  We establish an equivalency of the Corona problem (1962) and Gleason 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 including balls and polydiscs. Indeed, we extend our
recent work on Gleason 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 such as holomorphic Holder and Lipschitz spaces (left
open by Fornaess and Ovrelid 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 a, where a in (0, 1]; in particular, the Gelfand theory and the
maximal ideal spaces of these algebras are discussed.
