Embedded trace operator for infinite metric trees
arXiv:2303.04527 · doi:10.1002/mana.202300574
Abstract
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.
arXiv comment67 pages. Several wrong citations were corrected
References in corpus (14)
- Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
- Scattering on compact manifolds with infinitely thin horns
- Spectra of Schroedinger operators on equilateral quantum graphs
- Spectral Theory of Infinite Quantum Graphs
- Unitary dimension reduction for a class of self-adjoint extensions with applications to graph-like structures
- Geometric analysis on Cantor sets and trees
- Non-self-adjoint graphs
- Extension and trace results for doubling metric measure spaces and their hyperbolic fillings
- An example of unitary equivalence between self-adjoint extensions and their parameters
- Random walks and induced Dirichlet forms on self-similar sets
- Unbounded quantum graphs with unbounded boundary conditions
- New relations between discrete and continuous transition operators on (metric) graphs
- On the Decomposition of the Laplacian on Metric Graphs
- Resolvent expansions on hybrid manifolds