# Revista Matemática Iberoamericana

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**Published online first: 2019-10-07**

**DOI: 10.4171/rmi/1149**

Extensions of bounded holomorphic functions on the tridisk

Łukasz Kosiński^{[1]}and John E. McCarthy

^{[2]}(1) Jagiellonian University, Kraków, Poland

(2) Washington University in St. Louis, USA

A set $\mathcal{V}$ in the tridisk $\mathbb{D}^3$ has the polynomial extension property if for every polynomial $p$ there is a function $\phi$ on $\mathbb{D}^3$ so that $\| \phi \|_{\mathbb{D}^3} = \| p \|_{\mathcal{V}}$ and $\phi |_{\mathcal{V}} = p|_{\mathcal{V}}$. We study sets $\mathcal{V}$ that are relatively polynomially convex and have the polynomial extension property. If $\mathcal{V}$ is one-dimensional, and is either algebraic, or has polynomially convex projections, we show that it is a retract. If $\mathcal{V}$ is two-dimensional, we show that either it is a retract, or, for any choice of the coordinate functions, it is the graph of a function of two variables.

*Keywords: *Tridisk, retract, extension

Kosiński Łukasz, McCarthy John: Extensions of bounded holomorphic functions on the tridisk. *Rev. Mat. Iberoam.* Electronically published on October 7, 2019. doi: 10.4171/rmi/1149 (to appear in print)