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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] Follow this and additional works at: https://openscholarship.wustl.edu/math_facpubs Part of the Algebra Commons, and the Analysis Commons 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]. THE CORONA PROBLEM FOR KERNEL MULTIPLIER ALGEBRAS ERIC T. SAWYER† AND BRETT D. WICK‡ Abstract. We prove an alternate Toeplitz corona theorem for the algebras of pointwise kernel multipliers of Besov-Sobolev spaces on the unit ball in Cn, and for the algebra of bounded analytic functions on certain strictly pseudoconvex domains and polydiscs in higher dimensions as well. This alternate Toeplitz corona theorem extends to more general Hilbert function spaces where it does not require the complete Pick property. Instead, the kernel functions kx (y) of certain Hilbert function spaces H are assumed to be invertible multipliers on H, and then we continue a research thread begun by Agler and McCarthy in 1999, and continued by Amar in 2003, and most recently by Trent and Wick in 2009. In dimension n = 1 we prove the corona theorem for the kernel multiplier algebras of Besov-Sobolev Banach spaces in the unit disk, ∞ extending the result for Hilbert spaces H ∩ Qp by A. Nicolau and J. Xiao. Contents 1. Introduction 1 1.1. Organization of the paper 3 2. Preliminaries 4 2.1. Hilbert function spaces 4 2.2. Interpolation by rescalings 8 2.3. A characterization of rescaling 10 3. The Bezout kernel multiplier characterization 12 3.1. Kernel multiplier solutions 12 4. The alternate Toeplitz corona theorem 15 4.1. Corona Properties 15 4.2. Convex shifted spaces 16 4.3. Formulation and proof 18 5. The alternate Toeplitz corona theorem for Bergman and Hardy spaces in Cn 19 5.1. The Bergman and Hardy space on the unit ball 20 5.2. The Hardy and Bergman space on the unit polydisc 20 5.3. General domains 21 6. Invertible Multiplier Property for Hardy spaces 22 7. Smoothness spaces and nonholomorphic spaces 25 7.1. The kernel multiplier algebras for Besov-Sobolev spaces 25 arXiv:1410.8862v4 [math.CA] 10 Oct 2016 7.2. The Corona Property for kernel multiplier algebras on the disk 31 7.3. Bergman spaces of solutions to generalized Cauchy-Riemann equations 33 References 33 1. Introduction In 1962 L. Carleson [Car] proved the corona theorem for the algebra of bounded analytic functions on the unit disk. The proof used a beautiful ‘corona construction’ together with properties of Blaschke products. While there is a large literature on corona theorems for domains in one complex dimension (see e.g. [Nik]), progress in higher dimensions prior to 2011 had been limited to the Hp corona theorem on various bounded domains in Cn, and weaker results restricting N to 2 generators. In fact, apart from the simple cases in Date: October 11, 2016. † Research supported in part by a grant from the National Science and Engineering Research Council of Canada. ‡ Research supported in part by National Science Foundation DMS grant # 1603246 and #1560955. 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], Ess´en and Xiao [EsXi], Aulaskari, Stegenga and Xiao [AuStXi], or Xiao∩ [Xia, Xia3] for the definitions of these spaces.
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