The Corona Problem for Kernel Multiplier Algebras 3

Washington University in St. Louis Washington University Open Scholarship

Mathematics Faculty Publications Mathematics and Statistics

12-2016 The orC ona Problem for Kernel Multiplier Algebras Eric T. Sawyer

Brett .D Wick Washington University in St. Louis, [email protected]

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Recommended Citation Sawyer, Eric T. and Wick, Brett .,D "The orC ona Problem for Kernel Multiplier Algebras" (2016). Mathematics Faculty Publications. 39. https://openscholarship.wustl.edu/math_facpubs/39

This Article is brought to you for free and open access by the Mathematics and Statistics at Washington University Open Scholarship. It has been accepted for inclusion in Mathematics Faculty Publications by an authorized administrator of Washington University Open Scholarship. For more information, please contact [email protected]. arXiv:1410.8862v4 [math.CA] 10 Oct 2016 rgesi ihrdmnin ro o21 a enlmtdt the to limited one been in had domains 2011 for to theorems prior corona toget dimensions on construction’ higher literature ‘corona in large beautiful progress a a is used there proof While The disk. unit oan in domains n16 .Creo Cr rvdtecrn hoe o h algeb the for theorem corona the proved [Car] Carleson L. 1962 In .TeatraeTelt ooatermfrBrmnadHrys Hardy and Bergman for theorem corona Toeplitz alternate The proof and 5. Formulation spaces shifted 4.3. Convex Properties 4.2. theorem Corona corona Toeplitz alternate 4.1. The solutions multiplier 4. characterization Kernel multiplier kernel Bezout 3.1. The rescaling of characterization 3. A rescalings by 2.3. Interpolation spaces function 2.2. Hilbert 2.1. Preliminaries paper the of 2. Organization 1.1. Introduction 1. ..TeCrn rpryfrkre utpiragba nteds 3 eq Cauchy-Riemann disk generalized the to on solutions algebras of multiplier spaces References kernel Bergman for spaces Property Besov-Sobolev Corona 7.3. for The algebras multiplier kernel 7.2. spaces The nonholomorphic and spaces spaces 7.1. Hardy Smoothness for Property Multiplier polydisc 7. Invertible unit the on domains ball 6. space General unit Bergman the and on Hardy 5.3. space The Hardy and Bergman 5.2. The 5.1. ‡ † Date eerhspotdi atb ainlSineFudto D Foundation Scien Science National National the by from part grant in a supported by Research part in supported Research coe 1 2016. 11, October : netbemlileson multipliers invertible ooatermetnst oegnrlHletfnto spa function functions kernel Hilbert the general Instead, more property. hig to Pick in extends polydiscs theorem and corona domains pseudoconvex strictly certain xedn h eutfrHletspaces Hilbert for algebras multiplier result kernel the the extending for theorem corona the prove n otne yAa n20,adms eetyb rn n W and Trent by recently most and 2003, in Amar by continued and fBsvSblvsae nteui alin ball unit the on spaces Besov-Sobolev of Abstract. H OOAPOLMFRKRE UTPIRALGEBRAS MULTIPLIER KERNEL FOR PROBLEM CORONA THE C n n ekrrslsrestricting results weaker and , epoea lent opizcrn hoe o h algebr the for theorem corona Toeplitz alternate an prove We H n hnw otnearsac hedbgnb ge n McC and Agler by begun thread research a continue we then and , RCT SAWYER T. ERIC H ∞ 1. ∩ k Q x Introduction N C Contents p ( n y n o h ler fbuddaayi ucin on functions analytic bounded of algebra the for and , † yA ioa n .Xiao. J. and Nicolau A. by o2gnrtr.I at pr rmtesml ae in cases simple the from apart fact, In generators. 2 to fcranHletfnto spaces function Hilbert certain of ) N RT .WICK D. BRETT AND 1 Sgat#1026ad#1560955. and 1603246 # grant MS eadEgneigRsac oni fCanada. of Council Research Engineering and ce fBsvSblvBnc pcsi h ntdisk, unit the in spaces Banach Besov-Sobolev of e iesosa el hsatraeToeplitz alternate This well. as dimensions her e hr tde o eur h complete the require not does it where ces e ihpoete fBack products. Blaschke of properties with her c n20.I dimension In 2009. in ick H ao one nltcfntoso the on functions analytic bounded of ra p so onws enlmultipliers kernel pointwise of as ooatermo aiu bounded various on theorem corona ae in paces ope ieso seeg [Nik]), e.g. (see dimension complex ain 33 uations ‡ C H n r sue obe to assumed are rh n1999, in arthy n we 1 = 18 16 15 15 12 12 10 19 33 25 25 22 21 20 20 8 4 4 3 1 1 2 E. T. SAWYER AND B. D. WICK

which the maximal ideal space of the algebra can be identiﬁed with a compact subset of Cn, no complete corona theorem was proved in higher dimensions until the 2011 results of S. Costea and the authors in

2 which the corona theorem was established for the multiplier algebras MHn and MDn of the Drury-Arveson 2 Hardy space Hn and the Dirichlet space n on the ball in n dimensions. These latter results used the abstract Toeplitz corona theorem for HilbertD function spaces with a complete Pick kernel, see Ball, Trent and Vinnikov [BaTrVi] and Ambrosie and Timotin [AmTi]. The unresolved corona question for the algebra of bounded analytic functions on the ball in higher dimensions has remained a tantalizing problem for over half a century now (see e.g. [CoSaWi2] and [DoKrSaTrWi] for a more detailed history of this problem to date). We note in particular that Varopoulos [Var] gave an example of Carleson measure data in dimension n = 2 for which there is no bounded solution to the ∂ equation. This poses a significant obstacle to using the ∂ equation for the multiplier problem, and suggests a more operator theoretic approach akin to the Toeplitz corona theorem in order to solve the corona problem in higher dimensions. In this paper we prove in particular an alternate Toeplitz corona theorem for all of the algebras of σ B ∞ B kernel multipliers of Besov-Sobolev spaces B2 ( n) on the ball. These spaces include H ( n), and the alternate Toeplitz corona theorem also extends to the algebra H∞ (Ω) of bounded analytic functions on Ω, n n ∞ where Ω is either the unit polydisc D in C , a sufficiently small C perturbation of the unit ball Bn, or a bounded strictly pseudoconvex homogeneous complete circular domain in Cn. Moreover, this alternate Toeplitz corona theorem extends to the kernel multiplier space of certain Hilbert function spaces without assuming the complete Pick property. This essentially shows that whenever a Hilbert space has one of these special kernels, then the Corona Property for its kernel multiplier algebra reduces to what we call the Convex Poisson Property, a property which can be addressed by methods involving solution of the ∂ problem. To illustrate in a very special case, we show that for Ω as above, the algebra H∞ (Ω) has the Corona Property if and only if the Bergman space A2 (Ω) has the Convex Poisson Property if and only if the Hardy space H2 (Ω) has the Convex Poisson Property (in order to define the Hardy space we need to assume that ∂Ω is C2). The Corona Property for H∞ (Ω) is this: given ϕ ,...,ϕ H∞ (Ω) satisfying 1 N ∈ 1 max ϕ (z) 2 ,..., ϕ (z) 2 c2 > 0, z Ω, ≥ | 1 | | N | ≥ ∈ n ∞ o there are a positive constant C and f ,...,fN H (Ω) satisfying 1 ∈ 2 2 2 max f (z) ,..., fN (z) C , z Ω, | 1 | | | ≤ ∈ ϕ (z) f n(z)+ + ϕ (z) fN (zo) = 1, z Ω. 1 1 ··· N ∈ M M+1 2 2 For points a = (a ,...,aM ) Ω and θ = θ ,...,θM [0, 1] , and for h = A (Ω) or H (Ω), where 1 ∈ 0 ∈ ∈ H Ω is as above, set ka 1 by convention. Then we have 0 ≡ M 2 f 2 2 h Ha,θ θm h kam dv < . k k ≡ Ω | | ∞ m=0 Z X The Convex Poisson Property for the Hilbert space (either g Bergman or Hardy) is then this: given ϕ ,...,ϕ H∞ (Ω) satisfying H 1 N ∈ 1 max ϕ (z) 2 ,..., ϕ (z) 2 c2 > 0, z Ω, ≥ | 1 | | N | ≥ ∈ n o M there is a positive constant C such that for all points a = (a1,...,aM ) Ω and all θ = θ0,...,θM M+1 M ∈ ∈ [0, 1] with θm = 1, there are f ,...,fN satisfying m=0 1 ∈ H N P 2 2 fℓ a,θ C , k kH ≤ ℓ X=1 ϕ (z) f (z)+ + ϕ (z) fN (z) = 1, z Ω. 1 1 ··· N ∈ More generally we prove an alternate Toeplitz corona theorem for the kernel multiplier space KH of a Hilbert function space whose reproducing kernel k need not be a complete Pick kernel, but must have H the property (among others) that the kernel functions ka are invertible multipliers on . Here the Banach H kϕkakH space KH of kernel multipliers consists of those functions ϕ such that supa∈Ω kk k < . The roots ∈ H a H ∞ of this abstract result can be traced back to a research thread begun by Agler and McCarthy [AgMc] in THE CORONA PROBLEM FOR KERNEL MULTIPLIER ALGEBRAS 3

the bidisc using Ando’s theorem to reduce the corona problem to estimates on weighted Hardy spaces (see also [AgMc2, Chapters 11 and 13] for a nice survey of related prior work), and continued by work of Amar [Amar] who introduced the use of the von Neumann minimax theorem to circumvent Ando’s theorem and go beyond the bidisc, and more recently by work of Trent and the second author [TrWi] who further reduced matters to checking weights whose densities are the modulus squared of nonvanishing H∞ functions whose boundary values have bounded reciprocals. In this paper we extend and refine this approach to more general Hilbert function spaces and reduce matters to solving the Bezout equation ϕ f = 1 with solutions f in an analogue of a weighted L2 space - e.g. the method applies to Besov-Sobolev spaces· on the ball whose norms cannot be given as an L2 norm with respect to some measure. In dimension n = 1 we can prove new Corona theorems for the kernel multiplier algebras of the Besov- σ Sobolev spaces Bp of the unit disk (See Section 7 for definitions of these spaces). In the special case of σ D ∞ B2 ( ), the kernel multiplier algebra coincides with the Hilbert spaces H Q2σ, see Nicolau and Xiao [NiXi], Essén and Xiao [EsXi], Aulaskari, Stegenga and Xiao [AuStXi], or Xiao∩ [Xia, Xia3] for the definitions of these spaces.

1.1. Organization of the paper. In Section 2 of this paper we introduce background material on a Hilbert function space on a set Ω, as well as our new deﬁnition of the Banach space KH of kernel multipliers on a Hilbert function space . The space KH is often an algebra and plays the pivotal role in our alternate Toeplitz corona theorem,H given as Theorem 29 below. In Section 3 we use von Neumann’s minimax theorem, following the thread begun by Amar [Amar] and continued by Trent and Wick [TrWi], to characterize those vectors of corona data ϕ N L∞ for which solutions f N K exist to Bezout’s equation ϕ f = 1 in Ω. ∈ ⊕ ∈⊕ H · In Section 4 we use this characterization to obtain an alternate Toeplitz corona theorem for the spaces KH that uses a Convex Poisson Property instead of a Baby Corona Property as in the Toeplitz corona theorem for MH in [BaTrVi] and [AmTi]. Instead of requiring a complete Pick kernel we require the following four properties:

(1) the reproducing kernels ka for are invertible multipliers of , and the map a ka is lower H H → semicontinuous from Ω to MH, (2) the kernel multiplier space KH is an algebra, (3) the constant function 1 is in , and (4) the unit ball of enjoys a MontelH property. H This alternate Toeplitz corona theorem thus reduces the Corona Property for KH to the Convex Poisson Property for . As an application, we give examples of domains Ω in Cn for which the Corona Property for H∞ (Ω) isH equivalent to the Convex Poisson Property with taken to be either the Bergman or Hardy space on Ω. Section 5 is devoted to these higher dimensional exampleH s, which of course include the ball and polydisc. However, we are unable to obtain any new corona theorems in higher dimensions. Then in Section 6, we discuss the Invertible Multiplier Property, which when it holds, gives corona theorems in many situations as a corollary of our alternate Toeplitz corona theorem. However, the Invertible 1 Multiplier Property is known to hold only for the Szegökernel 1−az in dimension n = 1, where it can be used to prove a corona theorem in the disk, the annulus, and any other planar domain for which the reproducing 1 kernel is essentially the Szegökernel 1−az . In particular, we show the Invertible Multiplier Property fails for the Szegökernel on both the ball and polydisc in higher dimensions. Finally, in Section 7: σ (1) we prove that the space K is an algebra when is a Besov-Sobolev Hilbert space B (Bn) in H H 2 the ball, σ > 0, and conclude that KH has the Corona Property if and only if the Convex Poisson σ B Property holds for B2 ( n), (2) we provide a new proof of A. Nicolau and J. Xiao’s result that the Corona Property holds for the σ D algebras KH when = B2 ( ) is a Besov-Sobolev space in the disk with σ > 0, see [NiXi]; and H σ 1 we further demonstrate that the Corona Property holds for the algebra Kp when 0 <σ< p and 1

2. Preliminaries We begin with a quick review of reproducing kernel Hilbert spaces, also known as Hilbert function spaces.

