无限度量树上的嵌入迹算子
Embedded trace operator for infinite metric trees
AI总结：
本文研究无限加权度量树上的迹算子，统一了两种已有的边界迹构造，证明其等价性，并建立迹的Sobolev正则性，同时提出与任意紧致黎曼流形的等同方法。
摘要：
我们考虑一类无限加权度量树，它们是自相似正则树的扰动。通过将树的边界与曲面进行等同，我们讨论了此类结构上Sobolev空间中函数的边界迹的可能定义。我们的方法统一了Maury、Salort和Vannier（2009）针对二进离散加权树提出的构造（利用图上调和函数的正交基展开，并在表示边界的区域上使用Haar型基），以及Nicaise和Semin（2018）与Joly、Kachanovska和Semin（2019）针对分形度量树提出的构造（通过有限截面逼近并将边界等同于区间）：我们证明这两种机制给出相同的迹映射，并且在一组参数范围内，我们建立了迹的精确Sobolev正则性。此外，我们引入新的几何要素，提出与任意黎曼流形的等同。我们证明任何紧致流形都允许合适的多尺度分解，因此在迹定理的语境下可以被等同于度量树的边界。
英文摘要：
We consider a class of infinite weighted metric trees obtained as perturbations of self-similar regular trees. Possible definitions of the boundary traces of functions in the Sobolev space on such a structure are discussed by using identifications of the tree boundary with a surface. Our approach unifies some constructions proposed by Maury, Salort, Vannier (2009) for dyadic discrete weighted trees (expansion in orthogonal bases of harmonic functions on the graph and using Haar-type bases on the domain representing the boundary), and by Nicaise, Semin (2018) and Joly, Kachanovska, Semin (2019) for fractal metric trees (approximation by finite sections and identification of the boundary with a interval): we show that both machineries give the same trace map, and for a range of parameters we establish the precise Sobolev regularity of the traces. In addition, we introduce new geometric ingredients by proposing an identification with arbitrary Riemannian manifolds. It is shown that any compact manifold admits a suitable multiscale decomposition and, therefore, can be identified with a metric tree boundary in the context of trace theorems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
