Boundary Geometry and Surjective Linear Isometries of Weighted Hardy Spaces

Abstract

Let $D\subset\mathbb C^n$, $n\geq 2$, be a bounded pseudoconvex domain with smooth boundary, and let $H^p_\omega(D)$ be the Hardy space defined using a weighted boundary measure $\omega\,d\sigma$, where $\omega$ is bounded above and bounded away from zero. For every $0<p<\infty$, $p\neq2$, we prove that each surjective linear isometry $T$ of $H^p_\omega(D)$ has the rigid form \( Tf=T(1)(f\circ\varphi), \) where $\varphi\in\operatorname{Aut}(D)$. This extends the classical Forelli-type classification beyond highly symmetric or polynomially convex domains to arbitrary smoothly bounded pseudoconvex domains. The principal difficulty is not the construction of a holomorphic symbol, but proving that this symbol takes values in $D$ and is in fact biholomorphic. We overcome this difficulty by combining equimeasurability methods of Rudin and Schneider with boundary uniqueness, holomorphic approximation, plurisubharmonic exhaustion functions, and removable-singularity arguments across analytic sets. We also solve the complementary geometric problem of determining when an automorphism of $D$ gives rise to an isometry. The answer depends decisively on the boundary measure. We construct two natural measures for which every automorphism induces an isometry: one obtained from an invariant defining function when $\operatorname{Aut}(D)$ is compact, and the other given by Fefferman's invariant surface measure. In sharp contrast, we exhibit domains with noncompact automorphism group---including domains biholomorphic to the unit ball---for which the analogous conclusion fails for ordinary Euclidean surface measure. Thus the isometric structure of Hardy spaces detects not only the biholomorphic geometry of the domain, but also the finer interaction between that geometry and the chosen boundary measure.