2.1. Hilbert function spaces. A Hilbert space is said to be a Hilbert function space (also called a reproducing kernel Hilbert space) on a set Ω if theH elements of are complex-valued functions f on Ω with the usual vector space structure, such that each point evaluationH on is a nonzero continuous linear H functional, i.e. for every x Ω there is a positive constant Cx such that ∈ (2.1) f (x) Cx f , f , | |≤ k kH ∀ ∈ H and there is some f with f (x) = 0. Since point evaluation at x Ω is a continuous linear functional, there 6 ∈ is a unique element kx such that ∈ H f (x)= f, kx for all x Ω. h iH ∈ The element kx is called the reproducing kernel at x, and satisﬁes

ky (x)= ky, kx , x,y Ω. h iH ∈ In particular we have 2 kx = kx, kx = kx (x) , k kH h iH kx and so the normalized reproducing kernel is given by kx . ≡ √kx(x)

The function k (y, x) kx, ky H = kx (y) is self-adjoint (k (x, y) = k (y, x)), and for every ﬁnite subset N ≡ h i f xi of Ω, the matrix [k (xi, xj )] is positive semideﬁnite: { }i=1 1≤i,j≤N N N N N N 2

ξiξj k (xj , xi)= ξiξj kxi , kxj H = ξikxi , ξj kxj = ξikxi 0. * + ≥ i,j=1 i,j=1 i=1 j=1 H i=1 H X X X X X

Altogether we have shown that k is a kernel function in the following sense. Definition 1. A function k : Ω Ω C is a kernel function on Ω if k is self-adjoint and positive on × → N N the diagonal, and if for every finite subset x = xi Ω of Ω, the matrix [k (xi, xj )] is positive { }i=1 ∈ 1≤i,j≤N semidefinite, written [k (xi, xj )]1≤i,j≤N < 0, i.e. N N N (2.2) ξ ξ k (xi, xj ) 0, ξ C , x Ω ,N 1. i j ≥ ∈ ∈ ≥ i,j=1 X We write k < 0 if k is a kernel function. E. H. Moore discovered the following bijection between Hilbert function spaces and kernel functions. N Given a kernel function k on Ω Ω, define an inner product , on finite linear combinations ξ kx × h· ·iHk i=1 i i of the functions kx (ζ)= k (ζ, xi), ζ Ω, by i ∈ P N N N

(2.3) ξikxi , ηj kxj = ξiηj k (xj , xi) . *i=1 j=1 + i,j=1 X X Hk X

If the forms in (2.2) are positive definite, then finite collections of the kernel functions kxi are linearly independent, and the inner product in (2.3) is well-defined. See [AgMc2, page 19] for a proof that (2.3) is well-defined in general. Definition 2. Given a kernel function k : Ω Ω C on a set Ω, define the associated Hilbert function ×N → space k to be the completion of the functions i=1 ξikxi under the norm corresponding to the inner product (2.3).H P ′ Proposition 3. The Hilbert space k has kernel k. If and are Hilbert function spaces on Ω that have the same kernel function k, then thereH is an isometryH from Honto ′ that preserves the kernel functions H H kx, x Ω. ∈ THE CORONA PROBLEM FOR KERNEL MULTIPLIER ALGEBRAS 5

We will need the notion of rescaling a kernel as given in [AgMc2, page 25]. Given a nonvanishing complex- ρ valued function ρ :Ω C 0 and a kernel function k (y, x)= kx (y), deﬁne the ρ-rescaled kernel k by → \{ } kρ (y, x)= ρ (y) k (y, x) ρ (x), x,y Ω. ∈ It is easy to see that kρ is self-adjoint and positive semideﬁnite, and hence is a kernel function. We refer to the associated Hilbert function space kρ as the ρ-rescaling of the Hilbert function space k. A crucial H 1 ρ H choice of rescaling for us below is the point rescaling with ρ = that results in ka 1. We note that the δ ka ≡ used in [AgMc2] is our ρ. f

2.1.1. Multipliers. Let = (Ω) be a Hilbert function space on a set Ω. Let L∞ = L∞ (Ω) denote the space of bounded functionsH onH Ω normed by the supremum norm

h ∞ sup h (x) . k k ≡ x∈Ω | |

The supremum norm is relevant here as point evaluations are continuous in , and so h ∞ is a supremum of moduli of continuous linear functionals. We deﬁne the space H k k

∞ = ∞ (Ω) h : h < = L∞ (Ω) H H ≡{ ∈ H k k∞ ∞} H ∩ to consist of the bounded functions in , and we norm this space by H

h ∞ max h , h , k kH (Ω) ≡ {k kH k k∞} so that ∞ is a Banach space. A functionH ϕ is said to be a (pointwise) multiplier of if ϕf for all f . The collection of all H ∈ H ∈ H multipliers of is known to be a Banach algebra which we denote by MH. Indeed, (see e.g. [AgMc2]) if ϕ is a multiplierH of , and if we denote the linear operator of multiplication by H

ϕf ϕf, M ≡ then by the closed graph theorem

ϕf ϕ sup k kH < . MH ϕ H→H k k ≡ kM k ≡ f f f ∞ ∈H: 6=0 k kH If in addition 1 we have ϕ and ∈ H ∈ H

ϕ = ϕ1 ϕ 1 . k kH kM kH ≤ kM kH→H k kH

Finally and most importantly, ϕ is bounded in Ω by ϕ , i.e. kM kH→H

(2.4) ϕ ϕ . k k∞ ≤ kM kH→H Indeed, for all x Ω we have ∈ 2 2 ϕ (x) kx = ϕ (x) kx (x)= ϕkx, kx ϕ kx . | |k kH | | |h iH|≤k kMH k kH ∗ Moreover we have kx = ϕ (x)kx for all x Ω since Mϕ ∈

∗ f, ϕ (x)kx = ϕ (x) f, kx H = ϕ (x) f (x)= ϕf (x)= ϕf, kx H = f, ϕkx H h i M hM i M H D E for all f . Thus we have shown that M embeds in ∞ with ∈ H H H

ϕ ∞ max 1, 1 ϕ . k kH ≤ { k kH} k kMH 6 E. T. SAWYER AND B. D. WICK

2.1.2. Kernel multipliers. There is a Banach space K intermediate between M and ∞ (Ω) that plays a H H H major role in this paper, namely the Banach space KH of kernel multipliers consisting of all functions ϕ on Ω for which ϕ max ϕ1 , sup ϕk < , KH a k k ≡ H a∈Ω H ∞ 1 ka where 1 is the constant function k1k normalized toe have -norm f 1. Let ϕ KH. Clearly, from ka = H H ∈ √ka(a) and the reproducing property of ka, we have e f 1 ϕ (a) = ϕk , k = ϕk , k ϕk k ϕ , a a H a a a a KH | | ka (a) |h i | H ≤ H H ≤k k ∞ D E and so KH embeds in with f f f f H ϕ ∞ max 1, 1 ϕ . k kH ≤ { k kH} k kKH

Moreover, MH embeds in KH with ϕ K ϕ M since ϕka ϕ M ka = ϕ M if ϕ MH. k k H ≤ k k H H ≤ k k H H k k H ∈ Thus we have the embeddings

f , ֒ f M ֒ K ֒ ∞ H → H → H → H that show that the multiplier algebra MH is contained in the kernel multiplier space KH which is contained in the space ∞. Finally, we note that M multiplies the spaces K and ∞ as well as , i.e. that M is H H H H H H contained in both MK and M ∞ . Indeed, if ϕ M and f K , then H H ∈ H ∈ H ϕf = max ϕf1 , sup ϕfk ϕ max f1 , sup fk = ϕ f , KH a MH a MH KH k k H a∈Ω H ≤k k H a∈Ω H k k k k

and e f e f ϕf ∞ = max ϕf , ϕf ϕ max f , f = ϕ f ∞ . k kH {k kH k k∞}≤k kMH {k kH k k∞} k kMH k kH Of particular importance in this paper is the case when KH is an algebra. This occurs for example in the ∞ case = MH, as happens when is the classical Hardy or Bergman space on a bounded domain with 2 H n H ∞ C boundary in C . We also note that KH may be an algebra even if = MH. For example, KH is an σ H 6 algebra when is any of the Besov-Sobolev spaces B (Bn), σ > 0 and n 1, of analytic functions on the H 2 ≥ ball Bn. See Subsection 7.1 below for this. σ B 1 We recall at this point that the Corona Property has been proved for MH when = B2 ( n)for 0 σ 2 and n 1; the case n = 1 is in [Car], [Tol], [ArBlPa] and [Xia2], and the caseHn > 1 is in [CoSaWi].≤ ≤ In ≥ ∞ 0 D addition the Corona Property has been proved by Nicolau [Nic] for the algebra when = B2 ( ) is the classical Dirichlet space on the disk. In Subsection 7.2 we use the Peter JonesH solutionH [Jo], [Jo2] to the ∂-equation in the unit disk D, together with an adaptation of the argument of Arcozzi, Blasi and Pau [ArBlPa], to prove the Corona Property for the algebras K σ D . No other corona theorems for the algebras B2 ( ) ∞ σ B MH, KH or are currently known when = B2 ( n). However, we will reduce the Corona Property for H H σ B the algebra of kernel multipliers KH associated to = B2 ( n), to a simpler property we call the Convex Poisson Property for . See Theorem 29 where thisH is shown to hold for more general Hilbert function spaces and their associatedH kernel multiplier algebra K . H H 2.1.3. Shifted spaces and multiplier stability. Deﬁnition 4. If is a Hilbert function space on a set Ω with nonvanishing kernel function k, then for each H a 1 1 a Ω, we deﬁne the a-shifted Hilbert space to be δ where δ = , the -rescaling of , and where k k k ∈ H H fa fa H 1 ka = ka is the normalized reproducing kernel for . √ka(a) H fLemma 5. Let be a Hilbert function space on a set Ω with nonvanishing kernel function k. Then the a H a space consists of those complex-valued functions f on Ω such that kaf , and the inner product in is givenH by ∈ H H f f,g Ha = kaf, kag , f,g . h i H ∈ H D E In particular f Ha = kaf . f f k k H

f THE CORONA PROBLEM FOR KERNEL MULTIPLIER ALGEBRAS 7

Proof. Deﬁne a to be the linear space of functions on Ω having the form 1 h with h , and deﬁne an k G fa ∈ H a a inner product on by f,g Ga kaf, kag for f,g . We have G h i ≡ H ∈ G D k (ζ,E η) 1 k kδ (ζ, η)=f f and so kδ = η a. η ∈ G ka (ζ) ka (η) ka (η) ka It is easy to see that a is complete in the norm derived from this inner product, hence is a Hilbert space, f and we now show thatG point evaluationsf are continuousf on a. Indeed,f if f = 1 h with h then k G fa ∈ H 1 1 1 f (η) = h (η)= h, kη H = kaf, kη ka (η) ka (η) h i ka (η) H D E 1 = k f, k (η)k kδ = k f, k kδ f = f, kδ f a af a η af a η η Ga ka (η) H H D E D E and so f f f f f f

δ δ δ kη kη kη H f (η) f Ga kη a = f Ga kakη, kakη = f Ga , = f Ga k k . | |≤k k G k k H k k v k k r u*ka (η) ka (η) + ka (η) D E u H f f t Thus a is a Hilbert function space on Ω, and the above calculation shows that the reproducing kernel for f f f a is kGδ. By Proposition 3 a = a, and this completes the proof of the lemma. G G H J δ a We can give an alternate proof by computing that if f = xik (ζ) , then i=1 ηi ∈ H J J J P J J 2 δ δ δ δ δ k ηj , ηi f Ha = xikη , xj kη = xixj kη , kη = xixj k ηj , ηi = xixj k k i j i j Ha *i=1 j=1 + a i,j=1 i,j=1 i,j=1 ka (η )ka η X X H X D E X X i j and J J J J f f kη δ δ kηi j kaf, kaf = xikakη , xj kakη = xika , xj ka H i j *i=1 j=1 + *i=1 k (η )k j=1 k η k + D E X X H X a i a X a j a H f f J f fk J f f kηi ηj kηi ηj = xixj , = xixjf f , f f i,j=1 *k (η ) k η + i,j=1 k (η )k η X a i a j H X a i a j J δ a are equal. Now use that functions of the form f = xik (ζ) are dense in by deﬁnition. f f i=1 ηi f f H Despite the diﬀerence in norms of the shifted spacesP a, the multiplier algebras coincide and have identical norms. This is proved in [AgMc2, p.25] where it is shownH that rescaling a kernel leaves the multiplier algebra and the multiplier norms unchanged. We give the simple proof in our setting here.

Lemma 6. Let be a Hilbert function space on a set Ω with nonvanishing kernel function. Then MHa = MH with equality ofH norms for all a Ω. ∈ Proof. Fix a Ω. Suppose ﬁrst that ϕ M . We claim that ϕ M a with ϕ ϕ . Indeed, if H H MHa MH a ∈ 1 ∈ ∈ k k ≤k k f , then f = g where g with g = f a , and we have k H H ∈ H fa ∈ H k k k k 1 1 1 ϕf = ϕ g = ϕg = G, ka ka ka where G ϕg with G H ϕ M g H, and hence ≡ ∈ H k k ≤k k H k k f f f

ϕf Ha = kaϕf = G H ϕ M g H = ϕ M f Ha . k k H k k ≤k k H k k k k H k k

This proves the claimed inequality: ϕ ϕ . fk k MHa ≤k kMH Conversely, suppose that ϕ MHa . We claim that ϕ MH with ϕ ϕ . Indeed, if g , ∈ ∈ k kMH ≤ k kMHa ∈ H a then g = kaf where f with f a = g , and we have ∈ H k kH k kH ϕg = ϕk f = k ϕf = k F f a a a f f f 8 E. T. SAWYER AND B. D. WICK

a where F ϕf with F a ϕ f a , and hence ≡ ∈ H k kH ≤k kMHa k kH

ϕg H = kaϕf = F Ha ϕ M a f Ha = ϕ M a g H . k k H k k ≤k k H k k k k H k k

Hence we have: ϕ ϕ . These two inequalities show that MHa = MH with equality of norms. k kMH ≤k kM Hfa At this point we introduce the first main assumption needed for our alternate Toeplitz corona theorem. Definition 7. We say that the Hilbert function space is multiplier stable if H (1) the reproducing kernel functions kx are nonvanishing and are invertible multipliers on , i.e. kx 1 H ∈ MH and MH, for all x Ω, and kx ∈ ∈ (2) the map x kx from Ω to M is lower semicontinuous. → H 1 Note that we make no assumptions regarding the size of the norms of the multipliers kx and in this kx definition. We will see below that all the Besov-Sobolev spaces on the ball are multiplier stable, as well as the Bergman and Hardy spaces on strictly pseudoconvex domains with C2 boundary. A crucial consequence of the multiplier stable assumption is the -Poisson reproducing formula below. H Lemma 8. Suppose is a Hilbert function space on a set Ω with nonvanishing kernel and containing the H constant functions. Suppose furthermore that kx MH for all x Ω. Then for each a Ω we have the -Poisson reproducing formula ∈ ∈ ∈ H (2.5) f (a)= f, 1 a , f (Ω) , a Ω. h iH ∈ H ∈

Proof. Since ka MH by hypothesis, the function Fa (w) ka (w) f (w) is in for f , and so using Lemma 5, ∈ ≡ H ∈ H f ka (a)f (a)= ka (a) f (a)= Fa (a)= Fa, ka H = ka (a) kaf, ka = ka (a) f, 1 Ha , h i H h i p p D E p which gives (2.5). f f f The spaces a are in fact all equal to as sets, with diﬀerent but comparable norms (the constants of comparability needH not be bounded in a).H Lemma 9. Suppose is multiplier stable. For a Ω we have comparability of the norms for and a: H ∈ H H 1 a h H h Ha ka h H , h . 1 k k ≤k k ≤ MH k k ∈H∪H k fa MH f 1 Proof. Since ka, MH we see that ka ∈

f Ha = kaf ka f H , k k H ≤ MH k k

f1 f 1 1 g H = k ag kag = g Ha , k k k ≤ k H k k k a H a MH a MH

and these two inequalities prove the lemma.f f f f f 2.2. Interpolation by rescalings. Deﬁnition 10. Given a multiplier stable Hilbert function space on Ω with kernel k, and (a, θ) ΩM H ∈ × [0, 1]M+1, deﬁne the Hilbert function space a,θ to be with inner product given by H H M

f,g a,θ θ f,g + θm f,g a , f,g . h iH ≡ 0 h iH h iH m ∈ H m=1 X am We recall from Lemma 9 that all of the spaces are comparable, hence the inner product f,g Ha,θ a θ H h i is deﬁned for f,g , and all of the spaces , are comparable with . We will often use the convention a0 ∈ H H M H kη = kη in order to simplify the sum above to f,g Ha,θ = m=0 θm f,g Ham . We will refer to the spaces a,θ h i h i as the convex shifted spaces associated with . They are normed by f a,θ = f,f a,θ . H H P k kH h iH p THE CORONA PROBLEM FOR KERNEL MULTIPLIER ALGEBRAS 9

