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 $1<p<\infty$, consider the variable-coefficient Sobolev space on the nonhomogeneous flow-measure tree~$T$ \[ W_a^{1,p}(T,m) = \left\{ f\in L^p(T,m): \sum_{x\ne ρ}a(x)m(x)\lvert f(x)-f(π(x))\rvert^p<\infty \right\}, \] where $a$ is a positive variable coefficient, $\rho$ denotes the root and $\pi (x)$ 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 $p=2$.