Characterize boundary measures inducing Hardy-space isometries

Determine, for a bounded smoothly bounded pseudoconvex domain D, an automorphism φ ∈ Aut(D), and a boundary measure ω dσ, precisely when there exists a holomorphic multiplier g such that the weighted composition operator gCφ is an isometry of the weighted Hardy space H^p_ω(D).

Background

The paper proves that every surjective linear isometry of H^p_ω(D), for p ≠ 2, is induced by a biholomorphic automorphism of D together with a holomorphic multiplier. The converse problem is geometric: given an automorphism φ, one must determine whether the boundary-measure distortion associated with φ can be represented by the modulus of a holomorphic multiplier.

For ordinary Euclidean surface measure, the relevant boundary Jacobian generally depends on the extrinsic geometry of ∂D and need not be the modulus of a holomorphic function. The paper gives affirmative answers for two natural invariant measures—one derived from an invariant defining function when Aut(D) is compact, and Fefferman’s invariant surface measure—but the general characterization requested in Problem 1.2 is not stated as fully resolved.