Let M M M+1 ΣM θ = θm [0, 1] : θj =1 ≡  { }m=0 ⊂  j=0  X  denote the unit (M + 1)-dimensional simplex.   Assumption: We now make the standing assumption, in force for the remainder of the paper, that contains the constant functions on Ω, and upon multiplying by a positive constant, we may assumeH that 1 =1. k kH a,θ M Corollary 11. The norm of the constant function 1 in the space is 1 for all a Ω and θ ΣM . H ∈ ∈ Proof. We have M M M 2 1 Ha,θ = 1, 1 Ha,θ = θm 1, 1 Ham = θm kam , kam = θm =1. k k h i h i H m=0 m=0 m=0 X X D E X g g

2.2.1. Vector-valued norms. Let N 1 be fixed. Given Hilbert spaces ℓ for 1 ℓ N, define a complete N≥ H ≤ ≤ inner product on the direct sum ℓ by ⊕ℓ=1H N N N N f,g ⊕N H fℓ,gℓ H , f = (fℓ)ℓ=1 ,g = (gℓ)ℓ=1 ℓ=1 ℓ . h i ℓ=1 ℓ ≡ h i ℓ ∈⊕ H ℓ X=1 N N When all the spaces ℓ are equal to the same space , we write simply in place of ℓ=1 ℓ. H H ⊕ H N ⊕ H Given Banach spaces Bℓ for 1 ℓ N, define a complete norm on the direct sum Bℓ by ≤ ≤ ⊕ℓ=1 N N ϕ ⊕N B max ϕℓ B , ϕ = (ϕℓ)ℓ=1 ℓ=1Bℓ , k k ℓ=1 ℓ ≡ 1≤ℓ≤N k k ℓ ∈⊕ N N and again, when all the spaces Bℓ are equal to the same space B, we write simply B in place of Bℓ. ⊕ ⊕ℓ=1 In the case when the Bℓ are also Hilbert spaces, this definition differs from the previous one, but the intended definition should always be clear from the context. We will be mainly concerned with the cases B = MH and B = KH. When B = MH is the multiplier algebra of a Hilbert function space on a set Ω, there are two additional natural norms to consider, namely the row and column norms, to whHich we now turn. Let be a Hilbert function space on a set Ω. For ϕ N M define H ∈⊕ H N N ϕ : by ϕf = ϕ fα, M ⊕ H → H M α α=1 X N N Mϕ : by Mϕh = (ϕ h) . H→⊕ H α α=1 N ∗ ∗ Then we have ϕh = ϕ f , and in particular, M M α α=1 N N ∗ ∗ (2.6) ϕkz = ϕ kz = ϕℓ (z)kz . M M α α=1 α=1 We also define the row and column normse by e e M f ϕh N ϕ H M k k⊕ H ϕ op sup kM k and ϕ op sup . kM k ≡ f f N k k ≡ h h 6=0 k k⊕ H 6=0 k kH The two key inequalities involving these norms are:

1 N 2 2 ϕ L∞ ℓ2 sup ϕℓ (z) ϕ op , k k ( N ) ≡ z | | ≤ kM k ∈Ω ℓ ! X=1 ϕ N Mϕ 1 . k k⊕ H ≤ k kop k kH 10 E. T. SAWYER AND B. D. WICK

The ﬁrst inequality follows from (2.6), N N N 2 2 2 2 2 ∗ ϕℓ (z) = ϕℓ (z) kz = ϕℓ (z)kz = ϕkz | | | | H H M ⊕N H ℓ ℓ ℓ X=1 X=1 X=1 2 2 ∗ 2 e 2 e 2 e ϕ kz = ϕ op kz = ϕ op , ≤ M op H kM k H kM k

and the second inequality follows from e e N N N 2 2 2 2 M 2 ϕ N = ϕ = ϕ 1 = ϕ 1 = ϕ1 N k k⊕ H k ℓkH k ℓ kH M ℓ H k k⊕ H ℓ=1 ℓ=1 ℓ=1 X 2 2 X X Mϕ 1 . ≤ k kop k kH Then from our assumption that the norm of 1 in the space is 1, we have both of the inequalities ϕ L∞(ℓ2) H N k k ≤ ϕ and ϕ N Mϕ . We now have three norms on the Banach space M . kM kop k k⊕ H ≤k kop ⊕ H Definition 12. Given ϕ N M , define the three norms ∈⊕ H row column M max ϕ N M ϕ , ϕ N M ϕ , ϕ N M max ϕℓ M . k k⊕ H ≡ kM kop k k⊕ H ≡k kop k k⊕ H ≡ 1≤ℓ≤N k k H These norms are comparable since max row column ϕ N min ϕ N , ϕ N k k⊕ MH ≤ k k⊕ MH k k⊕ MH n row columno max max ϕ N , ϕ N √N ϕ N . ≤ k k⊕ MH k k⊕ MH ≤ k k⊕ MH n o When B = KH is the Banach space of kernel multipliers on , and in the presence of the standing assumption 1 = 1, we will use the following natural norm on the directH sum N K . k kH ⊕ℓ=1 H Definition 13. Let be a Hilbert function space on a set Ω. For N 1 and ϕ N K define the H ≥ ∈ ⊕ℓ=1 H following norm on N K : ⊕ H

ϕ N max ϕ N , sup ϕk . ⊕ KH ⊕ H a k k ≡ k k a∈Ω ⊕N H N In the event that KH = MH isometrically, then the norm just introducedf on KH is comparable to the three introduced on N M above: ⊕ ⊕ H column max (2.7) ϕ N ϕ N √N ϕ N √N ϕ N . k k⊕ KH ≤k k⊕ MH ≤ k k⊕ MH ≤ k k⊕ KH 2.3. A characterization of rescaling. We end this section on preliminaries with a characterization of when two kernels are rescalings of each other. For this we begin with a quick review of relevant properties of positive matrices. Recall that an n n self-adjoint matrix A of complex numbers is said to be positive, denoted A < 0, if all of its eigenvalues× are nonnegative; and said to be strictly positive, denoted A 0, if all of its eigenvalues are positive. Clearly sums of positive matrices are positive. Moreover, given≻ any λ1 0 0 ···. . 0 λ .. . self-adjoint A, there is a unitary matrix U such that UAU ∗ = Diag (λ ,...,λ ) =  2 , 1 n . . .  . .. .. 0     0 0 λn  2  ···  v v v  v vn  | 1| 1 2 ··· 1 2 .. . ∗  v2v1 v2 . .  with λj R. A dyad is a rank one matrix of the form v v = | | . Every ∈ ⊗ . .. ..  . . . vn−1vn   2   vnv vnvn vn   1 ··· −1 | |   I ∗  ∗ dyad is positive, and conversely, every positive matrix A is a sum αivi v of dyads vi v with i=1 ⊗ i ⊗ i nonnegative coeﬃcients αi. However, such decompositions are not in general unique. For example, any positive sum of dyads is a positive matrix A, and the spectral theoremP for A gives in general a diﬀerent decomposition into dyads with pairwise orthogonal vectors (the eigenvectors of the matrix A). One important THE CORONA PROBLEM FOR KERNEL MULTIPLIER ALGEBRAS 11

I ∗ consequence is that the Schur product of two positive matrices is positive. Indeed, if A = αivi v i=1 ⊗ i and B = J β w w∗, then A B = I J α β (v v∗) w w∗ , and it is easily veriﬁed that j=1 j j j i=1 j=1 i j i i j j P ∗ ∗ ⊗ ◦ ∗ ⊗ ◦ ⊗ (vi v ) wj w = (vi wi) (vj wj ) . ⊗ i ◦P ⊗ j ◦ ⊗ ◦ P P Now we turn to the problem of deciding when two self-adjoint nonvanishing functions K (x, y) and k (x, y) on a product set Ω Ω are rescalings of each other. The surprisingly simple answer depends only on the 2 2 and 3 3 principal× submatrices of the inﬁnite matrices [K (x, y)] and [k (x, y)] . × × (x,y)∈Ω×Ω (x,y)∈Ω×Ω Proposition 14. Suppose that K and k are two self-adjoint nonvanishing functions on a product set Ω Ω. Then K and k are rescalings of each other, i.e. there is a nonvanishing function ψ on Ω such that K× = (ψ ψ∗) k if and only if the following two conditions hold: ⊗ ◦ 2 2 (1) |K(x,y)| = |k(x,y)| for all x, y Ω, K(x,x)K(y,y) k(x,x)k(y,y) ∈ (2) arg K(x,z) = arg K(x,y) + arg K(y,z) (mod2π) for all x,y,z Ω. k(x,z) k(x,y) k(y,z) ∈ Proof. If K = (ψ ψ∗) k, then K (x, y)= ψ (x) k (x, y) ψ (y) and we have ⊗ ◦ 2 K (x, y) 2 ψ (x) k (x, y) ψ (y) k (x, y) 2 = = , | | | | K (x, x) K (y,y) ψ (x) k ( x, x) ψ (x)ψ (y) k (y,y ) ψ (y) k (x, x) k (y,y)

and K (x, z) arg = arg ψ (x) ψ (z) = arg ψ (x) ψ (y) 2 ψ (z) k (x, z) | | = arg ψ (x) ψ (y)+arg ψ (y) ψ (z) K (x, y) K (y,z) = arg + arg . k (x, y) k (y,z) J Conversely assume that both conditions (1) and (2) hold. Then given any ﬁnite set of points xj in { }j=1 Ω, deﬁne a matrix K(xi,xj ) J √K(xi,xi)√K(xj ,xj ) U [uij ] ; uij = . ≡ i,j=1 k(xi,xj ) √k(xi,xi)√k(xj ,xj ) and note that U is self-adjoint since K and k are, and that U is unimodular by condition (1). If we set iθij uij = e , then condition (2) says that

K (xi, xℓ) Kiℓ Kij Kjℓ (2.8) θiℓ = arg = arg = arg + arg k (xi, xℓ) kiℓ kij kjℓ

K (xi, xj ) K (xj , xℓ) = arg + arg = θij + θjℓ . k (xi, xj ) k (xj , xℓ)

Now define θ1 = 0 and θj = θ1j for 1 j N, so that θj = θj1 since U is self-adjoint. Then θ1i θ1j = ≤ ≤ K(x ,x− ) − −iθj √ j j θi1 θ1j = θij by (2.8), and so with ψ (xj ) e , we obtain − − − ≡ √k(xj ,xj ) K (x , x ) −i(θ1i−θ1j ) K (xi, xi) j j ψ (xi) k (xi, xj ) ψ (xj ) = e k (xi, xj ) (p k (xi, xi) p k (xj , xj ) ) 1 = eiθij K (xp, x ) = K (x , x ) . p u i j i j ij J Note that the vector ψ (xi) i=1 is determined uniquely up to a unimodular constant. Finally, we apply any{ appropriate} form of transfinite induction. From what we have done above, we get ∗ a consistent definition of ψ on an increasing maximal chain of subsets of Ω since two dyads ψ1 ψ1 and ψ ψ∗ are equal if and only if there is a unimodular constant eiθ such that ψ = eiθψ . Now we apply⊗ Zorn’s 2 ⊗ 2 1 2 Lemma to get a nonvanishing function ψ on Ω such that K (x, y) = ψ (x) k (x, y) ψ (y). If both kernels are K(x,y) holomorphic in their first variable x and antiholomorphic in their second variable y, then ψ (x) ψ (y)= k(x,y) 12 E. T. SAWYER AND B. D. WICK

is holomorphic and nonvanishing in x, which proves that ψ (x) is holomorphic and nonvanishing in x. Finally, √K(x,x) the supremum bounds follow from the formula ψ (x) = . | | √k(x,x)

Remark 15. Condition (1) is equivalent to the equality of distance functions dk (x, y)= dK (x, y), x, y Ω, ∈ where for any kernel k on Ω, the distance function dk is deﬁned by

k (x, y) 2 dk (x, y) 1 | | . ≡ s − k (x, x) k (y,y)

Note that dk (x, y) = sin θx,y where θx,y is the angle between kx and ky in the Hilbert function space k. H

3. The Bezout kernel multiplier characterization The following definition will be used to characterize when we can solve Bezout’s equation with a vector in the space N K of kernel multipliers of . ⊕ℓ=1 H H Definition 16. Let be a Hilbert function space on a set Ω with nonvanishing kernel, and let a be the H N ∞ H shifted Hilbert space for a Ω. We say that a vector ϕ ℓ=1L (Ω) satisfies the convex Poisson ∈ ∈ ⊕ H− M condition with positive constant C if for every finite collection of points a = (a1,...,aM ) Ω and every M ∈ θ M a,θ N collection of nonnegative numbers = θm m=0 summing to 1 = θm, there is a vector g ℓ=1 { } m=0 ∈ ⊕ H satisfying P a θ (3.1) ϕ (z) g , (z) = 1, z Ω, · ∈ M a,θ 2 a,θ 2 a,θ 2 2 g ⊕N Ha,θ = θ0 g ⊕N H + θm g ⊕N Ham C . ℓ=1 ℓ=1 ℓ=1 ≤ m=1 X We denote the smallest such constant C by ϕ . k kcPc 3.1. Kernel multiplier solutions. Here now is our abstract characterization of solutions to Bezout’s equation, which is of primary interest in those cases where the space of kernel multipliers KH is an algebra. But first we require additional structure on our Hilbert function space to substitute for Montel’s theorem in complex analysis. This leads to the second main assumption neededH for our alternate Toeplitz corona theorem. Definition 17. Let Ω be a topological space. A Hilbert function space of continuous functions on Ω is said to be have the Montel property if there is a dense subset S of Ω withH the property that for every sequence ∞ ∞ fn n=1 in the unit ball of , there are a subsequence fnk k=1 and a function g in the unit ball of , such that{ } H { } H

lim fnk (x)= g (x) , x S. k→∞ ∈ A main ingredient in the proof is the following minimax lemma of von Neumann, proved for example in [Gam]. This lemma was introduced in this context by Amar [Amar], and used subsequently by Trent and Wick [TrWi] as well. Lemma 18. Suppose that M is a convex compact subset of a normed linear space, and that P is a convex subset of a vector space. Let : M P [0, ) satisfy F × → ∞ (1) for each ﬁxed p P , the section p given by p (m)= (m,p) is concave and continuous, ∈ F F F

(2) for each ﬁxed m M, the section m given by m (p)= (m,p) is convex. ∈ F F F Then the following minimax equality holds: sup inf (m,p) = inf sup (m,p) . m∈M p∈P F p∈P m∈M F THE CORONA PROBLEM FOR KERNEL MULTIPLIER ALGEBRAS 13

Theorem 19. Let be a multiplier stable Hilbert function space of continuous functions on a separable topological space Ω,H and assume that has the Montel property. Suppose that ϕ N L∞ (Ω), and that H ∈ ⊕ℓ=1 C > 0 is a positive constant. Then there is a vector function f N K satisfying ∈⊕ H (3.2) ϕ (z) f (z) = 1, z Ω, · ∈ f N C, k k⊕ KH ≤ if and only if ϕ satisfies the convex Poisson condition in Definition 16 with constant ϕ C. H− k kH−cPc ≤ Porism: If we drop the lower semicontinuity assumption (2) in the definition of multiplier stability, then the proof below shows that if ϕ N L∞ (Ω) satisfies the convex Poisson condition in ∈ ⊕ℓ=1 H− Definition 16, then there is a vector function f N ∞ satisfying ∈⊕ H ϕ (z) f (z) = 1, z Ω, · ∈ f N ∞ ϕ . k k⊕ H ≤ k kH−cPc However, there is no converse assertion here in general.

