|z|)dx dy is Carleson, with its Carleson norm bounded byCf²_H∞(²), we get

D|h₁| h₂ f⁽ⁿ⁺¹⁾ 2log 1

|z|dx dy

≤

D|h₁|² f^{(n+1) 2}2log 1

|z|dx dy

_1/2

D

h₂ ²log 1

|z|dx dy _1/2

≤Ch_1H²h_2H²_(S2)

=Ch_H¹_(S2);

here the first integral in the second line is estimated using the fact that the measure is Carleson, and the second integral is simply the Littlewood–Paley representation of the normh_2H²_(S2). The integral

D|h₁| h₂ f⁽ⁿ⁺¹⁾ 2log(1/|z|)dx dyis es- timated similarly.

The terms in the decomposition (2.12) of Dⁿ((z)⁻¹Φ^∗) which contain only derivatives of order at most n are bounded. Therefore to estimate the rest of I₂ it is sufficient to estimate

D h log(1/|z|)dx dy. Decomposing as aboveh=h₁h₂ and using the fact that the measure log(1/|z|)dx dyis trivially Carleson, we get the estimate

D|h₁| h₂ log 1

|z|dx dy

≤

D|h₁|²log 1

|z|dx dy

_1/2

D

h₂ ²log 1

|z|dx dy _1/2

≤Ch_1H²h_2H²_(S2)

=Ch_H¹_(S2); The integral

D|h₁| h₂ log(1/|z|)dx dy, and thus the rest ofI₂ is estimated simi- larly.

2.5. Some remarks about Carleson measures

In this subsection we present for the convenience of the reader some well known facts about the Carleson measures, that we have used above in Section 2.

Let us recall that a measureµ in the unit disc D a Carleson measure if the embedding H²⊂L²(µ) holds, i.e. if the inequality

D|f(z)|²dµ(z)≤Cf²_H2 for allf∈D, (2.13)

holds for some C <∞. The best possible constantCin this inequality is called the Carleson norm of the measureµ.

There is a very simple geometric description of Carleson measures, cf. [3] or any other monograph about H^p spaces. Namely, a measure µ is Carleson if and only if

sup

ξ∈Tr>0

1

rµ{z∈D:|z−ξ|< r}<∞.

Moreover, the above supremum is equivalent (in the sense of a two-sided estimate) to the Carleson norm of the measureµ.

However, in this paper we will use the following simple and well-known fact about bounded analytic functions and Carleson measures.

Proposition 2.3. If F is a bounded analytic function in the unit disc with values in a Hilbert space, then the measure µ, dµ(z)=log(1/|z|) F(z) ²dx dy is Carleson with its Carleson norm bounded by CF²_∞.

Note, that this proposition is not true for functions with values in an arbitrary Banach space.

Note also, that in the scalar case this and even stronger propositions are well known and widely used, see for example the Garnett’s book [3].

There are several ways to prove this proposition, and it is easier for us to present the proof here and save the reader a trip to the library, than to give an exact reference.

Probably the simplest way to prove this proposition is to refer to the so-called Uchiyama lemma, cf. [6, Appendix 3, Lemma 6]. This lemma says that if u≥0 is a C²-smooth bounded subharmonic function (i.e. ∆u≥0) in D, then the measure

∆u(z) log(1/|z|)dx dy (where ∆ denotes the Laplacian) is Carleson with Carleson norm estimated by 2πeu²_∞. Noticing that for an analytic function F with val- ues in a Hilbert space ∆ u(z) ²=4∂∂ u(z) ²=4 u(z) ² we immediately get the proposition with the constantC=πe/2.

Another, more elementary way to prove the proposition is to use the Little- wood–Paley formula. Namely, if we apply the Green’s formula (see (G) in Sec- tion 2.4) to the function u(z)= f(z) ², wheref∈H²(E), E is a Hilbert space, we get the Littlewood–Paley identity

2 π

D

f(z) ²log 1

|z|dx dy=f²_H2(E)− f(0) ².

Thus, if we define the weightwonDbyw(z)=(2/π) log(1/|z|), then fL²(w)≤ fH²,

whereL²(w)=L²(E, w) is the weighted Lebesgue space of functions with values in E. Applying this estimate to a function f of the form f=F g, where F∈H^∞(E) andg is a scalar-valued function inH², we get using the triangle inequality Fg_L²_(w)≤ F g_L²_(w)+F g_H²≤ F∞g_L²_(w)+F∞g_H²≤2F∞g_H². But this implies that the measure (2/π) log(1/|z|)dx dy is Carleson with the Car- leson norm at most 4F²_∞.

We should also mention that if a measureµis Carleson, the embedding (2.13) holds (with the same constant) for the vector-valuedH²-spacesH²(X) with values in an arbitrary Banach space X. To see this it is sufficient to notice that f(z) ≤|h(z)| for all z∈D, where h is the scalar-valued outer function satisfying

|h(ξ)|= f(ξ) a.e. onT.

3. Estimates in the corona theorem for other algebras: preliminaries and the case of∂⁻ⁿA

3.1. Continuity of the best estimate

For a function algebra A (one should think about one of the algebras from Definition 1.2) letC(A, δ), δ >0, denote the best possible estimate on the norm of the solution of the Bezout equation,

C(A, δ) := sup

f inf

gA(²) :g·f:=

∞ k=1

g_kf_k≡1

,

where the supremum is taken over all f=(f₁, f₂, ..., f_m, ...)∈A(²), where f_A(²₎≤1 and such that

f(z) 2:=

_∞

k=1

|f_k(z)|² _1/2

≥δ.

We will show in the rest of the paper that for the function algebras from Definition 1.2 the constantsC(A, δ) coincide,

C(∂⁻ⁿH^∞, δ) =C(∂⁻ⁿA, δ) =C(∂⁻ⁿA_S, δ).