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arXiv · 2610.39425

The functional analytic approach establishes the Corona theorem in the theory of SCV

S. R. Patel
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Abstract

We give an affirmative solution of the original Corona problem (1962) for certain bounded pseudoconvex domains or polydomains in $\mathbb{C}^n$, including the balls and polydiscs, by using the Gleason solvability property of such domains for $H^\infty(G)$. In fact, we establish an equivalency of the Corona problem and Gleason's problem (1964) in the theory of several complex variables. Indeed, we extend our recent works on both problems based on the functional analytic approaches, and extend recent work of Clos, as well as apply the usual Banach algebra method. As an application, we establish the I. J. Schark's theorem for these $G$'s. We also obtain a welcome extension of a result of Krantz and Li (1995) for a bounded, strictly pseudoconvex domain in $\mathbb{C}^n$ with $C^3$ boundary.

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S. R. Patel. 2026-10-29. The functional analytic approach establishes the Corona theorem in the theory of SCV. https://arxiv.org/abs/2610.39425

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