Proof. Fix for the moment an integer M 1 and a positive constant εM to be chosen later. Now ﬁx M ≥ a = am Ω and consider the simplex { }m=1 ⊂ M M M+1 ΣM θ = θm [0, 1] : θj =1 . ≡  { }m=0 ⊂  j=0  X  a,θ a,θ For each θ ΣM pick g (Ω) as in the cPc Deﬁnition 16 for ϕ, i.e. ∈ ∈ H   a θ ϕ(z) g , (z) = 1 in Ω, a,θ · g ⊕N Ka,θ(Ω) ϕ cPc . ℓ=1 ≤ k k

Note that a,θ θ θ′ (3.3) g ⊕N Ha,θ′ (Ω) < , for all , ΣM , ℓ=1 ∞ ∈ a,θ since as observed earlier, it follows from Lemma 9 that all of the interpolating spaces (Ω) are comparable with . It is here that we use the full force of our assumption that is multiplierH stable, as opposed to H H merely assuming that the kernel functions ka are nonvanishing multipliers. Indeed, with only the latter 2 a 1 2 ka′ assumption, an element of has the form g for g , and then g a′ = f g could be inﬁnite. k H k H fa ∈ H k k fa H Thus a,θ , and we can deﬁne the set H ⊂ H a,θ a,M convex hull g : θ ΣM (εM ) C ≡ ∈ N a,θ to be the convex hull in ℓ=1 of these Bezout solutions g . We will apply the von Neumann minimax equality in Lemma 18 to⊕ the functionalH

M θ 2 2 2 a ( ,f) f ⊕N Ha,θ = θ0 f ⊕N H + θm f ⊕N Ham , F ≡k k ℓ=1 k k ℓ=1 k k ℓ=1 m=1 X deﬁned for θ ΣM , and for f a,M , noting that a (θ,f) is then ﬁnite by (3.3). ∈ ∈C F Both ΣM and a,M are convex, and in addition ΣM is compact. The functional a (θ,f) is linear and θ C L F continuous in , hence also concave. It is also convex in f since if λi 0 and i=1 λi = 1 and Ai a,M , then ≥ ∈C P L L 2 L 2 θ a , λiAi = λiAi λi Ai ⊕N Ha,θ(Ω) F ≤ k k ℓ=1 i=1 ! i=1 N a,θ i=1 ! ⊕ℓ=1H (Ω) X X X L L 2 θ λi Ai ⊕N Ha,θ(Ω) = λi a ( , Ai) . ≤ k k ℓ=1 F i=1 i=1 X X 14 E. T. SAWYER AND B. D. WICK

Thus the minimax equality in Lemma 18 applies to give

inf sup a (θ,f) = sup inf a (θ,f) a a f∈C ,M θ∈ΣM F θ∈ΣM f∈C ,M F a,θ sup a θ,g ≤ θ∈ΣM F a,θ 2 2 = sup g N a,θ ϕ cPc . ⊕ℓ=1H (Ω) θ∈ΣM ≤k k

(M) a,θ (M) So for each M 1, we can pick f a,M (Ω) so that f is almost optimal for inff∈Ca,M in the display above, more≥ precisely, ∈C ⊂ H θ (M) 2 sup a ,f < (1 + εM ) ϕ cPc . θ∈ΣM F k k (M) We also have ϕ(z) f (z)= 1. Let ej = (θ0,θ1,...,θM ) ΣM where θj = 1 and θk = 0 for k = j. Then we have · ∈ 6 2 (M) (M) (M) 2 (3.4) f = a e0,f sup a θ,f (1 + εM ) ϕ , N cPc ⊕ℓ=1H(Ω) F ≤ θ∈ΣM F ≤ k k and for each ℓ 1 we have, ≥ 2 2 (M) (M) (M) θ (M) 2 (3.5) f = f a eℓ,f sup a ,f (1 + εM ) ϕ cPc . N aℓ N aℓ ⊕ℓ=1H (Ω) ⊕ℓ=1H (Ω) ≤F ≤ θ∈ΣM F ≤ k k

Now we use the multiplier stable hypothesis together withe the -Poisson reproducing formula (2.5) in Lemma 8, and then the Cauchy-Schwarz inequality for inner products,H to obtain the pointwise estimates

2 2 (M) (M) (M) (M) f (am) = f , 1 f ,f 1, 1 ⊕N Ham ⊕N Ham ≤ ⊕N Ham h i ℓ=1 ℓ=1 ℓ=1 D 2 E D E (M) 2 = f kam , kam (1 + εM ) ϕ . N am N cPc ⊕ℓ=1H (Ω) ⊕ℓ=1H ≤ k k D E ∞ Now by separability of Ω, we can choose a dense setgSg= am in Ω as in Deﬁnition 17, and then { }m=1 choose the positive constants εM so small that

lim εM =0. M→∞ From (3.4) and the fact that , and so also N , has the Montel property, we conclude that there is a H ⊕ℓ=1H vector function f N with ∈⊕ℓ=1H 2 2 2 f ⊕N H lim (1 + εM ) ϕ cPc = ϕ cPc k k ℓ=1 ≤ M→∞ k k k k and (M) f (am) = lim f (am) , m 1. M→∞ ≥ Thus we have (M) (M) 1 = lim 1 = lim ϕ (am) f (am)= ϕ (am) lim f (am)= ϕ (am) f (am) M→∞ M→∞ · · M→∞ · N for all m 1, and hence f ℓ=1 is a solution to the Bezout equation ϕ (z) f (z) = 1 for z Ω since ϕ f is continuous≥ and S is dense.∈ ⊕ WeH also have from (3.5) the pointwise estimate· ∈ · (M) f (am) = lim f (am) lim sup (1 + εM ) ϕ cPc = ϕ cPc | | M→∞ ≤ M→∞ k k k k

N for each m 1. Since f satisﬁes f N ϕ and is continuous in Ω, and since S = ℓ=1 ⊕ℓ=1H cPc ∞ ≥ ∈ ⊕ H k k ≤ k k am is dense in Ω, we thus have { }m=1

f ⊕N H∞ = max f ⊕N H , f L∞ ℓ2 ϕ cPc , k k ℓ=1 k k ℓ=1 k k ( N ) ≤k k n o since f L∞ ℓ2 = supz∈Ω f (z) . k k ( N ) | | THE CORONA PROBLEM FOR KERNEL MULTIPLIER ALGEBRAS 15

Finally, using the lower semicontinuity assumption (2) in the deﬁnition of multiplier stability (which has not been needed until now), we also have the stronger estimate

(3.6) f N = max f , sup fk = max f , sup f a ϕ . ⊕ KH H a H H cPc k k ℓ=1 k k a∈Ω H k k a∈Ω k k ≤k k Indeed, upon letting M in (3.5), we obtain f → ∞

max f , sup f a ϕ . k kH k kH m ≤k kcPc 1≤m<∞

The map a ka is lower semicontinuous from Ω to M by assumption, and it follows that f a = → H k kH fka = k f is a lower semicontinuous function of a Ω. The proof of (3.6) is now completed using H a H M∞ff ∈ that S = a m is dense in Ω, so that f a lim infa a f a ϕ for all a Ω. { }m=1 k kH ≤ m→ k kH m ≤k kcPc ∈ fThe converse assertion is straightforward. Suppose that f N K solves ϕ f = 1 in Ω. Then ∈ ⊕ℓ=1 H · fkam f N and so N ⊕ℓ=1KH ⊕ℓ H ≤k k =1 M g 2 2 2 f ⊕N Ha,θ = θ0 f ⊕N H + θm f ⊕N Ham k k ℓ=1 k k ℓ=1 k k ℓ=1 m=1 X M 2 2 = θ0 f ⊕N H + θm fkam k k ℓ=1 ⊕N H m=1 ℓ=1 X M g 2 2 2 θ0 f ⊕N K + θm f ⊕N K = f ⊕N K . ≤ k k ℓ=1 H k k ℓ=1 H k k ℓ=1 H m=1 X

4. The alternate Toeplitz corona theorem We begin by establishing notation, in particular various corona properties. 4.1. Corona Properties. Fix N N throughout this subsection. Given a Hilbert function space on a N N ∈ H set Ω, denote by KH and MH the direct sum of the kernel multiplier spaces KH and the multiplier ⊕ ⊕ column algebras M respectively, equipped with the norms ϕ N and ϕ N respectively, as introduced in H ⊕ KH ⊕ MH k k column k k a previous subsection. We often write ϕ N = ϕ N from now on. Next we deﬁne two ‘baby’ ⊕ MH ⊕ MH properties: k k k k

(1) the first solves ϕ f = ka with f for all a Ω and all ϕ KH in the kernel multiplier space, (2) the second solves·ϕ f = h with f∈ H for all h∈ and all ϕ∈ M in the multiplier algebra. · ∈ H ∈ H ∈ H Note that we typically define conditions that a vector ϕ of corona data might have, and we typically define properties that a space might have in terms of these conditions. 4.1.1. Kernel Corona Property. Definition 20. Suppose is a Hilbert function space on a set Ω. H (1) Let C > 0. We say that a vector ϕ = (ϕ ,...,ϕ ) N L∞ (Ω) satisfies the kernel corona 1 N ∈ ⊕ H− condition with constant C if for each a Ω there are f ,...,fN satisfying ∈ 1 ∈ H 2 2 2 2 2 (4.1) f N = f + + fN C ka , k k⊕ H k 1kH ··· k kH ≤ k kH (ϕ f) (z) = ϕ (z) f (z)+ + ϕ (z) fN (z)= ka (z) , z Ω. · 1 1 ··· N ∈ (2) Let c,C > 0. We say that has the Kernel Corona Property with constants c, C if for every vector ϕ = (ϕ ,...,ϕ ) N K Hsatisfying both 1 N ∈⊕ H ϕ N 1 k k⊕ KH ≤ and (4.2) ϕ (z) 2 + + ϕ (z) 2 c2 > 0, z Ω, | 1 | ··· | N | ≥ ∈ the vector ϕ = (ϕ ,...,ϕ ) satisfies the -kernel corona condition with constant C. 1 N H 16 E. T. SAWYER AND B. D. WICK

4.1.2. Baby Corona Property. Definition 21. Suppose is a Hilbert function space on a set Ω. H (1) Let C > 0. We say that a vector ϕ = (ϕ ,...,ϕ ) N L∞ (Ω) satisfies the baby corona 1 N ∈ ⊕ H− condition with constant C if for each h there are f ,...,fN satisfying ∈ H 1 ∈ H 2 2 2 2 2 (4.3) f N = f + + fN C h , k k⊕ H k 1kH ··· k kH ≤ k kH (ϕ f) (z) = ϕ (z) f (z)+ + ϕ (z) fN (z)= h (z) , z Ω. · 1 1 ··· N ∈ (2) Let c,C > 0. We say that has the Baby Corona Property with constants c, C if for every vector ϕ = (ϕ ,...,ϕ ) N M Hsatisfying both 1 N ∈⊕ H ϕ N 1 k k⊕ MH ≤ and ϕ (z) 2 + + ϕ (z) 2 c2 > 0, z Ω, | 1 | ··· | N | ≥ ∈ the vector ϕ = (ϕ ,...,ϕ ) satisfies the -baby corona condition with constant C. 1 N H 4.1.3. Corona Property. Given a Hilbert function space on Ω, and an algebra contained in ∞ = L∞ (Ω), we define the Corona Property for the algebraH as follows. There is a smallA abuse of notationH H ∩ A N here since the definition we give will depend on N and the norm N that we use for the direct sum . k·k⊕ A ⊕ A Thus we should really define the Corona Property for the triple ,N, N , but we will often suppress A k·k⊕ A the dependence on N and N and only specify them when needed. k·k⊕ A Definition 22. Suppose N 2, is a Hilbert function space on a set Ω, is an algebra contained in ∞, N ≥ H A H and the direct sum is normed by N . ⊕ A k·k⊕ A (1) Let C > 0. We say that the vector ϕ = (ϕ ,...,ϕ ) N satisfies the -corona condition with 1 N ∈ ⊕ A A constant C if for each h there are f ,...,fN satisfying ∈ A 1 ∈ A 2 2 2 (4.4) f N C h , k k⊕ A ≤ k kA ϕ f (z) = h (z) , z Ω. · ∈ (2) Let c,C > 0. We say that has the Corona Property with constants c, C if for every vector N A ϕ = (ϕ1,...,ϕN ) satisfying both ϕ ⊕N A 1 and (4.2), the vector ϕ = (ϕ1,...,ϕN ) satisfies the -corona∈ ⊕ conditionA with constantk Ck . ≤ A Remark 23. If there are f ,...,fN satisfying (4.4) with h = 1, then we can multiply the equation 1 ∈ A through by h , and use that is an algebra to obtain f1h,...,fN h , and hence that satisfies the Corona Property.∈ A A ∈ A A 4.2. Convex shifted spaces. At this point we pause to note that (3.1) holds for all the extreme cases θ = em provided a slight strengthening of it holds for θ = e0. More precisely we have the following lemma. N ∞ Lemma 24. Suppose that a vector ϕ ℓ=1L (Ω) satisfies the -kernel corona condition (4.1) with ∈ ⊕ N a H 2 2 constant C. Then for every a Ω, there is g (depending on a) with g ⊕N Ha C such that ϕ g =1. ∈ ∈ ⊕ H k k ≤ · N Proof. Given a Ω there is by assumption a vector f = (f ,...,fN ) satisfying ∈ 1 ∈⊕ H ϕ f (z) = ϕ (z) f (z)+ + ϕ (z) fN (z)= ka (z) , z Ω, · 1 1 ··· N ∈ 2 2 2 2 f N = f1 + + fN C . k k⊕ H k kH ··· k kH ≤ f Now divide both sides of the Bezout equation above by ka to obtain

