AI 中文总结
本文在带点测地格罗莫夫（pmG）收敛语境下证明Hajlasz-Sobolev映射超极限存在及相关图收敛结论，将普赖斯现象推广到无限维巴拿赫空间目标映射及无加倍假设的切空间。
AI 中文摘要
本文证明了当空间序列$X_i$在带点测地格罗莫夫（pmG）意义下收敛到极限空间时，Hajlasz-Sobolev映射序列$f_i:X_i\to Y_i$的超极限存在，部分推广了[T. Ikonen和S. Wenger，2026年，arXiv:2603.05246]的最新结果。此外，本文还证明了这些映射的图$G(f_i)$在合适意义下pmG收敛到超极限的图，这一结论源于pmG收敛语境下的合适Arzela-Ascoli定理。作为应用，本文针对映射包$f:(X,\boldsymbol{\to}V$到任意巴拿赫空间建立了普赖斯现象的一个版本。除了将其推广到到无限维目标空间的映射外，本文的结果还通过在无加倍假设的带点测地格罗莫夫-豪斯多夫切空间上建立普赖斯现象，推广了现有版本的普赖斯现象[G. C. David，《几何与函数分析》，25（2015）]、[N. Gigli、A. Mondino和T. Rajala，《纯粹与应用数学杂志》，705（2015）]。
英文摘要
We show the existence of ultralimits of sequences of Hajlasz-Sobolev maps $f_i:X_i\to Y_i$ when $X_i$ converges to a limit space in the pointed measured Gromov (pmG) sense, partially extending recent results in [T. Ikonen and S. Wenger, (2026), arXiv:2603.05246]. We moreover demonstrate that the graphs $G(f_i)$ of the mappings pmG-converge to the graph of the ultralimit in a suitable sense. The latter fact stems from a suitable Arzela-Ascoli theorem in the context of pmG-convergence. As an application, we establish a version of Preiss's phenomenon for mapping packages $f:(X,μ)\to V$ into arbitrary Banach spaces. Besides extending it to maps into infinite dimensional targets, our result generalizes existing versions of Preiss's phenomenon [G. C. David, Geom. Funct. Anal., 25 (2015)], [N. Gigli, A. Mondino, and T. Rajala, J. Reine Angew. Math., 705 (2015)] by establishing it for pointed measured Gromov-Hausdorff tangents without a doubling assumption.
