Radial limit sets on the torus

Abstract:

Let ${U^N}$ denote the unit polydisc and ${T^N}$ the unit torus in the space of $N$ complex variables. A subset $A$ of ${T^N}$ is called an (RL)-set (radial limit set) if to each positive continuous function $\rho$ on ${T^N}$, there corresponds a function $f$ in ${H^\infty }({U^N})$ such that the radial limit $|f{|^ \ast }$ of the absolute value of $f$ equals $\rho$, a.e. on ${T^N}$ and everywhere on $A$. If $N > 1$, the question of characterizing (RL)-sets is open, but two positive results are obtained. In particular, it is shown that ${T^N}$ contains an (RL)-set which is homeomorphic to a cartesian product $K \times {T^{N - 1}}$, where $K$ is a Cantor set. Also, certain countable unions of “parallel” copies of ${T^{N - 1}}$ are shown to be (RL)-sets in ${T^N}$. In one variable, every subset of $T$ is an (RL)-set; in fact, there is always a zero-free function $f$ in ${H^\infty }(U)$ with the required properties. It is shown, however, that there exist a circle $A \subset {T^2}$ and a positive continuous function $\rho$ on ${T^2}$ to which correspond no zero-free $f$ in ${H^\infty }({U^2})$ with $|f{|^ \ast } = \rho$ a.e. on ${T^2}$ and everywhere on $A$.