f1 (z) fN (z) ϕ1 (z) + + ϕN (z)f =1, z Ω. ka (z) ··· ka (z) ∈

fℓ a a Then with gℓ for 1 ℓ N (here we use the deﬁnition of membership in ), we have k ≡ fa ∈ H ≤ ≤f f H 2 2 2 fℓ 2 gℓ Ha = gℓka = ka = fℓ H , k k H k k k a H

f f f THE CORONA PROBLEM FOR KERNEL MULTIPLIER ALGEBRAS 17

and so

ϕ g (z) = ϕ (z) g (z)+ + ϕ (z) gN (z)=1, z Ω, · 1 1 ··· N ∈ 2 2 2 2 2 2 g N a = g a + + gN a = f + + fN C . k k⊕ H k 1kH ··· k kH k 1kH ··· k kH ≤ We do not know if the -kernel corona condition (4.1) with constant C implies the ‘full’ convex Poisson condition with constant CH, or even with a larger positive constant C′. However, the convex Poisson N ∞ H− 2 D condition holds for every ϕ ℓ=1H (Ω) satisfying the -baby corona condition when = H ( ) is the classical Hardy space in the∈⊕ disk. This can be proved usingH an appropriate outer functionH in place of the reproducing kernel ka used in the proof above, and is carried out in Lemma 40 in Section 6 below. The outer function in question is actually an invertible multiplier on H2 (D), and this leads to the following deﬁnition. f Deﬁnition 25. Let = k be a multiplier stable Hilbert function space on a set Ω with reproducing kernel k, and containing theH constantH functions. We say that the kernel k has the Invertible Multiplier Property if M for every (a, θ) Ω ΣM (0), there is a normalized invertible multiplier ka,θ M such that ∈ × ∈ H

(4.5) f,g Ha,θ = ka,θf, ka,θg , f,g . h i H ∈ H g D E The normalized invertible multiplier ka,θ ing (4.5) isg uniquely determined up to a unimodular constant.

Lemma 26. If two normalized invertible multipliers ϕ1, ϕ2 MH satisfy g ∈ (4.6) ϕ f, ϕ g = ϕ f, ϕ g for all f,g , h 1 1 iH h 2 2 iH ∈ H then ϕ1 = Aϕ2 for some unimodular constant A. Proof. If equality holds in (4.6), then ϕ ϕ 1 1 1 1 1 f, 1 g = ϕ f , ϕ g = ϕ f , ϕ g = f,g , ϕ ϕ 1 ϕ 1 ϕ 2 ϕ 2 ϕ h iH 2 2 H 2 2 H 2 2 H and so in particular, ϕ1 ϕ1 f, kz = f, kz = f (z) , z Ω. ϕ ϕ h iH ∈ 2 2 H ϕ1(w) ϕ1(z) Thus the kernel Kz (w) ϕ (w) kz (w) satisﬁes ≡ 2 ϕ2(z)

ϕ1 (z) ϕ1 ϕ1 (z) ϕ1 ϕ2 ϕ1 ϕ1 (z) ϕ2 f,Kz = f, kz = f , kz = f (z)= f (z) h iH ϕ (z) ϕ ϕ (z) ϕ ϕ ϕ ϕ (z) ϕ 2 2 H 2 2 1 2 H 2 1 ϕ1(w) ϕ1(z) ϕ1(w) ϕ1(z) for all z Ω. By uniqueness of kernel functions, we obtain ϕ (w) kz (w)= kz (w), hence ϕ (w) = 1, ∈ 2 ϕ2(z) 2 ϕ2(z) since kz (w) is nonvanishing by multiplier stability. Thus we conclude that there is a nonzero constant A ϕ1(w) such that ϕ w = A for all w Ω. Finally, since both multipliers are normalized we have 1 = ϕ1 H = 2( ) ∈ k k Aϕ = A ϕ = A . k 2kH | |k 2kH | | The identity (4.5) is equivalent to the assertion that the reproducing kernels ka,θ and k of the Hilbert spaces a,θ and respectively, are rescalings of each other. Indeed, the reproducing kernel corresponding H H kw (z) to the inner product ka,θf, ka,θg is . It turns out that the Invertible Multiplier Property H ka θ (w)ka θ (z) g, g, is extremely rare - theD 1-dimensionalE Szegökernel has it, and this is essentially the only kernel we know with this property. Seeg Sectiong 6 below where we show that the Invertible Multiplier Property holds for the Hardy space on the disk, but fails for the Hardy space on both the ball and polydisc in higher dimensions. We now show that if satisfies the Invertible Multiplier Property, then the -baby corona condition is sufficient for the -convexH Poisson corona condition. H H Lemma 27. Let be a multiplier stable Hilbert function space on a set Ω, whose kernel k has the Invertible H N ∞ Multiplier Property. Then a vector ϕ ℓ=1L (Ω) satisfies the convex Poisson condition in Definition N∈⊕∞ H− 16 with positive constant C if ϕ ℓ=1L (Ω) satisfies the -baby corona condition (4.3) in Definition 21 with constant C. ∈⊕ H

Proof. Repeat the proof of Lemma 24 using the invertible multiplier ka,θ in place of ka and use (4.5).

g f 18 E. T. SAWYER AND B. D. WICK

4.3. Formulation and proof. In the case that MH is the multiplier algebra of a Hilbert function space with a complete Pick kernel k, there is a characterization of the Corona Property for MH in terms of matrix-valuedH kernel positivity conditions involving k. This results in the Toeplitz corona theorem which asserts the equivalence of the Baby Corona Property for and the Corona Property for its multiplier algebra H MH, and with the same constants c, C - see [BaTrVi], [AmTi] and also [AgMc2, Theorem 8.57]. Here is a special case of the Toeplitz corona theorem as given in [AgMc2, Theorem 8.57]. Toeplitz corona theorem: Let be a Hilbert function space in a set Ω with an irreducible complete Nevanlinna-Pick kernel. Let C >H 0 and N N and let ϕ = (ϕ ,...,ϕ ) N M . Then ϕ satisfies ∈ 1 N ∈⊕ H the MH-corona condition (4.4) with constant C in Definition 22 if and only if ϕ satisfies the -baby corona condition (4.3) with constant C in Definition 21. H In this paper we will use Theorem 19 to obtain an analogue of this theorem for the kernel multiplier space KH when it is an algebra. The role of the Baby Corona Property for will be played by the following property. H

Definition 28. Let be a Hilbert function space with kernel k on a set Ω, and let c,C > 0. We say that the space has the HConvex Poisson Property with positive constants c, C if for all vectors ϕ N K H ∈ ⊕ H satisfying ϕ N 1 and (4.2), the vector ϕ satisfies the convex Poisson condition in Definition 16 ⊕ KH with constantk kC. ≤ H−

Here is our alternate Toeplitz corona theorem.

Theorem 29. Suppose that is a multiplier stable Hilbert function space of continuous functions on Ω that contains the constant functions,H and enjoys the Montel property. Suppose further that the space of kernel multipliers KH is an algebra. N (1) Then K , with the direct sum K normed by N , satisfies the Corona Property with H H ⊕ KH positive constants c, C if and only⊕ if satisfies the Convexk·k Poisson Property with positive constants c, C. H (2) Suppose in addition that satisfies the Invertible Multiplier Property and that MH = KH isometri- H N cally. Equip the direct sum MH with the norm N . ⊕ k·k⊕ MH (a) Then satisfies the Baby Corona Property with constants c, C if MH satisfies the Corona PropertyH with the constants c, C. (b) Conversely, MH satisfies the Corona Property with constants c, C√N if satisfies the Baby Corona Property with constants c, C. H

Armed with Theorem 19, Lemma 27 and Corollary 11, it is now an easy matter to prove Theorem 29.

Proof of Theorem 29. The first assertion follows immediately from Theorem 19 and definitions. Now we turn to the second assertion. Clearly the Baby Corona Property with constants c, C holds for if the H multiplier algebra MH satisfies the Corona Property with constants c, C. Conversely, assume the Baby Corona Property with constants c, C holds for as above. We must show that the Corona Property holds √ H for MH with constants c, C N. So fix a vector ϕ MH with ϕ ⊕N M 1 and ϕ (z) c> 0 for z Ω. ∈ k k ℓ=1 H ≤ | |≥ ∈ Then the Baby Corona Property for with constants c, C implies that the vector ϕ satisfies the -baby corona condition with constant C, i.e.H for each h there is f N satisfying H ∈ H ∈⊕ H 2 2 2 f N C h and (ϕ f) (z)= h (z) , z Ω. k k⊕ H ≤ k kH · ∈ It follows from Lemma 27 that ϕ satisfies the -convex Poisson condition in Definition 16 with constant C. Now apply Theorem 19 to conclude that thereH is a vector function f N K such that ∈⊕ H

f N C and (ϕ f) (z)=1, z Ω. k k⊕ KH ≤ · ∈

By (2.7) we thus conclude that f N C√N, and this shows that MH satisﬁes the Corona Property k k⊕ MH ≤ with constants c, C√N. THE CORONA PROBLEM FOR KERNEL MULTIPLIER ALGEBRAS 19

Remark 30. In the event that = H2 (Ω) is the Hardy space on a bounded domain with C2 boundary in Cn (see Section 5 for deﬁnitions),H then we have

N 2 M h ϕ h 2 column ϕ ⊕N H α=1 M α H (Ω) ϕ N = Mϕ sup k k = sup ⊕ℓ MH op k k =1 k k ≡ h6=0 h h6=0 q h 2 k kH P k kH (Ω) N 2 2 ∂Ω α=1 ϕα h dσ N | | | | 2 r = sup = ϕα = ϕ L∞(ℓ2 ) h6=0 R P 2 | | k k N α=1 ∞ ∂Ω h dσ L (∂Ω) | | X q 2 RN 2 N ∂ α=1 ϕα ka dσ 2 Ω | | 2 = sup r = sup ϕα ka dσ ϕ N ⊕ℓ KH a∈Ω R P 2 a∈Ω v ∂Ω | | ! ≤k k =1 k dσ f uZ α=1 ∂Ω a u X r t f R column f 2 and so ϕ ⊕N M = ϕ ⊕N K = ϕ L∞ ℓ2 . Thus in the case = H (Ω), we can replace the constant k k ℓ=1 H k k ℓ=1 H k k ( N ) H C√N in part (2) of Theorem 29 with the constant C, which shows that the Corona Property for MH holds with the same constants c, C for which the Baby Corona Property holds for . H

5. The alternate Toeplitz corona theorem for Bergman and Hardy spaces in Cn Let Ω be a bounded domain in Cn. The Bergman space A2 (Ω) consists of those f H (Ω) such that ∈ f 2 dV < , | | ∞ ZΩ 1 e 2 where dV = |Ω| dV and dV is Lebesgue measure on Ω. We then have that f A2(Ω) Ω f dV deﬁnes 2 k k 2 ≡ | | a norm on A (Ω). Now point evaluations are bounded, hence continuous, on A (Ω)q byR the mean value property,e e

1 f (z) = f (w) dV (w) | | B (z, d∂Ω (z)) B(z,d∂Ω(z)) | | Z 1 2 1 2 −n f (w) dV (w) = cnd ∂ (z) f 2 . ≤ c d (z)n | | Ω k kA (Ω) n ∂Ω ZΩ 2 Thus the Bergman space A (Ω) is a reproducing kernel Hilbert space, and we denote by kz (w) the reproducing kernel for A2 (Ω) with respect to the inner product

2 f,g 2 fgdV, f,g A (Ω) . h iA (Ω) ≡ ∈ ZΩ We then have e 2 f (z)= f, kz 2 = f (w) kz (w)dV (w) , z Ω, f A (Ω) . h iA (Ω) ∈ ∈ ZΩ ∞ Note also that MA2(Ω) = H (Ω), the algebra of boundede analytic functions on Ω, and that in fact with = A2 (Ω), we have M = K = ∞ = H∞ (Ω). H H H H Remark 31. If Ω is strictly pseudoconvex, then by Theorem 2 in C. Fefferman [Fef] (see also Boutet de Monvel and Sjöstrand [BdMSj]) we have that ka is bounded for each a Ω, and moreover that ka kb = ∈ k − k∞ supz∈Ω ka (z) kb (z) tends to 0 as b a for each a Ω. Thus the map a ka is a continuous map from | − | ∞ → ∈ → Ω to the multiplier algebra H (Ω) = MA2(Ω). While the Bergman kernel functions are also nonvanishing in the ball, H. Boas has shown that in a generic sense, the Bergman kernels of strictly pseudoconvex domains have zeroes ([Boas]; see also Skwarczynski [Skw, Skw], and Boas [Boas2] for a nice survey on Lu Qi-Keng’s problem), and in a paper with Fu and Straube [BFS], they constructed specific examples of such domains, including even some strictly convex smooth Reinhardt domains. 20 E. T. SAWYER AND B. D. WICK

Now we recall the deﬁnition of the Hardy spaces H2 (Ω) for a bounded domain Ω in Cn with C2 boundary. The Hardy space H2 (Ω) consists of those f H (Ω) such that ∈ 2 sup f dσε < , | | ∞ ε>0 Z∂Ωε where Ωε z Ω: ρ (z) < ε and ρ is an appropriate deﬁning function for Ω, and where σε is surface ≡ { ∈ − } measure on ∂Ωε. We then have that

∗ 2 f 2 f dσ k kH (Ω) ≡ | | sZ∂Ω deﬁnes a norm on H2 (Ω), where σ is surface measure on ∂Ω, and where the nontangential boundary limits f ∗ exist a.e. [σ] on ∂Ω. We also note that

2 2 f dA C sup f dσε < | | ≤ ε> | | ∞ ZΩ 0 Z∂Ωε for some constant C depending only on Ω, and this shows that the Bergman space norm f 2 k kA (Ω) ≡ 2 f dA is dominated by a multiple of the Hardy space norm f 2 : Ω | | k kH (Ω) q R f 2 C f 2 . k kA (Ω) ≤ Ω k kH (Ω) Since point evaluations are continuous on A2 (Ω), it follows that they are also continuous on H2 (Ω). Thus 2 the Hardy space H (Ω) is a reproducing kernel Hilbert space, and we denote by kz (w) the reproducing kernel for H2 (Ω) with respect to the inner product

∗ ∗ 2 f,g 2 f g dσ, f,g H (Ω) . h iH (Ω) ≡ ∈ Z∂Ω We then have ∗ ∗ 2 f (z)= f, kz 2 = f (w) k (w)dσ (w) , z Ω,f H (Ω) . h iH (Ω) z ∈ ∈ Z∂Ω We will typically suppress the star in the superscript and simply write f,g 2 fgdσ and f, kz 2 = h iH (Ω) ≡ ∂Ω h iH (Ω) fk dσ. In the case of the Hardy space on the polydisc Dn, where ∂Dn is not C2, integration is restricted ∂Ω z R to the distinguished boundary Tn, a subset of measure zero of ∂Ω. R column The advantage of the Hardy space over the Bergman space is the equality of vector norms ϕ ⊕N H2(Ω) = row k k ϕ ⊕N H2(Ω) = ϕ L2 ℓ2 , while the advantage of the Bergman space over the Hardy space is the greater k k k k ( N ) generality of the domains Ω for which it is deﬁned.

