Boundary values of bounded pluriharmonic functions on Teichmüller space

Characterize the L∞ boundary functions on the projective measured lamination space PML_{g,m} that arise as radial limits of bounded pluriharmonic functions on the Teichmüller space T_{g,m}. Equivalently, determine necessary and sufficient conditions on a function f ∈ L∞(PML_{g,m}) for which there exists a bounded pluriharmonic function u on T_{g,m} whose boundary value u* equals f (with respect to the Thurston measure and the Poisson integral representation).

Background

The paper proves Poisson integral formulas that represent bounded pluriharmonic functions on Teichmüller space T_{g,m} via boundary data on PML_{g,m}. In contrast to the classical Fatou theorem for the disk, the authors note that the identification of which boundary functions actually occur as radial limits is not settled.

This issue is central to extending function-theoretic tools (e.g., Hardy space theory) to Teichmüller spaces, and it connects to understanding the classes of measurable functions on PML_{g,m} compatible with pluriharmonicity in the interior via the Teichmüller–Thurston Poisson kernel.