AI 中文总结
本文引入双曲空间上的对数$p$-拉普拉斯算子，证明其分数阶极限公式，建立逐点积分表示，验证其可作为延拓问题的解，并给出欧氏空间对应算子的延拓定理，填补相关研究空白。
AI 中文摘要
本文引入了$n\geq2$的双曲空间$\mathbb{H}^n$上的对数$p$-拉普拉斯算子$\log(-\Delta_{\mathbb{H}^n})_p$。我们证明：若$f$是$\mathbb{H}^n$上具有紧支集、指数为$\alpha\in(0,1)$的局部利普希茨函数，则对合适的常数$A_{n,p}>0$，有$\lim_{s\rightarrow0^+}(-\Delta_{\mathbb{H}^n})_p^s f(x)=A_{n,p}|f(x)|^{p-2}f(x)$，其中$x\in\mathbb{H}^n$，$(-\Delta_{\mathbb{H}^n})_p^s$表示$\mathbb{H}^n$上的$s$阶分数$p$-拉普拉斯算子。我们建立了算子$\log(-\Delta_{\mathbb{H}^n})_p=\frac{d}{ds}(-\Delta_{\mathbb{H}^n})_p^s|_{s=0}$的逐点积分表示。此外，我们证明该算子可实现为某类合适延拓问题的解，并给出了一个能得到$\mathbb{R}^n$中$\log(-\Delta)_p$算子的延拓定理。据我们所知，该性质尚未在欧几里得对数$p$-拉普拉斯算子$\log(-\Delta)_p$中得到证明。
英文摘要
In this paper, the logarithmic $p$-Laplacian operator $\log (-Δ_{\mathbb H ^n})_p$ on the hyperbolic space $\mathbb H^n$, with $n\geq 2$, is introduced. We prove that if $f$ is a locally Lipschitz function of exponent $α\in (0,1)$ with compact support in $\mathbb H^n$, then, for a suitable constant $A_{n,p}>0$, $$ \lim_{s\rightarrow 0^+}(-Δ_{\mathbb H ^n})_p^sf(x)=A_{n,p}|f(x)|^{p-2}f(x),\quad x\in \mathbb H^n, $$ where $(-Δ_{\mathbb H ^n})_p^s$ denotes the $s$-fractional $p$-Laplacian on $\mathbb H^n$. We establish a pointwise integral representation for the operator $\log (-Δ_{\mathbb H ^n})_p=\frac{d}{ds}(-Δ_{\mathbb H^n})_p^s\,_{|s=0}$. Furthermore, we show that $\log (-Δ_{\mathbb H ^n})_p$ can be realized as the solution of a suitable extension problem and provide an extension theorem that yields the operator $\log (-Δ)_p$ in $\mathbb R^n$. To the best of our knowledge, this property has not been established for the Euclidean logarithmic $p$-Laplacian $\log (-Δ)_p$.
