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arXiv · 2606.30927

Sobolev spaces on snowtrees

Efstathios Konstantinos Chrontsios Garitsis·Vyron Vellis
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Abstract

We introduce a discrete-energy Sobolev space $\mathcal{W}^{1,p}_{\mathscr V}(T)$ on Ahlfors regular snowtrees, a class of metric trees where every arc is a snowflake of the same type. Our main result shows that, for every partition $\mathscr V$ and every $1<p<\infty$, this discrete space coincides quantitatively with the Korevaar--Schoen space on $T$. This fact and the independence of the space on the particular partition used to define $\mathcal{W}^{1,p}_{\mathscr V}(T)$ are both novel even for the class of geodesic trees. We also determine the critical Korevaar-Schoen exponent for Ahlfors regular snowtrees and prove capacity attainment and upper estimates, which reveal the appropriate walk dimension needed for the corresponding probabilistic profile on these trees.

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BibTeXRIS

Efstathios Konstantinos Chrontsios Garitsis, Vyron Vellis. 2026-06-29. Sobolev spaces on snowtrees. https://arxiv.org/abs/2606.30927

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