Extension Problems for the Vladimirov--Taibleson Operator and the Hierarchical Laplacian

@misc{sun2026extension,
  title = {{Extension Problems for the Vladimirov--Taibleson Operator and the Hierarchical Laplacian}},
  author = {Yaojia Sun},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03647},
  eprint = {2609.03647},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03647},
}

TY  - UNPB
TI  - Extension Problems for the Vladimirov--Taibleson Operator and the Hierarchical Laplacian
AU  - Yaojia Sun
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.03647
UR  - https://arxiv.org/abs/2609.03647
AB  - We establish a non-Archimedean Caffarelli-Silvestre extension theory for the Vladimirov-Taibleson operator D^s for s>0 and, more generally, for hierarchical Laplacians L_C. For every f∈𝒮(ℚ_p^n), we equip the Bruhat-tits tree 𝒯_p^n with the weight w_v_k−1v_k=p^k(s−n) under the horocyclic coordinates. We prove that the resulting Dirichlet problem admits a unique bounded weighted-harmonic solution that is continuous on the end compactification. Its boundary traces converge to f uniformly and in L^2, while the associated deformed normal derivative converges pointwise and in L^2 to D^s f. We derive explicit Fourier and Poisson representations of the extension and establish an energy identity between its weighted tree energy and the quadratic form of D^s. Motivated by p-adic AdS/CFT, we also give an equivalent massive formulation on the unweighted tree, based on a renormalized trace and a scale-corrected normal derivative. Finally, under natural local-finiteness and scaling assumptions, we extend the construction to hierarchical Laplacians L_C on ultrametric spaces. We characterize the canonical edge conductances by requiring the cancellation Green operator on the ultrametric tree 𝒯_X to coincide with L_C^−1 on 𝒮_0(X). In the corresponding canonical flux class, the extension is unique, its Dirichlet-to-Neumann map is L_C, and it satisfies the associated energy identity. The Vladimirov-Taibleson construction is recovered as the homogeneous special case.
ER  -