Boundary interpolation sets for holomorphic functions smooth to the boundary and BMO

Bruna, Joaquim

doi:10.1090/S0002-9947-1981-0603770-5

Boundary interpolation sets for holomorphic functions smooth to the boundary and BMO
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by Joaquim Bruna PDF

Trans. Amer. Math. Soc. 264 (1981), 393-409 Request permission

Abstract:

Let ${A^p}$ denote the class of holomorphic functions on the unit disc whose first $p$-derivatives belong to the disc algebra. We characterize the boundary interpolation sets for ${A^p}$, that is, those closed sets $E \subset T$ such that every function in ${C^p}(E)$ extends to a function in ${A^p}$. We also give a constructive proof of the corresponding result for ${A^\infty }$ (see [1]). We show that the structure of these sets is in some sense related to BMO and that this fact can be used to obtain precise estimates of outer functions vanishing on $E$.

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