Carleson插值序列的连续保持者的刻画与边界正则性
Characterizations and boundary regularity of continuous preservers of Carleson interpolating sequences
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中文总结 AI 辅助
本文刻画了保持Carleson插值序列的连续圆盘映射，给出边界核成员资格的显式判据，并证明其等价于边界微分行列式为正，无需内部可微性。
中文摘要 AI 辅助
我们刻画了单位圆盘上保持每个Carleson插值序列的连续映射。单向保持等价于一个圆盘同胚，其逆映射在伪双曲度量下一致连续，且其相关的加权前推在正Carleson测度上有界。双向保持等价于一个圆盘同胚，其自身及其逆映射在该度量下均一致连续，并且具有强拟对称边界延拓。在此类中，边界缺陷的可比较性等价于一个双Lipschitz边界映射。基于我们先前工作中建立的典范分解，我们给出了连续边界核中成员资格的显式判据。对于具有恒等边界值和一阶一致边界扩张的闭圆盘同胚，核成员资格等价于实边界微分的行列式为正，或等价于边界缺陷的内向法向导数为正。内部不需要可微性。闭圆盘上的$C^1$判据作为推论得出，而一个反例表明逐点边界可微性并不充分。我们还刻画了具有拟共形核的分解，并记录了其径向因子的精确最大伸缩商。
英文摘要
We characterize the continuous mappings of the unit disk that preserve every Carleson interpolating sequence. Preservation in one direction is equivalent to being a disk homeomorphism whose inverse is uniformly continuous in the pseudohyperbolic metric and whose associated weighted pushforward is bounded on positive Carleson measures. Preservation in both directions is equivalent to being a disk homeomorphism that, together with its inverse, is uniformly continuous in that metric and has a strongly quasisymmetric boundary extension. Within this class, comparability of the boundary defects is equivalent to a bi-Lipschitz boundary map. Building on the canonical factorization established in our earlier work, we give explicit criteria for membership in the continuous boundary kernel. For a homeomorphism of the closed disk with identity boundary values and a uniform boundary expansion of first order, kernel membership is equivalent to positivity of the determinant of the real boundary differential, or equivalently of the inward normal derivative of the boundary defect. No differentiability in the interior is required. The $C^1$ criterion on the closed disk follows as a corollary, while a counterexample shows that pointwise boundary differentiability does not suffice. We also characterize factorizations with a quasiconformal kernel and record the exact maximal dilatation of its radial factor.
