加权哈代空间的边界几何与满射线性等距映射

中文摘要 AI 辅助

设 $D$ 是复空间 $\boldsymbol{\text{C}}^n$（$n\boldsymbol{\text{≥}}2$）中具有光滑边界的有界拟凸域，$H^p_\boldsymbol{\text{ω}}(D)$ 是使用加权边界测度 $\boldsymbol{\text{ω}}\boldsymbol{\text{d}}\boldsymbol{\text{σ}}$ 定义的哈代空间，其中 $\boldsymbol{\text{ω}}$ 是有上界且远离零的函数。对于所有满足 $0\boldsymbol{<}p\boldsymbol{<}\boldsymbol{\text{∞}}$ 且 $p\boldsymbol{≠}2$ 的情况，我们证明 $H^p_\boldsymbol{\text{ω}}(D)$ 的每个满射线性等距映射 $T$ 都具有刚性形式 $Tf=T(1)(f\boldsymbol{\text{∘}}\boldsymbol{\text{φ}})$，其中 $\boldsymbol{\text{φ}}\boldsymbol{\text{∈}}\boldsymbol{\text{Aut}}(D)$。这将经典的 Forelli 型分类从高度对称或多项式凸域扩展到任意光滑有界拟凸域。主要困难不在于全纯符号的构造，而在于证明该符号取值于 $D$ 且实际上是双全纯的。我们结合 Rudin 和 Schneider 的等测度方法、边界唯一性、全纯逼近、多重次调和穷竭函数以及解析集上的可去奇点论证，克服了这一困难。我们还解决了一个互补的几何问题：确定 $D$ 的自同构何时会产生等距映射，答案关键取决于边界测度。我们构造了两种自然测度，对于这两种测度，每个自同构都诱导出一个等距映射：一种是当 $\boldsymbol{\text{Aut}}(D)$ 是紧集时从不变定义函数得到的测度，另一种是 Fefferman 的不变曲面测度。与之形成鲜明对比的是，我们展示了具有非紧自同构群的域（包括与单位球双全纯的域），对于普通欧几里得曲面测度，类似的结论不成立。因此，哈代空间的等距结构不仅能检测域的双全纯几何，还能检测该几何与所选边界测度之间更精细的相互作用。

英文摘要

Let $D\subset\mathbb C^n$, $n\geq 2$, be a bounded pseudoconvex domain with smooth boundary, and let $H^p_ω(D)$ be the Hardy space defined using a weighted boundary measure $ω\,dσ$, where $ω$ is bounded above and bounded away from zero. For every $0<p<\infty$, $p\neq2$, we prove that each surjective linear isometry $T$ of $H^p_ω(D)$ has the rigid form \( Tf=T(1)(f\circφ), \) where $φ\in\operatorname{Aut}(D)$. 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.

发表机构

• Tianjin University（天津大学）
• University of California, Irvine（加州大学尔湾分校）
• Minjiang University（闽江学院）
• Fujian Normal University（福建师范大学）

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