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 $\varphi$ on ${\mathbb{D}}^3$ so that $\Vert \varphi {\Vert }_{{\mathbb{D}}^3}=\Vert p{\Vert }_{\mathcal{V}}$ and $\varphi {|}_{\mathcal{V}}=p{|}_{\mathcal{V}}$. 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.