5.1. The Bergman and Hardy space on the unit ball. First note that the reproducing kernels ka (w)= 1 2 B 2 B (1−a·w)n+1 for the standard inner product on A ( n) are invertible multipliers of A ( n) (with possibly large 1 B multiplier norm ka = n+1 , but this norm plays no role here) depending continuously on a n, k k∞ (1−|a|) ∈ ∞ 2 2 2 B B B that MA (Bn) = H ( n), that A ( n) contains the constants, and ﬁnally that A ( n) satisﬁes the Montel property precisely by Montel’s theorem for holomorphic functions. Similar considerations apply to the Hardy 2 space H (Bn). From part (2) of Theorem 29 we obtain the following Corollary.

∞ 2 Corollary 32. The Corona Property holds for H (Bn) if and only if A (Bn) satisfies the Convex Poisson 2 Property if and only if H (Bn) satisfies the Convex Poisson Property. 5.2. The Hardy and Bergman space on the unit polydisc. For the polydisc, we will again apply our alternate Toeplitz corona theorem to the Hardy space H2 (Dn) and the Bergman space A2 (Dn). First, we note that the multiplier algebra of H2 (Dn) is H∞ (Dn), and that the reproducing kernels k = n 1 a j=1 (1−aj zj ) for the standard inner product on H2 (Dn) are invertible multipliers of H2 (Dn) and that the map a k is Q a continuous from Dn to H∞ (Dn). Similar considerations apply to the Bergman space A2 (Dn). → Corollary 33. The corona theorem holds for H∞ (Dn) if and only if H2 (Dn) satisfies the Convex Poisson Property if and only if A2 (Dn) satisfies the Convex Poisson Property. THE CORONA PROBLEM FOR KERNEL MULTIPLIER ALGEBRAS 21

5.3. General domains. From part (2) of Theorem 29, we obtain the corona theorem for H∞ (Ω) for any bounded domain Ω in Cn for which the Convex Poisson Property holds for the Bergman space A2 (Ω), and 2 ∞ for which the Bergman space A (Ω) is multiplier stable (note that KA2(Ω) = H (Ω), contains constants and satisﬁes the Montel property). Greene and Krantz [GrKr] show that for domains Ω in a suﬃciently small C∞ neighbourhood of the unit ball Bn (or any other strictly pseudoconvex domain D with smooth boundary for which infz,w∈D kD (z, w) > 0), we have the lower bound infz,w∈Ω kΩ (z, w) > 0.

Corollary 34. Let D be a strictly pseudoconvex domain with smooth boundary for which infz,w∈D kD (z, w) > ∞ 0, where kD is the Bergman kernel for D. Then for Ω in a sufficiently small C neighbourhood of D, the corona theorem holds for H∞ (Ω) if and only if the Convex Poisson Property holds for A2 (Ω). If f Aut (Ω), then we have the Bergman kernel transformation law (see e.g. [Boas2]) ∈ K (z, w) = det f ′ (z) K (f (z) ,f (w)) det f ′ (w), z,w Ω. ∈ If Ω is a complete circular domain, then K (z, 0) is a nonzero constant K0 (see Bell [Bell]; see also [Boas2]). Now assume in addition that Ω is homogeneous and fix ζ Ω. Then with f Aut (Ω) such that f (ζ) = 0, the transformation law shows that ∈ ∈ ′ ′ ′ ′ K (z, ζ) = det f (z) K (f (z) ,f (ζ)) det f (ζ) = det f (z) K0det f (ζ). −1 ′ 1 If g = f we have det f (z) = det g′(z) , and from another application of the transformation law with η = f (0), ′ ′ ′ ′ K (z, η)= K (η,z) = det g (η) K (g (η) ,g (z)) det g (z) = det g (η) K0det g (z), we see that 1 det g′ (η)K K (z, ζ) = det f ′ (z) K det f ′ (ζ)= K det f ′ (ζ)= 0 K det f ′ (ζ) 0 det g′ (z) 0 K (z, η) 0 det g′ (η)det f ′ (ζ) = K2 , 0 K (z, η) and hence that ′ ′ 2 det g (η) det f (ζ) inf K (z, ζ) K0 | | > 0 z∈Ω | | ≥ | | Kη k k∞ since Kη ∞ < if Ω is a strictly pseudoconvex domain. The continuity of the map ζ K (z, ζ) also followsk fromk these∞ calculations. This gives the following corollary to Theorem 29. → Corollary 35. Let Ω be a bounded strictly pseudoconvex homogeneous complete circular domain with smooth boundary in Cn. Then the corona theorem holds for H∞ (Ω) if and only if the Convex Poisson Property holds for A2 (Ω). Remark 36. The above smoothness assumption on the boundaries of D and Ω can be relaxed, but we will not pursue that here. Remark 37. In all of the above corollaries, the Baby Corona Property for H2 (Ω) and A2 (Ω) is known to hold - see [KrLi] for strictly pseudoconvex domains as above, and see [Lin], [Li], [Lin2], [Tren], and [TrWi2] for the polydisc. Remark 38. For the pseudoconvex domain constructed by Sibony [Sib], in which the Corona Property for H∞ (Ω) fails, the Szegökernel functions cannot be invertible multipliers depending in a lower semicontinuous ∞ 2 way on the pole. Indeed, MH2(Ω) = H (Ω) and H (Ω) contains constants and satisfies the Montel property. Moreover, by a result of Andersson and Carlsson [AnCa], the baby corona theorem holds for H2 (Ω) on pseudoconvex domains with smooth boundary, and Theorem 29 now shows that the Szegökernels cannot be bounded away from both 0 and and satisfy the semicontinuity assumption on the Sibony domain. ∞ Remark 39. There is a partial result for certain ‘polydomains’ in Cn. The Bergman space A2 (Ω) satisfies ∞ KA2(Ω) = MA2(Ω) = H (Ω), and clearly contains the constants and is a Montel space. If the Bergman 2 nj 2 spaces A (Ωj ), Ωj C , are multiplier stable for 1 j J, then the Bergman space A (Ω) for the J ⊂ ≤ ≤ Cn J 2 polydomain Ω= Ωj , n = j=1 nj , is also multiplier stable (since the kernel function of A (Ω) is j=1 ⊂ Q P 22 E. T. SAWYER AND B. D. WICK

2 the product of the kernel functions for A (Ωj ), and the multiplier norm is the supremum norm). Thus in 2 this case MA2(Ω) satisﬁes the Corona Property if and only if A (Ω) satisﬁes the Convex Poisson Property.

6. Invertible Multiplier Property for Hardy spaces Here we begin by showing that the Invertible Multiplier Property holds for the Hardy space H2 (D) on the disk. Lemma 40. The Szeg¨okernel for the Hardy space = H2 (D) has the Invertible Multiplier Property. H

Proof. Take ka,θ to be the outer function

M 2π it 2 g 1 e + z it ka,θ (z) exp ln θm ka e dt , ≡ 2π eit z m ( Z0 − m=0 ! ) X g 2 g 2 it M it which by [Rud4, Theorem 17.16 (b)] satisﬁes ka,θ e = θm ka e for almost every 0 t< 2π m=0 m ≤

(the equality actually holds for all t by continuity of thePkam on the boundary). Here the normalized g 2 g √1−| am| 1 reproducing kernel kam (z) is given by kam (z)= . Then ka,θ and are bounded in the disk and 1−amz ka θ g g, we have g 2π g g it ∗ it it ∗ it ka,θf, ka,θg = ka,θ e f e ka,θ (e ) g (e ) dt H Z0 D E M 2π 2 2π 2 g g g∗ it ∗ it g it ∗ it ∗ it it = f e g (e ) ka,θ e dt = θm f e g (e ) kam e dt 0 m=0 0 Z X Z M M g g = θm kam f, kam g = θm f,g Ham = f,g Ha,θ . H2(D) h i h i m=0 m=0 X D E X g g In particular ka,θ = 1 Ha,θ = 1, and so we have shown that ka,θ is a normalized invertible multiplier H2(D) k k

satisfying (4.5). g g Now we show that the Invertible Multiplier Property fails for the Szeg¨okernel for the Hardy space on the ball in higher dimensions by showing that when at least two of the θm are positive, then there are no invertible 2 M multipliers whose real parts have boundary values equal to ln m=0 θm kam almost everywhere. This should be contrasted with results of Aleksandrov [Ale] and L/owP [Low] that yield nonvanishing multipliers 2 g M whose real parts do have boundary values equal to ln m=0 θm kam almost everywhere. In the proof of Theorem 19 above (see (3.3)) it was essential that Pka,θ was an invertible multiplier, as opposed to a g more general nonvanishing multiplier. But the reciprocals of the Aleksandrov and Lo /w multipliers are not bounded in the ball (only their boundary values are boundedg almost everywhere on the sphere), and in fact the reciprocals do not belong to any reasonable class of holomorphic functions on the ball. These nonvanishing multipliers can be thought of as ball analogues of products of singular inner functions with outer functions in the disk.

2 n Lemma 41. The Hardy space = H (Bn) on the unit ball in C fails to have the Invertible Multiplier H M Property when n> 1. The failure is spectacular in that for any (a, θ) Ω ΣM with at least two of the ∈∞ × θm positive, there is no normalized invertible multiplier ka,θ M = H (Bn) satisfying (4.5). ∈ H Proof. We prove the case when M = 1, θ = θ = 1 and a = αe = (α, 0,..., 0) B , and leave the general 0 1 2 g1 1 n case to the interested reader. We assume, in order to derive a contradiction, that∈ that there is a normalized invertible multiplier ϕ M such that ∈ H

f,g + ka f, ka g =2 f,g a,θ = ϕf, ϕg , f,g , h iH h 1 1 iH h iH h iH ∈ H THE CORONA PROBLEM FOR KERNEL MULTIPLIER ALGEBRAS 23

where we have absorbed the factor 2 into ϕ for convenience. Unraveling notation this becomes 2 2 2 B fg 1+ ka1 dσ = fg ϕ dσ, f,g H ( n) . ∂B ∂B | | ∈ Z n Z n

In particular we obtain that f 2 2 1+ ka = ϕ on ∂Bn 1 | | by the Stone-Weierstrass theorem, since the algebra generated by restrictions to ∂Bn of f and g for f A f,g A (Bn), is self-adjoint, separates points and contains the constants on ∂Bn; and thus is dense in ∈ A ∞ C (∂Bn). Since ϕ is an invertible multiplier, we claim it has a holomorphic logarithm F = log ϕ in H (Bn) whose boundary values satisfy 2 Re F = 2 Re log ϕ = ln ϕ 2 . | | ∞ Indeed, to see this we consider the dilates Fr (z) F (rz) of F which are clearly in H (Bn) (although not uniformly in 0

Re Fr (z)=Re F (rz) = Re log ϕ (rz) = ln ϕ (rz) ln ϕ < , | |≤k | |k∞ ∞ where the boundedness of ln ϕ follows from the maximum principle since ln ϕ is continuous in Bn and | | p | | harmonic in Bn. Then the Koranyi-Vagi theorem gives the uniform L estimate (see e.g. [Rud2, inequality (1) on page 125])

p p p Fr dσ Mp,n Re Fr dσ Mp,n ln ϕ ∞ , 1 holomorphic function on the ball if and only if u is harmonic in each slice and smooth→ near the origin). We have g (z)= ln A (λz) dm (λ) , Z|λ|=1 where 2n 2 n 2 2 1 α 2 −n −n A (z) 1+ − | | =1+ 1 α (1 αz) 1+ cα,n (1 αz) . ≡ q1 αz − | | − ≡ −

−

24 E. T. SAWYER AND B. D. WICK

Then we have 2 ∂ −n n+1 −n A (λz) = cα,n(1 αλz) ( n) (1 αλz) ( αλ)= cα,nnαλ (1 αλz) (1 αλz) ∂z − − − − − −

= cα,nnαλ (1 αλz) (A (λz) 1) , − − ∂ ∂ and of course ∂z A (λz)= ∂z A (λz) because A is real. Thus ∂ ∂ ∂ 1 ln A (λz)= cα,nnαλ (1 αλz) 1 ∂z ∂z ∂z − − A (λz) ∂ 1 1 ∂ = cα,nnαλ (1 αλz) = cα,nnαλ (1 αλz) A (λz) − − ∂z A (λz) − A (λz)2 ∂z

1 2 A (z) 1 = cα,nnαλ (1 αλz) nαλ (1 αλz) (A (λz) 1) = cα,n nαλ (1 αλz) − , − A (λz)2 { − − } | − | A (λz)2 which is strictly positive for λ = 1 if α = 0. This shows that | | 6 1 ∂ ∂ ∂ ∂ g (z)= F (z)= ln A (λz) dm (λ) > 0 4 △ ∂z ∂z ∂z ∂z Z|λ|=1 for z D, and this shows that g is not constant in the disk, providing the claimed contradiction. This completes∈ the proof of Lemma 41. Essentially the same argument as above shows that the Hardy space H2 (Dn) on the polydisc Dn also fails to have the Invertible Multiplier Property when n> 1. Lemma 42. The Hardy space = H2 (Dn) on the polydisc Dn in Cn fails to have the Invertible Multiplier Property when n> 1. H 1 Proof. We prove the Invertible Multiplier Property fails in the case when M = 1, θ0 = θ1 = 2 and a1 = n αe1 = (α, α, . . . , 0) D with α = 0. We assume, in order to derive a contradiction, that that there is a normalized invertible∈ multiplier ϕ6 M such that ∈ H f,g + ka f, ka g =2 f,g a,θ = ϕf, ϕg , f,g , h iH h 1 1 iH h iH h iH ∈ H where again we have absorbed the factor 2 into ϕ for convenience. Unraveling notation this becomes 2 2 2 Dn fg 1+ ka1 dm = fg ϕ dm, f,g H ( ) . Tn Tn | | ∈ Z Z In particular we obtain that f 2 2 n 1+ ka = ϕ on T 1 | | by the Stone-Weierstrass theorem, just as in the proof of Lemma 41. Since ϕ is an invertible multiplier,

the argument used in the proof of Lemma 41 showsf it has a holomorphic logarithm F = log ϕ in H2 (Dn) 2 2 n with boundary values ln ϕ = 2 Re log ϕ = 2Re F . Thus the restriction h of ln 1+ ka to T is | | 1 2 Dn the distinguished boundary value function of the real part Re F of a holomorphic function F in H ( ). n f It follows that for almost every ζ T , the slice function Fζ (λ) = F (λζ) deﬁned on the slice Sζ = n 2 ∈ 2 z D : z = λζ, λ D , is in H (Sζ ) H (D) and has boundary values equal to h Tn . In particular then, { ∈ ∈ } ≈ | the integral of h on the boundary of such a slice Sζ , with respect to Haar measure dm on T = ∂Sζ , must equal the constant ln ϕ (0) 2. Thus we have shown that the function | | 2 2 2 2 1 α 1 α G (ζ) ln 1+ k dm = ln 1+ − | | − | | dm (λ) a1   ≡ ∂S |λ|=1 q1 αλζ1 q1 αλζ2 Z ζ Z − −   2 f   equals the constant ln ϕ (0) almost everywhere on the distinguished boundary Tn. In particular the map | | 2 2 1 α 2 − | | g (z) G (z, z, 0,..., 0) = ln 1+  dm (λ) ≡ |λ|=1 q1 αλz  Z −        

THE CORONA PROBLEM FOR KERNEL MULTIPLIER ALGEBRAS 25

is the constant ln ϕ (0) 2 almost everywhere on D, and since g is clearly continuous in D, it is constant in D. But this function |g is the| same as the function g obtained in the previous proof for n = 2. We showed there that this function is not constant, and this completes the proof of Lemma 42. We emphasize that H2 (Ω) for certain bounded ﬁnitely connected planar domains are essentially the only spaces we currently know to have the Invertible Multiplier Property (H2 (Ω) has this property if Ω is simply connected by the Riemann mapping theorem).

