Extensions of bounded holomorphic functions on the tridisk

Abstract

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.