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Finite-Support Approximation in Variable-Coefficient Sobolev Spaces on Nonhomogeneous Flow-Measure Trees
Abstract
Let T be a nonhomogeneous rooted tree equipped with a positive flow measure m. For 1<p<∞, consider the variable-coefficient Sobolev space on the nonhomogeneous flow-measure tree~T Wa1,p(T,m) = { f∈ Lp(T,m): ∑x≠ ρa(x)m(x)lvert f(x)−f(π(x))rvertp<∞ }, where a is a positive variable coefficient, ρ denotes the root and π(x) denotes the parent of x. Using an averaging method over descendant levels, we prove the density of finitely supported functions in Wa1,p(T,m) for coefficients satisfying a weighted moment condition. As an application, the relationship between solutions on finite domains and global solutions is clarified for p=2.
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BibTeX
@misc{cheng2026finitesupport,
title = {{Finite-Support Approximation in Variable-Coefficient Sobolev Spaces on Nonhomogeneous Flow-Measure Trees}},
author = {Zeyu Cheng and Yichun Zhao and Jiang Zhou},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07350},
eprint = {2611.07350},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07350},
}
RIS
TY - UNPB
TI - Finite-Support Approximation in Variable-Coefficient Sobolev Spaces on Nonhomogeneous Flow-Measure Trees
AU - Zeyu Cheng
AU - Yichun Zhao
AU - Jiang Zhou
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07350
UR - https://arxiv.org/abs/2611.07350
AB - Let T be a nonhomogeneous rooted tree equipped with a positive flow measure m. For 1<p<∞, consider the variable-coefficient Sobolev space on the nonhomogeneous flow-measure tree~T W_a^1,p(T,m) = { f∈ L^p(T,m): ∑_x≠ ρa(x)m(x)lvert f(x)−f(π(x))rvert^p<∞ }, where a is a positive variable coefficient, ρ denotes the root and π(x) denotes the parent of x. Using an averaging method over descendant levels, we prove the density of finitely supported functions in W_a^1,p(T,m) for coefficients satisfying a weighted moment condition. As an application, the relationship between solutions on finite domains and global solutions is clarified for p=2.
ER -