7. Smoothness spaces and nonholomorphic spaces σ In the first subsection, we show that if is a Besov-Sobolev space B (Bn) on the ball for σ > 0 and H 2 n 1, then the kernel multiplier space KH is an algebra. Moreover, the reproducing kernels satisfy the hypotheses≥ of the alternate Toeplitz corona theorem 29, so that as a consequence of this subsection we obtain the following corollary. σ B Corollary 43. Let σ > 0 and n 1. The kernel multiplier algebra K2 ( n) satisfies the Corona Property ≥ σ B if and only if the Convex Poisson Property holds for B2 ( n). Then in the second subsection we make the obvious extension of kernel multiplier algebras for the Banach σ D spaces of analytic functions Bp ( ) and then prove the Corona Property for the kernel multiplier algebras 1 0 ∞ ∞ 0 KBσ D on the disk when 0 σ< . We recall here that the algebra B (D) = H (D) B (D) of bounded p ( ) ≤ p 2 ∩ 2 Dirichlet space functions was shown to have the Corona Property by Nicolau [Nic], who used the difficult theory of best approximation in V MO due to Peller and Hruscev [PeHr]. Finally, in the third subsection, we demonstrate that there are many Hilbert function spaces to which our alternate Toeplitz corona theorem applies, that are not spaces of holomorphic functions. 7.1. The kernel multiplier algebras for Besov-Sobolev spaces. We begin by extending some of the background material developed in [ArRoSa2] for the Besov spaces Bp (Bn) to the Besov-Sobolev spaces σ B (Bn), σ 0. Some of this material appears in [ArRoSa3]. p ≥ γ,t 7.1.1. Besov-Sobolev spaces. Recall the invertible “radial” operators R : H (Bn) H (Bn) given in [Zhu] by → ∞ Γ (n +1+ γ) Γ (n +1+ k + γ + t) Rγ,tf (z)= f (z) , Γ (n +1+ γ + t) Γ (n +1+ k + γ) k k=0 X ∞ provided neither n + γ nor n + γ + t is a negative integer, and where f (z)= k=0 fk (z) is the homogeneous γ,t expansion of f. If the inverse of R is denoted Rγ,t, then Proposition 1.14 of [Zhu] yields P γ,t 1 1 (7.1) R n+1+γ = n+1+γ+t , (1 w z) ! (1 w z) − · − · 1 1 Rγ,t n+1+γ+t = n+1+γ , (1 w z) ! (1 w z) − · − · γ,t for all w Bn. Thus for any γ, R is approximately differentiation of order t. From Theorem 6.1 and ∈ γ,m p m−1 (k) Theorem 6.4 of [Zhu] we have that the derivatives R f (z) are “L norm equivalent” to k=0 f (0) + f (m) (z) for m large enough. P

Proposition 44. (analogue of Theorem 6.1 and Theorem 6.4 of [Zhu]) Suppose that 0 ,m N, − | | ∈ p ∈ m+σ 2 (m) p n 1 z f (z) L (dλn) for all m + σ > ,m N, − | | ∈ p ∈ m+σ 2 γ,m p n 1 z R f (z) L (dλn) for some m + σ > ,m + n + γ / N, − | | ∈ p ∈− m+σ 2 γ,m p n 1 z R f (z) L (dλn) for all m + σ> ,m + n + γ / N. − | | ∈ p ∈− 26 E. T. SAWYER AND B. D. WICK

Moreover, with ρ (z)=1 z 2, we have for 1 , m1 + n + γ / N, m2 N, and where the constant C depends only on σ, m1, p ∈− ∈ m2, n, γ and p. σ B B Deﬁnition 45. We deﬁne the analytic Besov-Sobolev spaces Bp ( n) on the ball n by taking γ = 0 and n+1 m = p and setting

σ σ B B m+σ 0,m (7.3) Bp = Bp ( n)= f H ( n): ρ R f p < . ∈ L (dλn) ∞ n o We will indulge in the usual abuse of notation by using f Bσ B to denote any of the norms appearing k k p ( n) in (7.2).

7.1.2. Reproducing kernels. For α> 1, let , α denote the inner product for the weighted Bergman space 2 − h· ·i Aα:

2 f,g α = f (z) g (z)dνα (z) , f,g Aα, h i B ∈ Z n α 2 α α −n−1−α where dνα (z)= 1 z dV (z). Recall that K (z)= K (z, w)=(1 w z) is the reproducing − | | w − · 2 kernel for Aα (Theorem 2.7 in [Zhu]):

α α 2 f (w)= f,Kw α = f (z) Kw (z)dνα (z) , f Aα. h i B ∈ Z n p p This formula continues to hold as well for f Aα, 1 1, σ 0. − ≥ σ σ σ Deﬁnition 46. For α> 1, σ 0 and 1

σ α α α α f,g = n+1+α f, n+1+α g = n+1+α f (z) g (z)dν (z) α,p −σ −σ −σ n+1+α −σ α h i R p R p′ B R p R p′ α Z n n+1+α n+1+α 2 p α 2 p′ α = 1 z n+1+α f (z) 1 z g (z) dλ (z) . −σ n+1+α −σ n B − | | R p − | | R p′ Z n Clearly we have σ f,g α,p f Bσ g Bσ h i ≤k k p k k p′

THE CORONA PROBLEM FOR KERNEL MULTIPLIER ALGEBRAS 27

by H¨older’s inequality. By Theorem 2.12 of [Zhu], we also have that every continuous linear functional Λ on σ σ σ B is given by Λf = f,g for a unique g B ′ satisfying p h iα,p ∈ p σ (7.4) g Bσ = sup f,g α,p . p′ k k kfk σ =1 h i Bp

−1 σ ∗ α p ∗ p′ Indeed, if Λ B , then Λ n α (A ) , and by Theorem 2.12 of [Zhu], there is G A with p +1+ −σ α α ∈ ◦ R p ∈ ∈ −1 −1 α p α G p′ = Λ such that Λ n+1+α F = F, G for all F A . If we set g = n+1+α G, A −σ α α −σ k k α k k ◦ R p h i ∈ R p′ α then we have g σ = G p′ = Λ and with F = n+1+α f, we also have B ′ A −σ k k p k k α k k R p −1 α α α σ Λf =Λ n+1+α F = F, G = n+1+α f, n+1+α g = f,g −σ α −σ ′ −σ α,p ◦ R p h i R p R p α h i for all f Bσ. Then (7.4) follows from ∈ p σ g Bσ = Λ = sup Λ (f) = sup f,g α,p . p′ k k k k kfk σ =1 | | kfk σ =1 h i Bp Bp α 2 With Kw (z) the reproducing kernel for Aα, we now claim that the kernel −1 −1 σ,α,p α α α (7.5) kw (z)= n+1+α −σ n+1+α −σ Kw (z) R p′ R p σ satisﬁes the following reproducing formula for Bp :

σ,α,p σ α α σ,α,p σ (7.6) f (w)= f, k = n+1+α f (z) n α kw (z)dν (z) , f B . w α,p −σ +1+ −σ α p h i B R p R p′ ∈ Z n Indeed, for f a polynomial, we have −1 −1 α α α α α α α f (w) = f,Kw α = n+1+α −σ n+1+α −σf,Kw = n+1+α −σf, n+1+α −σ Kw h i R p R p α R p R p α −1 −1 α α α α α = n+1+α −σf, n+1+α −σ n+1+α −σ n+1+α −σ Kw R p R p′ R p′ R p * +α σ −1 −1 α α α = f, n+1+α −σ n+1+α −σ Kw . * R p′ R p + α,p σ σ,α,p σ We now obtain the claim since the polynomials are dense in Bp and the kernels kw are in Bp′ for each fixed w Bn. Thus we have proved the following theorem. ∈ σ σ Theorem 47. Let 1 1. Then the dual space of Bp can be identified with Bp′ σ ∞ ≥ − under the pairing , , and the reproducing kernel kσ,α,p for this pairing is given by (7.5). h· ·iα,p w From (7.5) and (7.1) we have −1 α σ,α,p α α (7.7) n+1+α −σkw (z) = n+1+α −σ Kw (z) R p′ R p −(n+1+α) = Rα− n+1+α +σ, n+1+α −σ (1 w z) p p − · − n+1+α −σ = (1 w z) p′ . − · σ B σ B 7.1.3. Kernel multiplier spaces. Define the Banach space Kp ( n) of kernel multipliers of Bp ( n) to consist of those holomorphic functions in the ball Bn such that

σ,p′ ϕka σ B Bp ( n) ϕ Kσ(B ) sup < , p n B σ,p′ k k ≡ a∈ n ka ∞ σ B Bp ( n)

28 E. T. SAWYER AND B. D. WICK

σ,p′ σ B σ,α,p′ where ka (z) is a reproducing kernel for Bp′ ( n), e.g., kw (z) - any admissible choice of α can be used σ,p′ σ B ∞ B here. Standard arguments using the reproducing kernel ka (z) show that Kp ( n) embeds in H ( n). Indeed,

1 σ,α,p′ σ,α,p ^σ,α,p′ ^σ,α,p ϕ (a) = ′ ϕka , ka ϕka , ka ϕ K , | | σ,α,p α ≈ ≤k k H ka (a) α D E Above we have used (see (7.8) below):

σ,α,p′ 1 σ,α,p′ k ka (a). a p B 2 σ Bσ ( n) ≈ (1 a ) ≈ − | | q σ B σ B Let W Cp ( n) consist of those holomorphic functions in f Bp ( n) that satisfy the weak or one-box σ-Carleson condition: ∈ σ B σ B W Cp ( n) f Bp ( n): f WCσ B < ; ≡ ∈ k k p ( n) ∞ n m+σ o p p 1 2 m f σ sup pσ 1 z f (z) dλ (z) , WC (Bn);m n k k p ≡ n − | | ∇ Q Q ZS(Q) | | and where Q is a nonisotropic ball on the sphere (see e.g. [Rud2, page 65]) and Q is its surface measure. n p | | Here m > σ and as usual, we will see below that the norms f σ B are equivalent provided p − k kWCp ( n);m p (m + σ) > n, and so we can drop the dependence on m. But ﬁrst we establish the following standard equivalence for one-box Carleson measures.

Lemma 48. For dµ a positive Borel measure on the ball Bn, σ> 0 and 1

We will call measures µ that satisfy Lemma 48 weak Carleson measures and norm them via either expres- sion. This can be contrasted with standard Carleson measures which are normed by:

1 p p µ σ = sup f(z) dµ(z) . CMp k k kfk σ B =1 Bn | | Bp ( n) Z Proof. The proof of Lemma 48 is standard using the pointwise bounds 2σ σ,p′ 1 (7.8) ka (z) & 2 , z Sa , 1 a ! ∈ − | | 2σ σ,p′ 1 k (z) . , z Bn , a 1 a z ∈ − · α σ,α,p′ 1 α which follow from (7.7), n+1+α k (z)= n+1+ α , since n+1+α is essentially diﬀerentiation − σ w +σ −σ R p (1 −w·z) p R p of order t = n+1+α σ, and so p − −1 σ,α,p′ α 1 1 1 k (z)= n+1+α = . w −σ n+1+α +σ n+1+α +σ−t 2σ R p (1 w z) p ≈ (1 w z) p (1 w z) − · − · − · From this we also have the use approximation that: 2 σ ]σ,p′ (1 a ) ka (z) − | | . ≈ (1 a z)2σ − · To show the inequality & we simply use the ﬁrst inequality in (7.8). Conversely, to show the inequality ., we break up the integral p p ]σ,p′ ]σ,p′ ka = ka (z) dµ (z) p B L (µ) Z n

THE CORONA PROBLEM FOR KERNEL MULTIPLIER ALGEBRAS 29

ℓ+1 ℓ into geometric annuli 2 Sa 2 Sa, where Sa is the usual Carleson box associated with a Bn. Then \ ℓ+1 ∈ we complete the proof using the one box condition on the Carleson boxes 2 Sa together with the second inequality in (7.8), and then summing up a geometric series.

σ ∞ σ Now we show that K (Bn) = H (Bn) W C (Bn) with comparable norms. The corresponding result p ∩ p σ B for the multiplier algebra MBp ( n) is due to Ortega and Fabrega [OrFa, Theorem 3.7], and the proof there carries over almost verbatim for weak Carleson measures in place of Carleson measures, which we provide for the ease of the reader. It follows as a corollary of this result that the weak Carleson measure condition is independent of m> n σ. p − ∞ B σ B n σ B Proposition 49. Let ϕ H ( n) Bp ( n), σ > 0 and m + σ > p . Then ϕ Kp ( n) if and only if σ ∈ ∩ ∈ ϕ W C (Bn) i.e. ∈ p m+σ p 2 (m) dµ (z) 1 z ϕ (z) dλn (z) ≡ − | |

σ B B is a weak Bp ( n)-Carleson measure on n.

