The logarithmic $p$-Laplacian on hyperbolic spaces

Abstract

In this paper, the logarithmic $p$-Laplacian operator $\log (-\hat{I}{”}_{{\mathbb{H}}^n}{)}_p$ on the hyperbolic space ${\mathbb{H}}^n$, with $n\ge 2$, is introduced. We prove that if $f$ is a locally Lipschitz function of exponent $\hat{I}\pm \in (0,1)$ with compact support in ${\mathbb{H}}^n$, then, for a suitable constant $A_{n,p}>0$, $$\underset{s\to 0^{+}}{\lim }(-\hat{I}{”}_{{\mathbb{H}}^n}{)}_p^{s}f(x)=A_{n,p}|f(x){|}^{p-2}f(x),x\in {\mathbb{H}}^n,$$ where $(-\hat{I}{”}_{{\mathbb{H}}^n}{)}_p^s$ denotes the $s$-fractional $p$-Laplacian on ${\mathbb{H}}^n$. We establish a pointwise integral representation for the operator $\log (-\hat{I}{”}_{{\mathbb{H}}^n}{)}_p=\frac{d}{ds}(-\hat{I}{”}_{{\mathbb{H}}^n}{)}_p^s{}_{|s=0}$. Furthermore, we show that $\log (-\hat{I}{”}_{{\mathbb{H}}^n}{)}_p$ can be realized as the solution of a suitable extension problem and provide an extension theorem that yields the operator $\log (-\hat{I}”{)}_p$ in ${\mathbb{R}}^n$. To the best of our knowledge, this property has not been established for the Euclidean logarithmic $p$-Laplacian $\log (-\hat{I}”{)}_p$.