Radial limit sets on the torus

Abstract

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