∞ σ p Proof. Suppose that ϕ H (Bn) W C (Bn). In [OrFa], Ortega and Fabrega use the notation A (Bn) ∈ ∩ p δ,k σ B n+δ to describe a space of holomorphic functions equivalent to Bp ( n) when σ = p k. They deﬁne the norm p B − on Aδ,k ( n) by 1 p δ−1 α p 2 f p B D f (z) 1 z dV (z) . Aδ,k( n) k k ≡  Bn | | − | |  |αX|≤k Z They show (bottom of page 66 of [OrFa]) that with R the radial derivative, 

′ ′ ′ (7.9) (I + R)k ϕkσ,p . kσ,p (I + R)k ϕ + Rℓϕ Rmkσ,p . a p B a p B a p B Aδ,0( n) Aδ,0( n) Aδ,0( n) m+ℓ≤k m>X0

Now the ﬁrst term on the right side of (7.9) is controlled by the weak Carleson norm ϕ WCσ B of ϕ. The k k p ( n) needed estimates for the other terms on the right hand side of (7.9) are obtained from the argument used to prove Theorem 3.5 in [OrFa] since only the weak Carleson condition is needed here. Indeed we quote from the bottom of page 66 in [OrFa]:

“The estimates of the other terms can be obtained following the same argument used to prove Theorem 3.5. Observe that the properties of ϕ used in this proof were that ϕ was a bounded holomorphic function and that for δ 1 N kp < 0 − − − p δ−1 k 2 (I + R) ϕ (w) 1 w δ−1−N−kp − | | dV (w) . 1 z 2 . n+1+N Bn 1 w z − | | Z | − · | ∞ It is clear that they are consequences of ϕ H (Bn) and just testing the measure dµ on the function n+1+N ∈ 1 p 1−w·z .” From this quote it is clear that we need only use the weak Carleson condition when testing the measure dµ for the other terms on the right side of (7.9).

σ Now we turn to showing that K (Bn) is an algebra for σ> 0 and 1 0 and 1 p . Note this choice is twice as large as need be, and this will play a role in the proof. σ B 2 2 2 To show that Kp ( n) is an algebra, it is enough by polarization, 2fg = (f + g) f g , to show that 2 2 2 2 σ − − ϕ ϕ σ . Now ϕ ϕ for ϕ K (B ) and using Lemma 48 it remains to show Kσ B K (Bn) ∞ p n p ( n) ≤ k k p ∞ ≤ k k ∈ that 2 2 (7.10) ϕ σ B . ϕ Kσ(B ) . WCp ( n) k k p n

30 E. T. SAWYER AND B. D. WICK

We have

1 m p −1 m+σ p 2 k 2 1 2 m 2 ϕ WCσ(B ) = ϕ (0) + sup pσ 1 z ϕ (z) dλn (z) , p n ∇ Q n − | | ∇ k=0 Q ZS(Q) ! X | |

and m m 2 m−k k ϕ (z)= cm,k ϕ (z) ϕ (z) . ∇ ∇ ∇ k=0 X Now

1 m p σp p 2 m 2 2 1 z ϕ (z) 1 z dλn (z) S(Q) − | | ∇ − | | ! Z 1 m p σp p 2 m p 2 C 1 z ϕ (z) ϕ (z) 1 z dλn (z) ≤ S(Q) − | | ∇ | | − | | ! Z 1 m 1 p − m−k p k p σp 2 m−k 2 k 2 +C 1 z ϕ (z) 1 z ϕ (z) 1 z dλn (z) S(Q) − | | ∇ − | | ∇ − | | ! k=1 Z X 1 m p σp p p 2 m 2 +C ϕ (z) 1 z ϕ (z) 1 z dλn (z) I + II + III. S(Q) | | − | | ∇ − | | ! ≡ Z

Terms I = III are easily controlled by

1 p p m+σ σ 2 m n I C ϕ 1 z ϕ (z) dλn (z) C ϕ ϕ WCσ B Q . ≤ k k∞ − | | ∇ ≤ k k∞ k k p ( n) | | ZS(Q) !

As for term II, ﬁx 1 k m 1 for the moment, and assume without loss of generality that k m k. Then we can use Cauchy’s≤ ≤ estimate− ≥ −

m−k 1 z 2 m−kϕ (z) . ϕ − | | ∇ k k∞

to obtain 1 m−k p k p σp p 2 m−k 2 k 2 1 z ϕ (z) 1 z ϕ (z) 1 z dλn (z) S(Q) − | | ∇ − | | ∇ − | | ! Z 1 p p k+σ σ 2 k n ϕ 1 z ϕ (z) dλn (z) ϕ ϕ WCσ B Q ≤ k k∞ − | | ∇ ≤k k∞ k k p ( n) | | ZS(Q) !

since k m k implies k m > n > n σ (this is where we use m> 2n ). Altogether then we have ≥ − ≥ 2 p p − p 1 p p m+σ σ 2 m 2 n 1 z ϕ (z) dλn (z) . ϕ ϕ σ B Q , − | | ∇ k k∞ k kWCp ( n) | | ZS(Q) !

which gives (7.10): 2 2 ϕ σ B . ϕ ∞ ϕ WCσ(B ) . ϕ Kσ(B ) . WCp ( n) k k k k p n k k p n

σ σ In particular, when = B (Bn) and σ > 0, the kernel multiplier space K = K (Bn) is an algebra. H 2 H 2 THE CORONA PROBLEM FOR KERNEL MULTIPLIER ALGEBRAS 31

7.2. The Corona Property for kernel multiplier algebras on the disk. Here we prove the Corona Property for the one-dimensional algebras of kernel multipliers Kσ(D) for 0 <σ< 1 , 1 0, z D. | 1 | | N | ≥ ∈ σ Then there are a positive constantnC and f ,...,fN K o(D) satisfying 1 ∈ p D max f1 Kσ D ,..., fN Kσ D C, z , k k p ( ) k k p ( ) ≤ ∈ ϕ (zn) f (z)+ + ϕ (z) fN (zo) = 1, z D. 1 1 ··· N ∈ σ D The Corona Property for the multiplier algebras MBp ( ) was obtained by Arcozzi, Blasi and Pau in [ArBlPa] using the Peter Jones solution to the ∂-equation, and we now adapt these methods to prove the σ D Corona Property for the algebras Kp ( ) of kernel multipliers. The key point is that in the arguments in [ArBlPa], which in turn are a generalization of the results from [Xia2] for the case p = 2, we are always able to substitute the weak or one-box Carlesons condition for the actual Carleson condition. 2 As in [ArBlPa] let dAs (z) = 1 z dA (z) where dA (z) is normalized area measure on the disk D − | | p (the connection between s there and σ here is s = pσ). If g (z) dAp(1+σ)−2 (z) is a weak Carleson measure σ D 1 | | for Bp ( ) with 0 <σ< p , then Lemma 6.3 in [ArBlPa] shows that g (z) dA (z) is a Carleson measure for 1 | | 2 D 2 D p D H ( )= B2 ( ) (or in fact H ( ) since Carleson measures for Hardy space are all the same!). Indeed, 2 2 σ+1− p −σ−1+ p g (z) dA (z) = g (z) 1 z 2 1 z 2 dA (z) | | | | − | | − | | ZS(I) ZS(I) 1 p−1 p p 2 p p(σ+1)−2 −(σ+1) p−1 + p−1 g (z) p 1 z 2 dA (z) 1 z 2 dA (z) ≤ S(I) | | − | | ! S(I) − | | ! Z Z p−1 pσ 1 (1−σ) p p . ( I ) p I p−1 = I . | | | | | | Now we obtain from the Peter Jones solution to the ∂-equation the following ‘weak’ version of Theorem 6.4 in [ArBlPa]. Theorem 52. Let 1

32 E. T. SAWYER AND B. D. WICK

Indeed, Lemma 4.5 proved the analogue

p(σ+1)−2 ∗ ∗ p 2 F p . F ∞ + F (z) 1 z dA (z) M(Lpσ) L (T) k k k k |∇ | − | | σ CMp where the norm on the left is the multiplier norm of F ∗, which is equivalent to

p(σ+1)−2 ∗ ∗ p 2 F L∞(T) + P F (z) 1 z dA (z) . k k |∇ | − | | CM σ p

The passage from Carleson norms to weak Carleson norms here is routine because one simply repeats the proof with a reproducing kernel in place of the generic function used in their proof.

Proof. To prove Theorem 52, we note that since dµ (z) g (z) dA (z) is a Carleson measure for H2 (D), we can invoke the Jones solution with ν = µ : ≡ kµkH2(D)−Car

u (z) = K (ν,z,ζ) dµ (ζ) , z D, D ∈ Z Z 2i 1 ζ 2 1+ ωz 1+ ωζ K (ν,z,ζ) − | | exp + dν (ω) . ≡ π (z ζ) 1 ζz |ω|≥|ζ| −1 ωz 1 ωζ − − (Z Z − − ) ∂f D ∗ ∞ T ∗ p This solution u (z) to ∂z = g on satisfies u L ( ). We now show that in addition u K Lpσ . For this purpose we consider the function ∈ ∈ 2i 1 ζ 2 1+ ωz 1+ ωζ v (z) = − | | 2 exp + g (ω) dA (ω) g (ζ) dA (ζ) , π D 1 ζz ( |ω|≥|ζ| −1 ωz 1 ωζ | | ) Z Z − Z Z − − −iθ p T which has the same boundary values as zu (z). Since e K Lpσ ( ) , it suffices by (7.11) to show that p(σ+1)−2 ∈ v∗ L∞ (T) and that v (z) p 1 z 2 dA (z) is a weak Carleson measure for Bσ (D). ∈ |∇ | − | | p We begin with an estimate of A. Nicolau and J. Xiao for this particular function v, see for example [NiXi] g (ω) (7.12) v (z) C | | 2 dA (ω) . |∇ |≤ D 1 ωz Z Z | − | p σ D Since g (z) dAp(σ+1)−2 (z) is a weak Carleson measure for B2 ( ), it now follows from (7.12) and a ‘weak’ modification| | of Lemma E in [ArBlPa], which is a Carleson spreading lemma of Arcozzi, Blasi and Pau, that p(σ+1)−2 v (z) p 1 z 2 dA (z) is a weak Carleson measure for Bσ (D) (see for example the analogous |∇ | − | | p lemmas in [ArRoSa2] and [CoSaWi2] and references given there). So it only remains to show that v∗ L∞ (T). But this is a consequence of the Jones trick that uses first ∈

1+ ωz 1+ ωζ 1 ζ 2 Re + g (ω) dA (ω) 2 − | | 2 g (ω) dA (ω) C, ( |ω|≥|ζ| −1 ωz 1 ωζ | | ) ≤ D 1 ζz | | ≤ Z Z − − Z Z − and then for z T that ∈ 2 ωz 2 1 ζz − R R 1−| | |g(ω)|dA(ω) |ω|≥|ζ| |1−ωz|2 v (z) . − 2 e g (ζ) dA (ζ) . 1, | | D 1 ζz | | Z Z −

which completes the proof of Theorem 52.

With Theorem 52 in hand, it is now a routine matter to use the Koszul complex, or the simpliﬁed version σ D 1 in dimension n = 1, to prove the Corona Theorem 51 for the algebras Kp ( ) when 0 <σ< p . This proof strategy for weak Carleson measures can be seen in [CoSaWi2]. THE CORONA PROBLEM FOR KERNEL MULTIPLIER ALGEBRAS 33

7.3. Bergman spaces of solutions to generalized Cauchy-Riemann equations. Let A (z) and B (z) be two smooth real-valued symmetric invertible n n matrices defined on a domain Ω in Cn = R2n, where × n Cn n n Rn Rn we identify z = (zj)j=1 with (xj )j=1 , (yj )j=1 under the correspondence zj xj + iyj. ∈ ∈ × n ↔ Then the complex first order partial differential operator P A (z) x + iB (z) y on C is elliptic, and we ≡ ∇ ∇ can consider the solution space P of complex-valued functions (that are necessarily smooth by ellipticity): N ∞ P f C (Ω; C): Pf = 0 in Ω . N ≡{ ∈ } Remark 53. If A = B is the n n identity matrix, then P = ∂. Thus in this case Pf =0 is the system of × Cauchy-Riemann equations on Ω, and P is the linear space of holomorphic functions in Ω. N The linear space P is actually an algebra since if f,g P , then P (αf + βg) = αP (f)+ βP (g)=0 N ∈ N and P (fg)= fP (g)+ gP (g) = 0. The real and imaginary parts u of the functions in P are also solutions of the following real elliptic smooth coefficient divergence form second order equation inN R2n: A∗A 0 P ∗Pu = div grad u =0. 0 B∗B In particular, from the Schauder interior estimates (see e.g. Theorem 6.9 in [GiTr]), solutions u to P ∗Pu =0 satisfy a local Lipschitz inequality 1 2 2 u (a) u (b) CK,A,B u (x) dx , a,b K, | − |≤ | | ∈ ZΩ for every compact subset K of Ω. 2 Now we define the P -Bergman space AP (Ω) on Ω by

2 1 2 A (Ω) f P : f (x) dx < , P ≡ ∈ N Ω | | ∞ | | ZΩ 1 2 with norm f A2 (Ω) = f (x) dx. The local Lipschitz inequality for real and imaginary parts of k k P |Ω| Ω | | 2 2 solutions shows that pointq evaluationsR on AP (Ω) are continuous, and so AP (Ω) is a pre-Hilbert function 2 space. Moreover the Montel property for AP (Ω) is a standard consequence of the local Lipschitz inequality 2 and the Arzela-Ascoli theorem, and we thus see that AP (Ω) is a Hilbert function space on Ω with the Montel 2 property. Using that P is an algebra, it is now easy to see with = AP (Ω), that the multiplier algebra N ∞ H MH of is isometrically isomorphic to (Ω) equipped with the sup norm h ∞ supa∈Ω h (a) . Thus 2H H k k ≡ | | = AP (Ω) will be multiplier stable if the kernel functions ka for are bounded away from 0 and and Hlower semicontinuous in a, i.e. H ∞ 1 (7.13) ka is lower semicontinuous in a, < , a Ω. k k∞ k ∞ ∈ a ∞

Thus provided (7.13) holds, our alternate Toeplitz corona theorem reduces the Corona Property for the Banach algebra ∞ (Ω) of bounded solutions f to Pf = 0, to the Convex Poisson Property for the P - H2 Bergman space AP (Ω). This generalizes the notion of holomorphicity for which such corona theorems may hold. We do not pursue this direction any further here.

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Eric T. Sawyer, Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4K1 Canada E-mail address: [email protected]

Brett D. Wick, Department of Mathematics, Washington University – St. Louis, One Brookings Drive, St. Louis, MO USA 63130 E-mail address: [email protected]