弗拉基米罗夫-泰布勒松算子与分层拉普拉斯算子的延拓问题
Extension Problems for the Vladimirov--Taibleson Operator and the Hierarchical Laplacian
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中文总结 AI 辅助
该研究针对弗拉基米罗夫-泰布勒松算子及分层拉普拉斯算子，建立非阿基米德型Caffarelli-Silvestre延拓理论，推导相关表示与能量恒等式，还将构造推广至超度量空间，为其提供了典范边电导的刻画。
中文摘要 AI 辅助
针对s>0的弗拉基米罗夫-泰布勒松算子D^s，以及更一般的分层拉普拉斯算子L_C，我们建立了非阿基米德型的Caffarelli-Silvestre延拓理论。对每个f∈𝒮(ℚ_p^n)，我们在水平坐标下赋予布吕阿-蒂茨树𝒯_{p^n}权重w_{v_{k-1}v_k}=p^{k(s-n)}，证明所得狄利克雷问题存在唯一有界的加权调和解，该解在端点紧化上连续，其边界迹以一致方式和L²意义收敛到f，关联的形变法向导数以逐点方式和L²意义收敛到D^s f。我们推导了该延拓的显式傅里叶表示与泊松表示，并建立了其加权树能量与D^s二次型之间的能量恒等式。受p进AdS/CFT启发，我们还基于重归一化迹与尺度修正法向导数，在无权树上给出了等价的质量型公式。最后，在自然的局部有限性与尺度假设下，我们将该构造推广到超度量空间上的分层拉普拉斯算子L_C，通过要求超度量树𝒯_X上的抵消格林算子在𝒮₀(X)上与L_C⁻¹一致来刻画典范边电导；在对应的典范通量类中，该延拓是唯一的，其狄利克雷-纽曼映射为L_C，且满足关联的能量恒等式，弗拉基米罗夫-泰布勒松构造作为齐次特例被包含在内。
英文摘要
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.
