The corona problem with two pieces of data

Krantz, Steven G.

doi:10.1090/s0002-9939-10-10462-6

Cited by 4 publications

(8 citation statements)

References 14 publications

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“…Also, we apply the same methods to solve the corona problem for the same familiar Banach spaces of bounded holomorphic functions on various types of domains in C n (including the ball and the polydisc). In the process, we also obtain extensions of certain results in some cases (e.g., see [KL2,K5]). In §5, we give some concluding remarks on our approaches, leading to future directions/open problems.…”

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confidence: 63%

On functional analytic approach for Gleason’s problem in the theory of SCV

Patel

2022

Banach J. Math. Anal.

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We establish an equivalency of the Corona problem (1962) and Gleason's problem (1964) in the theory of several complex variables. As an application, we give an affirmative solution of the Corona problem for certain bounded pseudoconvex domains or polydomains in C n including balls and polydiscs. Indeed, we extend our recent work on Gleason's problem based on the functional analytic approach, as well as extend recent work of Clos. We also use this equivalency or else other (functional analytic) methods to affirmatively solve both problems for various Banach spaces of bounded holomorphic functions (including certain holomorphic mixed-norm spaces) on various types of domains in C n such as holomorphic Hölder and Lipschitz spaces (left open by Fornaess and Øvrelid in 1983), holomorphic mean Besov-Lipschitz spaces, Besov-Lipschitz spaces, Hardy-Sobolev spaces, and a weighted Bergman space. The discussion goes via first studying Lipschitz algebras of holomorphic functions of order α, where α ∈ (0, 1]; in particular, the Gel'fand theory and the maximal ideal spaces of these algebras are discussed.[KL1]. Also, we apply the same methods to solve the corona problem for the same familiar Banach spaces of bounded holomorphic functions on various types of domains in C n (including the ball and the polydisc). In the process, we also obtain extensions of certain results in some cases (e.g., see [KL2,K5]). In §5, we give some concluding remarks on our approaches, leading to future directions/open problems. Throughout the paper, "algebra" will mean a complex, commutative algebra with identity.1 History of the two problems and formulation of an equivalency. Corona problem. In 1938, Gel'fand discussed Banach algebras in his thesis. In 1942, Kakutani asked the following fundamental question (known as Corona problem) in the subject: Let H ∞ (D) be the Banach algebra of all bounded, holomorphic functions on the open unit disc D in the complex plane C, equipped with the usual sup-norm, and let f j ∈ H ∞ (D), j = 1, 2, . . . , n, with the property that n j=1 |f j (z)| > ǫ for some positive constant ǫ and all z ∈ D (call it a finite set of corona data on the disc D), do there exist g j ∈H ∞ (D), j = 1, 2, . . . , n, such that n j=1 f j g j = 1 (i.e., n j=1 f j (z)g j (z) = 1

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confidence: 63%

On functional analytic approach for Gleason’s problem in the theory of SCV

Patel

2022

Banach J. Math. Anal.

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“…Our aim is to generalize in some way the results of Krantz [7] to an arbitrary number of pieces of corona data, using the same method (i.e. the Koszul complex) and obtain, in this setting, a solution to the problem.…”

Section: Main Resultmentioning

confidence: 99%

“…In the end, we shall make some remarks about the necessity of a stronger extra condition than the one used in [7], even for two pieces of data. We shall work on a bounded, strongly pseudoconvex domain Ω ⊂ C n .…”

Section: Main Resultmentioning

confidence: 99%

“…It is easy to see that on this set, f 1 , f 2 verify the Corona condition. Next, it goes just like in [7]: consider a partition of unity {ϕ 1 , ϕ 2 }, subordinated to the covering {U 1 (α), U 2 (α)} ∩ W k (α) ∩ . .…”

Section: Theoremmentioning

confidence: 99%

“…In fact, we will see that for m > 2, this is no longer possible, so the method used in [7] for proving the existence of bounded solutions to the ∂-equation fails.…”

Section: Remarksmentioning

confidence: 99%

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