Extension Problems for the Vladimirov--Taibleson Operator and the Hierarchical Laplacian
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 $f\in\mathcal{S}(\mathbb{Q}_p^n)$, we equip the Bruhat-tits tree $\mathcal{T}_{p^n}$ with the weight $w_{v_{k-1}v_k}=p^{k(s-n)}$ 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 $\mathcal{T}_X$ to coincide with $L_C^{-1}$ on $\mathcal{S}_0(X)$. 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.