Let $H^\infty $ be the algebra of bounded holomorphic functions on the open unit disk, and let $\mathfrak M$ be its maximal ideal space. Let $\mathfrak M_a$ be the union of nontrivial Gleason parts (analytic disks) of $\mathfrak M$. In this paper, we study the problem of extensions of bounded Banach-valued holomorphic functions and holomorphic maps with values in Oka manifolds from Gleason parts of $\mathfrak M_a\setminus \mathbb {D}$. The resulting extensions satisfy the uniform boundedness principle in the sense that their norms are bounded by constants that do not depend on the choice of the Gleason part. The results extend fundamental results of D. Suárez on the characterization of the algebra of restrictions of Gelfand transforms of functions in $H^\infty $ to Gleason parts of $ \mathfrak M_a\setminus \mathbb {D}$. The proofs utilize our recent advances on $\bar \partial $-equations on quasi-interpolating sets and Runge-type approximations.

令$H^\infty $为开单位圆盘上有界全纯函数的代数，并令$\mathfrak M$为其最大理想空间。令$\mathfrak M_a$为$\mathfrak M$的非平凡格里森部分（解析盘）的并集。在本文中，我们研究了$\mathfrak M_a\setminus \mathbb {D}$的 Gleason 部分的有界 Banach 值全纯函数和 Oka 流形中的值的全纯映射的扩展问题。由此产生的扩展满足统一有界原则，因为它们的范数受不依赖于格里森部分的选择的常数限制。这些结果将$H^\infty $中函数 Gelfand 变换限制代数表征的 D. Suárez 的基本结果扩展到$ \mathfrak M_a\setminus \mathbb {D}$的格里森部分。这些证明利用了我们在拟插值集和龙格型近似上的$\bar \partial $方程方面的最新进展。