Analytic and geometric properties of boundary functions satisfying the tangential CR-equation

Investigate the analytic and geometric properties of bounded measurable functions on the projective measured lamination space PML_{g,m} that satisfy the tangential CR-equation (Equation (CR)), which characterizes boundary values of bounded holomorphic functions on the Teichmüller space T_{g,m}.

Background

The boundary value (radial limit) of a bounded holomorphic function on T_{g,m} is characterized by a tangential CR-type integral condition (Equation (CR)). While this supplies a criterion for being a boundary value, the structure of functions satisfying this condition is not established.

Understanding these properties is a prerequisite for developing Hardy space theory and for bridging function theory on Teichmüller spaces with harmonic and pluriharmonic analysis on their Thurston boundaries.

References

The boundary value function (i.e., the radial limit function) of a bounded holomorphic function is essentially characterized by eq:CR. However, its analytic and geometric properties remain largely unknown.

— Function theory, Dynamics and Ergodic theory via Thurston theory (2507.20912 - Miyachi, 28 Jul 2025) in Subsection 5.2.1 (Herglotz-type formula and Hardy space); Equation (CR)